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

(x – 2) and (x + 2) are the factors of x³ + x² – 4x – 4. The third factor of the given polynomial is ______.

(x – 1)

(x – 4)

(x + 1)

(x + 4)

Radha deposited ₹ 400 per month in a recurring deposit account for 18 months. The qualifying sum of money for the calculation of interest is ______.

₹ 3600

₹ 7200

₹ 68,400

₹ 1,36,800

In the figure given below, AC is a diameter of the circle.

AP = 3 cm and PB = 4 cm and QP ⊥ AB. 

If the area of ΔAPQ is 18 cm², then the area of shaded portion QPBC is:

32 cm²

49 cm²

80 cm²

98 cm²

In the given diagram, the radius of the circle with center O is 3 cm. PA and PB are the tangents to the circle, which are at right angles to each other. The length of OP is:

`3/sqrt(2)` cm

3 cm

`3sqrt(2)` cm

`6sqrt(2)` cm

Assertion (A): If sec θ + tan θ = a and sec θ – tan θ = b then ab = 1.

Reason (R): sec² θ – tan² θ = 1

(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).

A solid sphere is cut into two identical hemispheres.

Assertion (A): The total volume of two hemispheres is equal to the volume of the original sphere.

Reason (R): The total surface area of two hemispheres together is equal to the surface area of the original sphere.

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

(A) is false, (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).

Given that the sum of the squares of the first seven natural numbers is 140, then their mean is ______.

20

70

280

980

A bag contains 3 red and 2 blue marbles. A marble is drawn at random. The probability of drawing a black marble is ______.

0

`1/5`

`2/5`

`3/5`

If matrix A = `[(-1, 2)]` and matrix B = `[(3),(4)]`, then matrix AB is equal to ______.

[–3]

[8]

[5]

`[(-1, 2),(3, 4)]`

A mixture of paint is prepared by mixing 2 parts of red pigments with 5 parts of the base. Using the given information in the following table, find the values of a, b and c to get the required mixture of paint.

a = 10, b = 10, c = 10

a = 5, b = 2, c = 5

a = 10, b = 5, c = 10

a = 10, b = 5, c = 15

An article which is marked at ₹ 1200 is available at a discount of 20% and the rate of GST is 18%. The amount of SGST is ______.

₹ 216.00

₹ 172.80

₹ 108.00

₹ 86.40

The sum of money required to buy 50, ₹ 40 shares at ₹ 38.50 is ______.

₹ 1920

₹ 1924

₹ 1925

₹ 1952

The roots of the quadratic equation x² – 1 = 0 are ______.

0, 0

1, 1

–1, –1

+1, –1

Which of the following equations represents a line equally inclined to the axes?

2x – 3y + 7 = 0

x – y = 7

x = 7

y = –7

Given, `x + 2 ≤ x/3 + 3` and x is a prime number. The solution set for x is ______.

∅

{0}

{1}

{0, 1}

While factorizing a given polynomial using the remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x³ + 7x² + 2x – 3.

1. Is the student’s solution correct in stating that (2x + 1) is a factor of the given polynomial?
2. Give a valid reason for your answer.

Also, factorize the given polynomial completely.

P is a point on the x-axis which divides the line joining A(–6, 2) and B(9, –4). Find:

1. the ratio in which P divides the line segment AB.
2. the coordinates of the point P.
3. equation of a line parallel to AB and passing through (–3, –2).

In the given figure, AC is the diameter of the circle with center O.

CD is parallel to BE.

∠AOB = 80° and ∠ACE = 20°

Calculate:

1. ∠BEC
2. ∠BCD
3. ∠CED

–11, –7, –3, ............., 49, 53 are the terms of a progression.

Answer the following:

1. What is the type of progression?
2. How many terms are there in all?
3. Calculate the value of middle most term.

In the diagram given below, a tilted right circular cylindrical vessel with a base diameter of 7 cm contains a liquid. When placed vertically, the height of the liquid in the vessel is the mean of two heights shown in the diagram. Find the area of the wet surface when the cylinder is placed vertically on a horizontal surface. `("Use"  π = 22/7)`.

