# ICSE Class 10 Mathematics Previous Year Question Paper 2019

Mathematics is one of the crucial and most scoring subjects in ICSE Class 10. TopperLearning presents study materials for ICSE Class 10 Mathematics which will enable students to score well in the board examination. The syllabus includes certain challenging concepts like Value Added Tax, Ratio and Proportion, Quadric Equations among others which require effective study materials. Our study materials for ICSE Class 10 consist of video lessons, question banks, sample papers, revision notes and past year papers which improve the quality of learning.

Q 1.

(a) Solve the following inequation and write down the solution set:

11x - 4 < 15x + 4 <13x + 14, x ∈ W

Represent the solution on a real number line.

(b) A man invests Rs. 4500 in shares of a company which is paying 7.5% dividend. If Rs. 100 shares are available at a discount of 10%.

Find:

(i) Number of shares he purchases.

(ii) His annual income.

(c) In class of 40 students, marks obtained by the students a class test (out of 10) are given below,

 Marks 1 2 3 4 5 6 7 8 9 10 Number of students 1 2 3 3 6 10 5 4 3 3

Calculate the following for the given distribution:

(i) Median

(ii) Mode

Q 2.

(a) Using the factor theorem, show that (x - 2) is a factor of x3 + x2 – x4 – 4, Hence factorise the polynomial completely.

(b) Prove that:

(cosec θ - sin θ)(sec θ – cos θ) (tan θ + cot θ) =1

(c) In an Arithmetic Progression (A.P.) the fourth and sixth terms are 8 and 14 respectively, Find the:

(i) first term

(ii) common difference

(iii) sum of the first 20 terms

Q 3.

(a) Simplify

(b) M and N are two points on the X axis and Y axis respectively.

P(3, 2) divides the line segment MN in the ratio 2 : 3.

Find:

(i) the coordinates of M and N

(ii) slope of the line MN.

(c) A solid metallic sphere of radius 6 cm is melted and made into a solid cylinder of height 32cm. Find the

(ii) curved surface area of the cylinder

Take Π = 3.1

Q 4.

(a) The following numbers, K + 3, K + 2, 3K - 7 and 2K – 3 are in proportion. Find K.

(b) Solve for x the quadratic equation x2 – 4x – 8 = 0.

(c) Use ruler and compass only for answering this question.

Draw a circle of radius 4 cm. Mark the centre as O. Mark a point P outside the circle at a distance of 7 cm from the centre, Construct two tangents to the circle from the external point P.

Measure and write down the length of any one tangent.

Q 5.

(a) There are 25 discs numbered 1 to 25. They are put in a closed box and shaken thoroughly. A disc is drawn at random from the box.

Find the probability that the number on the disc is:

(i) an odd number

(ii) divisible by 2 and 3 both.

(iii) a number less than 16.

(b) Rekha opened a recurring deposit account for 20 months. The rate of interest is 9% per annum and Rekha receives Rs. 441 as interest at the time of maturity.

Find the amount Rekha deposited each month.

(c) Use a graph sheet for this question.

Take 1 cm = 1 unit along both x and y axis.

(i) Plot the following points:

A(0,5), B(3,0), C(1,0), and D(1,-5)

(ii) Reflect the points B, C, and D on the y axis and name them as B’, C’ and D’ respectively.

(iii) Write down the coordinates of B’, C’ and D’.

(iv) Join the points A, B, C, D, D’, C’, B’, A in order and give a name to the closed figure ABCDD’C’B’.

Q 6.

(a) In the given figure PQR = PST = 90o, PQ = 5 cm and PS  = 2 cm.

(i) Prove that PQR = PST

(ii) Find Area of PQR: Area of quadrilateral SRQT.

(b) The first and last term of a Geometrical Progression (G.P.) are 3 and 96 respectively. If the common ratio is 2, find:

(i) ‘n’ the number of terms of the G.P.

(ii) Sum of the n terms.

(c) A hemispherical and a conical hole is scooped out of a solid wooden cylinder. Find the volume of the remaining solid where the measurements are as follows:

The height of the solid cylinder is 7 cm, radius of each of hemisphere, cone and cylinder is 3 cm. Height of cone is 3 cm.

