Maharashtra Board Class 10 Question Bank Maths
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Page 1
महाराष्ट्र शासन
शालेय शशक्षण व क्रीडा शवभाग
राज्य शैक्षशणक संशोधन व प्रशशक्षण पररषद, महाराष्ट्र
७०८ सदाशिव पेठ, कुमठे कर मार्ग, पुणे ४११०३०
संपकग क्रमांक (020) 2447 6938 E mail: [email protected]
-----------------------------------------------------------------------------------------------
Question Bank
Standard :- 10th
Subject :- Mathematics Part 1
सूचना
१. फक्त शवद्यार्थ्यांना प्रश्नप्रकारांचा सराव करून देण्यासाठीच
२. सदर प्रश्नसंचातील प्रश्न बोर्डाच्या प्रश्नपशिकेत येतीलच असे नाही
याची नोंद घ्यावी.
शालेय शशक्षण व क्रीडा शवभाग
राज्य शैक्षशणक संशोधन व प्रशशक्षण पररषद, महाराष्ट्र
७०८ सदाशिव पेठ, कुमठे कर मार्ग, पुणे ४११०३०
संपकग क्रमांक (020) 2447 6938 E mail: [email protected]
-----------------------------------------------------------------------------------------------
Question Bank
Standard :- 10th
Subject :- Mathematics Part 1
सूचना
१. फक्त शवद्यार्थ्यांना प्रश्नप्रकारांचा सराव करून देण्यासाठीच
२. सदर प्रश्नसंचातील प्रश्न बोर्डाच्या प्रश्नपशिकेत येतीलच असे नाही
याची नोंद घ्यावी.
Page 2
Standard: 10
Subject-Mathematics
Question bank
1.linear equation in two variables
------------------------------------------------------------------------------------
Q.1(A) MCQ
1. To draw the graph of 4x + 5y = 19, if x = 1 is taken then what will be the
value of y?
A) 4 B) 3 C) 2 D) −3
2) For the equations with variables x and y, if Dx = 26, Dy = −39 and D =
13 then x =?
A)2 B) − 3 C) − 2 D) 3
3) Which of the following is linear equation in two variables?
𝑥 5
A) + = 6 B) 2x2-3y=8-3y C) x+2y=5-3y D) 3 x²+y
3 𝑦
4) which of the following is not the solution of 3x+6y=12?
A) (-4,4) B) (0,2) C) (8, -2) D) (3,1)
3 5
5) | | = 2 ∴ 𝑥 = -------
2 𝑥
A) 3 B) 4 C) – 3 D) – 4
6) For equations 5x + 3y + 11 = 0 and 2x + 4y = - 10 find D.
A) 14 B) – 14 C) 26 D) – 26
7) If 49 x – 57 y = 172 and 57 x – 49 y = 252 then x + y =?
A) 80 B) 0 C) 10 D) 8
8) The solution of the equation 2x – y = 2 is ------.
A) (2,2) B) (5,2) C) (2,5) D)
(5,5)
Subject-Mathematics
Question bank
1.linear equation in two variables
------------------------------------------------------------------------------------
Q.1(A) MCQ
1. To draw the graph of 4x + 5y = 19, if x = 1 is taken then what will be the
value of y?
A) 4 B) 3 C) 2 D) −3
2) For the equations with variables x and y, if Dx = 26, Dy = −39 and D =
13 then x =?
A)2 B) − 3 C) − 2 D) 3
3) Which of the following is linear equation in two variables?
𝑥 5
A) + = 6 B) 2x2-3y=8-3y C) x+2y=5-3y D) 3 x²+y
3 𝑦
4) which of the following is not the solution of 3x+6y=12?
A) (-4,4) B) (0,2) C) (8, -2) D) (3,1)
3 5
5) | | = 2 ∴ 𝑥 = -------
2 𝑥
A) 3 B) 4 C) – 3 D) – 4
6) For equations 5x + 3y + 11 = 0 and 2x + 4y = - 10 find D.
A) 14 B) – 14 C) 26 D) – 26
7) If 49 x – 57 y = 172 and 57 x – 49 y = 252 then x + y =?
A) 80 B) 0 C) 10 D) 8
8) The solution of the equation 2x – y = 2 is ------.
A) (2,2) B) (5,2) C) (2,5) D)
(5,5)
Page 3
9) The solution of the equation x-y=10 and x+y=70 is --------.
A) (40,30) B) (30,40) C) (10,60) D) (50,20)
10) Find the value of Dx for the equation 4𝑥 + 3 𝑦 = 19 and 4 𝑥 −
3 𝑦 = −11
A) 24 B) 0 C) −24 D) 108
Q. 1 B) Each of 1 mark
1) State with reason whether the equation 3𝑥 2 − 7y = 13 is a linear
equation with two variables?
2) Show the condition using variable x and y: Two numbers differ by 3
3) For the equation 4x + 5y = 20 find y when x = 0
4) Write any two solutions of the equation x + y = 7.
5) Decide whether (0, 2) is the solution of the equation 5x + 3y = 6
6) Write any two solution of the equation a – b = - 3
7) If x+2y=5 and 2x+y=7 then find the value of x+y
8) If Dx = 24 and x = - 3 then find the value of D.
9) The cost of the book is 5 rupees more than twice the cost of a pen. Show
this using linear equation by taking Cost of book(x) and cost of a pen(y).
𝑎 𝑏
10) If + = 4, write the equation in standard form.
4 3
A) (40,30) B) (30,40) C) (10,60) D) (50,20)
10) Find the value of Dx for the equation 4𝑥 + 3 𝑦 = 19 and 4 𝑥 −
3 𝑦 = −11
A) 24 B) 0 C) −24 D) 108
Q. 1 B) Each of 1 mark
1) State with reason whether the equation 3𝑥 2 − 7y = 13 is a linear
equation with two variables?
2) Show the condition using variable x and y: Two numbers differ by 3
3) For the equation 4x + 5y = 20 find y when x = 0
4) Write any two solutions of the equation x + y = 7.
5) Decide whether (0, 2) is the solution of the equation 5x + 3y = 6
6) Write any two solution of the equation a – b = - 3
7) If x+2y=5 and 2x+y=7 then find the value of x+y
8) If Dx = 24 and x = - 3 then find the value of D.
9) The cost of the book is 5 rupees more than twice the cost of a pen. Show
this using linear equation by taking Cost of book(x) and cost of a pen(y).
𝑎 𝑏
10) If + = 4, write the equation in standard form.
4 3
Page 4
Q.2 A) Complete the activity (2 marks)
1) Complete the table to draw the graph of 2x – 3y = 3,
X −6
y 1
(x, y)
2. Solve the following to find the value of following determinant.
3 −2
| |= 3× − ×4= +8=
4 5
3) Complete the activity to find the value of 𝑥
3 𝑥 + 2𝑦 = 11 −---- (I) and 2 𝑥 + 3 𝑦 = 4 ------(II)
Solution: Multiply equation (I) by ----- and equation (II) by -----.
× (3 𝑥 + 2𝑦 = 11) ∴ 9 𝑥 + 6 𝑦 = 33
2 × (2 𝑥 + 3 𝑦 = 4) ∴ 4𝑥+ 6𝑦 =8
subtract (II) from (I),
----- 𝑥 = 25
∴ 𝑥 = ----
4) If (2, 0) is the solution of 2x + 3y = k then finds the value of k by
completing the activity
Solution: (2,0) is solution of the equation 2x +3y = k
Putting x = and y =
∴ 2( )+3×0=k
1) Complete the table to draw the graph of 2x – 3y = 3,
X −6
y 1
(x, y)
2. Solve the following to find the value of following determinant.
3 −2
| |= 3× − ×4= +8=
4 5
3) Complete the activity to find the value of 𝑥
3 𝑥 + 2𝑦 = 11 −---- (I) and 2 𝑥 + 3 𝑦 = 4 ------(II)
Solution: Multiply equation (I) by ----- and equation (II) by -----.
× (3 𝑥 + 2𝑦 = 11) ∴ 9 𝑥 + 6 𝑦 = 33
2 × (2 𝑥 + 3 𝑦 = 4) ∴ 4𝑥+ 6𝑦 =8
subtract (II) from (I),
----- 𝑥 = 25
∴ 𝑥 = ----
4) If (2, 0) is the solution of 2x + 3y = k then finds the value of k by
completing the activity
Solution: (2,0) is solution of the equation 2x +3y = k
Putting x = and y =
∴ 2( )+3×0=k
Page 5
∴4+0=k
∴k=
5) To find the values of x and y for the equations x- 2 y =5 and 2 x+ 3 y =10
complete the activity.
D= =3+4=7
D = x =
D= y =
By Cramer’s rule
Dx Dy
X= = ,y= =
D D
Q. 2 B) Each of 2 marks
1) The difference between an angle and its complement is 10° find measure
of larger angle.
5 2
2) Find the value of | |
0 −1
3) For the equation 𝑦 + 2𝑥 = 19 and 2𝑥 − 3𝑦 = − 3 Find the value of D
4) In the equation 2𝑥 − 𝑦 = 2 if 𝑥 = 3 then find y=?
5) If (2, −5) is the solution of the equation 2𝑥 − 𝑘 𝑦 = 14 then find k
=?
6) For the equation a + 2b = 7 find a when b = 4
∴k=
5) To find the values of x and y for the equations x- 2 y =5 and 2 x+ 3 y =10
complete the activity.
D= =3+4=7
D = x =
D= y =
By Cramer’s rule
Dx Dy
X= = ,y= =
D D
Q. 2 B) Each of 2 marks
1) The difference between an angle and its complement is 10° find measure
of larger angle.
5 2
2) Find the value of | |
0 −1
3) For the equation 𝑦 + 2𝑥 = 19 and 2𝑥 − 3𝑦 = − 3 Find the value of D
4) In the equation 2𝑥 − 𝑦 = 2 if 𝑥 = 3 then find y=?
5) If (2, −5) is the solution of the equation 2𝑥 − 𝑘 𝑦 = 14 then find k
=?
6) For the equation a + 2b = 7 find a when b = 4
Page 6
7) Decide whether x = 2 and y =−1 is the solution of the equation 2x + y =
3 or not?
8) Using variables 𝑎 𝑎𝑛𝑑 𝑏 write any two equations whose solution is
(0,2)
9. If 52 𝑥 + 65 𝑦 = 183 and 65 𝑥 + 52 𝑦 = 168 then find 𝑥 + 𝑦 =?
10. State with reason whether the point (3, −2) will lie on the graph of
the equation 5 𝑚 − 3 𝑛 = −21.
Q.3 A) Complete the activity.
1)
x x
I am an isosceles triangle. My base
is 3 cm more than twice my
congruent sides. My perimeter is
35 cm. Find the length of my base.
y
2) Complete the following table to draw the graph of 3 𝑥 − 2 𝑦 = 18
X 0 4 2 -1
Y -9 ------ ------- ----
x, y (0,-9) (--,--) (--,--) ----
3 or not?
8) Using variables 𝑎 𝑎𝑛𝑑 𝑏 write any two equations whose solution is
(0,2)
9. If 52 𝑥 + 65 𝑦 = 183 and 65 𝑥 + 52 𝑦 = 168 then find 𝑥 + 𝑦 =?
10. State with reason whether the point (3, −2) will lie on the graph of
the equation 5 𝑚 − 3 𝑛 = −21.
Q.3 A) Complete the activity.
1)
x x
I am an isosceles triangle. My base
is 3 cm more than twice my
congruent sides. My perimeter is
35 cm. Find the length of my base.
y
2) Complete the following table to draw the graph of 3 𝑥 − 2 𝑦 = 18
X 0 4 2 -1
Y -9 ------ ------- ----
x, y (0,-9) (--,--) (--,--) ----
Page 7
3) The sum of the two-digit number and the number obtained by
interchanging the digits is 132. The digit in the ten’s place is 2 more than the
digit in the unit’s place. Complete the activity to find the original number.
Activity: Let the digit in the unit’s place is y and the digit in the ten’s place is
𝑥.
∴ The number = 10 𝑥 + 𝑦
∴ The number obtained by interchanging the digits =
∴ The sum of the number and the number obtained by interchanging the
digits = 132
∴ 10 𝑥 + 𝑦 + 10 𝑦 + 𝑥 =
∴x+y= (I)
, By second condition,
Digit in the ten’s place = digit in the unit’s place + 2
∴x-y=2 . . . (II)
Solving equation (I) and (II)
∴x= y=
Ans: The original number =
interchanging the digits is 132. The digit in the ten’s place is 2 more than the
digit in the unit’s place. Complete the activity to find the original number.
Activity: Let the digit in the unit’s place is y and the digit in the ten’s place is
𝑥.
∴ The number = 10 𝑥 + 𝑦
∴ The number obtained by interchanging the digits =
∴ The sum of the number and the number obtained by interchanging the
digits = 132
∴ 10 𝑥 + 𝑦 + 10 𝑦 + 𝑥 =
∴x+y= (I)
, By second condition,
Digit in the ten’s place = digit in the unit’s place + 2
∴x-y=2 . . . (II)
Solving equation (I) and (II)
∴x= y=
Ans: The original number =
Page 8
Q.3 B) Each of 3 marks
1) Solve the given simultaneous equations graphically 𝑥 + 𝑦 = 5 and 𝑦 = 5,
2) Ajay is younger than Vijay by 3 years. The sum of their ages is 25 years,
what is the age of Ajay.
3) Solve by Cramer’s rule, 3x – 4y = 10; 4x + 3y = 5
4) Difference between two numbers is 3, the sum of three times the bigger
number and two times the smaller number is 19. Then find the numbers.
5) Solve: 4m – 2n = - 4; 4m + 3n =16
6) Solve: 99x + 101y = 499; 101x + 99y = 501
7) The length of the rectangle is 5 more than twice its breadth. The
perimeter of a rectangle is 52 cm then find the length of the rectangle.
8) The graph of the equations 2 𝑥 − 𝑦 − 4 = 0 and 𝑥 + 𝑦 + 1 = 0 intersect
each other in point P (a, b) then find the coordinates of P?
9) The solution of the equation 𝑎 𝑥 + b y + 5 = 0 and 𝑏 𝑥 − 𝑎 𝑦 − 12 = 0 is
(2, -3)
Find the values of a and b.
10) A person starts a job with some fixed salary and yearly increment.
After 4 years his salary is Rs.15000 and after 10 years it becomes Rs.
18000. Then find his monthly salary and increment.
11). For the equation 3 𝑥 − 2 𝑦 = 17 find the value of x when y = −1 and
find the value of y when x = 3.
1) Solve the given simultaneous equations graphically 𝑥 + 𝑦 = 5 and 𝑦 = 5,
2) Ajay is younger than Vijay by 3 years. The sum of their ages is 25 years,
what is the age of Ajay.
3) Solve by Cramer’s rule, 3x – 4y = 10; 4x + 3y = 5
4) Difference between two numbers is 3, the sum of three times the bigger
number and two times the smaller number is 19. Then find the numbers.
5) Solve: 4m – 2n = - 4; 4m + 3n =16
6) Solve: 99x + 101y = 499; 101x + 99y = 501
7) The length of the rectangle is 5 more than twice its breadth. The
perimeter of a rectangle is 52 cm then find the length of the rectangle.
8) The graph of the equations 2 𝑥 − 𝑦 − 4 = 0 and 𝑥 + 𝑦 + 1 = 0 intersect
each other in point P (a, b) then find the coordinates of P?
9) The solution of the equation 𝑎 𝑥 + b y + 5 = 0 and 𝑏 𝑥 − 𝑎 𝑦 − 12 = 0 is
(2, -3)
Find the values of a and b.
10) A person starts a job with some fixed salary and yearly increment.
After 4 years his salary is Rs.15000 and after 10 years it becomes Rs.
18000. Then find his monthly salary and increment.
11). For the equation 3 𝑥 − 2 𝑦 = 17 find the value of x when y = −1 and
find the value of y when x = 3.
Page 9
Q.4 Solve (Each of 4 marks)
1) Solve the following equations by graphical method, 𝑥 − 𝑦 = 1; 2 𝑥 + 𝑦 = 8
2) Using the determinants given below form two linear equations and solve
them.
5 7 5 4
D= | | Dy = | |
2 −3 2 −10
3) For an A.P, t = 54 and t = 30 find the first term(a) and common
17 9
difference(d).
4) A train covered a certain distance at a uniform speed. If the train would
have been 6 km/h faster, it would have taken 4 hours less than the
scheduled time. And ,if the train was slower by 6 km/h it would have taken 6
hours more than the scheduled time. Find the length of the journey.
5) Solve, 0.4 𝑥 + 0.3 𝑦 = 1.7; 0.7 𝑥 − 0.2 𝑦 = 0.8
6) The semi perimeter of a rectangular shape garden is 36 m. The length of
the garden is 4 m more than its breadth. Find the length and the breadth of
the garden.
Q. 5 Solve (Each of 3 marks)
1) Form the simultaneous linear equation using the determinants.
4 −3 5 −3 4 5
D= | | Dx = | | Dy =| |
2 5 9 5 2 9
2) I held a number 75 in my mind.
Write any condition showing the relation between their digits.
Write the condition showing relation between the number and the number
obtained by interchanging the digits.
1) Solve the following equations by graphical method, 𝑥 − 𝑦 = 1; 2 𝑥 + 𝑦 = 8
2) Using the determinants given below form two linear equations and solve
them.
5 7 5 4
D= | | Dy = | |
2 −3 2 −10
3) For an A.P, t = 54 and t = 30 find the first term(a) and common
17 9
difference(d).
4) A train covered a certain distance at a uniform speed. If the train would
have been 6 km/h faster, it would have taken 4 hours less than the
scheduled time. And ,if the train was slower by 6 km/h it would have taken 6
hours more than the scheduled time. Find the length of the journey.
5) Solve, 0.4 𝑥 + 0.3 𝑦 = 1.7; 0.7 𝑥 − 0.2 𝑦 = 0.8
6) The semi perimeter of a rectangular shape garden is 36 m. The length of
the garden is 4 m more than its breadth. Find the length and the breadth of
the garden.
Q. 5 Solve (Each of 3 marks)
1) Form the simultaneous linear equation using the determinants.
4 −3 5 −3 4 5
D= | | Dx = | | Dy =| |
2 5 9 5 2 9
2) I held a number 75 in my mind.
Write any condition showing the relation between their digits.
Write the condition showing relation between the number and the number
obtained by interchanging the digits.
Page 10
3) Write any two linear equations in two variables in which the value of
one variable is 12 and the other 10.
4) From the railway station I took a rickshaw to go home. It is decided
that I have to pay Rs. X for the first kilometre and for each kilometre Rs.
Y for the next. For 10 kilometres the fare is Rs. 40 and for 16 kilometres
fare is Rs. 58. Find the fare for the first kilometre.
one variable is 12 and the other 10.
4) From the railway station I took a rickshaw to go home. It is decided
that I have to pay Rs. X for the first kilometre and for each kilometre Rs.
Y for the next. For 10 kilometres the fare is Rs. 40 and for 16 kilometres
fare is Rs. 58. Find the fare for the first kilometre.
Page 11
2.Quadratic Equations
Q 1 A) Multiple choice questions (for 1 mark each) :
Choose the correct alternative answer for each of the following sub
questions and write the correct alphabet.
1) Which of the following is a quadratic equation?
A) X3+5X2+X+3=0 B) 4X2-3X-5=0 C) X+5=0 D) 4X5 = 0
2) Which of the following is not a quadratic equation?
A) 2X2-X+3=0 B) 4X2-3X=0 C) X3-5X +3 =0 D) 4X2 = 0
3) If the root of the given quadratic equation are real and equal then find the
value of ‘k’ X2 + 2X + k = 0 .
A) 1 B) -1 C) 2 D) -2
4) What is the value of discriminant for the quadratic equation X 2 – 2X - 3
= 0?
A) -16 B) 16 C) 8 D) 4
5) Which of the following quadratic equation has roots -3 and -5 ?
A) X2-8X+15=0 B) X2-8X-15=0 C) X2+8X+15=0 D) X2+8X-
15=0
6) If one of the roots of quadratic equation X2 – kX + 27 = 0 is 3 then find
the value of ‘k’.
A) 10 B) 12 C) -12 D) 16
7) Degree of quadratic equation is always ---------------.
A) 1 B) 2 C) 3 D) 4
Q 1 A) Multiple choice questions (for 1 mark each) :
Choose the correct alternative answer for each of the following sub
questions and write the correct alphabet.
1) Which of the following is a quadratic equation?
A) X3+5X2+X+3=0 B) 4X2-3X-5=0 C) X+5=0 D) 4X5 = 0
2) Which of the following is not a quadratic equation?
A) 2X2-X+3=0 B) 4X2-3X=0 C) X3-5X +3 =0 D) 4X2 = 0
3) If the root of the given quadratic equation are real and equal then find the
value of ‘k’ X2 + 2X + k = 0 .
A) 1 B) -1 C) 2 D) -2
4) What is the value of discriminant for the quadratic equation X 2 – 2X - 3
= 0?
A) -16 B) 16 C) 8 D) 4
5) Which of the following quadratic equation has roots -3 and -5 ?
A) X2-8X+15=0 B) X2-8X-15=0 C) X2+8X+15=0 D) X2+8X-
15=0
6) If one of the roots of quadratic equation X2 – kX + 27 = 0 is 3 then find
the value of ‘k’.
A) 10 B) 12 C) -12 D) 16
7) Degree of quadratic equation is always ---------------.
A) 1 B) 2 C) 3 D) 4
Page 12
Q 1 B) Examples for 1 mark :
1) Write the given quadratic equation in standard form and also write the
values of a, b and c .
4y2 – 3y = -7
2) Write the roots of following quadratic equation.
(p – 5 ) (p + 3 ) = 0
3) If a = 1, b = 4, c = -5 then find the value of b2 – 4ac.
4) If b2 – 4ac> 0 and b2 – 4ac< 0 then write the nature of roots of the
quadratic equation for each given case.
5) Write the given quadratic equation in standard form.
m (m – 6) = 9
Q 2 A) Activity based questions for 2 marks each:
1) Complete the following activity to solve the given quadratic equation
by factorization method.
Activity: X2 + 8 x – 20 = 0
X2 + (....) – 2x – 20 = 0
X (x+10) – (....) (x+10) = 0
(x + 10) (.......) = 0
X = …… or x = 2
2) Complete the following activity to find the value of discriminant for
quadratic equation 4x2 – 5x + 3 = 0.
Activity: 4x2 – 5x + 3 = 0
a= 4 , b = …… , c = 3
b - 4ac = (-5) 2 – (….) x 4 x 3
2
1) Write the given quadratic equation in standard form and also write the
values of a, b and c .
4y2 – 3y = -7
2) Write the roots of following quadratic equation.
(p – 5 ) (p + 3 ) = 0
3) If a = 1, b = 4, c = -5 then find the value of b2 – 4ac.
4) If b2 – 4ac> 0 and b2 – 4ac< 0 then write the nature of roots of the
quadratic equation for each given case.
5) Write the given quadratic equation in standard form.
m (m – 6) = 9
Q 2 A) Activity based questions for 2 marks each:
1) Complete the following activity to solve the given quadratic equation
by factorization method.