Use a ruler and compass to answer this question.

1. Construct a circle of radius 4.5 cm and draw a chord AB of length 6.5 cm. 
2. At A, construct ∠CAB = 75°, where C lies on the circumference of the circle. 
3. Construct the locus of all points equidistant from A and B. 
4. Construct the locus of all points equidistant from CA and BA. 
5. Mark the point of intersection of the two loci as P. Measure and write down the length of CP.

Ms. Kaur invested ₹ 8,000 in buying ₹ 100 shares of a company paying a 6% dividend at ₹ 80. After a year, she sold these shares at ₹ 75 each and invested the proceeds, including the dividend received during the first year, in buying ₹ 20 shares, paying a 15% dividend at ₹ 27 each. Find the:

1. dividend received by her during the first year. 
2. number of shares purchased by her using the total proceeds.

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

`5x - 21 < (5x)/7 - 6 ≤ -3 3/7 + x, x ∈ R`

Prove the following trigonometry identity:

(sin θ + cos θ)(cosec θ – sec θ) = cosec θ ⋅ sec θ – 2 tan θ

In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm.

1. Prove that ΔAPB ~ ΔCPD.
2. Find the length of CD. 
3. Find area ΔAPB : area ΔCPD.

Mr. Sam has a recurring deposit account and deposits ₹ 600 per month for 2 years. If he gets ₹ 15,600 at the time of maturity, find the rate of interest earned by him.

Using step-deviation method, find mean for the following frequency distribution:

Find the coordinates of the centroid P of the ΔABC, whose vertices are A(–1, 3), B(3, –1) and C(0, 0). Hence, find the equation of a line passing through P and parallel to AB.

In the given figure, the parallelogram ABCD circumscribe a circle, touching circle at P, Q, R and S.

1. Prove that: AB = BC 
2. What special name can be given to the parallelogram ABCD?

The following bill shows the GST rate and the marked price of articles:

Find the total amount to be paid (including GST) for the above bill.

Draw the necessary diagram for this question.

A man on the top of a lighthouse observes the angles of depression of two ships on the opposite sides of the lighthouse as 30° and 50°, respectively. If the height of the lighthouse is 80 m, find the distance between the two ships. Give your answer correct to the nearest meter.

The marks of 200 students in a test were recorded as follows:

Using graph sheet draw ogive for the given data and use it to find the,

1. median,
2. number of students who obtained more than 65% marks
3. number of students who did not pass, if the pass percentage was 35.

A box containing cards numbered between 0 and 100 are shuffled and a card is picked at random. Find the probability of getting a card which is:

1. divisible by 6. 
2. not divisible by 6.

If x, y and z are in continued proportion, Prove that:

`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`

A manufacturing company prepares spherical ball bearings, each of radius 7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box can have a maximum of 2156 cm³ of ball bearings. Find the:

1. maximum number of ball bearings that each box can have.
2. mass of each box of ball bearings in kg.
   (Use π = `22/7`)

Study the graph given below and answer the following:

1. Number of batsmen who scored 500 to 700 runs.
2. Modal class interval 
3. The value of mode.

An Arithmetic Progression (A.P.) has 3 as its first term. The sum of the first 8 terms is twice the sum of the first 5 terms. Find the common difference of the A.P.

The roots of equation (q – r)x² + (r – p)x + (p – q) = 0 are equal.

Prove that 2q = p + r; i.e., p, q, and r are in A.P.

The sum of the squares of three consecutive even numbers is 596. Find the numbers.

Given matrix X = `[(1, 1),(8, 3)]` and I = `[(1, 0),(0, 1)]`, prove that X² = 4X + 5I.

Use a graph sheet for this question. Take 1 cm = 1 unit along both the x and y axis. Plot ABCDE, where A(4, 0), B(4, 2), C(2, 2), D(2, 4) and E(0, 4).

1. Reflect the points A, B, C and D on the y-axis and name them as F, G, H and I, respectively.
2. Join the points A, B, C, D, E, I, H, G and F in order. Reflect the figure ABCDEIHGF on the x-axis and name it as AMNPQRSTF.
3. Give the geometrical name of the closed figure AEFQ.