# Mathematics Set 1 2021-2022 ICSE Class 10 Question Paper Solution

Mathematics [Set 1]
Date: April 2022
Duration: 1h30m

Note :

1. Answers to this Paper must be written on the paper provided separately.
2. You will not be allowed to write during the first 10 minutes.
3. This time is to be spent in reading the question paper.
4. The time given at the head of this Paper is the time allowed for writing the answers.
5. Attempt all questions from Section A and any three questions from Section B.
6. The intended marks for questions or parts of questions are given in brackets [ ].

SECTION - A : (Attempt all questions from this Section)
[10] 1 | Choose the correct answers to the questions from the given options. (Do not copy the question, Write the correct answer only.)
[1] 1.i

The point (3, 0) is invariant under reflection in:

The origin

x-axis

y-axis

both x and y axes

Concept: Reflection Concept
Chapter: [0.09] Reflection
[1] 1.ii

In the given figure, AB is a diameter of the circle with centre ‘O’. If ∠COB = 55⁰ then the value of x is:

27.5°

55°

110°

125°

Concept: Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle
Chapter: [0.02] Circles
[1] 1.iii

If a rectangular sheet having dimensions 22 cm x 11 cm is rolled along its shorter side to form a cylinder. Then the curved surface area of the cylinder so formed is:

968 cm2

424 cm2

121 cm2

242 cm2

Concept: Surface Area of Cylinder
Chapter: [0.04] Mensuration
[1] 1.iv

If the vertices of a triangle are (1, 3), (2, - 4) and (-3, 1). Then the co-ordinate of its centroid is:

(0, 0)

(0, 1)

(1, 0)

(1, 1)

Concept: The Mid-point of a Line Segment (Mid-point Formula)
Chapter: [0.01] Co-ordinate Geometry Distance and Section Formula
[1] 1.v

tan θ × sqrt(1 - sin^2 θ) is equal to:

cos θ

sin θ

tan θ

cot θ

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[1] 1.vi

The median class for the given distribution is:

 Class Interval 1 - 5 6 - 10 11 - 15 16 - 20 Cumulative Frequency 2 6 11 18

1 – 5

6 – 10

11 – 15

11 – 20

Concept: Concept of Median
Chapter: [0.06] Statistics
[1] 1.vii

If the lines 7y = ax + 4 and 2y = 3 − x, are parallel to each other, then the value of ‘a’ is:

- 1

(-7)/2

(-2)/7

14

Concept: Slope of a Line
Chapter: [0.1] Co-ordinate Geometry Equation of a Line
[1] 1.viii

Volume of a cylinder is 330 cm3. The volume of the cone having same radius and height as that of the given cylinder is:

330 cm3

165 cm3

110 cm3

220 cm3

Concept: Volume of a Cylinder
Chapter: [0.04] Mensuration
[1] 1.ix

In the given graph, the modal class is the class with frequency:

72

21

48

36

Concept: Finding the Mode from the Histogram
Chapter: [0.06] Statistics
[1] 1.x

If the probability of a player winning a game is 0.56. The probability of his losing this game is:

0.56

1

0.44

0

Concept: Probability - A Theoretical Approach
Chapter: [0.07] Probability
SECTION - B : (Attempt any three questions from this Section)
[10] 2
[2] 2.i

Find the ratio in which the x-axis divides internally the line joining points A (6, -4) and B ( -3, 8).

Concept: Section Formula
Chapter: [0.01] Co-ordinate Geometry Distance and Section Formula
[2] 2.ii

Three rotten apples are accidentally mixed with twelve good ones. One apple is picked at random. What is the probability that it is a good one?

Concept: Probability - A Theoretical Approach
Chapter: [0.07] Probability
[3] 2.iii

In the given figure, AC is a tangent to circle at point B. ∆EFD is an equilateral triangle and ∠CBD = 40°. Find:

1. ∠BFD
2. ∠FBD
3. ∠ABF
Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments
Chapter: [0.02] Circles
[3] 2.iv

A drone camera is used to shoot an object P from two different positions R and S along the same vertical line QRS. The angle of depression of the object P from these two positions is 35° and 60° respectively as shown in the diagram. If the distance of the object P from point Q is 50 metres, then find the distance between R and S correct to the nearest meter.

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
Chapter: [0.05] Trigonometry
[10] 3
[2] 3.i

In the given figure, PT is a tangent to the circle at T, chord BA is produced to meet the tangent at P. Perpendicular BC bisects the chord TA at C. If PA = 9 cm and TB = 7 cm, find the lengths of:

1. AB
2. PT

Concept: Tangent to a Circle
Chapter: [0.02] Circles
[2] 3.ii

How many solid right circular cylinders of radius 2 cm and height 3 cm can be made by melting a solid right circular cylinder of diameter 12 cm and height 15 cm?