Activity: X2 + 8 x – 20 = 0
X2 + (....) – 2x – 20 = 0
X (x+10) – (....) (x+10) = 0
(x + 10) (.......) = 0
X = …… or x = 2
2) Complete the following activity to find the value of discriminant for
quadratic equation 4x2 – 5x + 3 = 0.
Activity: 4x2 – 5x + 3 = 0
a= 4 , b = …… , c = 3
b - 4ac = (-5) 2 – (….) x 4 x 3
2
Page 13
= (….) – 48
b - 4ac = ……
2
3) If one of the root of quadratic equation X2 + kx + 54 = 0 is -6 then
complete the following activity to find the value of ‘k’.
Activity: one of the roots of the quadratic equation X2 + kx + 54 = 0 is
-6
Therefore let’s take x =.......
(-6)2+ k(-6) + 54= 0
(......) -6 k + 54= 0
-6k+ ...... = 0
k= ........
4) To decide whether 1 is a root of quadratic equation X2 + 4x -5 = 0 or
not complete the following activity.
Activity: when x= (....)
L.H.S.
= 12+ 4 (……) – 5
=1+4–5
= (….) – 5
= ……..
= R.H.S.
Therefore x = 1 is a root of quadratic equation X2 + 4x -5 = 0.
Q 2 B) Examples for 2marks each:
1) Solve the following quadratic equation by factorization method.
3p2 + 8p+ 5= 0
2) If one of the roots of quadratic equation X2 – kx – 15 = 0 is -3 then
find the value of ‘k’.
3) If the Roots of a quadratic equation are 4 and -5 then form the
quadratic equation.
b - 4ac = ……
2
3) If one of the root of quadratic equation X2 + kx + 54 = 0 is -6 then
complete the following activity to find the value of ‘k’.
Activity: one of the roots of the quadratic equation X2 + kx + 54 = 0 is
-6
Therefore let’s take x =.......
(-6)2+ k(-6) + 54= 0
(......) -6 k + 54= 0
-6k+ ...... = 0
k= ........
4) To decide whether 1 is a root of quadratic equation X2 + 4x -5 = 0 or
not complete the following activity.
Activity: when x= (....)
L.H.S.
= 12+ 4 (……) – 5
=1+4–5
= (….) – 5
= ……..
= R.H.S.
Therefore x = 1 is a root of quadratic equation X2 + 4x -5 = 0.
Q 2 B) Examples for 2marks each:
1) Solve the following quadratic equation by factorization method.
3p2 + 8p+ 5= 0
2) If one of the roots of quadratic equation X2 – kx – 15 = 0 is -3 then
find the value of ‘k’.
3) If the Roots of a quadratic equation are 4 and -5 then form the
quadratic equation.
Page 14
4) If roots of a quadratic equation 3y2+ ky+ 12= 0 are real and equal then
find the value of ‘k’.
5) Roots of a quadratic equation are 5 and -4 then form the quadratic
equation.
Q 3 A) Examples for 3 marks each:
1) Complete the following activity to solve the given quadratic equation by
formula method.
2x2 + 13x + 15 = 0
Activity :2x2 + 13x + 15 = 0
a= (.....) , b = 13, c = 15
b – 4ac = (13)2 - 4 x 2 x (…….)
2
= 169 – 120
b – 4ac = 49
2
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
−(… … ) ± √49
𝑥=
4
−13 ± (… . )
𝑥=
4
6 −20
𝑥= 𝑜𝑟 𝑥=
4 4
X = (….) or x = (….)
2) Complete the following activity to solve the given word problem.
Sum of squares of two consecutive even natural numbers is 244 then find
those numbers.
find the value of ‘k’.
5) Roots of a quadratic equation are 5 and -4 then form the quadratic
equation.
Q 3 A) Examples for 3 marks each:
1) Complete the following activity to solve the given quadratic equation by
formula method.
2x2 + 13x + 15 = 0
Activity :2x2 + 13x + 15 = 0
a= (.....) , b = 13, c = 15
b – 4ac = (13)2 - 4 x 2 x (…….)
2
= 169 – 120
b – 4ac = 49
2
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
−(… … ) ± √49
𝑥=
4
−13 ± (… . )
𝑥=
4
6 −20
𝑥= 𝑜𝑟 𝑥=
4 4
X = (….) or x = (….)
2) Complete the following activity to solve the given word problem.
Sum of squares of two consecutive even natural numbers is 244 then find
those numbers.
Page 15
Activity: let the first even natural number be X,
Therefore its consecutive even natural number will be = (.....)
By the given condition,
X2 + (x+ 2)2 = 244
X2+X2+4x+4 – (....) = 0
2x2 + 4x – 240 = 0
X2 + 2x – 120 = 0
X2+ (……..) – (……..) – 120 = 0
X (x+12) – (…..) (x+12) = 0
(x + 12) (x – 10) = 0
X = (……) / X = 10
But natural number cannot be negative x = -12 is not possible.
Therefore first even natural number is x = 10.
Second even consecutive natural number = x+ 2= 10+ 2= 12.
Q 3 B) Examples for 3 marks each:
1) If the roots of the given quadratic equations are real and equal then find
the value of ‘k’.
kx (x-2) + 6 = 0
2) Mukund has Rs. 50 more than Sagar. If the product of the amount they
have is 15,000 then find the amount each has.
3) Solve the following quadratic equation.
√3 x2 + √2 x – 2√3 =0
3) Solve the following quadratic equations by formula method.
a) 5m2 – 4m -2 = 0
Therefore its consecutive even natural number will be = (.....)
By the given condition,
X2 + (x+ 2)2 = 244
X2+X2+4x+4 – (....) = 0
2x2 + 4x – 240 = 0
X2 + 2x – 120 = 0
X2+ (……..) – (……..) – 120 = 0
X (x+12) – (…..) (x+12) = 0
(x + 12) (x – 10) = 0
X = (……) / X = 10
But natural number cannot be negative x = -12 is not possible.
Therefore first even natural number is x = 10.
Second even consecutive natural number = x+ 2= 10+ 2= 12.
Q 3 B) Examples for 3 marks each:
1) If the roots of the given quadratic equations are real and equal then find
the value of ‘k’.
kx (x-2) + 6 = 0
2) Mukund has Rs. 50 more than Sagar. If the product of the amount they
have is 15,000 then find the amount each has.
3) Solve the following quadratic equation.
√3 x2 + √2 x – 2√3 =0
3) Solve the following quadratic equations by formula method.
a) 5m2 – 4m -2 = 0
Page 16
1
b) Y2 + y = 2
3
4) Form a quadratic equation if the roots of the quadratic equation are 2 +
√7 and 2 - √7 .
Q 4) Examples for 4 marks each :
1) Present age of mother of Manish is 1 year more than 5 times the present
age of Manish. Four years before If the product of their ages was 22 then
find the present age of Manish and his mother.
2) In an orchard there are total 200 trees. If the number of trees in each
column is more by 10 than the number of trees in each row then find the
number of trees in each row.
3) If the roots of the given quadratic equation are real and equal then find
the value of ‘m’.
(m-12) x2 + 2 (m-12) x + 2 = 0
4) Solve the following quadratic equation.
1 1 1
- =
4−𝑝 2+𝑝 4
b) Y2 + y = 2
3
4) Form a quadratic equation if the roots of the quadratic equation are 2 +
√7 and 2 - √7 .
Q 4) Examples for 4 marks each :
1) Present age of mother of Manish is 1 year more than 5 times the present
age of Manish. Four years before If the product of their ages was 22 then
find the present age of Manish and his mother.
2) In an orchard there are total 200 trees. If the number of trees in each
column is more by 10 than the number of trees in each row then find the
number of trees in each row.
3) If the roots of the given quadratic equation are real and equal then find
the value of ‘m’.
(m-12) x2 + 2 (m-12) x + 2 = 0
4) Solve the following quadratic equation.
1 1 1
- =
4−𝑝 2+𝑝 4
Page 17
5) Sum of the roots of the quadratic equation is 5 and sum of their cubes is
35 then find the quadratic equation.
Q 5) Examples of 3 marks each:
1) Form a quadratic equation such that one of its roots is 5. Form a
quadratic equation for it and write. ( For the formation of word
problem you can use quantities like age, rupees or natural numbers.)
(sample solution for the above example is given below students can
take another number to form another example)
Solution: We need one of the solutions of quadratic equation as 5. Then
we can take another root as any number like positive or negative number
or zero. Here I am taking another root of quadratic equation as 2.
Then we can form a word problem as below,
Smita is younger than her sister Mita by 3 years (5-2= 3). If the product
of their ages is (5 x 2 = 10). Then find their present ages. (to form a word
problem 1 mark)
Let the age of Mita be x,
Therefore age of Smita = x-3 (1 mark for this)
By the given condition,
X(x-3) = 10
X2 – 3x – 10 = 0 (to form a quadratic equation 1 mark)
35 then find the quadratic equation.
Q 5) Examples of 3 marks each:
1) Form a quadratic equation such that one of its roots is 5. Form a
quadratic equation for it and write. ( For the formation of word
problem you can use quantities like age, rupees or natural numbers.)
(sample solution for the above example is given below students can
take another number to form another example)
Solution: We need one of the solutions of quadratic equation as 5. Then
we can take another root as any number like positive or negative number
or zero. Here I am taking another root of quadratic equation as 2.
Then we can form a word problem as below,
Smita is younger than her sister Mita by 3 years (5-2= 3). If the product
of their ages is (5 x 2 = 10). Then find their present ages. (to form a word
problem 1 mark)
Let the age of Mita be x,
Therefore age of Smita = x-3 (1 mark for this)
By the given condition,
X(x-3) = 10
X2 – 3x – 10 = 0 (to form a quadratic equation 1 mark)
Page 18
3.Arithmetic Progression
• Question for 1 Mark.
A)Choose the correct alternative answer for each of the following sub
questions.
1. In an Arithmetic Progression 2,4,6,8,……..the common difference d
is………..
(A) 8 (B) 6 (C) 2 (D) -2
2. What is the common difference of the sequence 0,-4,-8,-12 ?
(A) 4 (B) -4 (C) 8 (D) -8
3. For an A.P. 5,12,19,26,........ a=?
(A) 12 (B) 26 (C) 19 (D) 5
4. A set of numbers where the numbers are arranged in a definite order,
like the natural numbers, is called a .........
(A) index (B) numbers (C) line (D) sequence
5. First four terms of an A.P., are……………………whose first term is -2 and
common difference is -2.
(A)-2,0,2,4 (B)-2,4,-8,16 (C)-2,-4,-6,-8 (D)-2,-4,-8,-16
6. 1,4,7,10,13...... Next two terms of this A.P. are………………………….
(A) 16,19 (B) 10,7 (C) 19,22 (D) 16,18
7. Find d of an A.P. whose first two terms are -3 and -4.
(A) 7 (B) 4 (C) -7 (D) -3
8. If third term and fifth term of an A.P. are 13 and 25 respectively, find
its 7th term.
(A) 30 (B) 33 (C) 37 (D) 38
9. Find t3 = ? in an A.P. 9,15,21,27
(A) 27 (B) 21 (C) 15 (D) 9
• Question for 1 Mark.
A)Choose the correct alternative answer for each of the following sub
questions.
1. In an Arithmetic Progression 2,4,6,8,……..the common difference d
is………..
(A) 8 (B) 6 (C) 2 (D) -2
2. What is the common difference of the sequence 0,-4,-8,-12 ?
(A) 4 (B) -4 (C) 8 (D) -8
3. For an A.P. 5,12,19,26,........ a=?
(A) 12 (B) 26 (C) 19 (D) 5
4. A set of numbers where the numbers are arranged in a definite order,
like the natural numbers, is called a .........
(A) index (B) numbers (C) line (D) sequence
5. First four terms of an A.P., are……………………whose first term is -2 and
common difference is -2.
(A)-2,0,2,4 (B)-2,4,-8,16 (C)-2,-4,-6,-8 (D)-2,-4,-8,-16
6. 1,4,7,10,13...... Next two terms of this A.P. are………………………….
(A) 16,19 (B) 10,7 (C) 19,22 (D) 16,18
7. Find d of an A.P. whose first two terms are -3 and -4.
(A) 7 (B) 4 (C) -7 (D) -3
8. If third term and fifth term of an A.P. are 13 and 25 respectively, find
its 7th term.
(A) 30 (B) 33 (C) 37 (D) 38
9. Find t3 = ? in an A.P. 9,15,21,27
(A) 27 (B) 21 (C) 15 (D) 9
Page 19
10. In an A.P., 0,-4,-8,-12...................find t2 = ?
(A) -8 (B) -4 (C) -12 (D) 0
B) Solve the following sub questions.
1. Decide whether the given sequence 2,4,6,8........is an A.P.
2. Find a and d for an A.P., 1,4,7,10.......
3. Write the formula of the sum of first n terms for an A.P.
4. Find tn if a=20 आणि d=3
5. Find t5 if a=3 आणि d=-3
6. tn = 2n-5 in a sequence, find its first two terms.
7. Find first term of the sequence tn=2n+1
8. Find two terms of the sequence tn=3n-2
9. Find common difference of an A.P.,0.9,0.6,0.3......
10. Find d if t9=23 व a=7.
• Question for 2 Marks.
A) complete the following activity
1)Find the sum of first 1000 positive integers.
Activity :- Let 1+2+3+------+1000
Using formula for the sum of first n terms of an A.P.,
Sn=
S1000 = 2 (1+1000)
= 500 × 1001
=
Therefore, Sum of the first 1000 positive integer is
2) Which term of following A.P. is -940.
50,40,30,20.......
(A) -8 (B) -4 (C) -12 (D) 0
B) Solve the following sub questions.
1. Decide whether the given sequence 2,4,6,8........is an A.P.
2. Find a and d for an A.P., 1,4,7,10.......
3. Write the formula of the sum of first n terms for an A.P.
4. Find tn if a=20 आणि d=3
5. Find t5 if a=3 आणि d=-3
6. tn = 2n-5 in a sequence, find its first two terms.
7. Find first term of the sequence tn=2n+1
8. Find two terms of the sequence tn=3n-2
9. Find common difference of an A.P.,0.9,0.6,0.3......
10. Find d if t9=23 व a=7.
• Question for 2 Marks.
A) complete the following activity
1)Find the sum of first 1000 positive integers.
Activity :- Let 1+2+3+------+1000
Using formula for the sum of first n terms of an A.P.,
Sn=
S1000 = 2 (1+1000)
= 500 × 1001
=
Therefore, Sum of the first 1000 positive integer is
2) Which term of following A.P. is -940.
50,40,30,20.......
Page 20
Activity:- Here a= d= tn= -940
According to formula, tn= a+(n-1)d
-940 =
n =
3) For an A.P., If t1= 1 and tn= 149 then find Sn.
Activitry :- Here t1= 1, tn= 149, Sn = ?
Sn = 2 ( + )
n
= 2 ×
n
= n
4) t19 = ? for the given A.P., 9,4,-1,-6......
Activity :- Here a=9, d=
tn = a+(n-1)d
t19= 9+(19-1)
= 9+
=
5) Common difference, d = ? for the given A.P., 7,14,21,28.........
Activity :- Here t1=7, t2=14, t3=21, t4=
t2 - t1=
t3 – t2= 7
t4 – t3=
Therefore, common difference d=
B) Solve the following.
1. Decide whether the following sequence is an A.P. or not.
According to formula, tn= a+(n-1)d
-940 =
n =
3) For an A.P., If t1= 1 and tn= 149 then find Sn.
Activitry :- Here t1= 1, tn= 149, Sn = ?
Sn = 2 ( + )
n
= 2 ×
n
= n
4) t19 = ? for the given A.P., 9,4,-1,-6......
Activity :- Here a=9, d=
tn = a+(n-1)d
t19= 9+(19-1)
= 9+
=
5) Common difference, d = ? for the given A.P., 7,14,21,28.........
Activity :- Here t1=7, t2=14, t3=21, t4=
t2 - t1=
t3 – t2= 7
t4 – t3=
Therefore, common difference d=
B) Solve the following.
1. Decide whether the following sequence is an A.P. or not.
Page 21
3,5,7,9,11…….
2. Find first four terms of an A.P., whose first term is 3 and common
difference is 4.
3. 1,6,11,16......Find the 18th term of this A.P.
4. In an A.P. a=2 and d=3, then find S12.
5. Find first four terms of the sequence tn=n+2.
6. In an A.P., a=10 and d= -3 then find its first four terms.
7. 1,7,13,19......find 18th term of this A.P.
8. In an A.P. a=4 and d=0, then find first five terms.
9. If a=6 and d=10, then find S10.
10. Decide whether the given sequence 24,17,10,3......is an A.P.? If yes
find its common term (tn) .
• Question for 3 Marks
A) complete the following activity
1) how many two-digit numbers are divisible by 5?
Activity :- Two-digit numbers divisible by 5 are, 10,15,20........95.
Here, d=5, therefore this sequence is an A.P.
Here a=10, d=5, Tn=95, n=?
tn = a+(n-1 )
= 10+(n-1)×5
= (n-1)×5
= (n-1)
therefore n=
there are two-digit numbers divisible by 5.
2. Find first four terms of an A.P., whose first term is 3 and common
difference is 4.
3. 1,6,11,16......Find the 18th term of this A.P.
4. In an A.P. a=2 and d=3, then find S12.
5. Find first four terms of the sequence tn=n+2.
6. In an A.P., a=10 and d= -3 then find its first four terms.
7. 1,7,13,19......find 18th term of this A.P.
8. In an A.P. a=4 and d=0, then find first five terms.
9. If a=6 and d=10, then find S10.
10. Decide whether the given sequence 24,17,10,3......is an A.P.? If yes
find its common term (tn) .
• Question for 3 Marks
A) complete the following activity
1) how many two-digit numbers are divisible by 5?
Activity :- Two-digit numbers divisible by 5 are, 10,15,20........95.
Here, d=5, therefore this sequence is an A.P.
Here a=10, d=5, Tn=95, n=?
tn = a+(n-1 )
= 10+(n-1)×5
= (n-1)×5
= (n-1)
therefore n=
there are two-digit numbers divisible by 5.
Page 22
2) Kalpana saves some amount every month. In first three months she
saves Rs.100, Rs.150 and Rs.200 respectively. In how many months will
she save Rs.1200?
Activity :- Kalpana’s monthly saving is Rs.100,Rs.150,Rs.200........Rs.1200
Here d=50.Therefore this sequence is an A.P.
a=10, d=50, tn= n=?
tn = a+(n-1)
= 100+(n-1)×50
= n-1
50
n =
therefore, she saves Rs.1200 in months.
3) Determine the sum of first 100 terms of given A.P. 12,14,16,18,20........
Activity :- here, a=12, d= n=100, S100=?
Sn= 2 [ + ( n - 1 ) d ]
n
S100= 2 [ 24+( 100 - 1 ) d ]
= 50 ( 24 + )
= 5
=
4) Find the sum of natural numbers between 1 to 140, which are
divisible by 4.
Activity :- Natural numbers between 1 to 140 divisible by 4 are,
4,8,12,16,......136
Here d=4, therefore this sequence is an A.P.
a=4, d=4, tn=136, Sn = ?
saves Rs.100, Rs.150 and Rs.200 respectively. In how many months will
she save Rs.1200?
Activity :- Kalpana’s monthly saving is Rs.100,Rs.150,Rs.200........Rs.1200
Here d=50.Therefore this sequence is an A.P.
a=10, d=50, tn= n=?
tn = a+(n-1)
= 100+(n-1)×50
= n-1
50
n =
therefore, she saves Rs.1200 in months.
3) Determine the sum of first 100 terms of given A.P. 12,14,16,18,20........
Activity :- here, a=12, d= n=100, S100=?
Sn= 2 [ + ( n - 1 ) d ]
n
S100= 2 [ 24+( 100 - 1 ) d ]
= 50 ( 24 + )
= 5
=
4) Find the sum of natural numbers between 1 to 140, which are
divisible by 4.
Activity :- Natural numbers between 1 to 140 divisible by 4 are,
4,8,12,16,......136
Here d=4, therefore this sequence is an A.P.
a=4, d=4, tn=136, Sn = ?
Page 23
tn = a+(n-1)d
= 4+(n-1)×4
= (n-1)×4
n=
Now,
Sn = 2 + [a + tn]
n
Sn =17 ×
Sn =
Therefore, the sum of natural numbers between 1 to 140, which are
divisible by 4 is
5) Decide whether 301 is term of given sequence 5,11,17,23,.......
Activity :- Here, d = therefore this sequence is an A.P.
a= 5, d=
Let nth term of this A.P. be 301.
tn = a+(n-1)
301 = 5+(n-1)×6
301 = 6n-1
302
n = =
6
But n is not positive integer
Therefore, 301 is the term of sequence 5,11,17,23.........
B)Solve the following sub questions.
1. Find S10 if a=6 and d=3
2. 12,16,20,24………Find 25th term of this A.P.
3. If tn=2n-5 is the nth term of an A.P., then find its first five terms.
= 4+(n-1)×4
= (n-1)×4
n=
Now,
Sn = 2 + [a + tn]
n
Sn =17 ×
Sn =
Therefore, the sum of natural numbers between 1 to 140, which are
divisible by 4 is
5) Decide whether 301 is term of given sequence 5,11,17,23,.......
Activity :- Here, d = therefore this sequence is an A.P.
a= 5, d=
Let nth term of this A.P. be 301.
tn = a+(n-1)
301 = 5+(n-1)×6
301 = 6n-1
302
n = =
6
But n is not positive integer
Therefore, 301 is the term of sequence 5,11,17,23.........
B)Solve the following sub questions.
1. Find S10 if a=6 and d=3
2. 12,16,20,24………Find 25th term of this A.P.
3. If tn=2n-5 is the nth term of an A.P., then find its first five terms.
Page 24
4. Find the sum of three-digit natural numbers, which are divisible by 4.
5. Merry got a job with salary Rs.15000 per month. If her salary
increases by Rs.100 per month, how much would be her salary after
20 months?
6. The nth term of an A.P 5,8,11,14...... is 68. Find n=?
7. What is the sum of an odd numbers between 1 to 50.
8. For an A.P., t4=12 and its common difference d=-10, then find tn.
9. Find 27th and nth term of given A.P. 5,2,-1,-4.............
10. Find the first terms and common difference of an A.P. whose t 8=3 and
t12=52.
• Question for 4 marks
Solve the following sub questions.
1. Sum of first 55 terms of an A.P. is 3300. Then find its 28th term.
2. Find the sum of numbers between 1 to 140, divisible by 4.
3. In a ‘Mahila Bachat Gat’, Sharvari invested Rs.2 on first day, Rs.4 on
second day and Rs.6 on third day. If She saves like this, then what
would be her total savings in the month of February 2010?
4. Find the sum of odd natural numbers from 1 to 101.
5. Shubhankar invested in a national savings certificate scheme. In the
first year he invested Rs.500, in the second year Rs.700, in the third
year Rs.900 and so on. Find the total amount that he invested in 12
years.
