MBOSE Class 10 Question Paper 2023 for Maths

Total No. of Printed Pages—15
X/23/M

2023

MATHEMATICS

( FOR CANDIDATES WITH INTERNAL ASSESSMENT )

Full Marks : 80
Pass Marks : 24

( FOR CANDIDATES WITHOUT INTERNAL ASSESSMENT )

Full Marks : 100
Pass Marks : 30

Time : 3 hours

( FOR ALL CATEGORIES OF CANDIDATES )

General Instructions :

(i) This Question Paper comprises of 32 questions divided
into six Sections A, B, C, D, E and F.

(ii) Marks allocated to every question are indicated against
each.

(iii) Question Nos. 1 to 30 (Section—A to Section—E) are to
be answered by all candidates.

(iv) Question Nos. 31 and 32 of Section—F are to
be answered by Candidates without Internal
Assessment.

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( 2 )

(v) In question on construction, the drawing should be
neat and exactly as per the given measurements.

(vi) Questions, which are meant for Visually Handicapped
(Blind) Students, should be answered by them only.

(vii) Use of Calculator/Mobile Phone is not permitted.

SECTION—A

( Marks : 8 )

( Question Nos. 1 to 8 carry 1 mark each )

1. Write the exponent of 2 in the prime factorization of 288. 1

n
2. What is the 2nd term of the sequence an  ? 1
n 1

1
3. Check whether the equation x  2, x  0 is a
x
quadratic equation or not. 1

4. Express cos 65  tan 65 in terms of angles between
0° and 30°. 1

5. State AA-similarity criterion. 1

6. Find the distance between two parallel tangents of a
circle of radius 6 cm. 1

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7. Write the formula of the total surface area of a right
circular cylinder whose radius of the base is r units
and its height is h units. 1

8. Define modal class. 1

SECTION—B
( Marks : 14 )
( Question Nos. 9 to 15 carry 2 marks each )

9. Find the value of k so that 8k + 4, 6k – 2 and 2k – 7
are the three consecutive terms of an A.P. 2

10. Prove that
cos 48 cosec42  sec 48 sin 42  2 2

11. Find the value of x (0  x  90) in
cos x  cos 60 cos 30  sin 60 sin 30 2
Or

If x  a cos3  and y  b sin3  , then prove that
2 2
 x 3  y 3 2
     1
a  b 

12. Find the distance between the pair of points (a, 0)
and (0, b). 2
Or
Find the coordinates of the midpoint of the line segment
joining the points P (7, 0) and Q (–5, 4). 2

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13. Find the third vertex of a triangle ABC, if two of its
vertices are B (–3, 1) and C(0, –2), and its centroid is at
the origin. 2
14.

In the above figure, D and E are the points on the
sides AB and AC respectively of ABC. Determine whether
DE || BC or not if AD = 5·7 cm, DB = 9·5 cm, AE = 4·8 cm
and EC = 8 cm. 2

[ For Visually Handicapped (Blind) Students only,
instead of Question No. 14 given above ]

14. (a) Two geometric figures having the same shape but
different sizes are said to be _____ figures.
( Fill in the blank ) 1

(b) If a line divides any two sides of a triangle in the same
ratio, then the line must be parallel to the third side.
( State whether True or False ) 1
15.

In the above figure, PT and PT  are tangents from P to
the circle with centre O. R is a point on the circle.
Prove that PA + AR = PB + BR. 2
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[ For Visually Handicapped (Blind) Students only,
instead of Question No. 15 given in Page No. 4 ]

15. (a) Define a tangent. 1

(b) How many tangents can be drawn through a point
lying on the circle? 1

SECTION—C
( Marks : 24 )

( Question Nos. 16 to 23 carry 3 marks each )

16. Using ruler and compass only, construct a triangle ABC
such that each of its sides is 2
3
rd of the corresponding
side of ABC. It is given that AB = 4 cm, BC = 5 cm and
AC = 6 cm. (Only traces of constructions are required.) 3

