NBSE Class 10 Question Paper 2022 Maths

Total number of printed pages : 6 NB-T/M/1

2022
MATHEMATICS
Total marks : 80 Time : 3 hours

General Instructions:
i) Approximately 15 minutes is allotted to read the question paper and revise the answers.
ii) The question paper consists of 18 questions.
iii) All questions are compulsory.
iv) Internal & general choice have been provided in some questions.
v) Marks allocated to every question are indicated against it.
N.B: Check to ensure that all pages of the question paper is complete as indicated on the top
left side.

Section – A
1. Choose the correct answer from the given alternatives.
(a) If α and β are the zeroes of the polynomial f  x   px 2  2 x  3 p and     
then the value of p is 1
2 2 1 1
(i)  (ii) (iii) (iv) 
3 3 3 3
(b) The pair of equations x  y  40  0 and x  2 y  14  0 have 1
(i) a unique solution. (ii) exactly two solutions.
(iii) infinitely many solutions. (iv) no solution.

(c) If x  1 is a common root of ax 2  ax  2  0 and x 2  x  b  0 , then ab = 1
(i) 1 (ii) 2 (iii) 3 (iv) 4

(d) If sum of the first five terms of an AP is 30 and the 5th term is 10, then the
sum of the first four terms of the AP is 1
(i) 5 (ii) 10 (iii) 20 (iv) 30

(e) If tan 2  cot   15 , where 2 and   15 are acute, the value of  is 1
(i) 22 (ii) 25 (iii) 30 (iv) 35

(f) y-axis divides the join of P(4, 2) and Q(8, 3) in the ratio 1
(i) 3 : 1 (ii) 1 : 3 (iii) 2 : 1 (iv) 1 : 2

(g) If tangents PA and PB from a point P to a circle with centre O are inclined to
each other at angle of 80, then POA is equal to 1
(i) 50 (ii) 60 (iii) 70 (iv) 80

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(h) An arc of a circle is of length 5 cm and the sector it bounds has an area of 20
cm2. The radius of the circle is 1
(i) 16 cm (ii) 12 cm (iii) 8 cm (iv) 4 cm

(i) The ratio of the total surface area to the lateral surface area of a cylinder with
base radius 80 cm and height 20 cm is 1
(i) 2 : 1 (ii) 3 : 1 (iii) 4 : 1 (iv) 5 : 1

(j) Two dice are rolled once. The probability of getting such numbers on two dice,
whose product is a perfect square, is 1
1 9 2
(i) 8 (ii) (iii) (iv)
8 2 9

Section – B
2. An army contingent of 616 members is to march behind an army band of 32
members in a parade. The two groups are to march in the same number of
columns. What is the maximum number of columns in which they can march? 2

3. Find the value of k for the quadratic equation kx x  2   6  0 has equal roots. 2

4. Find the values of y for which the distance between the points P(2, 3) and Q(10, y)
is 10 units. 2

5. In the adjoining figure, ABC is a right triangle in which B
B = 90 and BD  AC. If AB = x units, BC = y units y
x
and AC = z units, then find BD.
A z D C
2
6. The length of the minute hand of a clock is 14 cm. Find the area swept by the
minute hand in 5 minutes. 2

Section – C
7. Answer any three from the following questions (a) to (e). 3×3=9

(a) On dividing x 3  3x 2  x  2 by a polynomial g(x), the quotient and remainder
were  x  2  and  2 x  4 respectively. Find g(x).

(b) Meena went to a bank to withdraw 2000. She asked the cashier to give her
50 and 100 notes only. Meena got 25 notes in all. Find how many notes of
50 and 100 she received.

(c) Solve the following pair of linear equations by cross-multiplication method:

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x  3y  7  0
3x  3 y 15  0

(d) A train travels 360 km at a uniform speed. If the speed had been 5 km/h
more, it would have taken 1 hour less for the same journey. Find the speed of
the train.
(e) A sum of 700 is to be used to give seven cash prizes to students of a school
for their overall academic performance. If each prize is 20 less than its
preceding prize, find the value of each of the prizes.

8. Answer any two from the following questions (a) to (d). 2×3=6
1
(a) In triangle ABC, right angled at B, if tan A  , find the value of
3
cos A cos C  sin A sin C , with the help of a right triangle.

1 1
(b) If sin A  B  , cos A  B  , 0  A  B  90 , A  B , find A and B.
2 2
1  sec A sin 2 A
(c) Prove that:  , where angle A is an acute angle.
sec A 1  cos A
(d) A tree breaks due to storm and the broken part bends so that the top of the
tree touches the ground making an angle 30 with it. The distance between
the foot of the tree to the point where the top touches the ground is 8 m. Find
the height of the tree.