6. A merchant borrows Rs.1000 and agrees to repay its interest Rs.140
with principal in 12 monthly instalments. Each instalment being less
than the preceding one by Rs.10. Find the amount of the first first
instalment.
7. Find t21, if S41 = 4510 in an A.P.
8. In an A.P. t10=57 and t15=87 then find t21.
5. Merry got a job with salary Rs.15000 per month. If her salary
increases by Rs.100 per month, how much would be her salary after
20 months?
6. The nth term of an A.P 5,8,11,14...... is 68. Find n=?
7. What is the sum of an odd numbers between 1 to 50.
8. For an A.P., t4=12 and its common difference d=-10, then find tn.
9. Find 27th and nth term of given A.P. 5,2,-1,-4.............
10. Find the first terms and common difference of an A.P. whose t 8=3 and
t12=52.
• Question for 4 marks
Solve the following sub questions.
1. Sum of first 55 terms of an A.P. is 3300. Then find its 28th term.
2. Find the sum of numbers between 1 to 140, divisible by 4.
3. In a ‘Mahila Bachat Gat’, Sharvari invested Rs.2 on first day, Rs.4 on
second day and Rs.6 on third day. If She saves like this, then what
would be her total savings in the month of February 2010?
4. Find the sum of odd natural numbers from 1 to 101.
5. Shubhankar invested in a national savings certificate scheme. In the
first year he invested Rs.500, in the second year Rs.700, in the third
year Rs.900 and so on. Find the total amount that he invested in 12
years.
6. A merchant borrows Rs.1000 and agrees to repay its interest Rs.140
with principal in 12 monthly instalments. Each instalment being less
than the preceding one by Rs.10. Find the amount of the first first
instalment.
7. Find t21, if S41 = 4510 in an A.P.
8. In an A.P. t10=57 and t15=87 then find t21.
Page 25
9. If Rs.3900 will have to repay In 12 monthly instalments such that
each instalment being more than the preceding one by Rs.10, then
find the amount of the first and last instalment.
10. Find the next 4 terms of the sequence 6 , , also find Sn.
1 1 1
4 3
each instalment being more than the preceding one by Rs.10, then
find the amount of the first and last instalment.
10. Find the next 4 terms of the sequence 6 , , also find Sn.
1 1 1
4 3
Page 26
4.Probability
Q.1 (A) There are four alternative answers for each of the following sub-
questions. Choose the correct alternative answer for each of the following
questions and write the alphabet. (1 mark question)
1) Which of the following number cannot represent a probability?
A) 0.66 B) 1.5 C) 0.15 D) 0.7
1
2) If n(A) = 5 , P(A)= then n(S) = ?
2
3 4 1
A) 10 B) C) D)
5 5 3
3) When a dice is thrown the number of sample points in the sample space are
..........
A) 4 B) 6 C) 2 D) 52
4) In how many ways a card can be drawn from a well shuffled pack of playing
cards,
A) 4 B) 1 C) 26 D) 52
5) What is the probability of the event that a number chosen from 1 to 50 is a
prime number ?
3 1 1 3
A) B) C) D)
10 2 4 25
6) Which of the following options shows the highest probability.
4 1
A) B) 0.83 C)%58 D)
5 2
7) When two dice are thrown the number of sample points in the sample space
are
........
A) 6 B) 𝟏𝟐 C) 36 D) 52
Q.1 (B) Solve the following sub-questions. (1 mark question)
1) Write a sample space if two coins are tossed simultaneously.
2) Write a sample space when a die is thrown.
3) In a set of 25 cards , each card bears only one number from 1 to 25. One
card
Q.1 (A) There are four alternative answers for each of the following sub-
questions. Choose the correct alternative answer for each of the following
questions and write the alphabet. (1 mark question)
1) Which of the following number cannot represent a probability?
A) 0.66 B) 1.5 C) 0.15 D) 0.7
1
2) If n(A) = 5 , P(A)= then n(S) = ?
2
3 4 1
A) 10 B) C) D)
5 5 3
3) When a dice is thrown the number of sample points in the sample space are
..........
A) 4 B) 6 C) 2 D) 52
4) In how many ways a card can be drawn from a well shuffled pack of playing
cards,
A) 4 B) 1 C) 26 D) 52
5) What is the probability of the event that a number chosen from 1 to 50 is a
prime number ?
3 1 1 3
A) B) C) D)
10 2 4 25
6) Which of the following options shows the highest probability.
4 1
A) B) 0.83 C)%58 D)
5 2
7) When two dice are thrown the number of sample points in the sample space
are
........
A) 6 B) 𝟏𝟐 C) 36 D) 52
Q.1 (B) Solve the following sub-questions. (1 mark question)
1) Write a sample space if two coins are tossed simultaneously.
2) Write a sample space when a die is thrown.
3) In a set of 25 cards , each card bears only one number from 1 to 25. One
card
Page 27
is drawn randomly. Write the sample space for this random experiment?
4) A two digit number is formed with digits 2, 3, 5 without repetition, Write the
sample space ?
5) write the event in the set form for the following random experiment.
‘ If one die is thrown, the number obtained on the upper face is even.’
Q.2 (A) Complete the following activity. (2 marks question)
1) If one die is rolled then find the probability of the following event by
completing the activity.
Event A: Number on the upper face is prime.
Activity : Let ‘S’ is the sample space.
S = {1, 2, 3, 4, 5, 6} ∴ n(S) = 6
Event A : Prime number on the upper face.
A = { …………..} ∴ n(A) = 3
P(A) = ………… (Formula)
𝑛(𝑆)
=
6
1
∴ P(A) =
2) Two coins are tossed simultaneously. Write the sample space (S) and
expected
sample points in the given events by completing the activity.
i) Event A : to get at least one head.
ii) Event B : to get no head.
Activity : Let ‘S’ is the sample space , when two coins are tossed
simultaneously.
∴ { = S ,HT, TH, }
Event A : to get at least one head.
∴ A { = HH, , TH }
Event B : to get no head.
∴ B{= }
3) A card is drawn from a well shuffled pack of 52 playing cards. Find the
4) A two digit number is formed with digits 2, 3, 5 without repetition, Write the
sample space ?
5) write the event in the set form for the following random experiment.
‘ If one die is thrown, the number obtained on the upper face is even.’
Q.2 (A) Complete the following activity. (2 marks question)
1) If one die is rolled then find the probability of the following event by
completing the activity.
Event A: Number on the upper face is prime.
Activity : Let ‘S’ is the sample space.
S = {1, 2, 3, 4, 5, 6} ∴ n(S) = 6
Event A : Prime number on the upper face.
A = { …………..} ∴ n(A) = 3
P(A) = ………… (Formula)
𝑛(𝑆)
=
6
1
∴ P(A) =
2) Two coins are tossed simultaneously. Write the sample space (S) and
expected
sample points in the given events by completing the activity.
i) Event A : to get at least one head.
ii) Event B : to get no head.
Activity : Let ‘S’ is the sample space , when two coins are tossed
simultaneously.
∴ { = S ,HT, TH, }
Event A : to get at least one head.
∴ A { = HH, , TH }
Event B : to get no head.
∴ B{= }
3) A card is drawn from a well shuffled pack of 52 playing cards. Find the
Page 28
probability of i) Event A : Card drawn is a red card.
Activity : Let ‘S’ is the sample space. ∴ n(S) = 52
Event A : Card drawn is a red card.
∴ Total red cards = ( ) hearts + 13 diamonds
∴ n(A) = ( )
∴ P(A) = ………… Formula
𝑛(𝑆)
26
P(A) =
52
P(A) =
4) In Adarsh High School , out of 30 students in a class 3 students wear
glasses(spectacles). If a student in the class is randomly selected, find the
probability that he or she wears glasses(spectacles) by completing the
following activity.
Activity : There are a total of 30 students in the class.
∴ n(S) =
Event : A Selected student wears glasses(spectacles)
∴ n(A) =
P(A) = 𝑛(𝑆) …………Formula
P (A) =
Q.2 (B) Solve the following sub-questions. (2 marks question)
1) A card is drawn at random from a pack of well shuffled 52 playing cards.
Find the probability that the card drawn is a spade.
2) If two coins are tossed, find the probability of event getting head on both
the
coins.
3) If one die is rolled then find the probability of event that the number on the
upper face is greater than 6?
4) If three coins are tossed simultaneously, find the probability of the event to
get no head
5) There are 30 cards in a box, each bearing one of the numbers from 1 to 30.
Activity : Let ‘S’ is the sample space. ∴ n(S) = 52
Event A : Card drawn is a red card.
∴ Total red cards = ( ) hearts + 13 diamonds
∴ n(A) = ( )
∴ P(A) = ………… Formula
𝑛(𝑆)
26
P(A) =
52
P(A) =
4) In Adarsh High School , out of 30 students in a class 3 students wear
glasses(spectacles). If a student in the class is randomly selected, find the
probability that he or she wears glasses(spectacles) by completing the
following activity.
Activity : There are a total of 30 students in the class.
∴ n(S) =
Event : A Selected student wears glasses(spectacles)
∴ n(A) =
P(A) = 𝑛(𝑆) …………Formula
P (A) =
Q.2 (B) Solve the following sub-questions. (2 marks question)
1) A card is drawn at random from a pack of well shuffled 52 playing cards.
Find the probability that the card drawn is a spade.
2) If two coins are tossed, find the probability of event getting head on both
the
coins.
3) If one die is rolled then find the probability of event that the number on the
upper face is greater than 6?
4) If three coins are tossed simultaneously, find the probability of the event to
get no head
5) There are 30 cards in a box, each bearing one of the numbers from 1 to 30.
Page 29
One card is drawn at random from the box. Find the probability of event
that the card drawn shows a number which is a multiple of 5.
Q.3 (A) Complete the following activity. (3 marks question)
1) A box contains 5 strawberry chocolates, 6 coffee chocolates and 2
peppermint chocolates. If one of the chocolates is picked from the box at
randomly, Find the probability of the following events by completing the
activity. (i) Event A : it is a coffee chocolate.
(ii) Event B : it is a peppermint chocolate.
Activity : Let ‘S’ is the sample space.
∴ n(S) = 5+6+2 = 13
(i) Event A : it is a coffee chocolate
∴ n(A) =
∴ P(A) = 𝑛(𝑆) ………… Formula
P(A) = 13
(ii) Event B : it is a peppermint chocolate.
∴ n(B) =
∴ P(B) = 𝑛(𝑆) …………Formula
P(B) = 13
Q.3 (B) Solve the following sub-questions. (3 marks question)
1) If two dice are rolled simultaneously, find the probability of the
following events.
i) Event A : The sum of the digits on the upper faces is at least 10.
ii) Event B : The sum of the digits on the upper faces is 33.
2) If Three coins are tossed simultaneously, find the probability of the
following events.
i) Event A : To get no heads.
ii) Event B : To get at least two heads.
3) If One coin and one die are thrown simultaneously, find the probability of
the following events.
that the card drawn shows a number which is a multiple of 5.
Q.3 (A) Complete the following activity. (3 marks question)
1) A box contains 5 strawberry chocolates, 6 coffee chocolates and 2
peppermint chocolates. If one of the chocolates is picked from the box at
randomly, Find the probability of the following events by completing the
activity. (i) Event A : it is a coffee chocolate.
(ii) Event B : it is a peppermint chocolate.
Activity : Let ‘S’ is the sample space.
∴ n(S) = 5+6+2 = 13
(i) Event A : it is a coffee chocolate
∴ n(A) =
∴ P(A) = 𝑛(𝑆) ………… Formula
P(A) = 13
(ii) Event B : it is a peppermint chocolate.
∴ n(B) =
∴ P(B) = 𝑛(𝑆) …………Formula
P(B) = 13
Q.3 (B) Solve the following sub-questions. (3 marks question)
1) If two dice are rolled simultaneously, find the probability of the
following events.
i) Event A : The sum of the digits on the upper faces is at least 10.
ii) Event B : The sum of the digits on the upper faces is 33.
2) If Three coins are tossed simultaneously, find the probability of the
following events.
i) Event A : To get no heads.
ii) Event B : To get at least two heads.
3) If One coin and one die are thrown simultaneously, find the probability of
the following events.
Page 30
i) Event A : To get a tail and an even number.
ii) Event B: To get head and an odd number.
4) A balloon vendor has 2 red, 3 blue and 4 green balloons. He wants to
choose one of them at random to give it to Pranali. What is the
probability of the event that Pranali gets,
(i) a red balloon
(ii) a blue balloon.
Q.4 Solve the following sub-questions. (4 marks question)
1) What is the probability that an ordinary year has 53 Sundays?
2) What is the probability that a leap year has 53 Sundays?
3) A box contains 36 cards, bearing only one number from 1 to 36 on each.
If one card is drawn at random, find the probability of an event that the
card drawn bears,
(i) a complete square number.
(ii) a prime number.
(iii)a number divisible by 3.
Q.5 Creative questions of 3 marks
1) A bag contains 5 white balls and some blue balls. If the probability of
drawing a blue ball is double that of a white ball, determine the number of
blue balls in the bag .
2) The faces of a die bear numbers 0, 1, 2, 3, 4, 5. If the die is rolled twice,
then find the probability that the product of digits on the upper face is zero.
2) A missing helicopter is reported to have crashed somewhere in the
rectangular region shown in the figure .
ii) Event B: To get head and an odd number.
4) A balloon vendor has 2 red, 3 blue and 4 green balloons. He wants to
choose one of them at random to give it to Pranali. What is the
probability of the event that Pranali gets,
(i) a red balloon
(ii) a blue balloon.
Q.4 Solve the following sub-questions. (4 marks question)
1) What is the probability that an ordinary year has 53 Sundays?
2) What is the probability that a leap year has 53 Sundays?
3) A box contains 36 cards, bearing only one number from 1 to 36 on each.
If one card is drawn at random, find the probability of an event that the
card drawn bears,
(i) a complete square number.
(ii) a prime number.
(iii)a number divisible by 3.
Q.5 Creative questions of 3 marks
1) A bag contains 5 white balls and some blue balls. If the probability of
drawing a blue ball is double that of a white ball, determine the number of
blue balls in the bag .
2) The faces of a die bear numbers 0, 1, 2, 3, 4, 5. If the die is rolled twice,
then find the probability that the product of digits on the upper face is zero.
2) A missing helicopter is reported to have crashed somewhere in the
rectangular region shown in the figure .
Page 31
What is the probability that it crashed inside the lake shown in the figure ?
LAKE
4) Three horses A ,B and C are in a race. A is twice as likely to win as B and
B is twice as likely to win as C What is their individual probability of
winning?
5) A bag contains 8 red balls and some blue balls. If one ball is drawn
randomly the probability of drawing a red ball to a blue ball are in the
ratio 5 : 2, determine the probability of drawing a blue ball from the
bag.
LAKE
4) Three horses A ,B and C are in a race. A is twice as likely to win as B and
B is twice as likely to win as C What is their individual probability of
winning?
5) A bag contains 8 red balls and some blue balls. If one ball is drawn
randomly the probability of drawing a red ball to a blue ball are in the
ratio 5 : 2, determine the probability of drawing a blue ball from the
bag.
Page 32
महाराष्ट्र शासन
शालेय शशक्षण व क्रीडा शवभाग
राज्य शैक्षशणक संशोधन व प्रशशक्षण पररषद, महाराष्ट्र
७०८ सदाशिव पेठ, कुमठे कर मार्ग, पुणे ४११०३०
संपकग क्रमांक (020) 2447 6938 E mail: [email protected]
-----------------------------------------------------------------------------------------------
Question Bank
Standard :- 10th
Subject :- Mathematics Part 2
सूचना
१. फक्त शवद्यार्थ्यांना प्रश्नप्रकारांचा सराव करून देण्यासाठीच
२. सदर प्रश्नसंचातील प्रश्न बोर्डाच्या प्रश्नपशिकेत येतीलच असे नाही
याची नोंद घ्यावी.
शालेय शशक्षण व क्रीडा शवभाग
राज्य शैक्षशणक संशोधन व प्रशशक्षण पररषद, महाराष्ट्र
७०८ सदाशिव पेठ, कुमठे कर मार्ग, पुणे ४११०३०
संपकग क्रमांक (020) 2447 6938 E mail: [email protected]
-----------------------------------------------------------------------------------------------
Question Bank
Standard :- 10th
Subject :- Mathematics Part 2
सूचना
१. फक्त शवद्यार्थ्यांना प्रश्नप्रकारांचा सराव करून देण्यासाठीच
२. सदर प्रश्नसंचातील प्रश्न बोर्डाच्या प्रश्नपशिकेत येतीलच असे नाही
याची नोंद घ्यावी.
Page 33
Class-10
Mathematics part-2
Question bank
1.Similarity
Q.1 A) MCQ ( 1 Mark)
1.If ∆ABC~ ∆PQR and AB: PQ = 3: 4 then A(∆ABC): A(∆PQR) =?
(A)9:25 (B) 9:16 (C) 16:9 (D)25:9
2.Which of the following is not a test of similarity?
(A)AAA (B)SAS (C) SAA (D)SSS
3.If ∆XYZ ~ ∆PQR and A(∆XYZ ) = 25 𝑐𝑚2 , A(∆PQR) = 4 𝑐𝑚2 then XY: PQ =?
(A) 4:25 (B)2:5 (C) 5:2 (D)25:4
4.Ratio of areas of two similar tringles is 9:25. _____ is the ratio of their
corresponding sides.
(A)3:4 (B)3 :5 (C) 5:3 (D)25:81
5. 𝐺𝑖𝑣𝑒𝑛 ∆ABC~ ∆DEF, if ⦟A = 45° and ⦟E = 35° then ⦟B =?
(A) 45° (B)35° (C)25° (D) 40°
6. In fig,seg DE ⃦ seg BC, identify correct statement.
AD AE AD AB
(A)DB =AC (B)DB =AC
AD EC AD AE
(C) DB =AC (D) DB =EC
XY YZ
7.If ∆XYZ~ ∆PQR then PQ=QR =?
Mathematics part-2
Question bank
1.Similarity
Q.1 A) MCQ ( 1 Mark)
1.If ∆ABC~ ∆PQR and AB: PQ = 3: 4 then A(∆ABC): A(∆PQR) =?
(A)9:25 (B) 9:16 (C) 16:9 (D)25:9
2.Which of the following is not a test of similarity?
(A)AAA (B)SAS (C) SAA (D)SSS
3.If ∆XYZ ~ ∆PQR and A(∆XYZ ) = 25 𝑐𝑚2 , A(∆PQR) = 4 𝑐𝑚2 then XY: PQ =?
(A) 4:25 (B)2:5 (C) 5:2 (D)25:4
4.Ratio of areas of two similar tringles is 9:25. _____ is the ratio of their
corresponding sides.
(A)3:4 (B)3 :5 (C) 5:3 (D)25:81
5. 𝐺𝑖𝑣𝑒𝑛 ∆ABC~ ∆DEF, if ⦟A = 45° and ⦟E = 35° then ⦟B =?
(A) 45° (B)35° (C)25° (D) 40°
6. In fig,seg DE ⃦ seg BC, identify correct statement.
AD AE AD AB
(A)DB =AC (B)DB =AC
AD EC AD AE
(C) DB =AC (D) DB =EC
XY YZ
7.If ∆XYZ~ ∆PQR then PQ=QR =?
Page 34
XZ XZ XZ YZ
(A) PR (B)PQ (C) QR (D) PQ
8. If ∆ABC~ ∆LMN and ⦟A = 60° then ⦟L =?
(A) 45° (B)60° (C)25° (D) 40°
DE FE
9. In ∆DEF and ∆XYZ , = & ⦟E ≅ ⦟Y ______ test gives similarity
XY YZ
between ∆DEF & ∆XYZ.
(A)AAA (B)SAS
(C) SAA (D)SSS
A(∆ABC)
10. In fig BD=8, BC=12 B-D-C then =?
A(∆ABD)
(A)2:3 (B)3:2
(C) 5:3 (D)3:4
Q.1 B) Solve 1 mark
B.1 Are triangles in figure similar ? If yes then write the test of similarity.
2. In fig line BC ⃦ line DE, AB=2 ,BD=3 ,AC=4 and CE= x , then find the value of x.
(A) PR (B)PQ (C) QR (D) PQ
8. If ∆ABC~ ∆LMN and ⦟A = 60° then ⦟L =?
(A) 45° (B)60° (C)25° (D) 40°
DE FE
9. In ∆DEF and ∆XYZ , = & ⦟E ≅ ⦟Y ______ test gives similarity
XY YZ
between ∆DEF & ∆XYZ.
(A)AAA (B)SAS
(C) SAA (D)SSS
A(∆ABC)
10. In fig BD=8, BC=12 B-D-C then =?
A(∆ABD)
(A)2:3 (B)3:2
(C) 5:3 (D)3:4
Q.1 B) Solve 1 mark
B.1 Are triangles in figure similar ? If yes then write the test of similarity.
2. In fig line BC ⃦ line DE, AB=2 ,BD=3 ,AC=4 and CE= x , then find the value of x.
Page 35
3.State whether the following triangles are similar or not : If yes , then write the test of
similarity.
P x ⦟P = 35° , ⦟x = 35° and ⦟Q = 60°, ⦟Y =
Y Z
Q 60° R
4. If ∆ABC~ ∆LMN & ⦟B = 40° then ⦟M =? Give reason .
5.Areas of two simlar triangles are in the ratio 144:49. Find the ratio of their
corresponding sides.
6. ∆PQR~ ∆SUV write pair of congruent angle.
7. ∆ABC~ ∆DEF write ratio of their corresponding sides.
8. R In fig. TP =10 cm PS=6 cm
A∆(RTP)
=?
A(∆RPS)
T P S
9.Ratio of corresponding sides of two similar triangles is 4:7 then find the ratio of
their areas = ?
similarity.
P x ⦟P = 35° , ⦟x = 35° and ⦟Q = 60°, ⦟Y =
Y Z
Q 60° R
4. If ∆ABC~ ∆LMN & ⦟B = 40° then ⦟M =? Give reason .
5.Areas of two simlar triangles are in the ratio 144:49. Find the ratio of their
corresponding sides.
6. ∆PQR~ ∆SUV write pair of congruent angle.
7. ∆ABC~ ∆DEF write ratio of their corresponding sides.
8. R In fig. TP =10 cm PS=6 cm
A∆(RTP)
=?
A(∆RPS)
T P S
9.Ratio of corresponding sides of two similar triangles is 4:7 then find the ratio of
their areas = ?
Page 36
10. Write the test of similarity for triangles given in figure.
Q.2 A.Complete the activity 2marks
1. A in fig. BP AC,CQ AB A-P-C
P
Q & A-Q-B then show that
∆APB & ∆AQC are similar
B C In ∆APB & ∆AQC ⦟APB = [ ]0 … (𝐼)
⦟AQC = [ ]0 … (𝐼𝐼)
⦟APB ≅ ⦟AQC (I) & (II)
⦟PAB ≅ ⦟QAC [...........]
∆APB~ ∆AQC [..........]
2.Observe the figure & complete following activity.
in fig⦟B = 750 , ⦟D = 750
⦟B ≅ [… . ] each of 750
⦟C ≅ ⦟C [....]
