Mathematics Official Board Paper 2025-2026 (English Medium) ICSE Class 10 Question Paper Solution

−1

1

−5

5

5%

12%

18%

28%

1

6

8

9

A and B opened a recurring deposit account in a bank which is paying simple interest at 9% per annum. A deposited ₹ 1,500 for one year and B deposited ₹ 1,200 for 15 months. The amount invested by ______.

A is ₹ 27 more than B

A is ₹ 300 more than B

A is ₹ 300 less than B

Both A and B are same (₹ 18,000)

Find the equation of a line whose y-intercept is 6 and is parallel to x-axis.

y = 6

x = 6

x + y = 6

y − x = 6

Asha buys ₹ 20 shares of a company which pays 9% dividend at such a price that she gets a return of 12% on her investment. At what price did she buy each share?

₹ 20

₹ 15

₹ 25

₹ 18

1 : 1

2 : 1

`sqrt3 : 2`

`2 : sqrt3`

In the given diagram, O is the centre of the circle and ABCD is a cyclic quadrilateral. If ∠CDE = 65°, then the value of x is:

32.5°

65°

115°

130°

The nature of roots of quadratic equation 3x² − 6x − 3 = 0 are ______.

real and equal

real, distinct and rational

real, distinct and irrational

no real roots

Assertion (A): If a die is rolled, the probability of getting a number greater than 6 is `1/6`

Reason (R): There are six possible outcomes when rolling a die, {1, 2, 3, 4, 5, 6}.

(A) is true and (R) is false.

(A) is false and (R) is true.

Both (A) and (R) are true and (R) is the correct explanation of (A).

Both (A) and (R) are true but (R) is not the correct explanation of (A).

If the areas of two similar triangles are in the ratio 9 : 64, then the ratio of their corresponding altitudes is ______.

3 : 8

2 : 1

9 : 64

8 : 3

What must be added to x³ + 7x² + 3x + 2 so that the result is completely divisible by (x + 2)?

−40

−16

16

40

In the given diagram, ΔAOB is a right-angled triangle and C is the mid-point of AB. The coordinates of the point which is equidistant from the three vertices of ΔAOB is:

(x, y)

(y, x)

`(x/2, y/2)`

`((2x)/3, (2y)/3)`

Given matrix A = `[(2, 3), (1, 2)]` and matrix B = [2 −4]. Product AB is a matrix of order:

2 × 2

2 × 1

1 × 2

Product AB is not possible.

Assertion (A): The 9^th term of a Geometric Progression (G.P.) 6, −12, 24, −48, ... is a positive term.

Reason (R): The value of (−2)⁸ is always positive.

(A) is true and (R) is false.

(A) is false and (R) is true.

Both (A) and (R) are true and (R) is the correct explanation of (A).

Both (A) and (R) are true but (R) is not the correct explanation of (A).

The fourth and seventh terms of an Arithmetic Progression (A.P.) are 60 and 114 respectively. Find the:

1. first term and common difference.
2. sum of its first 10 terms.

Given, `A = [(3, 1), (5, 3)] and B = [(-1, a), (3, -5)]` and product AB = `[(b, 7), (4, 5)]`. Find the values of ‘a’ and ‘b’.

In the given diagram, O is the centre of the circle and the tangent DE touches the circle at B. If ∠ADB = 32°. Find the values of x and y.

The polynomial kx³ + 3x² − 11x − 6 when divided by (x + 1), leaves a remainder of 6.

1. Find the value of k.
2. Using the value of k factorise completely the polynomial kx³ + 3x² − 11x − 6.

An eye drop bottle is prepared consisting of a hemisphere, a cylinder and a conical cap, as shown in the given diagram. Height of the cylindrical and conical parts are each, equal to the diameter (7 cm). Find the:

1. minimum height of the cylindrical box required to pack this bottle.
2. volume of the liquid medicine (shaded part) in the bottle. Give your answer to the nearest whole number. (Use π = `22/7`)

Use ruler and compass for the following construction:

1. Construct an equilateral triangle ABC of side 5 cm.
2. Construct the circumcircle of ΔABC.
3. Construct the locus of points which are equidistant from AB and BC. Mark the point where the circumcircle and locus meet, as D.
4. Give the geometrical name of quadrilateral ABCD.