[ For Visually Handicapped (Blind) Students only,
instead of Question No. 16 given above ]

16. (a) Define an equilateral triangle. 1

(b) When are two triangles said to be equiangular? 1

(c) The ratio of the areas of two similar triangles
is equal to the ratio of the squares of their
corresponding sides.
( State whether True or False ) 1

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17. In the figure below, OA and OB are tangents to the circle
drawn from an external point O. CD is a third tangent
touching the circle at P. If OB = 10 cm and CP = 2 cm,
then find the length of OC. 3

Or

1
In the figure below, PQ || BC such that AQ  AC .
If AB = 6 cm, find AP. 4 3

[ For Visually Handicapped (Blind) Students only,
instead of Question No. 17 given above ]

17. (a) Define a circle. 1

(b) Can a triangle have two right angles? 1

(c) If two triangles are congruent, then their areas
are _____.
( Fill in the blank ) 1

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18. A race track is in the form of a ring whose inner
circumference is 352 m and the outer circumference is
396 m. Find the width of the track. (Use   22 ) 3
7

Or
Find the area of the minor sector of a circle of radius
4 cm and of angle 30°. (Take  = 3·14) 3

19. Find the smallest number which when divided by 35, 56
and 91 leaves remainder of 7 in each case. 3

Or
Using Euclid’s division algorithm, find the HCF of
105 and 120. 3

20. Find a quadratic polynomial whose zeroes are 3  5
and 3  5 . 3

21. The 9th term of an A.P. is equal to 6 times its second
term. If its 5th term is 22, find the A.P. 3

Or
Find the sum of the first 100 natural numbers. 3

22. Prove that
1 1
  2sec2  3
1  sin  1  sin 
Or
If sin( A  B )  1 and cos( A  B )  1, 0  A  B  90 , A  B ,
find A and B. 3

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23. A bag contains 5 red, 8 white and 7 black balls. A ball
is drawn at random from the bag. Find the probability
that the drawn ball is—
(a) red or white;
(b) neither black nor white. 3

SECTION—D
( Marks : 16 )

( Question Nos. 24 to 27 carry 4 marks each )

24. The sum of the numerator and denominator of a fraction
is 4 more than twice the numerator. If the numerator
and denominator are increased by 3, they are in the
ratio 2 : 3. Determine the fraction. 4

Or

Find two consecutive natural numbers whose product
is 20. 4

25. An angle of elevation of a ladder leaning against a wall
is 60° and the foot of the ladder is 9·5 m away from the
wall. Find the length of the ladder. 4

Or

The angle of elevation of the top of a hill from the foot of
a tower is 60° and the angle of elevation of the top of the
tower from the foot of the hill is 30°. If the tower is 50 m
high, then what is the height of the hill? 4

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[ For Visually Handicapped (Blind) Students only,
instead of Question No. 25 given in Page No. 8 ]

cos17
25. (a) Evaluate . 1
sin 73

(b) If cos   1, then  = _____.
( Fill in the blank ) 1

(c) If sec   tan   m and sec   tan   n , then prove
that mn = 1. 2

26. Find the ratio in which the point P (m, 6) divides the line
segment joining the points A (–1, 3) and B (2, 8). Also,
find the value of m. 4
Or
Find the value of p for which the points A (–1, 3), B (2, p)
and C (5, –1) are collinear. 4

27. Prove that in a triangle, if the square of one side is equal
to the sum of the squares of the other two sides, then the
angle opposite to the first side is a right angle. 4

[ For Visually Handicapped (Blind) Students only,
instead of Question No. 27 given above ]

27. (a) What is the area of an equilateral triangle having
side a units? 2

(b) The greatest side of a right-angled triangle is called
hypotenuse.
( State whether True or False ) 1