∆ABC~ ∆[........]
....[.....]similarity test
3. ∆ABC~ ∆PQR , A( ∆ABC)= 80sqcm A(∆PQR) = 125 sqcm then complete
Q.2 A.Complete the activity 2marks
1. A in fig. BP AC,CQ AB A-P-C
P
Q & A-Q-B then show that
∆APB & ∆AQC are similar
B C In ∆APB & ∆AQC ⦟APB = [ ]0 … (𝐼)
⦟AQC = [ ]0 … (𝐼𝐼)
⦟APB ≅ ⦟AQC (I) & (II)
⦟PAB ≅ ⦟QAC [...........]
∆APB~ ∆AQC [..........]
2.Observe the figure & complete following activity.
in fig⦟B = 750 , ⦟D = 750
⦟B ≅ [… . ] each of 750
⦟C ≅ ⦟C [....]
∆ABC~ ∆[........]
....[.....]similarity test
3. ∆ABC~ ∆PQR , A( ∆ABC)= 80sqcm A(∆PQR) = 125 sqcm then complete
Page 37
A(∆ABC) 80 [….] AB […..]
= 125 = [….] hence PQ=[…..]
A( ∆PQR)
4.in fig.PM=10 cm A( ∆PQS)= 100sqcm A( ∆QRS) = 110sqcm then NR?
∆PQS &∆QRS having seg QS common base
Areas of two triangles whose base are common, are in proportion of
their corresponding [.......]
A(∆PQS) [….] 100 [….]
= NR , = NR , NR = [.....] cm
A( ∆QRS) 110
Q.2 B A 1. In figAB BC and DC BC AB=6, DC=4
A(∆ABC)
𝑡ℎ𝑒𝑛 =?
A(∆BCD)
B C
D
2. In fig seg AC & seg BD intersect each other at point p
AP BP
=PD then prove that ∆ABP~ ∆DPC
PC
3. ∆ABP~ ∆DEF & A( ∆ABP): A(∆DEF) = 144: 81 then AB: DE =?
4. From given information is PQ ⃦ BC ?
A AP=2, PB=4 AQ=3,QC=6
p P Q
= 125 = [….] hence PQ=[…..]
A( ∆PQR)
4.in fig.PM=10 cm A( ∆PQS)= 100sqcm A( ∆QRS) = 110sqcm then NR?
∆PQS &∆QRS having seg QS common base
Areas of two triangles whose base are common, are in proportion of
their corresponding [.......]
A(∆PQS) [….] 100 [….]
= NR , = NR , NR = [.....] cm
A( ∆QRS) 110
Q.2 B A 1. In figAB BC and DC BC AB=6, DC=4
A(∆ABC)
𝑡ℎ𝑒𝑛 =?
A(∆BCD)
B C
D
2. In fig seg AC & seg BD intersect each other at point p
AP BP
=PD then prove that ∆ABP~ ∆DPC
PC
3. ∆ABP~ ∆DEF & A( ∆ABP): A(∆DEF) = 144: 81 then AB: DE =?
4. From given information is PQ ⃦ BC ?
A AP=2, PB=4 AQ=3,QC=6
p P Q
Page 38
B C
5. Areas of two similar triangles are 225 𝑐𝑚2 and ,81 𝑐𝑚2 if side of smaller
triangle is 12cm. find corresponding side of major triangle
6. D from adjoining figure
A ⦟ABC = 90° ⦟DCB = 90° AB = 6,
A(∆ABC)
6 8 DC=8 then =?
A(∆BCD)
B C
Q.3A) Complete the following activity 3 marks
.
1. ∆ABC APpendicular BC & BQ perpendicular AC , B-P-C,A-Q-C
then show that ∆CPA~ ∆CQB if AP=7,BQ=8 BC=12
0
then AC=? 𝐼𝑛∆CPAand ∆CQB ⦟CPA ≅ [⦟ … ].(each 90 )
⦟ACP ≅ [⦟ … ].(common angle)
∆CPA~ ∆CQB (..........similarity test )
AP [….]
= BC (corresponding sides of similar triangle)
BQ
7 [….]
8
= 12
ACx[.....]=7x12 AC=10.5
2. A line is parallel to one side of triangle which intersects remaining two sides in two
distinct point then that line divdes sides in same proportion.
5. Areas of two similar triangles are 225 𝑐𝑚2 and ,81 𝑐𝑚2 if side of smaller
triangle is 12cm. find corresponding side of major triangle
6. D from adjoining figure
A ⦟ABC = 90° ⦟DCB = 90° AB = 6,
A(∆ABC)
6 8 DC=8 then =?
A(∆BCD)
B C
Q.3A) Complete the following activity 3 marks
.
1. ∆ABC APpendicular BC & BQ perpendicular AC , B-P-C,A-Q-C
then show that ∆CPA~ ∆CQB if AP=7,BQ=8 BC=12
0
then AC=? 𝐼𝑛∆CPAand ∆CQB ⦟CPA ≅ [⦟ … ].(each 90 )
⦟ACP ≅ [⦟ … ].(common angle)
∆CPA~ ∆CQB (..........similarity test )
AP [….]
= BC (corresponding sides of similar triangle)
BQ
7 [….]
8
= 12
ACx[.....]=7x12 AC=10.5
2. A line is parallel to one side of triangle which intersects remaining two sides in two
distinct point then that line divdes sides in same proportion.
Page 39
Given :In ∆ABC line l II side BC & line l intersect side AB in P & side
AC in Q A
AP AQ
P Q Q Given: PB = QC construction :draw CP & BQ
𝑃𝑟𝑜𝑜𝑓: ∆APQ&∆PQB have equal
height
A(∆APQ) [….]
B C A( ∆PQB)
=
PB
(areas in proportion of base)I
A(∆APQ) [….]
= (areas in proportion of baseII
A( ∆PQC) QC
∆PQC&∆PQB have [.....]is common base
SegPQ II Seg BC hence height of:
∆APQ&∆PQB
𝐴(∆PQC)=A( ∆......)...........(III)
A(∆APQ) A(∆ …….)
= ..............[(I),(II)&(III)]
A( ∆PQB) A( ∆ ……… )
AP AQ
PB
= QC ............[(I) & (II)
,
From fig.seg PQ II side BC
AP= x +3 ,PB=x -3,AQ= x +5 ,QC=x-2
then complete the activity to find the
3. A value of x
in∆PQB, PQ II side BC
x+3 x+5
AP AQ
= [… ] ..........([...........])
PB
P Q
x+3 x+5
=
x−3 [… ]
x-3 x-2
(x+3)[......]=(x+5)(x-3)
B C
𝑥 2 +x-[....]=𝑥 2 +2x-15
x=[....]
AC in Q A
AP AQ
P Q Q Given: PB = QC construction :draw CP & BQ
𝑃𝑟𝑜𝑜𝑓: ∆APQ&∆PQB have equal
height
A(∆APQ) [….]
B C A( ∆PQB)
=
PB
(areas in proportion of base)I
A(∆APQ) [….]
= (areas in proportion of baseII
A( ∆PQC) QC
∆PQC&∆PQB have [.....]is common base
SegPQ II Seg BC hence height of:
∆APQ&∆PQB
𝐴(∆PQC)=A( ∆......)...........(III)
A(∆APQ) A(∆ …….)
= ..............[(I),(II)&(III)]
A( ∆PQB) A( ∆ ……… )
AP AQ
PB
= QC ............[(I) & (II)
,
From fig.seg PQ II side BC
AP= x +3 ,PB=x -3,AQ= x +5 ,QC=x-2
then complete the activity to find the
3. A value of x
in∆PQB, PQ II side BC
x+3 x+5
AP AQ
= [… ] ..........([...........])
PB
P Q
x+3 x+5
=
x−3 [… ]
x-3 x-2
(x+3)[......]=(x+5)(x-3)
B C
𝑥 2 +x-[....]=𝑥 2 +2x-15
x=[....]
Page 40
Q.3 B 3 marks
1. There are two poles having heights 8m & 4m on
plane ground as shown in fig. Because of sunlight
shadow of smaller pole is 6m long then find the length
of shadow of longer pole.
2.In ∆ABC B-D-C & BD=7, BC=20 then find the following ratio
A(∆ABD)
1) A( ∆ADC)
A(∆ABD)
2)
A( ∆ABC)
A(∆ADC)
3)
A( ∆ABC)
3. In given fig.quadrilateral PQRS side PQ II ⃦side SR ,AR=5 AP,
then prove that , SR=5PQ
4.
In triangle ABC point D is on side BC (B-D-C) such that
⦟BAC = ⦟ADC then prove that CA2 = CBxCD
5. A D In Quadrlateral ABCD Side AD II BC diagonal AC &
AP PC
BD intersct in point P then prove that PD = BP
P
B C
1. There are two poles having heights 8m & 4m on
plane ground as shown in fig. Because of sunlight
shadow of smaller pole is 6m long then find the length
of shadow of longer pole.
2.In ∆ABC B-D-C & BD=7, BC=20 then find the following ratio
A(∆ABD)
1) A( ∆ADC)
A(∆ABD)
2)
A( ∆ABC)
A(∆ADC)
3)
A( ∆ABC)
3. In given fig.quadrilateral PQRS side PQ II ⃦side SR ,AR=5 AP,
then prove that , SR=5PQ
4.
In triangle ABC point D is on side BC (B-D-C) such that
⦟BAC = ⦟ADC then prove that CA2 = CBxCD
5. A D In Quadrlateral ABCD Side AD II BC diagonal AC &
AP PC
BD intersct in point P then prove that PD = BP
P
B C
Page 41
Q.4 4 marks
1. Side of eqilateral triangle PQR is 8 cm then find the area of triangle whose side is
half of side of triangle PQR
2.Areas of two similar triangle are equal then prove that triangles are congruent
3.Two triangles are similar .Smaller triangle sides are 4 cm ,5 cm,6 cm perimter of
larger triangle is 90 cm then find the sides of larger triangle.
Q.5 3 marks
1. ln fig , PS = 2, SQ=6 QR = 5, PT = x & TR = y. then find the pair of value of x&y
such that ST ll side QR.
P
S T
Q 5 R
2 .An architecture have model of building, length of building is 1m then length of
model is 0.75cm then find length & height of model building whose actual length is
22.5m& heght is 10m.
1. Side of eqilateral triangle PQR is 8 cm then find the area of triangle whose side is
half of side of triangle PQR
2.Areas of two similar triangle are equal then prove that triangles are congruent
3.Two triangles are similar .Smaller triangle sides are 4 cm ,5 cm,6 cm perimter of
larger triangle is 90 cm then find the sides of larger triangle.
Q.5 3 marks
1. ln fig , PS = 2, SQ=6 QR = 5, PT = x & TR = y. then find the pair of value of x&y
such that ST ll side QR.
P
S T
Q 5 R
2 .An architecture have model of building, length of building is 1m then length of
model is 0.75cm then find length & height of model building whose actual length is
22.5m& heght is 10m.
Page 42
2. PYTHAGORAS THEOREM
Que. 1 (A). Choose the correct alternative from those given below
(1 mark each )
1. Out of given triplets, which is a Pythagoras triplet ?
(A) (1,5,10) (B) (3,4,5) (C) (2,2,2) (D) (5,5,2)
2. Out of given triplets, which is not a Pythagoras triplet ?
(A) (5,12,13) (B) (8,15,17) (C) (7,8,15) (D) (24,25,7)
3. Out of given triplets, which is not a Pythagoras triplet ?
(A) (9,40,41) (B) (11,60,61) (C) (6,14,15) (D) (6,8,10)
4. In right angled triangle, if sum of square of sides of right angle is
169 then what is the length of hypotenuse?
(A) 15 (B) 13 (C) 5 (D) 12
5. A rectangle having length of a side is 12 and length of diagonal is
20 then what is length of other side?
(A)2 (B) 13 (C) 5 (D) 16
6. If the length of diagonal of square is √2 then what is the length of
each side ?
(A)2 (B)√3 (C) 1 (D) 4
7. If length of both diagonals of rhombus are 60 and 80 then what is
the length of side?
Que. 1 (A). Choose the correct alternative from those given below
(1 mark each )
1. Out of given triplets, which is a Pythagoras triplet ?
(A) (1,5,10) (B) (3,4,5) (C) (2,2,2) (D) (5,5,2)
2. Out of given triplets, which is not a Pythagoras triplet ?
(A) (5,12,13) (B) (8,15,17) (C) (7,8,15) (D) (24,25,7)
3. Out of given triplets, which is not a Pythagoras triplet ?
(A) (9,40,41) (B) (11,60,61) (C) (6,14,15) (D) (6,8,10)
4. In right angled triangle, if sum of square of sides of right angle is
169 then what is the length of hypotenuse?
(A) 15 (B) 13 (C) 5 (D) 12
5. A rectangle having length of a side is 12 and length of diagonal is
20 then what is length of other side?
(A)2 (B) 13 (C) 5 (D) 16
6. If the length of diagonal of square is √2 then what is the length of
each side ?
(A)2 (B)√3 (C) 1 (D) 4
7. If length of both diagonals of rhombus are 60 and 80 then what is
the length of side?
Page 43
(A)100 (B)50 (C) 200 (D) 400
8. If length of sides of triangle are a ,b, c and a2 + b2 = c2 then which
type of triangle it is ?
(A)Obtuse angled triangle (B) Acute angled triangle
(C) Equilateral triangle (D)Right angled triangle
9. In ∆ABC, AB = 6√3 cm, AC = 12 cm, and BC = 6 cm then mA
=?
(A)300 (B) 600 (C) 900 (D) 450
10. The diagonal of a square is 10 √2 cm then its perimeter is ......... .
(A)10 cm. (B) 40√2 cm. (C) 20 cm. (D) 40 cm.
11. Out of all numbers from given dates, which is a Pythagoras triplet
?
(A)15/8/17 (B)16/8/16 (C) 3/5/17 (D) 4/9/15
Que. 1 (B). Solve the following questions : (1 mark each )
1.Height and base of a right angled triangle are 24 cm and 18 cm find
the length of its hypotenus ?
2. From given figure, In ∆ ABC, AB⊥ BC, AB =BC then m A = ?
A
B C
3. From given figure, In ∆ ABC, AB⊥ BC, AB =BC, AC = 2√2 then
Ɩ (AB) = ? A
8. If length of sides of triangle are a ,b, c and a2 + b2 = c2 then which
type of triangle it is ?
(A)Obtuse angled triangle (B) Acute angled triangle
(C) Equilateral triangle (D)Right angled triangle
9. In ∆ABC, AB = 6√3 cm, AC = 12 cm, and BC = 6 cm then mA
=?
(A)300 (B) 600 (C) 900 (D) 450
10. The diagonal of a square is 10 √2 cm then its perimeter is ......... .
(A)10 cm. (B) 40√2 cm. (C) 20 cm. (D) 40 cm.
11. Out of all numbers from given dates, which is a Pythagoras triplet
?
(A)15/8/17 (B)16/8/16 (C) 3/5/17 (D) 4/9/15
Que. 1 (B). Solve the following questions : (1 mark each )
1.Height and base of a right angled triangle are 24 cm and 18 cm find
the length of its hypotenus ?
2. From given figure, In ∆ ABC, AB⊥ BC, AB =BC then m A = ?
A
B C
3. From given figure, In ∆ ABC, AB⊥ BC, AB =BC, AC = 2√2 then
Ɩ (AB) = ? A
Page 44
B C
4. From given figure, In ∆ ABC, AB⊥BC, AB =BC, AC = 5√2 then
what is the height of ∆ ABC ?
A
B C
5. Find the height of an equilateral triangle having side 4 cm. ?
6. From given figure, In ∆ ABQ, If AQ = 8 cm. then AB = ?
A
300
B Q
7. In right angled triangle, if length of hypotenuse is 25 cm. and
height is 7 cm. then what is the length of its base ?
8. If a triangle having sides 50 cm., 14 cm, and 48 cm., then state
wheather given triangle is right angled triangle or not.
9. If a triangle having sides 8 cm., 15 cm., and 17 cm., then state
wheather given triangle is right angled triangle or not.
10. A rectangle having dimensions 35 m X 12 m, then what is the
length of its diagonal ?
Que. 2 (A). Complete the following activities ( 2 marks each )
* ( Write complete answers, don’t just fill the boxes )
1. From given figure, In ∆ ABC, If AC = 12 cm. then AB = ?
4. From given figure, In ∆ ABC, AB⊥BC, AB =BC, AC = 5√2 then
what is the height of ∆ ABC ?
A
B C
5. Find the height of an equilateral triangle having side 4 cm. ?
6. From given figure, In ∆ ABQ, If AQ = 8 cm. then AB = ?
A
300
B Q
7. In right angled triangle, if length of hypotenuse is 25 cm. and
height is 7 cm. then what is the length of its base ?
8. If a triangle having sides 50 cm., 14 cm, and 48 cm., then state
wheather given triangle is right angled triangle or not.
9. If a triangle having sides 8 cm., 15 cm., and 17 cm., then state
wheather given triangle is right angled triangle or not.
10. A rectangle having dimensions 35 m X 12 m, then what is the
length of its diagonal ?
Que. 2 (A). Complete the following activities ( 2 marks each )
* ( Write complete answers, don’t just fill the boxes )
1. From given figure, In ∆ ABC, If AC = 12 cm. then AB = ?
Page 45
A
300
B C
Activity : From given figure, In ∆ ABC, ABC = 900, ACB = 300
∴ BAC =
∴ ∆ABC is 300– 600– 900 ∆.
∴ In ∆ABC by Property of 300– 600– 900 ∆.
1 √3
∴AB = AC and = AC.
2 2
1 √3
∴ = × 12 And BC = × 12
2 2
∴ = 6 व BC = 6 √3 .
2. From given figure, In ∆ABC, AD⊥BC, then prove that
AB2+CD2 =BD2 +AC2 by completing activity.
B
D
C A
Activity : From given figure, In ∆ ABC, By pythagoras theorem
AC2 =AD2 +
∴ AD2 =AC2–CD2….. (I)
Also, In ∆ABD, by pythagoras theorem,
AB2 = +BD2
∴ AD2 =AB2–BD2…..….. (II)
300
B C
Activity : From given figure, In ∆ ABC, ABC = 900, ACB = 300
∴ BAC =
∴ ∆ABC is 300– 600– 900 ∆.
∴ In ∆ABC by Property of 300– 600– 900 ∆.
1 √3
∴AB = AC and = AC.
2 2
1 √3
∴ = × 12 And BC = × 12
2 2
∴ = 6 व BC = 6 √3 .
2. From given figure, In ∆ABC, AD⊥BC, then prove that
AB2+CD2 =BD2 +AC2 by completing activity.
B
D
C A
Activity : From given figure, In ∆ ABC, By pythagoras theorem
AC2 =AD2 +
∴ AD2 =AC2–CD2….. (I)
Also, In ∆ABD, by pythagoras theorem,
AB2 = +BD2
∴ AD2 =AB2–BD2…..….. (II)
Page 46
∴ - BD2 = AC2-
∴ AB2+CD2 =AC2+ BD2
3. From given figure, In ∆ ABC, If ABC = 900 CAB=300 , AC =
14 then for finding value of AB and BC, complete the following
activity.
A
30°
B C
Activity : In ∆ ABC, If ABC = 900 CAB=300
∴ BCA =
By theorem of 300– 600– 900 ∆le,
1 √3
∴ = AC and = AC
2 2
1 √3
∴ BC = × & AB = × 14
2 2
∴ BC = 7 & AB =7 √3 .
4. From given figure, In ∆ MNK, If MNK = 900 M=450 , MK = 6
then for finding value of MK and KN, complete the following
activity.
∴ AB2+CD2 =AC2+ BD2
3. From given figure, In ∆ ABC, If ABC = 900 CAB=300 , AC =
14 then for finding value of AB and BC, complete the following
activity.
A
30°
B C
Activity : In ∆ ABC, If ABC = 900 CAB=300
∴ BCA =
By theorem of 300– 600– 900 ∆le,
1 √3
∴ = AC and = AC
2 2
1 √3
∴ BC = × & AB = × 14
2 2
∴ BC = 7 & AB =7 √3 .
4. From given figure, In ∆ MNK, If MNK = 900 M=450 , MK = 6
then for finding value of MK and KN, complete the following
activity.
Page 47
M
45°
N K
Activity : In ∆ MNK, If MNK = 900 M=450 ...( given )
∴K= .... ( remaining angles of ∆ MNK )
By theorem of 450– 450– 900 ∆le,
1 1
∴ = MK and = MK
√2 √2
1 1
∴ MN = × & KN = × 6
√2 √2
∴ MN = 3 √2 . & KN = 3√2 .
5. A ladder 10 m long reaches a window 8m above the ground. Find
the distance of the foot of the ladder from the base of wall. Complete
the given activity.
Activity : as shown in fig. suppose
P
Q R
45°
N K
Activity : In ∆ MNK, If MNK = 900 M=450 ...( given )
∴K= .... ( remaining angles of ∆ MNK )
By theorem of 450– 450– 900 ∆le,
1 1
∴ = MK and = MK
√2 √2
1 1
∴ MN = × & KN = × 6
√2 √2
∴ MN = 3 √2 . & KN = 3√2 .
5. A ladder 10 m long reaches a window 8m above the ground. Find
the distance of the foot of the ladder from the base of wall. Complete
the given activity.
Activity : as shown in fig. suppose
P
Q R
Page 48
PR is the length of ladder = 10 m
At P – window, At Q – base of wall, At R – foot of ladder
∴ PQ = 6 m
∴ QR = ?
In ∆PQR , m PQR = 900
By Pythagoras Theorem,
∴ PQ2 + = PR2…… (I)
Here, PR =10 , PQ =
From equation (I)
82 + QR2 =102
QR2 = 102- 82
QR2 = 100– 64
QR2 =
QR =6
∴ The distance of foot of the ladder from the base of wall is 6 m.
6. From the given figure, In ∆ ABC, If AD⊥BC, C = 450, AC =
8√2, BD = 5 then for finding value of AD and BC, complete the
following activity.
At P – window, At Q – base of wall, At R – foot of ladder
∴ PQ = 6 m
∴ QR = ?
In ∆PQR , m PQR = 900
By Pythagoras Theorem,
∴ PQ2 + = PR2…… (I)
Here, PR =10 , PQ =
From equation (I)
82 + QR2 =102
QR2 = 102- 82
QR2 = 100– 64
QR2 =
QR =6
∴ The distance of foot of the ladder from the base of wall is 6 m.
6. From the given figure, In ∆ ABC, If AD⊥BC, C = 450, AC =
8√2, BD = 5 then for finding value of AD and BC, complete the
following activity.
Page 49
A
B C
D
Activity : In ∆ ADC, If ADC = 900 C=450 ... ( given )
∴ DAC = .... ( remaining angles of ∆ ADC )
By theorem of 450– 450– 900 ∆le,
1 1
∴ = AC and = AC
√2 √2
1 1
∴ AD = × & DC = × 8√2
√2 √2
∴ AD = 8 & DC = 8
∴BC =BD +DC = 5 + 8 = 13
7. Complete the following activity to find the length of hypotenuse of
right angled triangle, if sides of right angle are 9 cm and 12 cm.