Prove that:

(sec θ − cos θ)(cosec θ − sin θ) = sin θ cos θ

In the given diagram, DE || BC and AD : DB = 2 : 3. 

1. Prove that: ΔADE ~ ΔABC and hence find DE : BC.
2. Prove: ΔDFE ~ ΔCFB
3. Given, area of ΔDFE = 16 square units, find the area of ΔCFB.

The histogram drawn on the graph represents the number of students of different heights (in cm).

Using the graph, answer the following:

1. The number of students whose height is 150 cm and above.
2. The modal height.
3. The total number of students.

A(−10, −2) and B(2, 10) are two end points of a line segment. If AB intersects the x-axis at P, find the:

1.  ratio in which ‘P’ divides AB.
2. coordinates of point P.

Solve the quadratic equation (x − 2)² − 5x − 3 = 0 and give your answer correct to 3 significant figures.

Kabir bought 120 shares of a company with nominal value ₹ 100, available at a premium of ₹ 25. Find:

1. The money invested by Kabir in buying these shares.
2. The rate of dividend, if he received ₹ 1,080 as dividend from these shares after one year.
3. His rate of return.

Find the mean of the following frequency distribution using step-deviation method.

Take assumed mean = 28

The difference of two natural numbers is 5 and sum of their reciprocals is `3/10`​. Find the two numbers.

A flagpole is erected at the top of a building. The angle of elevation of the top and foot of the flagpole from a point 100 m away, on the same level as that of the foot of the building, are 33° and 31° respectively. Find the height of the flagpole. Give your answer correct to the nearest metre.

Using a graph paper, draw an ogive for the following distribution which shows a record of weight in kilograms of 100 students. 

Use your ogive to estimate the following:

1. The median weight of the students.
2. Percentage of students whose weight is 60 kg or more.
3. The weight above which 20% of the students lie.

Rohit and Vinay both opened a recurring deposit account in a bank for 2 years at 8% simple interest. Vinay deposited ₹ 300 per month. On maturity, Rohit’s interest was ₹ 800 more than Vinay’s interest. Find:

1. interest earned by Vinay.
2. sum deposited by Rohit every month.

The fourth term of a Geometric Progression (G.P.) is 16 and its seventh term is 128. Find its:

1. common ratio
2. first term

Use graph sheet for this question. Take 2 cm = 1 unit along both x and y axis. Graphically represent parallelogram OABC, where O(0, 0), A(2, 3), B(5, 3) and C(3, 0).

Reflect OABC:

1. on the x-axis and name its image as ODEC.
2. through the origin and name its image as OIJH.
3. on the y-axis and name its image as OFGH.

Solve the following inequation, write the solution set and represent it on the real number line.

`−1< (2x − 3)/3 − x/5 ≤ 1, x ∈ R​`

Use the following graph and answer the given questions:

1. Write the co-ordinates of points A, B and C.
2. Find the equation of a line passing through the mid-point of AC and parallel to AB.

A solid wooden toy is prepared by joining a cone, a cylinder and a sphere, as shown in the given diagram. The radius of each of the three solids is 7 cm and heights of each of the cone and the cylinder is 24 cm. Find:

1. The total surface area of the given solid.
2. The cost of painting the total surface at the rate of ₹ 0.50 per cm². 
   (Use `pi = 22/7`)

If x = `(5ab)/(a − b), a ≠ b,`

1. Find: `x/a`
2. Using properties of proportion, find: `(x + a)/(x - a)`

A survey was conducted on 300 families having 2 children each. The results obtained are given below.

If one family is selected at random, find the probability that it will have:

1. one girl child
2. one or more girl child
3. no boy child

In the given figure ‘O’ is the centre of the circle. PQ is a tangent to the circle at B and AB = AC. If ∠CBQ = 40°, find the unknown angles x, y, z and w.