9. Case Study based question: 3
Y
6
B
4
2 C
A
X 8 6 4 2 0 2 4 6 8 10 X
2
4 D
6

Y
Four friends are seated at the points A, B, C and D on a lawn keeping social
distancing, as shown in the figure above. One more friend wants to join them and
sit exactly at the middle position E on a straight line between A and C. Based on
the above information, answer the following questions (i) to (iii).

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(i) The distance between A and D is
(a) 72 units (b) 73 units (c) 74 units (d) 75 units

(ii) The coordinates of the position of E are
3   3
(a)  , 3  (b)  3,  (c) (6, 3) (d) (3, 6)
2   2
(iii) x-axis divides the distance/length between B and D in the ratio
(a) 3 : 4 (b) 4 : 3 (c) 5 : 4 (d) 4 : 5

10. a. Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then
1
another triangle whose sides are 1 times the corresponding sides of the
2
isosceles triangle. (Traces of construction only is required.)
Or 3
b. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of
radius 4 cm and taking B as centre, draw another circle of radius 3 cm.
Construct tangents to each circle from the centre of the other circle. (Traces
of construction only is required.)

11. a. A chord of a circle of radius 15 cm subtends an angle of 60 at the centre.
Find the areas of the corresponding minor and major segments of the circle.
[Use  = 3.14 and 3  1.73 ]
Or 3
b. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be
melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?

12. Answer any two from the following questions (a) to (c). 2×3=6

(a) A student noted the number of cars passing through a spot on a road for 100
periods each of 3 minutes and summarised it in the table given below. Find the
mode of the data:
Number of cars 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Frequency 7 14 13 12 20 11 15 8

(b) The lengths of 40 leaves of a plant are measured correct to the nearest
millimeter, and the data obtained is represented in the following table:
Length (in mm) 118-126 127-135 136-144 145-153 154-162 163-171 172-180
Number of leaves 3 5 9 12 5 4 2
Find the median length of the leaves.

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(c) A box contains 90 discs which are numbered from 1 to 90. If one disc is
drawn at random from the box, find the probability that it bears: (i) a two-
digit number, (ii) a perfect square number, (iii) a number divisible by 5.

Section – D
13. a. Draw the graphs of the equations x  y  1  0 and 3x  2 y  12  0 .
Determine the coordinates of the vertices of the triangle formed by these
lines and the x-axis and shade the triangular region.
Or 5
b. Places A and B are 100 km apart on a highway. One car starts from A and
another from B at the same time. If the cars travel in the same direction at
different speeds, they meet in 5 hours. If they travel towards each other, they
meet in 1 hour. What are the speeds of the two cars?

14. a. A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the
ground, the angle of elevation of the top of the statue is 60 and from the same
point the angle of elevation of the top of the pedestal is 45. Find the height of
the pedestal.
Or 5
b. A 1.2 m tall girl spots a balloon moving
with the wind in a horizontal line at a
height of 88.2 m from the ground. The
angle of elevation of the balloon from the
eyes of the girl at any instant is 60. After
sometime, the angle of elevation reduces 88.2m
to 30. Find the distance travelled by the 60
30
balloon during the interval.

15. a. State and Prove Pythagoras Theorem.
Or 5
b. Sides AB and BC and median AD of a P
triangle ABC are respectively proportional A
to sides PQ and QR and median PM of
PQR. Show that ABC  PQR.
B D CQ M R

16. a. In the adjoining figure, XY and XY are two X P AY
parallel tangents to a circle with centre O and
another tangent AB with point of contact C O C
intersecting XY at A and XY at B. Prove
that AOB = 90
X Q B Y
Or 5

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b. In the adjoining figure, O is the centre of the P
circle and TP is the tangent to the circle from
an external point T. If PBT = 30, prove that 30
BA : AT = 2 : 1 B O A T

17. a. A gulab jamun, contains sugar syrup up to about
30% of its volume. Find approximately how much
syrup would be found in 45 gulab jamuns, each
shaped like a cylinder with two hemispherical
ends with length 5 cm and diameter 2.8 cm.
Or 5
b. A container, opened from the top and made up of a metal sheet, is in the form
of a frustrum of a cone of height 16 cm with radii of its lower and upper ends
as 8 cm and 20 cm, respectively. Find the cost of the milk which can
completely fill the container @ of 20 per litre. Also find the cost of metal
sheet used to make the container, if it costs 8 per 100 cm2. [Take  = 3.14]

18. a. The following distribution shows the daily pocket allowance of children of a
locality. Find the mean pocket allowance by step-deviation method.
Daily pocket More More More More More More More
allowance than 11 than 13 than 15 than 17 than 19 than 21 than 23
(in )
Number of 64 57 51 42 29 9 4
children
Or 5
b. The distribution of heights (in cm) of 96 children is given below:
Height (in cm) Number of children
124-128 5
128-132 8
132-136 17
136-140 24
140-144 16
144-148 12
148-152 6
152-156 4
156-160 3
160-164 1
Draw a less than type cumulative frequency curve for this data and use it to
compute median height of the children.

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