Activity : In ∆PQR , m PQR = 900
P
Q R
B C
D
Activity : In ∆ ADC, If ADC = 900 C=450 ... ( given )
∴ DAC = .... ( remaining angles of ∆ ADC )
By theorem of 450– 450– 900 ∆le,
1 1
∴ = AC and = AC
√2 √2
1 1
∴ AD = × & DC = × 8√2
√2 √2
∴ AD = 8 & DC = 8
∴BC =BD +DC = 5 + 8 = 13
7. Complete the following activity to find the length of hypotenuse of
right angled triangle, if sides of right angle are 9 cm and 12 cm.
Activity : In ∆PQR , m PQR = 900
P
Q R
Page 50
By Pythagoras Theorem,
∴ PQ2 + = PR2…… (I)
∴ PR2 = 92 + 122
∴ PR2 = + 144
∴ PR2 =
∴ PR = 15
∴ Length hypotenuse of triangle PQR is cm.
8. From given figure, In ∆ PQR, If QPR = 900, PM ⊥QR, PM =
10, QM = 8 then for finding the value of QR, complete the following
activity.
P
Q M R
Activity : In ∆ PQR, If QPR = 900, PM ⊥QR, ...,.. ( given )
In ∆ PMQ, By Pythagoras Theorem,
∴ PM2 + = PQ2…… (I)
∴ PQ2 = 102 + 82
∴ PQ2 = + 64
∴ PQ2 + = PR2…… (I)
∴ PR2 = 92 + 122
∴ PR2 = + 144
∴ PR2 =
∴ PR = 15
∴ Length hypotenuse of triangle PQR is cm.
8. From given figure, In ∆ PQR, If QPR = 900, PM ⊥QR, PM =
10, QM = 8 then for finding the value of QR, complete the following
activity.
P
Q M R
Activity : In ∆ PQR, If QPR = 900, PM ⊥QR, ...,.. ( given )
In ∆ PMQ, By Pythagoras Theorem,
∴ PM2 + = PQ2…… (I)
∴ PQ2 = 102 + 82
∴ PQ2 = + 64
Page 51
∴ PQ2 =
∴ PQ = √164
Here, ∆QPR ~∆QMP ~∆PMR
∴ ∆QMP ~∆PMR
PM QM
∴ =
RM PM
∴PM2=RM X QM
∴102= RM X 8
100
RM = =
8
And,
QR =QM + MR
25 41
QR = + =
2 2
9. Find the diagonal of a rectangle whose length is 16 cmand area is
192sq.cm. Complete the following activity.
Activity : T N
L M
As shown in fig. LMNT is rectangle
∴ Area of rectangle = length X breadth
∴ Area of rectangle = X breadth
∴ 192 = X breadth
∴ Breadth = 12 cm.
Also, TLM = 900 ..... ( each angle of rectangle is right angle )
In ∆TLM, By Pythagoras theorem
∴ PQ = √164
Here, ∆QPR ~∆QMP ~∆PMR
∴ ∆QMP ~∆PMR
PM QM
∴ =
RM PM
∴PM2=RM X QM
∴102= RM X 8
100
RM = =
8
And,
QR =QM + MR
25 41
QR = + =
2 2
9. Find the diagonal of a rectangle whose length is 16 cmand area is
192sq.cm. Complete the following activity.
Activity : T N
L M
As shown in fig. LMNT is rectangle
∴ Area of rectangle = length X breadth
∴ Area of rectangle = X breadth
∴ 192 = X breadth
∴ Breadth = 12 cm.
Also, TLM = 900 ..... ( each angle of rectangle is right angle )
In ∆TLM, By Pythagoras theorem
Page 52
∴ TM2 = TL2 +
∴ TM2 = 122 +
∴ TM2 = 144 +
∴TM2 = 400
∴ TM = 20
10. In ∆ LMN, l = 5, m = 13, n = 12 then complete the activity to
show that wheather given traingle is right angled traingle or not.
* ( l , m, n are opposite sides of L, M, N respectively )
Activity :
In ∆LMN मध्ये, l = 5, m = 13, n =
∴ l2 = ; m2= 169 ; n2= 144.
∴ l2 + n2 = 25 + 144 =
∴ + l 2= m2
∴By Converse of Pythagoras theorem, ∆LMN is right angled triangle.
Que. 3 (B). Solve the following questions : (3 marks each )
1. As shwon in figure, DFE = 900, FG⊥ED, If GD = 8, FG = 12,
then (1) EG = ? (2) FD = ? (3) EF = ?
G D
E F
2. A congruent side of an isosceles right angled triangle is 7 cm ,Find
its perimetre .
∴ TM2 = 122 +
∴ TM2 = 144 +
∴TM2 = 400
∴ TM = 20
10. In ∆ LMN, l = 5, m = 13, n = 12 then complete the activity to
show that wheather given traingle is right angled traingle or not.
* ( l , m, n are opposite sides of L, M, N respectively )
Activity :
In ∆LMN मध्ये, l = 5, m = 13, n =
∴ l2 = ; m2= 169 ; n2= 144.
∴ l2 + n2 = 25 + 144 =
∴ + l 2= m2
∴By Converse of Pythagoras theorem, ∆LMN is right angled triangle.
Que. 3 (B). Solve the following questions : (3 marks each )
1. As shwon in figure, DFE = 900, FG⊥ED, If GD = 8, FG = 12,
then (1) EG = ? (2) FD = ? (3) EF = ?
G D
E F
2. A congruent side of an isosceles right angled triangle is 7 cm ,Find
its perimetre .
Page 53
Que. 4. Solve the following questions : (Challenging question 4
marks each )
1. As shwon in figure, LK = 6 √2 then 1) MK = ? 2) ML = ? 3) MN
=?
N
450
M
L K
300
marks each )
1. As shwon in figure, LK = 6 √2 then 1) MK = ? 2) ML = ? 3) MN
=?
N
450
M
L K
300
Page 54
3 Circle.
Q.1. Four alternative answers for each of the following questions are given.
Choose the correct alternative.
1) Two circles intersect each other such that each circle passes through the
centre of the other. If the distance between their centres is 12, what is the
radius of each circle ?
(A) 6 cm (B) 12 cm (C) 24 cm (D) can’t say
2) A circle touches all sides of a parallelogram. So the parallelogram must be a,
......... .
(A) rectangle (B) rhombus (C) square (D) trapezium
3) ∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ ACB = 65°,
find m(arc ACB).
(A) 65° (B) 130° (C) 295° (D) 230°
4) In a cyclic ⃞ ABCD, twice the measure of ∠A is thrice the measure of ∠C.
Find the measure of ∠C?
(A) 36 (B) 72 (C) 90 (D) 108
5) How many circles can drawn passing through three non -collinear points?
(A) 0 (B) Infinite (C) 2 (D) One and only
one(unique)
6) Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the
distance between their centres
(A)9.7 (B) 1.3 (C) 2.6 (D) 4.6
7) What is the measurement of angle inscribed in a semicircle?
(A) 90° (B) 120° (C) 100° (D) 60°
8) Two circles having diameters 8 cm and 6 cm touch each other internally.
Find
the distance between their centres.
(A) 2 (B) 14 (C) 7 (D) 1
9) Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No
point, except point B, is common to the arcs. Which is the type of ∆ ABC?
(A) Equilateral triangle (B) Scalene triangle
(C) Right angled triangle (D) Isosceles triangle
10) In PQRS if ∠RSP = 80° then find ∠RQT ?
(A) 100° (B) 80°
(C) 70° (D) 110°
Q.1. Four alternative answers for each of the following questions are given.
Choose the correct alternative.
1) Two circles intersect each other such that each circle passes through the
centre of the other. If the distance between their centres is 12, what is the
radius of each circle ?
(A) 6 cm (B) 12 cm (C) 24 cm (D) can’t say
2) A circle touches all sides of a parallelogram. So the parallelogram must be a,
......... .
(A) rectangle (B) rhombus (C) square (D) trapezium
3) ∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ ACB = 65°,
find m(arc ACB).
(A) 65° (B) 130° (C) 295° (D) 230°
4) In a cyclic ⃞ ABCD, twice the measure of ∠A is thrice the measure of ∠C.
Find the measure of ∠C?
(A) 36 (B) 72 (C) 90 (D) 108
5) How many circles can drawn passing through three non -collinear points?
(A) 0 (B) Infinite (C) 2 (D) One and only
one(unique)
6) Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the
distance between their centres
(A)9.7 (B) 1.3 (C) 2.6 (D) 4.6
7) What is the measurement of angle inscribed in a semicircle?
(A) 90° (B) 120° (C) 100° (D) 60°
8) Two circles having diameters 8 cm and 6 cm touch each other internally.
Find
the distance between their centres.
(A) 2 (B) 14 (C) 7 (D) 1
9) Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No
point, except point B, is common to the arcs. Which is the type of ∆ ABC?
(A) Equilateral triangle (B) Scalene triangle
(C) Right angled triangle (D) Isosceles triangle
10) In PQRS if ∠RSP = 80° then find ∠RQT ?
(A) 100° (B) 80°
(C) 70° (D) 110°
Page 55
Q.2 Solve the following sub-questions. (1 mark question)
1) How many circles can be drawn passing through a point?
2)
Segment DP and segment DQ are tangent
segments to the circle with center A,
If DP = 7 cm. So find the length of the
segment DQ?
3) Two circles having radii 3.5 cm and 4.8 cm touch each other internally.
Find the distance between their centres.
4) What is the measure of a semi circular arc?
5)
A, B, C are any points on the circle with centre
O. If m arc (BC) = 110° and m arc (AB) =
125°, find measure arc AC
6) In the figure if ∠PQR = 50° then find ∠PSR
7)
In the adjoining figure the radius of a circle with
centre C is 6 cm, line AB is a tangent at A. What is
the measure of ∠ CAB? Why?
8) In the figure quadrilateral ABCD is a cyclic , if
∠DAB = 75° then find measure of ∠DCB
1) How many circles can be drawn passing through a point?
2)
Segment DP and segment DQ are tangent
segments to the circle with center A,
If DP = 7 cm. So find the length of the
segment DQ?
3) Two circles having radii 3.5 cm and 4.8 cm touch each other internally.
Find the distance between their centres.
4) What is the measure of a semi circular arc?
5)
A, B, C are any points on the circle with centre
O. If m arc (BC) = 110° and m arc (AB) =
125°, find measure arc AC
6) In the figure if ∠PQR = 50° then find ∠PSR
7)
In the adjoining figure the radius of a circle with
centre C is 6 cm, line AB is a tangent at A. What is
the measure of ∠ CAB? Why?
8) In the figure quadrilateral ABCD is a cyclic , if
∠DAB = 75° then find measure of ∠DCB
Page 56
9)
In the adjoining figure, seg DE is the chord of
the circle with center C. seg CF⊥ seg DE and
DE = 16 cm, then find the length of DF?
In the figure, if ∠ABC = 35° then find
10)
m(𝑎𝑟𝑐 AXC ) ?
Q.3 Complete the following activities (2 marks each).
The chords corresponding to congruent arcs of a circle are congruent.Prove the
theorem by completing following activity.
Given : In a circle with centre B
arc APC ≅ arc DQE
To Prove : Chord AC ≅chord DE
Proof : In ∆ ABC and ∆ DBE,
side AB ≅ side DB
side BC ≅ side
∠ ABC ≅ ∠ DBE (measure of congruent arcs)
∆ ABC ≅ ∆ DBE
chord AC ≅ chord DE (c.s.c.t)
2)
In figure , points G, D, E, F
are concyclic points of a circle with centre C.
∠ ECF = 70°, m(arc DGF) = 200°
find m(arc DEF) by completing activity.
In the adjoining figure, seg DE is the chord of
the circle with center C. seg CF⊥ seg DE and
DE = 16 cm, then find the length of DF?
In the figure, if ∠ABC = 35° then find
10)
m(𝑎𝑟𝑐 AXC ) ?
Q.3 Complete the following activities (2 marks each).
The chords corresponding to congruent arcs of a circle are congruent.Prove the
theorem by completing following activity.
Given : In a circle with centre B
arc APC ≅ arc DQE
To Prove : Chord AC ≅chord DE
Proof : In ∆ ABC and ∆ DBE,
side AB ≅ side DB
side BC ≅ side
∠ ABC ≅ ∠ DBE (measure of congruent arcs)
∆ ABC ≅ ∆ DBE
chord AC ≅ chord DE (c.s.c.t)
2)
In figure , points G, D, E, F
are concyclic points of a circle with centre C.
∠ ECF = 70°, m(arc DGF) = 200°
find m(arc DEF) by completing activity.
Page 57
m(arc EF) = ∠ECF ..... (Definition of measure of arc )
∴ m(arc EF) =
But; m(arc DE)+ m(arc EF)+ m(arc DGF) = (measure of a complete circle)
∴ m(arc DE) =
∴ m(arc DEF) = m(arc DE) + m(arc EF)
∴ m(arc DEF) =
3)
In the figure if the chord PQ and chord RS intersect at point T Prove that :
𝟏
𝐦∠STQ = 𝟐 [m(arc PR) + m(arc SQ)] for any measure of ∠STQ by filling out the
boxes.
Proof: 𝐦∠𝐒𝐓𝐐 = 𝐦∠𝐒𝐏𝐐 + .. (Theorem of the external angle of a
triangle)
m(कंस SQ) +
𝟏
= 𝟐 …… (inscibed angle
theorem)
𝟏
= [ + ]
𝟐
∴ m(arc EF) =
But; m(arc DE)+ m(arc EF)+ m(arc DGF) = (measure of a complete circle)
∴ m(arc DE) =
∴ m(arc DEF) = m(arc DE) + m(arc EF)
∴ m(arc DEF) =
3)
In the figure if the chord PQ and chord RS intersect at point T Prove that :
𝟏
𝐦∠STQ = 𝟐 [m(arc PR) + m(arc SQ)] for any measure of ∠STQ by filling out the
boxes.
Proof: 𝐦∠𝐒𝐓𝐐 = 𝐦∠𝐒𝐏𝐐 + .. (Theorem of the external angle of a
triangle)
m(कंस SQ) +
𝟏
= 𝟐 …… (inscibed angle
theorem)
𝟏
= [ + ]
𝟐
Page 58
4) In figure, chord EF || chord GH. Prove that,
chord EG≅ chord FH. Fill in the blanks and write the proof.
Proof : Draw seg GF.
∠𝐄𝐅𝐆 = ∠𝐅𝐆𝐇 ...... (I)
∠𝐄𝐅𝐆 = .....( inscribed angle theorem ) (II)
∠𝐅𝐆𝐇 = .....( inscribed angle theorem) (III)
∴ m(arc EG) = .....[ By (I) , (II) व (III) ]
chord EG ≅ chord FH ......(corresponding chords of congruent arcs )
5)
The angle inscribed in the semicircle is a right angle Prove the result by completing the
following activity .
Given: ∠ABC is inscribed angle in a
semicircle with center M.
To prove : ∠ABC is a right angle.
Proof: segment AC is a diameter of the circle.
∴ m(arc AXC) =
Arc AXC is intercepted by the inscribed angle ∠ABC .
∠ABC = .....(Inscribed angle theorem)
𝟏
= ×
𝟐
∴ m ∠ABC =
∴ ∠ABC is a right angle.
chord EG≅ chord FH. Fill in the blanks and write the proof.
Proof : Draw seg GF.
∠𝐄𝐅𝐆 = ∠𝐅𝐆𝐇 ...... (I)
∠𝐄𝐅𝐆 = .....( inscribed angle theorem ) (II)
∠𝐅𝐆𝐇 = .....( inscribed angle theorem) (III)
∴ m(arc EG) = .....[ By (I) , (II) व (III) ]
chord EG ≅ chord FH ......(corresponding chords of congruent arcs )
5)
The angle inscribed in the semicircle is a right angle Prove the result by completing the
following activity .
Given: ∠ABC is inscribed angle in a
semicircle with center M.
To prove : ∠ABC is a right angle.
Proof: segment AC is a diameter of the circle.
∴ m(arc AXC) =
Arc AXC is intercepted by the inscribed angle ∠ABC .
∠ABC = .....(Inscribed angle theorem)
𝟏
= ×
𝟐
∴ m ∠ABC =
∴ ∠ABC is a right angle.
Page 59
6) Prove that angles inscribed in the same arc are congruent.
Given: In a circle with centre C, ∠PQR
and ∠PSR is inscribed in same arc
PQR.Arc PTR is intercepted by the angles.
To prove : ∠PQR ≅ ∠PSR.
𝟏
Proof : m∠PQR = × [m(arc PTR)] ........ (i)
𝟐
𝟏
m∠ = × [m(arc PTR)] ....... (ii)
𝟐
m∠ = m∠PSR ........By(i) &(ii)
∴ ∠PQR ≅ ∠PSR
7) If O is the center of the circle in the figure alongside , then complete the table from
the given information.
The type of arc
Type of circular arc Name of circular arc Measure of circular arc
Minor arc
Major arc
Q.4. Solve the following sub-questions. (2 marks question)
1)
In the adjoining figure circle with Centre D
touches the sides of ∠ACB at A and B. If
∠ ACB = 52°, find measure of ∠ ADB.
Given: In a circle with centre C, ∠PQR
and ∠PSR is inscribed in same arc
PQR.Arc PTR is intercepted by the angles.
To prove : ∠PQR ≅ ∠PSR.
𝟏
Proof : m∠PQR = × [m(arc PTR)] ........ (i)
𝟐
𝟏
m∠ = × [m(arc PTR)] ....... (ii)
𝟐
m∠ = m∠PSR ........By(i) &(ii)
∴ ∠PQR ≅ ∠PSR
7) If O is the center of the circle in the figure alongside , then complete the table from
the given information.
The type of arc
Type of circular arc Name of circular arc Measure of circular arc
Minor arc
Major arc
Q.4. Solve the following sub-questions. (2 marks question)
1)
In the adjoining figure circle with Centre D
touches the sides of ∠ACB at A and B. If
∠ ACB = 52°, find measure of ∠ ADB.
Page 60
2)
In the adjoining figure, the line MN touches the
circle with center A at point M. If AN = 13 and
MN = 5 then find the radius of the circle?
3) What is the distance between two parallel tangents of a circle having radius
4.5 cm? Justify your answer.
4)
In figure, m(arc NS) = 125°, m(arc EF) = 37°,
find the measure ∠NMS.
5) Length of a tangent segment drawn from a point which is at a distance 15 cm
from the centre of a circle is 12 cm, find the diameter of the circle?
6) In the figure a circle with center C has
m (arc AXB) = 100° then find central ∠ACB and
measure m (arc AYB).
7)
In figure , M is the centre of the circle and seg KL
is a tangent segment. If MK = 12, KL = 6√3 then
find (1) Radius of the circle.
(2) Measures of ∠K and ∠M.
8)
In figure, chords AC and DE intersect at B.
If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC) .
In the adjoining figure, the line MN touches the
circle with center A at point M. If AN = 13 and
MN = 5 then find the radius of the circle?
3) What is the distance between two parallel tangents of a circle having radius
4.5 cm? Justify your answer.
4)
In figure, m(arc NS) = 125°, m(arc EF) = 37°,
find the measure ∠NMS.
5) Length of a tangent segment drawn from a point which is at a distance 15 cm
from the centre of a circle is 12 cm, find the diameter of the circle?
6) In the figure a circle with center C has
m (arc AXB) = 100° then find central ∠ACB and
measure m (arc AYB).
7)
In figure , M is the centre of the circle and seg KL
is a tangent segment. If MK = 12, KL = 6√3 then
find (1) Radius of the circle.
(2) Measures of ∠K and ∠M.
8)
In figure, chords AC and DE intersect at B.
If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC) .
Page 61
Q. 5. Complete the following activity. (3 marks each)
1) Tangent segments drawn from an external point to a circle are congruent , prove
this theorem.Complete the following activity.
Given :
To Prove:
Proof : Draw radius AP and radius AQ and complete the following proof of
the theorem.
In ∆PAD and ∆QAD ,
Seg PA ≅ .... ( radii of the same circle. )
Seg AD ≅ Seg AD .... ( )
∠APD ≅ ∠AQD = 90° ....(tangent theorem )
∴ ∆PAD ≅ ∆QAD .... ( )
∴seg DP ≅ seg DQ .... ( )
2)
MRPN is cyclic, ∠R = (5x - 13) °, ∠ N = (4x + 4) °. Find measures of ∠
R and ∠ N, by completing the following activity.
Solution : MRPN is cyclic
The opposite angles of a cyclic square are
∠R + ∠N =
∴(5x-13)° + (4x+4°) =
∴ 9x = 189
∴ x =
∴ ∠R = (5x-13)° =
∴ ∠N = (4x+4)° =
1) Tangent segments drawn from an external point to a circle are congruent , prove
this theorem.Complete the following activity.
Given :
To Prove:
Proof : Draw radius AP and radius AQ and complete the following proof of
the theorem.
In ∆PAD and ∆QAD ,
Seg PA ≅ .... ( radii of the same circle. )
Seg AD ≅ Seg AD .... ( )
∠APD ≅ ∠AQD = 90° ....(tangent theorem )
∴ ∆PAD ≅ ∆QAD .... ( )
∴seg DP ≅ seg DQ .... ( )
2)
MRPN is cyclic, ∠R = (5x - 13) °, ∠ N = (4x + 4) °. Find measures of ∠
R and ∠ N, by completing the following activity.
Solution : MRPN is cyclic
The opposite angles of a cyclic square are
∠R + ∠N =
∴(5x-13)° + (4x+4°) =
∴ 9x = 189
∴ x =
∴ ∠R = (5x-13)° =
∴ ∠N = (4x+4)° =
Page 62
3) In figure , seg AB is a diameter of
a circle with centre O . The bisector
of ∠ACB intersects the circle at point D.
Prove that, seg AD ≅ seg BD.
Complete the following proof by filling
in the blanks.
Proof Draw seg OD.
∠ACB = .......... angle inscribed in semicircle
∠DCB = .......... CD is the bisector of ∠C
m(arc DB) = .......... inscribed angle theorem
∠DOB = .......... definition of measure of an arc (I)
seg OA ≅ seg OB .......... (II)
∴line OD is of seg AB .......... From (I) and (II)
∴seg AD ≅ seg BD
4)
In the adjoining figure circles with centres X and Y
touch each other at point Z. A secant passing
through Z intersects the circles at points A and
B respectively.
Prove that , radius XA || radius YB.
Fill in the blanks and complete the proof.
Construction : Draw segments XZ and YZ.
Proof :By theorem of touching circles, points X, Z, Y are
∴ ∠XZA ≅ .......... opposite angles
Let ∠XZA = ∠BZY = a ..... (I)
Now, seg XA ≅ seg XZ ........ (radii of the same circle.)
∴ ∠XAZ = .......... = a ........ (isosceles triangle theorem) (II)
similarly, seg YB ≅ seg YZ ........ (radii of the same circle.)
∴ ∠BZY = .......... = a ........ (isosceles triangle theorem.) (III)
a circle with centre O . The bisector
of ∠ACB intersects the circle at point D.