Concept: Surface Area of Cylinder
Chapter: [0.04] Mensuration
[3] 3.iii

Prove that:

(cos^2 "A")/(cos "A" - sin "A") + (sin "A")/(1 - cot "A") = sin A + cos A

Concept: Trigonometry
Chapter: [0.05] Trigonometry
[3] 3.iv

Use graph paper for this question, take 2 cm = 10 marks along one axis and 2 cm = 10 students along the other axis.

The following table shows the distribution of marks in a 50 marks test in Mathematics:

 Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No. of. Students 6 10 13 7 4

Draw the ogive for the above distribution and hence estimate the median marks.

Concept: Graphical Representation of Ogives
Chapter: [0.06] Statistics
[10] 4
[2] 4.i

Find the equation of the perpendicular dropped from the point (−1, 2) onto the line joining the points (1, 4) and (2, 3)

Concept: Equation of a Line
Chapter: [0.1] Co-ordinate Geometry Equation of a Line
[2] 4.ii

Find the mean for the following distribution:

 Class Interval 20 - 40 40 - 60 60 - 80 80 - 100 Frequency 4 7 6 3
Concept: Mean of Grouped Data
Chapter: [0.06] Statistics
[3] 4.iii

A solid piece of wooden cone is of radius OP = 7 cm and height OQ = 12 cm. A cylinder whose radius and height equal to half of that of the cone is drilled out from this piece of wooden cone. Find the volume of the remaining piece of wood. (Use, pi = 22/7)

Concept: Volume of a Combination of Solids
Chapter: [0.04] Mensuration
[3] 4.iv

Use a graph sheet for this question, take 2 cm = 1 unit along both x and y-axis:

1. Plot the points A (3, 2) and B (5, 0). Reflect point A on the y-axis to A΄. Write co-ordinates of A΄.
2. Reflect point B on the line AA΄ to B΄. Write the co-ordinates of B΄.
3. Name the closed figure A’B’AB.
Concept: Simple Applications of All Co-ordinate Geometry.
Chapter: [0.1] Co-ordinate Geometry Equation of a Line
[10] 5
[2] 5.i

In the given figure, the sides of the quadrilateral PQRS touches the circle at A, B, C and D. If RC = 4 cm, RQ = 7 cm and PD = 5 cm. Find the length of PQ:

Concept: Cyclic Properties
Chapter: [0.02] Circles
[2] 5.ii

Prove that:

(sin^3 theta + cos^3 theta)/(sin theta + cos theta) = 1 - sin theta cos theta

Concept: Trigonometry
Chapter: [0.05] Trigonometry
[3] 5.iii

In the given diagram, OA = OB, ∠OAB = 𝜃 and the line AB passes through point P (-3, 4).

Find:

1. Slope and inclination (𝜃) of the line AB
2. Equation of the line AB
Concept: Equation of a Line
Chapter: [0.1] Co-ordinate Geometry Equation of a Line
[3] 5.iv

Use graph paper for this question. Estimate the mode of the given distribution by plotting a histogram. [Take 2 cm = 10 marks along one axis and 2 cm = 5 students along the other axis]

 Daily wages (in ₹) 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 No. of Workers 6 12 20 15 9
Concept: Graphical Representation of Data as Histograms
Chapter: [0.06] Statistics [0.06] Statistics
[10] 6
[2] 6.i

A box contains tokens numbered 5 to 16. A token is drawn at random. Find the probability that the token drawn bears a number divisible by:

1. 5
2. Neither by 2 nor by 3
Concept: Random Experiments
Chapter: [0.07] Probability
[2] 6.ii

Point M (2, b) is the mid-point of the line segment joining points P (a, 7) and Q (6, 5). Find the values of ‘a’ and ‘b’.

Concept: The Mid-point of a Line Segment (Mid-point Formula)
Chapter: [0.01] Co-ordinate Geometry Distance and Section Formula
[3] 6.iii

An aeroplane is flying horizontally along a straight line at a height of 3000 m from the ground at a speed of 160 m/s. Find the time it would take for the angle of elevation of the plane as seen from a particular point on the ground to change from 60⁰ to 45⁰. Give your answer correct to the nearest second.

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
Chapter: [0.05] Trigonometry
[3] 6.iv

Given that the mean of the following frequency distribution is 30, find the missing frequency ‘f’.

 Class Interval 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 Frequency 4 6 10 f 6 4
Concept: Mean of Grouped Data
Chapter: [0.06] Statistics

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