Q 1.

(a) Solve the following inequation and write down the solution set:

11x - 4 < 15x + 4 <13x + 14, x ∈ W

Represent the solution on a real number line.

(b) A man invests Rs. 4500 in shares of a company which is paying 7.5% dividend. If Rs. 100 shares are available at a discount of 10%.

Find:

(i) Number of shares he purchases.

(ii) His annual income.

(c) In class of 40 students, marks obtained by the students a class test (out of 10) are given below,

 Marks 1 2 3 4 5 6 7 8 9 10 Number of students 1 2 3 3 6 10 5 4 3 3

Calculate the following for the given distribution:

(i) Median

(ii) Mode

Solution:

(a) We have,

11x - 4 < 15x + 4 13x + 14

So, we can say

11x - 4 < 15x + 4

(b) Total investment = Rs. 4500

Dividend percentage = 7.5%

Nominal value = Rs. 100

Discount % = 10%

(c)

 Marks xi No. of Students fi xifi 1 1 1 2 2 4 3 3 9 4 3 12 5 6 30 6 10 60 7 5 35 8 4 32 9 3 27 10 3 30 Total 40 240

Q 2.

(a) Using the factor theorem, show that (x - 2) is a factor of x3 + x2 – x4 – 4, Hence factorise the polynomial completely.

(b) Prove that:

(cosec θ - sin θ)(sec θ – cos θ) (tan θ + cot θ) =1

(c) In an Arithmetic Progression (A.P.) the fourth and sixth terms are 8 and 14 respectively, Find the:

(i) first term

(ii) common difference

(iii) sum of the first 20 terms

Solution:

(a) The given polynomial is p(x) = x3 + x2 - 4x - 4

Using factor theorem, we know that

(x - a) is a factor of p(x) if p(a) = 0.

So to prove that (x - 2) is a factor of p(x), we need to prove that p(2) = 0

Consider, p(2) = 23 + 22 - 4 × 2 - 4

= 8 + 4 - 8 - 4

= 0

Therefore, (x - 2) is a factor of p(x).

Now,

x3 + x2 - 4x - 4

= x3 - 4x +  x2 - 4

= x(x2 - 4) + 1(x2 - 4)

= (x2 - 4) (x + 1)

= (x- 22) (x + 1)

= (x - 2) (x + 2) (x + 1)

(b)

Consider

(c) Let  be the nth term of A.P.

Given: Fourth and sixth terms are 8 and 14

i.e. t4 = 8 and t6 = 14

Where, a and d are first term and common difference of an A.P respectively.

i. Using (1), we get

a + 3d = 8   ....(2) and a + 5d = 14  ......(3)

Solving (2) and (3) simultaneously, we get

a = -1

ii. Substituting the value of 'a' in equation (2), we get

d = 3

iii. Sum of first n terms of an A.P. is given by

Q 3.

(a) Simplify

(b) M and N are two points on the X axis and Y axis respectively.

P(3, 2) divides the line segment MN in the ratio 2 : 3.

Find:

(i) the coordinates of M and N

(ii) slope of the line MN.

(c) A solid metallic sphere of radius 6 cm is melted and made into a solid cylinder of height 32cm. Find the

(ii) curved surface area of the cylinder

Take Π = 3.1

Solution:

(a)

(b)

i. Let M and N be the points be (x, 0) and (0, y) on the x – axis and y - axis respectively.

P(3, 2) divides MN in the ratio 2:3.

ii. Slope of MN =

(c)

i. Let r and R be the radius of sphere and cylinder respectively and h be the height of the cylinder.
r = 6 cm, h = 32 cm
According to the question,
Volume of sphere = Volume of cylinder

R = 3 cm

The radius of the sphere is 3 cm.

ii. For cylinder, h = 32 cm and R = 9 cm

Curved surface area of a cylinder = ΠR2h = 3.1 × 32 × 32 = 892.8 cm2

Q 4.

(a) The following numbers, K + 3, K + 2, 3K - 7 and 2K – 3 are in proportion. Find K.