Prove that, seg AD ≅ seg BD.
Complete the following proof by filling
in the blanks.
Proof Draw seg OD.
∠ACB = .......... angle inscribed in semicircle
∠DCB = .......... CD is the bisector of ∠C
m(arc DB) = .......... inscribed angle theorem
∠DOB = .......... definition of measure of an arc (I)
seg OA ≅ seg OB .......... (II)
∴line OD is of seg AB .......... From (I) and (II)
∴seg AD ≅ seg BD
4)
In the adjoining figure circles with centres X and Y
touch each other at point Z. A secant passing
through Z intersects the circles at points A and
B respectively.
Prove that , radius XA || radius YB.
Fill in the blanks and complete the proof.
Construction : Draw segments XZ and YZ.
Proof :By theorem of touching circles, points X, Z, Y are
∴ ∠XZA ≅ .......... opposite angles
Let ∠XZA = ∠BZY = a ..... (I)
Now, seg XA ≅ seg XZ ........ (radii of the same circle.)
∴ ∠XAZ = .......... = a ........ (isosceles triangle theorem) (II)
similarly, seg YB ≅ seg YZ ........ (radii of the same circle.)
∴ ∠BZY = .......... = a ........ (isosceles triangle theorem.) (III)
Page 63
∴from (I), (II), (III),
∠XAZ =
∴ radius XA || radius YB .......... ( )
5) An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to
its adjacent interior angle, to prove the theorem complete the activity .
Given : ABCD is cyclic ,
is the exterior angle of ABCD
To prove : ∠DCE ≅ ∠BAD
Proof : + ∠BCD = ....(Angles in linear pair) (I)
ABCD is a cyclic .
+ ∠BAD = .....(Theorem of cyclic quadrilateral) (II)
By (I) and (II)
∠DCE + ∠BCD = + ∠BAD
∠DCE ≅ ∠BAD
6)
Seg RM and seg RN are tangent segments
of a circle with centre O. Prove that seg
OR bisects ∠MRN as well as ∠MON with
the help of activity.
∠XAZ =
∴ radius XA || radius YB .......... ( )
5) An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to
its adjacent interior angle, to prove the theorem complete the activity .
Given : ABCD is cyclic ,
is the exterior angle of ABCD
To prove : ∠DCE ≅ ∠BAD
Proof : + ∠BCD = ....(Angles in linear pair) (I)
ABCD is a cyclic .
+ ∠BAD = .....(Theorem of cyclic quadrilateral) (II)
By (I) and (II)
∠DCE + ∠BCD = + ∠BAD
∠DCE ≅ ∠BAD
6)
Seg RM and seg RN are tangent segments
of a circle with centre O. Prove that seg
OR bisects ∠MRN as well as ∠MON with
the help of activity.
Page 64
Proof : In ∆RMO and ∆RNO ,
∠RMO ≅ ∠RNO = 90° ....( )
hypt OR ≅ hypt OR .... ( )
seg OM ≅ seg .... ( radii of the same circle )
∴ ∆RMO ≅ ∆RNO .... ( )
∠MOR ≅ ∠NOR
Similarly ∠MRO ≅ .... ( )
7)
In figure , O is the centre of the circle.
Seg AB, seg AC are tangent segments.
Radius of the circle is r and ℓ(AB) = r ,
Prove that, ABOC is a square.
Proof : Draw segment OB and OC.
𝓵(AB) = r ..... (Given) (I)
AB=AC .....( ) (II)
But OB = OC = r ..... ( ) (III)
From (I),(II) and (III)
AB= = OB = OC = r
∴ Quadrilateral ABOC is a .
Similarly ∠OBA = ....( Tangent Theorem )
If one angle of is right angle ,then it is a square.
∴ Quadrilateral ABOC is a suqare.
∠RMO ≅ ∠RNO = 90° ....( )
hypt OR ≅ hypt OR .... ( )
seg OM ≅ seg .... ( radii of the same circle )
∴ ∆RMO ≅ ∆RNO .... ( )
∠MOR ≅ ∠NOR
Similarly ∠MRO ≅ .... ( )
7)
In figure , O is the centre of the circle.
Seg AB, seg AC are tangent segments.
Radius of the circle is r and ℓ(AB) = r ,
Prove that, ABOC is a square.
Proof : Draw segment OB and OC.
𝓵(AB) = r ..... (Given) (I)
AB=AC .....( ) (II)
But OB = OC = r ..... ( ) (III)
From (I),(II) and (III)
AB= = OB = OC = r
∴ Quadrilateral ABOC is a .
Similarly ∠OBA = ....( Tangent Theorem )
If one angle of is right angle ,then it is a square.
∴ Quadrilateral ABOC is a suqare.
Page 65
Q.6. Solve the following sub-questions. (3 marks question)
1) Prove the following theorems:
i) Opposite angles of a cyclic quadrilateral are supplementry.
ii) Tangent segments drawn from an external point to a circle are congruent.
iii) Angles inscribed in the same arc are congruent.
2)
Line ℓ touches a circle with centre O at point P. If
radius of the circle is 9 cm, answer the following.
(i) What is d(O, P) = ? Why ?
(ii) If d(O, Q) = 8 cm, where does the point Q lie ?
(iii) If d(PQ) = 15 cm, How many locations of point R
are line on line ℓ? At what distance will each of
them be from point P ?
3) In the adjoining figure, O is the centre of the
circle. From point R, seg RM and seg RN are
tangent segments touching the circle at M and
N. If (OR) = 10 cm and radius of the circle = 5
cm, then
(1) What is the length of each tangent segment ?
(2) What is the measure of ∠MRO ?
(3) What is the measure of ∠ MRN ?
4)
In figure ,chord AB ≅ chord CD,
Prove that, arc AC ≅ arc BD
1) Prove the following theorems:
i) Opposite angles of a cyclic quadrilateral are supplementry.
ii) Tangent segments drawn from an external point to a circle are congruent.
iii) Angles inscribed in the same arc are congruent.
2)
Line ℓ touches a circle with centre O at point P. If
radius of the circle is 9 cm, answer the following.
(i) What is d(O, P) = ? Why ?
(ii) If d(O, Q) = 8 cm, where does the point Q lie ?
(iii) If d(PQ) = 15 cm, How many locations of point R
are line on line ℓ? At what distance will each of
them be from point P ?
3) In the adjoining figure, O is the centre of the
circle. From point R, seg RM and seg RN are
tangent segments touching the circle at M and
N. If (OR) = 10 cm and radius of the circle = 5
cm, then
(1) What is the length of each tangent segment ?
(2) What is the measure of ∠MRO ?
(3) What is the measure of ∠ MRN ?
4)
In figure ,chord AB ≅ chord CD,
Prove that, arc AC ≅ arc BD
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5)
In figure , in a circle with centre O, length of
chord AB is equal to the radius of the circle. Find
measure of each of the following.
(1) ∠AOB (2) ∠ACB
(3) arc AB
6)
In figure , chord LM ≅ chord LN , ∠L = 35°
find (i) m(arc MN)
(ii) m(arc LN)
7) Prove that, any rectangle is a cyclic quadrilateral.
8) In figure , PQRS is cyclic.
side PQ ≅ side RQ. ∠ PSR = 110°,
Find- (1) measure of ∠ PQR
(2) m(arc PQR)
(3) m(arc QR)
9)
In figure , line ℓ touches the circle with
centre O at point P. Q is the mid point of
radius OP. RS is a chord through Q such that
chords RS || line ℓ. If RS = 12 find the radius
of the circle
10)
In figure , O is the centre of a circle,
chord PQ ≅chord RS If ∠ POR = 70°
and (arc RS) = 80°, find (1) m(arc PR) (2)
m(arc QS) (3) m(arc QSR)
In figure , in a circle with centre O, length of
chord AB is equal to the radius of the circle. Find
measure of each of the following.
(1) ∠AOB (2) ∠ACB
(3) arc AB
6)
In figure , chord LM ≅ chord LN , ∠L = 35°
find (i) m(arc MN)
(ii) m(arc LN)
7) Prove that, any rectangle is a cyclic quadrilateral.
8) In figure , PQRS is cyclic.
side PQ ≅ side RQ. ∠ PSR = 110°,
Find- (1) measure of ∠ PQR
(2) m(arc PQR)
(3) m(arc QR)
9)
In figure , line ℓ touches the circle with
centre O at point P. Q is the mid point of
radius OP. RS is a chord through Q such that
chords RS || line ℓ. If RS = 12 find the radius
of the circle
10)
In figure , O is the centre of a circle,
chord PQ ≅chord RS If ∠ POR = 70°
and (arc RS) = 80°, find (1) m(arc PR) (2)
m(arc QS) (3) m(arc QSR)
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11)
In the adjoining figure circle with Centre Q
touches the sides of ∠MPN at M and N. If
∠ MPN = 40°, find measure of ∠ MQN.
12)
In the figure if O is the center of the circle
and two chords of the circle EF and GH
are parallel to each other. Show that
∠𝐄𝐎𝐆 ≅ ∠𝐅𝐎𝐇
Q. 7. Solve the following sub-questions. (4 marks question)
1)
In the figure segment PQ is the diameter of
the circle with center O. The tangent to the
tangent circle drawn from point C on it ,
intersects the tangents drawn from points P
and Q at points A and B respectively ,
prove that ∠AOC = 90°
2) The chords AB and CD of the circle intersect at point M in the interior of
the same circle then prove that CM × BD = BM × AC.
3)
A circle with centre P is inscribed in the
∆ABC. Side AB, side BC and side AC touches
the circle at points L, M and N respectively.
Radius of the circle is r.
1
Prove that : A(∆ABC) = (𝐴𝐵 + 𝐵𝐶 + 𝐴𝐶) × r
2
In the adjoining figure circle with Centre Q
touches the sides of ∠MPN at M and N. If
∠ MPN = 40°, find measure of ∠ MQN.
12)
In the figure if O is the center of the circle
and two chords of the circle EF and GH
are parallel to each other. Show that
∠𝐄𝐎𝐆 ≅ ∠𝐅𝐎𝐇
Q. 7. Solve the following sub-questions. (4 marks question)
1)
In the figure segment PQ is the diameter of
the circle with center O. The tangent to the
tangent circle drawn from point C on it ,
intersects the tangents drawn from points P
and Q at points A and B respectively ,
prove that ∠AOC = 90°
2) The chords AB and CD of the circle intersect at point M in the interior of
the same circle then prove that CM × BD = BM × AC.
3)
A circle with centre P is inscribed in the
∆ABC. Side AB, side BC and side AC touches
the circle at points L, M and N respectively.
Radius of the circle is r.
1
Prove that : A(∆ABC) = (𝐴𝐵 + 𝐵𝐶 + 𝐴𝐶) × r
2
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4)
In the figure ABCD is a cyclic
quadrilateral. If m(arc ABC) = 230°.then
find ∠ABC , ∠CDA , ∠CBE
5)
The figure∆ABC is an isosceles triangle with a
perimeter of 44 cm. The sides AB and BC are
congruent and the length of the base AC is 12
cm. If a circle touches all three sides as shown
in the figure, then find the length of the tangent
segment drawn to the circle from the point B
6)
In the figure ∆ABC is an equilateral
triangle.The angle bisector of ∠𝐁 will
intersect the circumcircle ∆ABC at point P.
Then prove that : CQ = CA.
7)
In the figure quadrilateral ABCD is cyclic
, If m(arc BC) = 90° and ∠DBC = 55°.
Then find the measure of ∠BCD .
In the figure ABCD is a cyclic
quadrilateral. If m(arc ABC) = 230°.then
find ∠ABC , ∠CDA , ∠CBE
5)
The figure∆ABC is an isosceles triangle with a
perimeter of 44 cm. The sides AB and BC are
congruent and the length of the base AC is 12
cm. If a circle touches all three sides as shown
in the figure, then find the length of the tangent
segment drawn to the circle from the point B
6)
In the figure ∆ABC is an equilateral
triangle.The angle bisector of ∠𝐁 will
intersect the circumcircle ∆ABC at point P.
Then prove that : CQ = CA.
7)
In the figure quadrilateral ABCD is cyclic
, If m(arc BC) = 90° and ∠DBC = 55°.
Then find the measure of ∠BCD .
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8)
Given : A circle inscribed in a right
angled ∆ABC. If ∠ACB = 90° and the
radius of the circle is r.
To prove : 2 r = 𝒶 + b – c
9) In a circle with centre P , chord AB is parallel to a tangent and intersects the
radius drawn from the point of contact to its midpoint. If AB = 16√3 then
find the radius of the circle.
10)
In the figure, O is the center of the circle.
Line AQ is a tangent. If OP = 3
m(arc PM) = 120°
then find the length of AP?
Q. 8. Solve the following sub-questions (3 marks each)
1)
In the figure, O is the centre of the circle
and ∠AOB = 90° , ∠ABC = 30°
Then find ∠CAB?
Given : A circle inscribed in a right
angled ∆ABC. If ∠ACB = 90° and the
radius of the circle is r.
To prove : 2 r = 𝒶 + b – c
9) In a circle with centre P , chord AB is parallel to a tangent and intersects the
radius drawn from the point of contact to its midpoint. If AB = 16√3 then
find the radius of the circle.
10)
In the figure, O is the center of the circle.
Line AQ is a tangent. If OP = 3
m(arc PM) = 120°
then find the length of AP?
Q. 8. Solve the following sub-questions (3 marks each)
1)
In the figure, O is the centre of the circle
and ∠AOB = 90° , ∠ABC = 30°
Then find ∠CAB?
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2) In the figure a circle with center P
touches the semicircle at points Q
and C having center O. if diameter
AB = 10, AC = 6 then find the
radius 𝓍 of the smaller circle?
3) In the figure a circle touches all the sides of
quadrilateral ABCD from the inside. The
center of the circle is O. If AD⊥ DC and
BC = 38 , QB = 27, DC = 25 then find the
radius of the circle?
4)
If AB and CD are the common
tangents in the circles of two unequal
(different) radii then show that
seg AB ≅ seg CD
5) Circles with centres A, B and C touch each other externally. If AB = 36,
BC = 32, CA = 30, then find the radii of each circle.
touches the semicircle at points Q
and C having center O. if diameter
AB = 10, AC = 6 then find the
radius 𝓍 of the smaller circle?
3) In the figure a circle touches all the sides of
quadrilateral ABCD from the inside. The
center of the circle is O. If AD⊥ DC and
BC = 38 , QB = 27, DC = 25 then find the
radius of the circle?
4)
If AB and CD are the common
tangents in the circles of two unequal
(different) radii then show that
seg AB ≅ seg CD
5) Circles with centres A, B and C touch each other externally. If AB = 36,
BC = 32, CA = 30, then find the radii of each circle.
Page 71
4. Geometric Constructions
Question 1) (A) choose the correct alternative answer for each of the following sub
question. Write the correct alphabet.
1) …………… number of tangents can be drawn to a circle from the point on the
circle.
A) 3 B) 2 C) 1 D) 0
2) The tangents drawn at the end of a diameter of a circle are…………..
A) Perpendicular B) parallel C) congruent D) can’t say
𝐿𝑀 2
3) ∆LMN ~∆HIJ and = then
𝐻𝐼 3
A) ∆ LMN is a smaller triangle.
B) ∆ HIJ is a smaller triangle.
C) Both triangles are congruent.
D) Can’t say.
4) ……………….number of tangents can be drawn to a circle from the point
outside the circle.
A) 2 B) 1 C) one and only one D) 0
Question 1) (A) choose the correct alternative answer for each of the following sub
question. Write the correct alphabet.
1) …………… number of tangents can be drawn to a circle from the point on the
circle.
A) 3 B) 2 C) 1 D) 0
2) The tangents drawn at the end of a diameter of a circle are…………..
A) Perpendicular B) parallel C) congruent D) can’t say
𝐿𝑀 2
3) ∆LMN ~∆HIJ and = then
𝐻𝐼 3
A) ∆ LMN is a smaller triangle.
B) ∆ HIJ is a smaller triangle.
C) Both triangles are congruent.
D) Can’t say.
4) ……………….number of tangents can be drawn to a circle from the point
outside the circle.
A) 2 B) 1 C) one and only one D) 0
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5)
In the figure ∆ ABC ~∆ ADE then the ratio of their corresponding sides is
--------.
3 1 3 4
A) B) C) D)
1 3 4 3
6) Which theorem is used while constructing a tangent to the
circle by using center of a circle?
A) tangent – radius theorem.
B) Converse of tangent – radius theorem.
C) Pythagoras theorem
D) Converse of Pythagoras theorem.
7) ∆PQR ~ ∆ABC, 𝐴𝐶 = 7 then
𝑃𝑅 5
A) ∆ABC is greater.
B) ∆ PQR is greater.
C) Both triangles are congruent.
In the figure ∆ ABC ~∆ ADE then the ratio of their corresponding sides is
--------.
3 1 3 4
A) B) C) D)
1 3 4 3
6) Which theorem is used while constructing a tangent to the
circle by using center of a circle?
A) tangent – radius theorem.
B) Converse of tangent – radius theorem.
C) Pythagoras theorem
D) Converse of Pythagoras theorem.
7) ∆PQR ~ ∆ABC, 𝐴𝐶 = 7 then
𝑃𝑅 5
A) ∆ABC is greater.
B) ∆ PQR is greater.
C) Both triangles are congruent.
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D) Can’t say.
8) ∆ABC ~∆AQR. = 5 then which of the following option is true.
𝐴𝐵 7
𝐴𝑄
A) A-Q-B B) A-B-Q C) A-C-B D) A-R-B
Question 1 (B) solve the following examples (1 mark each)
1) Construct ∠ABC =60 0 and bisect it.
2) Construct ∠PQR = 115 and divide it into two equal parts.
0
3) Draw Seg AB of lenght 9.7cm. Take point P on it such that AP =
3.5 cm and A-P-B. Construct perpendicular to seg AB from point
P.
4) Draw seg AB of length 4.5 cm and draw its perpendicular bisector.
5) Draw seg AB of length 9 cm and divide it in the ratio 3:2.
6) Draw a circle of radius 3 cm and draw a tangent to the circle
from point P on the circle.
Question 2) (A) Solve the following examples as per the instructions given
in the activity. (2 marks each)
1) Draw a circle and take any point P on the circle. Draw ray OP
Draw perpendicular to ray OP from point P.
8) ∆ABC ~∆AQR. = 5 then which of the following option is true.
𝐴𝐵 7
𝐴𝑄
A) A-Q-B B) A-B-Q C) A-C-B D) A-R-B
Question 1 (B) solve the following examples (1 mark each)
1) Construct ∠ABC =60 0 and bisect it.
2) Construct ∠PQR = 115 and divide it into two equal parts.
0
3) Draw Seg AB of lenght 9.7cm. Take point P on it such that AP =
3.5 cm and A-P-B. Construct perpendicular to seg AB from point
P.
4) Draw seg AB of length 4.5 cm and draw its perpendicular bisector.
5) Draw seg AB of length 9 cm and divide it in the ratio 3:2.
6) Draw a circle of radius 3 cm and draw a tangent to the circle
from point P on the circle.
Question 2) (A) Solve the following examples as per the instructions given
in the activity. (2 marks each)
1) Draw a circle and take any point P on the circle. Draw ray OP
Draw perpendicular to ray OP from point P.
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2) Draw a circle with center O and radius 3cm
Take any point P on the circle.
Draw ray OP.
Draw perpendicular to ray OP from point P
1) To draw tangents to the circle from the end points of the
diameter of the circle.
Construct a circle with center O. Draw any diameter AB of
it.
Draw ray OA and OB
Construct perpendicular to ray OA from point A
Take any point P on the circle.
Draw ray OP.
Draw perpendicular to ray OP from point P
1) To draw tangents to the circle from the end points of the
diameter of the circle.
Construct a circle with center O. Draw any diameter AB of
it.
Draw ray OA and OB
Construct perpendicular to ray OA from point A
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Construct perpendicular to Ray OB from point B
Question 2) (B) Solve the following examples (2 marks each)
1) Draw a circle of radius 3.4 cm take any point P on it. Draw tangent to the circle from point
P.
2) Draw a circle of radius 4.2 cm take any point M on it. Draw tangent to the circle from
point M.
3) Draw a circle of radius 3 cm. Take any point K on it. Draw a tangent to the circle from
point K without using center of the circle.
4) Draw a circle of radius 3.4 cm. Draw a chord MN 5.7 cm long in a circle. Draw a tangent to
the circle from point M and point N.
5) Draw a circle of 4.2 cm. Draw a tangent to the point P on the circle without using the
center of the circle.
6) Draw a circle with a diameter AB of length 6 cm. Draw a tangent to the circle from the
endpoints of the diameter.
7) Draw seg AB = 6.8 cm. Draw a circle with diameter AB. Draw points C on the circle apart
from A and B. Draw line AC and line CB Write the measure of angle ACB.
Question 3) (A) Do the activity as per the given instructions. (3 marks
each)
1) Complete the following activity to draw tangents to the circle.
a) Draw a circle with radius 3.3 cm and center O. Draw chord PQ of length 6.6cm..
Draw ray OP and ray OQ.
b) Draw a line perpendicular to the ray OP from P.
Question 2) (B) Solve the following examples (2 marks each)
1) Draw a circle of radius 3.4 cm take any point P on it. Draw tangent to the circle from point
P.
2) Draw a circle of radius 4.2 cm take any point M on it. Draw tangent to the circle from
point M.
3) Draw a circle of radius 3 cm. Take any point K on it. Draw a tangent to the circle from
point K without using center of the circle.
4) Draw a circle of radius 3.4 cm. Draw a chord MN 5.7 cm long in a circle. Draw a tangent to
the circle from point M and point N.
5) Draw a circle of 4.2 cm. Draw a tangent to the point P on the circle without using the
center of the circle.
6) Draw a circle with a diameter AB of length 6 cm. Draw a tangent to the circle from the
endpoints of the diameter.
7) Draw seg AB = 6.8 cm. Draw a circle with diameter AB. Draw points C on the circle apart
from A and B. Draw line AC and line CB Write the measure of angle ACB.
Question 3) (A) Do the activity as per the given instructions. (3 marks
each)
1) Complete the following activity to draw tangents to the circle.
a) Draw a circle with radius 3.3 cm and center O. Draw chord PQ of length 6.6cm..
Draw ray OP and ray OQ.
b) Draw a line perpendicular to the ray OP from P.
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c) Draw a line perpendicular to the ray OQ from Q.
2) Draw a circle with center O. Draw an arc AB of 1000 measure.
Perform the following steps to draw tangents to the circle from point A
and B.
a) Draw a circle with any radius and center P.
b) Take any point A on the circle.
c) Draw ray PB such ∠ APB = 1000.
d) Draw perpendicular to ray PA from point A.
e) Draw perpendicular to ray PB from point B.
3) Do the following activity to draw tangents to the circle without using
center of the circle.
a) Draw a circle with radius 3.5 cm and take any point C on it.
b) Draw chord CB and an inscribed angle CAB
c) With the center A and any convenient radius draw an arc intersecting
the sides of angle BAC in points M and N.
d) Using the same radius draw and center C, draw an arc intersecting
the chord CB at point R.
e) Taking the radius equal to d(MN) and center R, draw an arc
intersecting the arc drawn in the previous step. Let D be the point of
intersection of these arcs. Draw line CD. Line CD is the required
tangent to the circle.
