MBOSE Class 10 Question Paper 2022 for Maths

Total No. of Printed Pages—12
X/22/M

2022

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. Express the decimal 11·225 as a rational number. 1

2. What is the degree of a biquadratic polynomial? 1

3. Find the first term ‘a’ and common difference ‘d’ of the
A.P. : 45, 50, 55, 60, .... . 1

4. When are two triangles said to be similar? 1

sin10
5. Find the value of . 1
cos 80

6. How many tangents can be drawn to a circle from a point
outside the circle? 1

7. Write the formula for the area of a sector of angle ‘’ of
a circle of radius ‘r’ units. 1

8. Find the class mark of the class 35–55. 1

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

9. Find the discriminant of the quadratic equation
2x 2  16x  30  0 and hence write the nature of its roots. 2

10. If A  30º and B  60 , verify that
sin( A  B )  sin A cos B  cos A sin B 2
11. Find the value of x (0º  x  90º ) in 2sin 2x  3 . 2
Or
In  ABC , right angled at A, if AB  4 , AC  3 and
BC  5 , then find cos B and cosec B. 2
12. Find the distance between the pair of points (5, 8)
and (–3, 2). 2
13. Find the coordinates of the centroid of the triangle
whose vertices are (–3, 0), (5, –2) and (–8, 5). 2
Or
Find the ratio in which the point (2, y) divides the line
segment joining the points A (–2, 2) and B (3, 7). 2
14. The triangles ABC and DEF are similar. If the
2
ar (ABC )  9 cm , ar (DEF )  64 cm2 and DE  5·1 cm,
then find AB. 2
15. In the given figure, CP and CQ are tangents to a circle with
centre O. ARB is another tangent touching the circle at R.
If CP = 11 cm and BC = 7 cm, then find the length of BR. 2

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[ For Visually Handicapped (Blind) Students only,
instead of Question No. 15 given in Page No. 3 ]

15. (a) One and only one tangent can be drawn through a
point lying on the circle.
( State whether True or False ) 1

(b) Can we draw infinite number of tangents to a circle? 1

SECTION—C
( Marks : 24 )
( Question Nos. 16 to 23 carry 3 marks each )

16. Using ruler and compass only, construct two tangents
to a circle of radius 3·5 cm from a point P at a distance
of 6·2 cm from its centre. (Only traces of construction
are required.) 3

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

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

(b) The lengths of the tangents drawn from an external
point to a circle are equal.
( State whether True or False ) 1

(c) Find the radius of a circle whose diameter is 35 cm. 1

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17. In the adjoining figure, ABC is circumscribing a circle.
Find the length of BC. 3

Or
In the figure below, OAB ~ OCD . When AB = 8 cm,
BO = 6·4 cm, OC = 3·5 cm and CD = 5 cm, find OA. 3

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

17. (a) If the corresponding sides of two triangles are equal,
then they are similar.
( State whether True or False ) 1

(b) The distance between two parallel tangents to a
circle of radius 6 cm is _____.
( Fill in the blank ) 2

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18. A sheet of paper is in the form of a rectangle ABCD
in which AB = 40 cm and AD = 28 cm as shown in
the adjoining figure. A semi-circular portion with AD
as diameter is cut off. Find the area of the remaining
paper. (Use   22 ) 3
7

Or
What is the perimeter of a sector of angle 45° of a circle
with radius 7 cm? (Use   22
7
) 3

19. A die is thrown once. Find the probability of getting—
(a) an even number;
(b) a number less than 5. 3

20. Using Euclid’s division algorithm, find the HCF of
714 and 924. 3
Or

Find the HCF and LCM of the pair of integers 26
and 91 by prime factorisation method and verify that
LCM × HCF = product of the two numbers. 3

2 2
21. If one zero of the polynomial (a  9)x  15 x  6a is
reciprocal of the other, then find the value of ‘a’. 3

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22. Is 310 a term of the A.P. : 3, 8, 13, 18, ... ? 3
Or
The nth term of a sequence is 2n  3 . Is the sequence
an A.P.? If so, find its 6th term. 3

23. Prove that
1  sin 
 (sec   tan )2 3
1  sin 
Or
a2 b2
If x cosec  a and y cot   b , prove that   1. 3
x2 y2
SECTION—D
( Marks : 16 )
( Question Nos. 24 to 27 carry 4 marks each )

24. Six years hence a man’s age will be three times as old as
his son and three years ago he was nine times as old as
his son. Find their present ages. 4
Or
Find two consecutive positive integers, the sum of whose
squares is 365. 4