(c) All equilateral triangles are _____.
( Fill in the blank ) 1

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SECTION—E
( Marks : 18 )
( Question Nos. 28 to 30 carry 6 marks each )
28. Solve the following system of linear equations graphically :
4x  y  4  0
3x  2y  14  0
Find the vertices of the triangle formed by the lines and
the y-axis. (Plot at least three points for each graph.) 6

[ For Visually Handicapped (Blind) Students only,
instead of Question No. 28 given above ]
28. Solve the following system of linear equations : 6
2x  y  7
4x  3y  1  0

29. The dimensions of a metallic cuboid are 100 cm × 80 cm
× 64 cm. It is melted and recast into a cube. Find the
total surface area of the cube. 6
Or
The volume of a hemisphere is 2425 12 cm3. Find its
curved surface area. (Use   22 ) 6
7
30. Find the mean of the following data : 6
Class Interval 0–100 100–200 200–300 300–400 400–500

Frequency 6 9 15 12 8

Or
Find the mode of the following frequency distribution : 6
Age (in years) 0–5 5–10 10–15 15–20 20–25 25–30 30–35
Number of Patients 6 11 18 24 17 13 5

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SECTION—F
( Marks : 20 )
[ For Candidates without Internal Assessment (WIM) ]
31. Answer the following as directed (any eight) : 1×8=8
(a) Every composite number can be expressed as a
product of
(A) coprimes
(B) primes
(C) twin primes
(D) None of the above
( Choose the correct option )

(b) The number of zeroes of a polynomial of degree n is
(A) equal to n
(B) greater than n
(C) less than n
(D) less than or equal to n
( Choose the correct option )

(c) A quadratic equation ax 2  bx  c  0, a  0 has real
roots if
(A) discriminant = 0
(B) discriminant > 0

(C) discriminant  0

(D) discriminant  0
( Choose the correct option )

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(d) The total surface area of a hemisphere of radius
r units is

(A) r 2 sq. units

(B) 2r 2 sq. units

(C) 3r 2 sq. units

(D) 4r 2 sq. units
( Choose the correct option )

(e) The degree of a constant polynomial is _____.
( Fill in the blank )

(f) Abscissa of any point on the y-axis is zero.
( State whether True or False )

(g) The different numbers in a sequence are called
terms.
( State whether True or False )

(h) tan2   sec2   1
( State whether True or False )

(i) Two angles are said to be complementary if their
sum is _____.
( Fill in the blank )

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343
(j) The number has terminating decimal
2  7  52
representation.
( State whether True or False )

(k) Write the first term of 2n.

(l) Write the maximum value of cos .

(m) The lengths of two tangents drawn from an external
point to a circle are _____.
( Fill in the blank )

(n) Ogives are cumulative frequency curves.

( State whether True or False )

32. Answer any six from the following : 2×6=12

(a) Find the prime factorization of 2025.

(b) Without performing long division method, state
whether 23
8
has a terminating decimal expansion or
a non-terminating repeating decimal expansion.

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(c) Find the zeroes of the polynomial x 2  2x  3.

(d) Solve the quadratic equation

2
 1
x    4
 2

(e) Write the first two terms of the sequence

3n
tn 
2n  1

(f) Determine whether the following given sides of a
triangle form a right triangle or not :
a = 3 cm, b = 4 cm and c = 7 cm

(g) If A = 30°, verify that

cos 2A  1  2sin2 A

(h) Eliminate ‘’ for the equations :

x  a sec  and y  b tan 

(i) If P (E )  0·05, what is the probability of ‘not E ’?

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(j) State whether the following pair of linear equations
has a unique solution, no solution or infinitely many
solutions :
3x  5y  7
6x  10y  3

(k) Find the coordinates of the centroid of the triangle
whose vertices are (–2, 3), (2, –1) and (4, 0).

(l) Write the discriminant of the quadratic equation

x2  x  2  0

  

X/23/M/3 K23—56280