Question 3 B) Solve the following examples (3 marks each):
2) Draw a circle with center O. Draw an arc AB of 1000 measure.
Perform the following steps to draw tangents to the circle from point A
and B.
a) Draw a circle with any radius and center P.
b) Take any point A on the circle.
c) Draw ray PB such ∠ APB = 1000.
d) Draw perpendicular to ray PA from point A.
e) Draw perpendicular to ray PB from point B.
3) Do the following activity to draw tangents to the circle without using
center of the circle.
a) Draw a circle with radius 3.5 cm and take any point C on it.
b) Draw chord CB and an inscribed angle CAB
c) With the center A and any convenient radius draw an arc intersecting
the sides of angle BAC in points M and N.
d) Using the same radius draw and center C, draw an arc intersecting
the chord CB at point R.
e) Taking the radius equal to d(MN) and center R, draw an arc
intersecting the arc drawn in the previous step. Let D be the point of
intersection of these arcs. Draw line CD. Line CD is the required
tangent to the circle.
Question 3 B) Solve the following examples (3 marks each):
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1) △ ABC ~ △ PBQ, In △ ABC, AB = 3 cm, ∠ B = 900, BC = 4 cm.
Ratio of the corresponding sides of two triangles is 7:4. Then construct
△ ABC and △ PBQ
2) ∆RHP ~∆NED,𝐼𝑛 ∆NED,NE=7 cm ,∠D=30 0 , ∠N=20 0 and 𝐸𝐷 =5. Then
𝐻𝑃 4
construct ∆RHP and ∆NED.
3) ∆PQR~∆ABC, In ∆PQR PQ=3.6cm, QR=4 cm, PR=4.2 cm ratio of
the corresponding sides of triangle is 3:4 then construct ∆PQR 𝑎𝑛𝑑 ∆ABC.
4) Construct an equilateral △ ABC with side 5cm. △ ABC ~ △ LMN, ratio
the corresponding sides of triangle is 6:7 then construct ∆LMN 𝑎𝑛𝑑 ∆ABC
5) Draw a circle with center O and radius 3.4. Draw a chord MN of
length 5.7 cm in a circle. Draw a tangent to the circle from point M
and N.
6) Draw a circle with center O and radius 3.6 cm. draw a tangent to
the circle from point B at a distance of 7.2 cm from the center of the
circle.
7) Draw a circle with center C and radius 3.2 cm. Draw a tangent to
the circle from point P at a distance of 7.5 cm from the center of the
circle.
Ratio of the corresponding sides of two triangles is 7:4. Then construct
△ ABC and △ PBQ
2) ∆RHP ~∆NED,𝐼𝑛 ∆NED,NE=7 cm ,∠D=30 0 , ∠N=20 0 and 𝐸𝐷 =5. Then
𝐻𝑃 4
construct ∆RHP and ∆NED.
3) ∆PQR~∆ABC, In ∆PQR PQ=3.6cm, QR=4 cm, PR=4.2 cm ratio of
the corresponding sides of triangle is 3:4 then construct ∆PQR 𝑎𝑛𝑑 ∆ABC.
4) Construct an equilateral △ ABC with side 5cm. △ ABC ~ △ LMN, ratio
the corresponding sides of triangle is 6:7 then construct ∆LMN 𝑎𝑛𝑑 ∆ABC
5) Draw a circle with center O and radius 3.4. Draw a chord MN of
length 5.7 cm in a circle. Draw a tangent to the circle from point M
and N.
6) Draw a circle with center O and radius 3.6 cm. draw a tangent to
the circle from point B at a distance of 7.2 cm from the center of the
circle.
7) Draw a circle with center C and radius 3.2 cm. Draw a tangent to
the circle from point P at a distance of 7.5 cm from the center of the
circle.
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8) Draw a circle with a radius of 3.5 cm. Take the point K anywhere
on the circle. Draw a tangent to the circle from K (without using the
center of the circle).
9) Draw a circle of radius 4.2 cm. Draw arc PQ measuring 1200
Draw a tangent to the circle from point P and point Q.
10) Draw a circle of radius 4.2 cm. Draw a tangent to the circle from
a point 7 cm away from the center of the circle.
11) Draw a circle of radius 3 cm and draw chord XY 5 cm long. Draw
the tangent of the circle passing through point X and point Y (without
using the center of the circle).
Question 4) solve the following examples. (4 marks each)
1) ∆AMT ~∆AHE, In ∆AMT, AM =6.3 cm
=
AM 7
∠MAT= 120 , AT = 4.9 cm, then construct ∆AMT and ∆AHE .
0
HA 5
0 𝐻𝑃
=
4
2) ∆RHP~∆NED, In ∆NED, NE=7 cm. ∠D=30 , ∠N=20 , then construct
0
𝐸𝐷 5
∆RHP and △NED .
3) ∆ABC. ~∆PBR, BC=8 cm, AC=10 cm , ∠B=90 0 ,
=
𝐵𝐶 5
then construct △ABC and ∆PBR
𝐵𝑅 4
on the circle. Draw a tangent to the circle from K (without using the
center of the circle).
9) Draw a circle of radius 4.2 cm. Draw arc PQ measuring 1200
Draw a tangent to the circle from point P and point Q.
10) Draw a circle of radius 4.2 cm. Draw a tangent to the circle from
a point 7 cm away from the center of the circle.
11) Draw a circle of radius 3 cm and draw chord XY 5 cm long. Draw
the tangent of the circle passing through point X and point Y (without
using the center of the circle).
Question 4) solve the following examples. (4 marks each)
1) ∆AMT ~∆AHE, In ∆AMT, AM =6.3 cm
=
AM 7
∠MAT= 120 , AT = 4.9 cm, then construct ∆AMT and ∆AHE .
0
HA 5
0 𝐻𝑃
=
4
2) ∆RHP~∆NED, In ∆NED, NE=7 cm. ∠D=30 , ∠N=20 , then construct
0
𝐸𝐷 5
∆RHP and △NED .
3) ∆ABC. ~∆PBR, BC=8 cm, AC=10 cm , ∠B=90 0 ,
=
𝐵𝐶 5
then construct △ABC and ∆PBR
𝐵𝑅 4
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= , then
𝐴𝑀 7
4) ∆AMT. ~∆AHE, 𝐼𝑛 ∆AMT AM=6.3 cm, ∠TAM=50 ,AT=5.6cm,
0
𝐴𝐻 5
construct △AMT and ∆AHE.
5) Draw a circle with radius 3.3cm. Draw a chord PQ of length 6.6cm .
Draw tangents to the circle at points P and Q. Write your observation
about the tangents.
6) Draw a circle with center O and radius 3 cm. Take the point P and the
point Q at a distance of 7 cm from the center of the circle on the
opposite side of the circle at the intersection passing through the center
of the circle Draw a tangent to the circle from the point P and the point
Q.
Question 5) Solve the following examples (3 marks each)
1) Draw a circle with radius 4cm and construct two tangents to a circle
such that when those two tangents intersect each other outside the
circle they make an angle of 600 with each other.
2) AB = 6 cm, ∠BAQ = 500. Draw a circle passing through A and B so
that AQ is the tangent to the circle.
3) Draw a circle with radius 3 cm. Construct a square such that each of its
side will touch the circle from outside.
4) Take points P and Q on the same side of line AB Draw a circle
passing through point P and point Q so that it touches line AB.
𝐴𝑀 7
4) ∆AMT. ~∆AHE, 𝐼𝑛 ∆AMT AM=6.3 cm, ∠TAM=50 ,AT=5.6cm,
0
𝐴𝐻 5
construct △AMT and ∆AHE.
5) Draw a circle with radius 3.3cm. Draw a chord PQ of length 6.6cm .
Draw tangents to the circle at points P and Q. Write your observation
about the tangents.
6) Draw a circle with center O and radius 3 cm. Take the point P and the
point Q at a distance of 7 cm from the center of the circle on the
opposite side of the circle at the intersection passing through the center
of the circle Draw a tangent to the circle from the point P and the point
Q.
Question 5) Solve the following examples (3 marks each)
1) Draw a circle with radius 4cm and construct two tangents to a circle
such that when those two tangents intersect each other outside the
circle they make an angle of 600 with each other.
2) AB = 6 cm, ∠BAQ = 500. Draw a circle passing through A and B so
that AQ is the tangent to the circle.
3) Draw a circle with radius 3 cm. Construct a square such that each of its
side will touch the circle from outside.
4) Take points P and Q on the same side of line AB Draw a circle
passing through point P and point Q so that it touches line AB.
Page 80
5) Draw any circle with radius greater than 1.8 cm and less than 3 cm.
Draw a chord AB 3.6 cm long in this circle. Tangent to the circle passing
through A and B without using the center of the circle
6) Draw a circle with center O and radius 3 cm. Take point P outside the
circle such that d (O, P) = 4.5 cm. Draw tangents to the circle from point P.
7) Draw a circle with center O and radius 2.8 cm. Take point P in the exterior
of a circle such that tangents PA and PB drawn from point P make an angle
∠APB of measure 70 0 .
8) Point P is at a distance of 6 cm from line AB. Draw a circle of radius
4cm passing through point P so that line AB is the tangent to the circle.
………………………………………………………………………………………………………………………………………………
…
Draw a chord AB 3.6 cm long in this circle. Tangent to the circle passing
through A and B without using the center of the circle
6) Draw a circle with center O and radius 3 cm. Take point P outside the
circle such that d (O, P) = 4.5 cm. Draw tangents to the circle from point P.
7) Draw a circle with center O and radius 2.8 cm. Take point P in the exterior
of a circle such that tangents PA and PB drawn from point P make an angle
∠APB of measure 70 0 .
8) Point P is at a distance of 6 cm from line AB. Draw a circle of radius
4cm passing through point P so that line AB is the tangent to the circle.
………………………………………………………………………………………………………………………………………………
…
Page 81
Coordinate Geometry
Q. 1 A) MCQ
1) Point P is midpoint of segment AB where A(- 4,2) and B(6,2) then the
coordinates of P are ---------
A) ( -1, 2 ) B) ( 1, 2 ) C) (1, - 2) D) ( -1, - 2)
2) The distance between Point P ( 2 , 2 ) and Q ( 5, x ) is 5 cm then the
value of x = ----------
A) 2 B) 6 C) 3 D) 1
3) The distance between points P ( -1 , 1 ) and Q(5, -7 ) is ------------..
A) 11 cm B) 10 cm C) 5 cm D) 7 cm
4) If the length of the segment joining point L (x , 7 ) and point
M( 1, 15 ) is 10 cm then the value of x is ---------
A) 7 B) 7 or -5 C) - 1 D) 1
5) Find distance between point A ( -3 , 4 ) and origin O.
A) 7 cm B) 10 cm C) 5 cm D) -5 cm
6) If point P ( 1 , 1 ) divide segment joining point A and point B ( -1 , -1 )
in the ratio 5 : 2 then the coordinates of A are ---------
A)( 3 ,3 ) B)( 6, 6 ) C)(2, 2 ) D)(1, 1 )
7) If segment AB is parallel Y-axis and coordinates of A are (1, 3) then
the coordinates of B are -------------
A)( 3 ,1 ) B)( 5, 3) C)(3, 0) D)(1, -3)
8) If point P is midpoint of segment joining point A (-4, 2) and point
B(6, 2 ) then the coordinates of P are ------------
A)( -1, 2 ) B)( 1 , 2 ) C)(1 , -2) D) (-1, - 2)
Q. 1 A) MCQ
1) Point P is midpoint of segment AB where A(- 4,2) and B(6,2) then the
coordinates of P are ---------
A) ( -1, 2 ) B) ( 1, 2 ) C) (1, - 2) D) ( -1, - 2)
2) The distance between Point P ( 2 , 2 ) and Q ( 5, x ) is 5 cm then the
value of x = ----------
A) 2 B) 6 C) 3 D) 1
3) The distance between points P ( -1 , 1 ) and Q(5, -7 ) is ------------..
A) 11 cm B) 10 cm C) 5 cm D) 7 cm
4) If the length of the segment joining point L (x , 7 ) and point
M( 1, 15 ) is 10 cm then the value of x is ---------
A) 7 B) 7 or -5 C) - 1 D) 1
5) Find distance between point A ( -3 , 4 ) and origin O.
A) 7 cm B) 10 cm C) 5 cm D) -5 cm
6) If point P ( 1 , 1 ) divide segment joining point A and point B ( -1 , -1 )
in the ratio 5 : 2 then the coordinates of A are ---------
A)( 3 ,3 ) B)( 6, 6 ) C)(2, 2 ) D)(1, 1 )
7) If segment AB is parallel Y-axis and coordinates of A are (1, 3) then
the coordinates of B are -------------
A)( 3 ,1 ) B)( 5, 3) C)(3, 0) D)(1, -3)
8) If point P is midpoint of segment joining point A (-4, 2) and point
B(6, 2 ) then the coordinates of P are ------------
A)( -1, 2 ) B)( 1 , 2 ) C)(1 , -2) D) (-1, - 2)
Page 82
9) If point P divides segment AB in the ratio 1:3 where A(-5 , 3) and
B(3 , -5) then the coordinates of P are -----------------
A)( -2, -2 ) B)( -1 , -1 ) C) (-3 , 1 ) D) ( 1, - 3 )
10) If the sum of x-coordinates of the vertices of a triangle is 12 and the
sum of Y-coordinates is 9 then the coordinates of centroid are ----------
A)(12 , 9) B)(9 , 12) C)(4, 3 ) D)(3 ,4 )
Q. 1 B. Solve the following (1 mark each)
1) Find the coordinates of the point of intersection of the graph of the
equation X = 2 and y = -3.
2) Find distance between point A (7, 5) and B (2, 5).
3) The coordinates of diameter AB of a circle are A (2, 7) and
B (4 , 5) then find the coordinates of the centre.
4) Write the X-coordinate and Y-coordinate of point P(- 5 , 4).
5) What are the coordinates of origin?
6) Find distance of point A( 6, 8 ) from origin:
7) Find coordinates of midpoint joining ( -2 ,6 ) and ( 8 ,2 )
8) Find the coordinates of centroid of a triangle whose vertices are
(4, 7) , (8, 4) and (7 ,11).
9) Find distance between point O(0, 0) and B (-5 , 12).
10) Find coordinates of midpoint of point (0, 2) and (12, 14).
Q. 2 A) Complete the activity (each of 2 mark)
1) Find distance between point Q (3 , - 7) and point R ( 3, 3)
Solution: Suppose Q (x1 , y1 ) and point R ( x2 , y2 )
X1 = 3 , y1 = -7 and x2 = 3 , y2 = 3
Using distance formula,
d ( Q, R ) = √
B(3 , -5) then the coordinates of P are -----------------
A)( -2, -2 ) B)( -1 , -1 ) C) (-3 , 1 ) D) ( 1, - 3 )
10) If the sum of x-coordinates of the vertices of a triangle is 12 and the
sum of Y-coordinates is 9 then the coordinates of centroid are ----------
A)(12 , 9) B)(9 , 12) C)(4, 3 ) D)(3 ,4 )
Q. 1 B. Solve the following (1 mark each)
1) Find the coordinates of the point of intersection of the graph of the
equation X = 2 and y = -3.
2) Find distance between point A (7, 5) and B (2, 5).
3) The coordinates of diameter AB of a circle are A (2, 7) and
B (4 , 5) then find the coordinates of the centre.
4) Write the X-coordinate and Y-coordinate of point P(- 5 , 4).
5) What are the coordinates of origin?
6) Find distance of point A( 6, 8 ) from origin:
7) Find coordinates of midpoint joining ( -2 ,6 ) and ( 8 ,2 )
8) Find the coordinates of centroid of a triangle whose vertices are
(4, 7) , (8, 4) and (7 ,11).
9) Find distance between point O(0, 0) and B (-5 , 12).
10) Find coordinates of midpoint of point (0, 2) and (12, 14).
Q. 2 A) Complete the activity (each of 2 mark)
1) Find distance between point Q (3 , - 7) and point R ( 3, 3)
Solution: Suppose Q (x1 , y1 ) and point R ( x2 , y2 )
X1 = 3 , y1 = -7 and x2 = 3 , y2 = 3
Using distance formula,
d ( Q, R ) = √
Page 83
∴ d( Q,R ) = √ + 100
∴ d( Q,R ) = √
∴ d ( Q,R )=
2) Find distance between point A(-1 , 1) and point B (5, -7) :
Solution : - Suppose A( x1 , y1 ) and B(x2, y2)
X1 = -1, y1 = 1 and x2 = 5 , y2 = -7
Using distance formula,
d ( A, B ) = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2
∴d ( A , B ) = √ + ((−7)− )2
∴ d ( A,B ) = √
∴ d( A,B )=
3) Find coordinates of the midpoint of a segment joining point A(-1 ,
1 ) and point B( 5 , -7).
Solution: - Suppose A( x1 , y1 ) and B(x2, y2 )
X1 = -1, y1 = 1 and x2 = 5 , y2 = -7
Using midpoint formula,
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 AB =
𝑥1+𝑥2 𝑦1+𝑦2
( , ) = ( , )
2 2 2 2
4
∴ Coordinates of the midpoint = ( , )
2 2
∴ Coordinates of the midpoint = (2, )
∴ d( Q,R ) = √
∴ d ( Q,R )=
2) Find distance between point A(-1 , 1) and point B (5, -7) :
Solution : - Suppose A( x1 , y1 ) and B(x2, y2)
X1 = -1, y1 = 1 and x2 = 5 , y2 = -7
Using distance formula,
d ( A, B ) = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2
∴d ( A , B ) = √ + ((−7)− )2
∴ d ( A,B ) = √
∴ d( A,B )=
3) Find coordinates of the midpoint of a segment joining point A(-1 ,
1 ) and point B( 5 , -7).
Solution: - Suppose A( x1 , y1 ) and B(x2, y2 )
X1 = -1, y1 = 1 and x2 = 5 , y2 = -7
Using midpoint formula,
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 AB =
𝑥1+𝑥2 𝑦1+𝑦2
( , ) = ( , )
2 2 2 2
4
∴ Coordinates of the midpoint = ( , )
2 2
∴ Coordinates of the midpoint = (2, )
Page 84
4) The coordinates of the vertices of a triangle ABC are A (-7, 6) ,B(2
, -2) and C( 8 , 5) find coordinates of its centroid.
Solution : - Suppose A( x1 , y1 ) and B( x2,y2) and C ( x3, y3 )
X1 = -7, y1 = 6 and x2 = 2 , y2 = -2 and x3 = 8 , y3 = 5
Using Centroid formula
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝑥1+𝑥2+𝑥3 𝑦1+𝑦2+𝑦3
ABC= ( , ) = ( , )
3 3 3 3
3
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ABC = ( , )
3
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ABC= ( 1, )
Q. 2 Solve (Each of 2 marks)
1) The point Q divides segment joining A (3, 5) and B (7, 9) in the ratio
2 : 3. Find the X-coordinate of Q.
2) If the distance between point L ( x , 7) and point M ( 1, 15 ) is 10
then find the value of X.
3) Find the coordinates of midpoint of segment joining (22, 20) and (0
,16)
4) Find distance CD where C(-3a , a), D(a, -2a).
5) Show that the point(11, -2) is equidistant from (4, -3) and (6, 3).
Q. 3 A) Complete the activity ( Each of 3 marks)
1) If the point P (6,7) divides the segment joining A (8, 9) and B(1, 2) in
some ratio. Find that ratio.
Solution : Point P divides segment AB in the ratio m : n.
A ( 8, 9 ) = ( x1 , y1 ) , B (1, 2 ) = ( x2 , y2 ) and P (6, 7 ) = (x,
y)
Using Section formula of internal division,
, -2) and C( 8 , 5) find coordinates of its centroid.
Solution : - Suppose A( x1 , y1 ) and B( x2,y2) and C ( x3, y3 )
X1 = -7, y1 = 6 and x2 = 2 , y2 = -2 and x3 = 8 , y3 = 5
Using Centroid formula
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝑥1+𝑥2+𝑥3 𝑦1+𝑦2+𝑦3
ABC= ( , ) = ( , )
3 3 3 3
3
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ABC = ( , )
3
∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ABC= ( 1, )
Q. 2 Solve (Each of 2 marks)
1) The point Q divides segment joining A (3, 5) and B (7, 9) in the ratio
2 : 3. Find the X-coordinate of Q.
2) If the distance between point L ( x , 7) and point M ( 1, 15 ) is 10
then find the value of X.
3) Find the coordinates of midpoint of segment joining (22, 20) and (0
,16)
4) Find distance CD where C(-3a , a), D(a, -2a).
5) Show that the point(11, -2) is equidistant from (4, -3) and (6, 3).
Q. 3 A) Complete the activity ( Each of 3 marks)
1) If the point P (6,7) divides the segment joining A (8, 9) and B(1, 2) in
some ratio. Find that ratio.
Solution : Point P divides segment AB in the ratio m : n.
A ( 8, 9 ) = ( x1 , y1 ) , B (1, 2 ) = ( x2 , y2 ) and P (6, 7 ) = (x,
y)
Using Section formula of internal division,
Page 85
𝑚( ) + 𝑛( 9)
∴ 7 =
𝑚+𝑛
∴7m +7n = + 9n
∴7m - = 9n -
∴ = 2n
𝑚
∴
𝑛 = A ( -1 , 1 )
1) From the figure given alongside find the length of
the median AD of triangle ABC .
Complete the activity.
C(3 ,5 )
Solution :- Here A (-1 , 1),B(5, -3), C (3, 5) and B ( 5 , -3 ) D
suppose D (x,y) are coordinates of point D.
Using midpoint formula,
5+3 −3+5
X= y=
2 2
∴ 𝑥 = ∴ 𝑦=
Using distance formula,
∴ 𝐴𝐷 = √(4 − )2 + ( 1 − 1 )2
∴ 𝐴𝐷 = √ ( ) 2 + ( 0 )2
∴ 𝐴𝐷 = √
∴ 𝑇ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑒𝑑𝑖𝑎𝑛 𝐴𝐷 =
Q. 3 B) Solve the following (Each of 3 marks)
1) Show that P(-2 , 2 ), Q (2, 2) and R ( 2, 7) are vertices of a right
angled triangle.
2) Show that the point (0 , 9) is equidistant from the points (-4,1) and
(4 , 1).
∴ 7 =
𝑚+𝑛
∴7m +7n = + 9n
∴7m - = 9n -
∴ = 2n
𝑚
∴
𝑛 = A ( -1 , 1 )
1) From the figure given alongside find the length of
the median AD of triangle ABC .
Complete the activity.
C(3 ,5 )
Solution :- Here A (-1 , 1),B(5, -3), C (3, 5) and B ( 5 , -3 ) D
suppose D (x,y) are coordinates of point D.