(b) Solve for x the quadratic equation x2 – 4x – 8 = 0.

(c) Use ruler and compass only for answering this question.

Draw a circle of radius 4 cm. Mark the centre as O. Mark a point P outside the circle at a distance of 7 cm from the centre, Construct two tangents to the circle from the external point P.

Measure and write down the length of any one tangent.

Solution:

(a) K + 3, K + 2, 3K - 7 and 2K – 3 are in proportion.

(b) Comparing with ax2 + bx + c = 0 we get

a = 1, b = -4 and c = -8

b2 – 4ac = (-4)2 – 4 × 1 × (-8)

= 16 + 32

= 48

∴ x = 2 ± 2 × 1.732

∴ x = 2 ± 3.464

∴ x = 5.464 or -1.464

(c)

1. Take measure 4 cm in compass and draw a circle, with center as O.
2. Draw a straight line from O to P, such that OP = 7cm
3. Now find the midpoint of OP by drawing a perpendicular bisector
4. Mark the midpoint as X
5. Take measure of XO in the compass and cut arcs at S and T on the Circle
6. Join PS and PT
7. Measure of PS comes out to be 5.74 cm

Q 5.

(a) There are 25 discs numbered 1 to 25. They are put in a closed box and shaken thoroughly. A disc is drawn at random from the box.

Find the probability that the number on the disc is:

(i) an odd number

(ii) divisible by 2 and 3 both.

(iii) a number less than 16.

(b) Rekha opened a recurring deposit account for 20 months. The rate of interest is 9% per annum and Rekha receives Rs. 441 as interest at the time of maturity.

Find the amount Rekha deposited each month.

(c) Use a graph sheet for this question.

Take 1 cm = 1 unit along both x and y axis.

(i) Plot the following points:

A(0,5), B(3,0), C(1,0), and D(1,-5)

(ii) Reflect the points B, C, and D on the y axis and name them as B’, C’ and D’ respectively.

(iii) Write down the coordinates of B’, C’ and D’.

(iv) Join the points A, B, C, D, D’, C’, B’, A in order and give a name to the closed figure ABCDD’C’B’.

Solution:

(a) There are 25 discs numbered from 1 to 25.
Hence, the number of possible outcomes in the sample space is n(S) = 25

i. Let A be the event of getting an odd number

A = {1, 3, 5, 7, …. 25}

n(A) = 13

Thus, the probability of getting an odd number is .

ii. Let B be the event of getting a number divisible by 2 and 3.
To find the numbers which are divisible by 2 and 3 both, we need to find the number which are divisible by 6.

B = {6, 12, 18, 24}

n(B) = 4

Thus, the probability of getting an odd number is .

iii. Let C be the event of getting a number less than 16.

C = {1, 2, 3, 4, 5, 6, …..15}

n(C) = 15

Thus, the probability of getting a number less than 16 is .

(b) Given that n = 20 months, r = 9% per annum, Interest = Rs. 441

According to the question,

Hence, the amount Rekha deposited each month is Rs. 280.

(c)

i. Reflected points of B, C and D on the y-axis are B’(-3, 0), C’(-1, 0) and D’(-1, -5) respectively.

ii. After joining the points A, B, C, D, D’, C’, B’, A in order gives us the closed figure as arrow.

Q 6.

(a) In the given figure PQR = PST = 90o, PQ = 5 cm and PS  = 2 cm.

(i) Prove that PQR = PST

(ii) Find Area of PQR: Area of quadrilateral SRQT.

(b) The first and last term of a Geometrical Progression (G.P.) are 3 and 96 respectively. If the common ratio is 2, find:

(i) ‘n’ the number of terms of the G.P.

(ii) Sum of the n terms.

(c) A hemispherical and a conical hole is scooped out of a solid wooden cylinder. Find the volume of the remaining solid where the measurements are as follows:

The height of the solid cylinder is 7 cm, radius of each of hemisphere, cone and cylinder is 3 cm. Height of cone is 3 cm.

Solution:

(a) In ΔPQR and ΔPST,

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