25. A vertical pole stands on the level ground. From a point
on the ground, 25 m away from the foot of the pole, the
angle of elevation of its top is found to be 60°. Find the
height of the pole. (Use 3  1732 ) 4
Or
The length of a string between a kite and a point on the
ground is 90 metres. If the string makes an angle ‘’ with
the ground level such that tan   3 , how high is the
kite? Assume that there is no slack in the string.
(Use 3  1732 ) 4

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[ For Visually Handicapped (Blind) Students only,
instead of Question No. 25 given in Page No. 7 ]

25. (a) Prove that (1  sin2 )sec2   1 . 2

(b) The value of sec 60° is 1 .
2
( State whether True or False ) 1
(c) cos (90  ) = _____. ( Fill in the blank ) 1

26. If P (x, y) is a point equidistant from the points A (6, –1)
and B (2, 3), then show that x – y = 3. 4
Or
Find the area of the triangle whose vertices are
(5, –7), (–4, –5) and (4, 5). 4

27. Prove that a tangent to a circle is perpendicular to the
radius through the point of contact. 4

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

27. (a) Define a circle. 2

(b) A line, which intersects the circle in two points, is
called a _____.
( Fill in the blank ) 1

(c) The portion of a circular region enclosed between a
chord and the corresponding arc is called a segment
of the circle.
( State whether True or False ) 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 :
3x  y  3
x  2y  4
Also, shade the area of the region bounded by the lines
and x-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  3y  0
3x  4y  5

29. The largest possible sphere is carved out of a wooden
solid cube of side 7 cm. Find the volume of the wood
left. (Use   22
7
) 6
Or
A solid metallic sphere of radius 5·6 cm is melted and
solid cones each of radius 2·8 cm and height 3·2 cm are
made. Find the number of such cones formed. 6
30. Find the mean of the following data : 6
Class Interval 10–20 20–30 30–40 40–50 50–60 60–70

Frequency 11 15 20 30 14 10

Or
Find the median of the following frequency distribution : 6
Class Interval 0–100 100–200 200–300 300–400 400–500
Frequency 40 32 48 22 8

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SECTION—F
( Marks : 20 )
[ For Candidates without Internal Assessment ]
31. Answer the following as directed (any eight) : 1×8=8
(a) If p and q are two prime numbers, then their LCM is
(A) pq
(B) p
(C) q
(D) 1 ( Choose the correct option )
(b) Which of the following is not a quadratic equation?
(A) x 2  5x  3  0
(B) x 2  4x  x 2  2x
2
(C) x  2x  5  0
1
(D) 3x 2  5 x   0
3
( Choose the correct option )
(c) The first term of the sequence an  n (n  2) is
(A) 1
(B) 2
(C) 3
(D) 4 ( Choose the correct option )
(d) If a pair of linear equations a1x  b1y  c1  0 and
a2x  b2y  c 2  0 represents intersecting lines, then
a1 b1
(A) 
a2 b2
a1 b1 c1
(B)  
a2 b2 c 2
a1 b1 c1
(C)  
a2 b2 c 2
(D) None of the above
( Choose the correct option )
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(e) A polynomial having three terms is called _____.
( Fill in the blank )

(f) Write the value of sin 37  cos 53.

(g) sin2   cos2   1.
( State whether True or False )

(h) Write the coordinates of the origin.

(i) Zero (0) is the smallest natural number.
( State whether True or False )

(j) Write the formula of the volume of a cylinder of
radius ‘r ’ and height ‘h’.

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

(l) Define modal class.

(m) Write the standard form of a quadratic equation.

(n) If sin   cos  , then the value of ‘’ is _____.
( Fill in the blank )

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

(a) Is x = 1 a solution of the quadratic equation
x 2  3x  2  0?

(b) Find the zeroes of the polynomial 3x 2  2x  1 .

(c) Express the number 49896 as a product of their
primes.

(d) Evaluate sin 60 cos 30  cos 60 sin 30.

(e) Find the coordinates of the midpoint of the line
segment joining the points A(–7, 6) and B (9, –10).

(f) Find the sum of first n natural numbers.

(g) Find the value of ‘k’ for which the quadratic
equation 9x 2  24x  k  0 has real and equal roots.

(h) A man goes 15 m due west and then 8 m due north.
How far is he from the starting point?

(i) The circumference of a circle is 39·6 cm. Find its
radius. (Use   22
7
)

(j) The probability that it will rain today is 0·87. What is
the probability that it will not rain today?

(k) State the converse of Pythagoras Theorem.

  

X/22/M/3 22K—62630