Using midpoint formula,
5+3 −3+5
X= y=
2 2
∴ 𝑥 = ∴ 𝑦=
Using distance formula,
∴ 𝐴𝐷 = √(4 − )2 + ( 1 − 1 )2
∴ 𝐴𝐷 = √ ( ) 2 + ( 0 )2
∴ 𝐴𝐷 = √
∴ 𝑇ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑒𝑑𝑖𝑎𝑛 𝐴𝐷 =
Q. 3 B) Solve the following (Each of 3 marks)
1) Show that P(-2 , 2 ), Q (2, 2) and R ( 2, 7) are vertices of a right
angled triangle.
2) Show that the point (0 , 9) is equidistant from the points (-4,1) and
(4 , 1).
Page 86
3) Point P(-4 , 6) divides point A (-6 , 10) and B (m , n) in the ratio
2:1 then find the coordinates of point B.
Q. 4 Solve (Each of 4 marks)
1) Show that points A(-4 , -7), B(-1,2 ), C (8, 5) and D (5 , -4) are the
vertices of a parallelogram ABCD.
2) Show that the points (0 , -1), (8, 3) , (6 , 7) and (-2 , 3) are vertices of a
rectangle.
3) Show that the points (2, 0 ), (-2 , 0) and (0, 2 ) are vertices of a triangle.
State the type of triangle with reason.
4) If A(5,4 ) , B (-3 , -2) and C (1 -8) are the vertices of a ∆ 𝐴𝐵𝐶 . Segment
AD is median. Find the length of seg AD:
5) Show that A (1, 2) , (1, 6) , C (1 + 2√3 , 4 ) are vertices of an equilateral
triangle.
Q.5) Solve (Each of 3 marks)
A (0,2)
1)
Seg OA is the radius of a circle with centre O.
O
The coordinates of point A is (0 , 2) then
decide whether the point B(1, 2) is on the circle?
2) Find the ratio in which Y-axis divides the point A( 3, 5) and point B(-
6 , 7). Find the coordinates of that point.
3) The points (7, -6) , (2, K) and (h,18) are the vertices of triangle. If
(1,5) are the coordinates of centroid. Find the value of h and k..
4) Using distance formula decide whether the points (4, 3), (5, 1) and
(1, 9) are collinear or not ?
2:1 then find the coordinates of point B.
Q. 4 Solve (Each of 4 marks)
1) Show that points A(-4 , -7), B(-1,2 ), C (8, 5) and D (5 , -4) are the
vertices of a parallelogram ABCD.
2) Show that the points (0 , -1), (8, 3) , (6 , 7) and (-2 , 3) are vertices of a
rectangle.
3) Show that the points (2, 0 ), (-2 , 0) and (0, 2 ) are vertices of a triangle.
State the type of triangle with reason.
4) If A(5,4 ) , B (-3 , -2) and C (1 -8) are the vertices of a ∆ 𝐴𝐵𝐶 . Segment
AD is median. Find the length of seg AD:
5) Show that A (1, 2) , (1, 6) , C (1 + 2√3 , 4 ) are vertices of an equilateral
triangle.
Q.5) Solve (Each of 3 marks)
A (0,2)
1)
Seg OA is the radius of a circle with centre O.
O
The coordinates of point A is (0 , 2) then
decide whether the point B(1, 2) is on the circle?
2) Find the ratio in which Y-axis divides the point A( 3, 5) and point B(-
6 , 7). Find the coordinates of that point.
3) The points (7, -6) , (2, K) and (h,18) are the vertices of triangle. If
(1,5) are the coordinates of centroid. Find the value of h and k..
4) Using distance formula decide whether the points (4, 3), (5, 1) and
(1, 9) are collinear or not ?
Page 87
Trigonometry
Que.) 1 A) .Choose the correct alternative from those given below
each question : (1 mark for each MCQ )
1. cos θ . sec θ = ?
1
A) 1 B) 0 C) D) √2
2
2. sec 600 = ?
1 2
A) B) 2 C) D) √2
2 √3
3. 1+ cot 2 θ = ?
A) tan2 θ B) sec 2 θ C) cosec 2 θ D)
cos2 θ
4. cot θ . tan θ = ?
A) 1 B) 0 C) 2 D) √2
5. sec 2 θ - tan2 θ = ?
A) 0 B) 1 C) 2 D) √2
6. sin2 θ + sin2 (90 − θ) = ?
A) 0 B) 1 C) 2 D) √2
1+ cot2 A
7. =?
1+ tan2 A
A) tan2 θ B) sec 2 θ C) cosec 2 θ D) cot 2 θ
1
8. sin θ = then θ = ?
2
A) 300 B) 450 C) 600 D) 900
Que.) 1 A) .Choose the correct alternative from those given below
each question : (1 mark for each MCQ )
1. cos θ . sec θ = ?
1
A) 1 B) 0 C) D) √2
2
2. sec 600 = ?
1 2
A) B) 2 C) D) √2
2 √3
3. 1+ cot 2 θ = ?
A) tan2 θ B) sec 2 θ C) cosec 2 θ D)
cos2 θ
4. cot θ . tan θ = ?
A) 1 B) 0 C) 2 D) √2
5. sec 2 θ - tan2 θ = ?
A) 0 B) 1 C) 2 D) √2
6. sin2 θ + sin2 (90 − θ) = ?
A) 0 B) 1 C) 2 D) √2
1+ cot2 A
7. =?
1+ tan2 A
A) tan2 θ B) sec 2 θ C) cosec 2 θ D) cot 2 θ
1
8. sin θ = then θ = ?
2
A) 300 B) 450 C) 600 D) 900
Page 88
9. tan (90-θ) = ?
A) sin θ B) cos θ C) cot θ D)
tan θ
10. cos 450 = ?
A) sin 450 B) sec 450 C) cot 450 D)
tan 450
3
11. If sin θ = then cos θ = ?
5
5 3 4 5
A) B) C) D)
3 5 5 4
12. Which is not correct formula ?
A) 1 + tan2 θ = sec 2 θ
B) 1 + sec 2 θ = tan2 θ
C) cosec 2 θ − cot 2 θ = 1
D) sin2 θ + cos2 θ = 1
13. If A = 300 then tan 2A = ?
1
A) 1 B) 0 C) D) √3
√3
Que.) 1 B). Solve the following questions : (1 mark each )
1− tan2 450
1. =?
1+ tan2 450
13
2. If tan θ = then cot θ = ?
12
3. Prove that cosec θ X √1 − cos2 θ = 1.
4. If tan θ = 1 then sin θ . cos θ = ?
5. If 2 sin θ = 3 cos θ then tan θ = ?
A) sin θ B) cos θ C) cot θ D)
tan θ
10. cos 450 = ?
A) sin 450 B) sec 450 C) cot 450 D)
tan 450
3
11. If sin θ = then cos θ = ?
5
5 3 4 5
A) B) C) D)
3 5 5 4
12. Which is not correct formula ?
A) 1 + tan2 θ = sec 2 θ
B) 1 + sec 2 θ = tan2 θ
C) cosec 2 θ − cot 2 θ = 1
D) sin2 θ + cos2 θ = 1
13. If A = 300 then tan 2A = ?
1
A) 1 B) 0 C) D) √3
√3
Que.) 1 B). Solve the following questions : (1 mark each )
1− tan2 450
1. =?
1+ tan2 450
13
2. If tan θ = then cot θ = ?
12
3. Prove that cosec θ X √1 − cos2 θ = 1.
4. If tan θ = 1 then sin θ . cos θ = ?
5. If 2 sin θ = 3 cos θ then tan θ = ?
Page 89
6. If cot ( 90 – A ) = 1 then A = ?
1
7. If 1 − cos2 θ = then θ = ?
4
cos ( 90 – A ) sin ( 90 – A )
8. Prove that = .
sin A cos A
9. If tan θ X = sin θ then =?
10. (sec θ + tan θ) . (sec θ - tan θ) = ?
sin 750
11. =?
cos 150
Que.) 2 A). Complete the following activities ( 2 marks each )
* ( Write complete answers, don’t just fill the boxes )
1. Prove that cos2 θ . (1 + tan2 θ ) = 1. Complete the activity given
below.
Activity L . H . S. =
= cos2 θ X ...(1 + tan2 θ =
)
= (cos θ X )2
= 12
=1
= R .H .S.
5
2. − 5 cot 2 θ, Complete the activity given below.
sin2 θ
5
Activity − 5 cot 2 θ
sin2 θ
1
7. If 1 − cos2 θ = then θ = ?
4
cos ( 90 – A ) sin ( 90 – A )
8. Prove that = .
sin A cos A
9. If tan θ X = sin θ then =?
10. (sec θ + tan θ) . (sec θ - tan θ) = ?
sin 750
11. =?
cos 150
Que.) 2 A). Complete the following activities ( 2 marks each )
* ( Write complete answers, don’t just fill the boxes )
1. Prove that cos2 θ . (1 + tan2 θ ) = 1. Complete the activity given
below.
Activity L . H . S. =
= cos2 θ X ...(1 + tan2 θ =
)
= (cos θ X )2
= 12
=1
= R .H .S.
5
2. − 5 cot 2 θ, Complete the activity given below.
sin2 θ
5
Activity − 5 cot 2 θ
sin2 θ
Page 90
1
= ( − cot 2 θ)
sin2 θ
1
=5( - cot 2 θ ) ......... ( =
sin2 θ
)
= 5(1)
=
3. If sec θ + tan θ = √3 . Complete the activity to find the value of
sec θ − tan θ
Activity = 1+ tan2 θ ……(Fundamental trigonometric
identity)
- tan2 θ = 1
(sec θ + tan θ) . (sec θ - tan θ) =
√3 . (sec θ - tan θ) = 1
(sec θ - tan θ) =
9
4. If tan θ = . Complete the activity to find the value of sec θ .
40
Activity sec 2 θ = 1+ ........ (Fundamental trigonometric identity)
sec 2 θ = 1+ 2
sec 2 θ = 1+
sec θ =
Que.) 2 B). Solve the following questions : (2 marks each )
24
1. If cos θ = then sin θ = ?
25
= ( − cot 2 θ)
sin2 θ
1
=5( - cot 2 θ ) ......... ( =
sin2 θ
)
= 5(1)
=
3. If sec θ + tan θ = √3 . Complete the activity to find the value of
sec θ − tan θ
Activity = 1+ tan2 θ ……(Fundamental trigonometric
identity)
- tan2 θ = 1
(sec θ + tan θ) . (sec θ - tan θ) =
√3 . (sec θ - tan θ) = 1
(sec θ - tan θ) =
9
4. If tan θ = . Complete the activity to find the value of sec θ .
40
Activity sec 2 θ = 1+ ........ (Fundamental trigonometric identity)
sec 2 θ = 1+ 2
sec 2 θ = 1+
sec θ =
Que.) 2 B). Solve the following questions : (2 marks each )
24
1. If cos θ = then sin θ = ?
25
Page 91
sin2 θ
2. Prove that + cos θ = sec θ .
cos θ
1
3. Prove that = cosec θ + cot θ .
cosec θ−cot θ
4. If cos (450 + 𝑥 ) = sin 300 then 𝑥 = ?
5. If tan θ + cot θ = 2 then tan2 θ + cot 2 θ = ?
6. Prove that sec 2 θ + cosec 2 θ = sec 2 θ X cosec 2 θ .
7. Prove that cot 2 θ X sec 2 θ = cot 2 θ + 1 .
8. If 3 sin θ = 4 cos θ then sec θ = ?
9. If sin 3A = cos 6 A then A = ?
10. Prove that sec 2 θ − cos2 θ = tan2 θ + sin2 θ .
tan A sec2 A
11. Prove that = .
cot A cosec2 A
sin θ+ tan θ
12. Prove that = tan θ ( 1 + sec θ) .
cos θ
cos2 θ
13. Prove that + sin θ = cosec θ .
sin θ
cos θ 1−sin θ
14. Prove that = .
1+sin θ cos θ
Que.) 3 A). Complete the following activities ( 3 marks each )
* ( Write complete answers, don’t just fill the boxes )
1. sin4 A − cos4 A = 1 − 2cos2 A, For proof of this complete the
activity given below.
Activity L . H . S. =
= ( sin2 A + cos 2 A ) ( )
= 1 ( ) ................ ( sin2 A + =1)
2. Prove that + cos θ = sec θ .
cos θ
1
3. Prove that = cosec θ + cot θ .
cosec θ−cot θ
4. If cos (450 + 𝑥 ) = sin 300 then 𝑥 = ?
5. If tan θ + cot θ = 2 then tan2 θ + cot 2 θ = ?
6. Prove that sec 2 θ + cosec 2 θ = sec 2 θ X cosec 2 θ .
7. Prove that cot 2 θ X sec 2 θ = cot 2 θ + 1 .
8. If 3 sin θ = 4 cos θ then sec θ = ?
9. If sin 3A = cos 6 A then A = ?
10. Prove that sec 2 θ − cos2 θ = tan2 θ + sin2 θ .
tan A sec2 A
11. Prove that = .
cot A cosec2 A
sin θ+ tan θ
12. Prove that = tan θ ( 1 + sec θ) .
cos θ
cos2 θ
13. Prove that + sin θ = cosec θ .
sin θ
cos θ 1−sin θ
14. Prove that = .
1+sin θ cos θ
Que.) 3 A). Complete the following activities ( 3 marks each )
* ( Write complete answers, don’t just fill the boxes )
1. sin4 A − cos4 A = 1 − 2cos2 A, For proof of this complete the
activity given below.
Activity L . H . S. =
= ( sin2 A + cos 2 A ) ( )
= 1 ( ) ................ ( sin2 A + =1)
Page 92
= - cos2 A ................ ( sin2 A = 1 − cos 2 A )
=
= R. H. S.
2. tan2 θ − sin2 θ = tan2 θ X sin2 θ .For proof of this complete the
activity given below.
Activity L . H . S. =
sin2 θ
= (1− )
tan2 θ
= tan2 θ ( 1 − sin2 θ
)
cos2 θ
sin2 θ cos2 θ
= tan2 θ ( 1− X )
1
= tan2 θ ( 1 − )
= tan2 θ X ............... ( 1 − cos2 θ = sin2 θ )
= R. H. S.
7
3. If tan θ = then To find value of cos θ complete the activity
24
given below.
Activity sec 2 θ = 1+ ............(Fundamental tri.
identity)
sec 2 θ = 1+ 2
sec 2 θ = 1+
576
sec 2 θ =
576
sec θ =
=
= R. H. S.
2. tan2 θ − sin2 θ = tan2 θ X sin2 θ .For proof of this complete the
activity given below.
Activity L . H . S. =
sin2 θ
= (1− )
tan2 θ
= tan2 θ ( 1 − sin2 θ
)
cos2 θ
sin2 θ cos2 θ
= tan2 θ ( 1− X )
1
= tan2 θ ( 1 − )
= tan2 θ X ............... ( 1 − cos2 θ = sin2 θ )
= R. H. S.
7
3. If tan θ = then To find value of cos θ complete the activity
24
given below.
Activity sec 2 θ = 1+ ............(Fundamental tri.
identity)
sec 2 θ = 1+ 2
sec 2 θ = 1+
576
sec 2 θ =
576
sec θ =
Page 93
1
cos θ = ....................( cos θ = )
sec θ
4. To prove cot θ + tan θ = cosec θ X sec θ . Complete the activity
given below.
Activity L . H . S. =
sin θ
= =
sin θ cos θ
cos2 θ+ sin2 θ
=
1
= ............( cos2 θ + sin2 θ = )
sin θ . cos θ
1 1
= X
sin θ
=
= R. H. S.
Que.) 3 B). Solve the following questions : (3 marks each )
41
1. If sec θ = then find values of sin θ, cot θ, cosec θ .
40
2. If 5 sec θ − 12 cosec θ = 0 then find values of sin θ, sec θ .
tan (90− θ ) + cot (90− θ )
3. Prove that = sec A .
cosec θ
4. Prove that cot 2 θ − tan2 θ = cosec 2 θ − sec 2 θ .
1+ sin θ
5. Prove that = (sec θ + tan θ )2 .
1− sin θ
sin θ sin θ
6. Prove that + = 2 cot θ .
sec θ+ 1 sec θ− 1
sec A
7. Prove that = sin A .
tan A +cot A
sin θ+cosec θ
8. Prove that = 2 + cot 2 θ .
sin θ
cos θ = ....................( cos θ = )
sec θ
4. To prove cot θ + tan θ = cosec θ X sec θ . Complete the activity
given below.
Activity L . H . S. =
sin θ
= =
sin θ cos θ
cos2 θ+ sin2 θ
=
1
= ............( cos2 θ + sin2 θ = )
sin θ . cos θ
1 1
= X
sin θ
=
= R. H. S.
Que.) 3 B). Solve the following questions : (3 marks each )
41
1. If sec θ = then find values of sin θ, cot θ, cosec θ .
40
2. If 5 sec θ − 12 cosec θ = 0 then find values of sin θ, sec θ .
tan (90− θ ) + cot (90− θ )
3. Prove that = sec A .
cosec θ
4. Prove that cot 2 θ − tan2 θ = cosec 2 θ − sec 2 θ .
1+ sin θ
5. Prove that = (sec θ + tan θ )2 .
1− sin θ
sin θ sin θ
6. Prove that + = 2 cot θ .
sec θ+ 1 sec θ− 1
sec A
7. Prove that = sin A .
tan A +cot A
sin θ+cosec θ
8. Prove that = 2 + cot 2 θ .
sin θ
Page 94
cot A tan A
9. Prove that + =−1.
1−cot A 1−tan A
1+cos A
10. Prove that √ = cosec A + cot A .
1−cos A
11. Prove that sin4 A − cos4 A = 1 − 2cos2 A .
12. Prove that sec 2 θ − cos2 θ = tan2 θ + sin2 θ .
sin θ
13. Prove that cosec θ – cot θ = .
1+ cos θ
12
14. In ∆ ABC, cos C = and BC = 24 then AC = ?
13
1+ sec A sin2 A
15. Prove that = .
sec A 1−cos A
3
16. If sin A = then show that 4 tan A + 3 tan A = 6 cos A
5
1+sin B cos B
17. Prove that + = 2 sec B .
cos B 1 +sin B
Que. 4 Solve the following questions : (Challenging questions, 4
marks each )
1. Prove that
sin2 A . tan A + cos2 A . cot A + 2 sin A . cos A = tan A + cot A
.
2sin2 A−1
2
2. Prove that sec A − cosec A = 2
.
sin2 A . cos2 A
cot A + cosec A − 1 1+cos A
3. Prove that = .
cot A− cosec A + 1 sin A
4. Prove that sin θ ( 1 – tan θ ) − cos θ ( 1 − cot θ ) = cosec θ − sec θ
.
.
2 √𝑚 𝑚+1
5. If cos A = then Prove that cosec A =
𝑚+1 𝑚−1
9. Prove that + =−1.
1−cot A 1−tan A
1+cos A
10. Prove that √ = cosec A + cot A .
1−cos A
11. Prove that sin4 A − cos4 A = 1 − 2cos2 A .
12. Prove that sec 2 θ − cos2 θ = tan2 θ + sin2 θ .
sin θ
13. Prove that cosec θ – cot θ = .
1+ cos θ
12
14. In ∆ ABC, cos C = and BC = 24 then AC = ?
13
1+ sec A sin2 A
15. Prove that = .
sec A 1−cos A
3
16. If sin A = then show that 4 tan A + 3 tan A = 6 cos A
5
1+sin B cos B
17. Prove that + = 2 sec B .
cos B 1 +sin B
Que. 4 Solve the following questions : (Challenging questions, 4
marks each )
1. Prove that
sin2 A . tan A + cos2 A . cot A + 2 sin A . cos A = tan A + cot A
.
2sin2 A−1
2
2. Prove that sec A − cosec A = 2
.
sin2 A . cos2 A
cot A + cosec A − 1 1+cos A
3. Prove that = .
cot A− cosec A + 1 sin A
4. Prove that sin θ ( 1 – tan θ ) − cos θ ( 1 − cot θ ) = cosec θ − sec θ
.
.
2 √𝑚 𝑚+1
5. If cos A = then Prove that cosec A =
𝑚+1 𝑚−1
Page 95
1 1
6. If sec A = 𝑥 + then show that sec A + tan A = 2𝑥 or .
4𝑥 2𝑥
7. In ∆ ABC , √2 AC = BC, sin A = 1, sin2 A + sin2 B + sin2 C = 2
then A = ? B = ? C= ?
8. Prove that sin6 A + cos6 A = 1 – 3 sin2 A . cos2 A .
9. Prove that 2 (sin6 A + cos6 A) – 3 (sin4 A + cos4 A ) + 1 = 0 .
cot A tan A
10. Prove that + = 1+ tanA + cotA = secA . cosecA
1−tan A 1−co t A
+1
Que. 5 Solve the following questions : (Creative questions, 3
marks each )
1. If 3 sin A + 5 cos A = 5 then show that 5 sin A – 3 cos A = ±3.
2. If cos A + cos 2 A = 1 then sin2 A + sin4 A = ?
3. If cosec A – sin A = p आणि sec A – cos A = q then prove that
2 2
2 2 3
(𝑝 𝑞) + (𝑝𝑞 ) = 1
3
4. Show that tan 70 X tan 230 X tan 600 X tan 670 X tan 830 = √3.
5. If sin θ + cos θ = √3 then show that tan θ + cot θ = 1 .
1
6. If tan θ - sin2 θ = cos 2 θ then show that sin2 θ = .
2
7. Prove that
( 1 − cos2 A ) . sec 2 B + tan2 B ( 1− sin2 A ) = sin2 A + tan2 B
6. If sec A = 𝑥 + then show that sec A + tan A = 2𝑥 or .
4𝑥 2𝑥
7. In ∆ ABC , √2 AC = BC, sin A = 1, sin2 A + sin2 B + sin2 C = 2
then A = ? B = ? C= ?
8. Prove that sin6 A + cos6 A = 1 – 3 sin2 A . cos2 A .
9. Prove that 2 (sin6 A + cos6 A) – 3 (sin4 A + cos4 A ) + 1 = 0 .
cot A tan A
10. Prove that + = 1+ tanA + cotA = secA . cosecA
1−tan A 1−co t A
+1
Que. 5 Solve the following questions : (Creative questions, 3
marks each )
1. If 3 sin A + 5 cos A = 5 then show that 5 sin A – 3 cos A = ±3.
2. If cos A + cos 2 A = 1 then sin2 A + sin4 A = ?
3. If cosec A – sin A = p आणि sec A – cos A = q then prove that
2 2
2 2 3
(𝑝 𝑞) + (𝑝𝑞 ) = 1
3
4. Show that tan 70 X tan 230 X tan 600 X tan 670 X tan 830 = √3.
5. If sin θ + cos θ = √3 then show that tan θ + cot θ = 1 .
1
6. If tan θ - sin2 θ = cos 2 θ then show that sin2 θ = .
2
7. Prove that
( 1 − cos2 A ) . sec 2 B + tan2 B ( 1− sin2 A ) = sin2 A + tan2 B
Document Details
| Board / Org | Maharashtra Board |
| Exam | Class 10 |
| Type | Question Bank |
| Pages | 95 |
| Language | english |
| Updated | 22 Jul 2026 |