Mathematics Specimen Paper 2024-2025 (English Medium) ICSE Class 10 Question Paper Solution

A polynomial in ‘x’ is divided by (x – a) and for (x – a) to be a factor of this polynomial, the remainder should be ______.

– a

0

a

2a

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 adjoining figure, 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 adjoining diagram, O is the centre of the circle and PT is a tangent. The value of x is ______.

20°

40°

55°

70°

In the adjoining diagram the length of PR is ______.

`3sqrt(3)` cm

`6sqrt(3)` cm

`9sqrt(3)` cm

18 cm

A solid sphere is cut into two identical hemispheres.

Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere.

Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere.

Which of the following is valid?

Both the statements are true.

Both the statements are false.

Statement 1 is true and Statement 2 is false.

Statement 1 is false and Statement 2 is true.

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 A = `[(3, -2)]` and B = `[(-1, 4),(2, 0)]`

Assertion (A): Product AB of the two matrices A and B is possible.

Reason (R): Number of columns of matrix A is equal to number of rows in matrix B.

A is true, R is false.

A is false, R is true.

Both A and R are true and R is the correct reason for A.

Both A and R are true and R is incorrect reason for A.

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 quadratic equation x² – 1 = 0 are ______.

0

1

–1

±1

Which of the following equation 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 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 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.

A line segment joining P(2, –3) and Q(0, –1) is cut by the x-axis at the point R. A line AB cuts the y-axis at T(0, 6) and is perpendicular to PQ at S.

Find the:

1. equation of line PQ
2. equation of line AB
3. coordinates of points R and S.

In the given figure AC is the diameter of the circle with centre O. CD is parallel to BE.

∠AOB = 80° and ∠ACE = 20°.

Calculate

1. ∠BEC
2. ∠BCD
3. ∠CED

In a Geometric Progression (G.P.) the first term is 24 and the fifth term is 8. Find the ninth term of the G.P.

In the adjoining diagram, a tilted right circular cylindrical vessel with base diameter 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 wet surface, when the cylinder is placed vertically on a horizontal surface. (Use π = `22/7`).

Study the graph and answer each of the following:

1. Write the coordinates of points A, B, C and D.
2. Given that, point C is the image of point A. Name and write the equation of the line of reflection.
3. Write the coordinates of the image of the point D under reflection in y-axis.
4. Whats the name given to a point whose image is the point itself?
5. On joining the points A, B, C, D and A in order, a figure is formed. Name the closed figure.

A man buys 250, ten-rupee shares each at ₹ 12.50. If the rate of dividend is 7%, find the:

1. dividend he receives annually.
2. percentage return on his investment.

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:

(sinθ + cosθ)(cosecθ – secθ) = cosecθ.secθ – 2 tanθ

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. Sameer has a recurring deposit account and deposits ₹ 600 per month for 2 years. If he gets ₹ 15600 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 PT is a tangent to the circle. Chord BA produced meets the tangent PT at P.

Given PT = 20 cm and PA = 16 cm.

1. Prove ΔPTB ~ ΔPAT
2. Find the length of AB.

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.

A vertical tower standing on a horizontal plane is surmounted by a vertical flagstaff. At a point 100 m away from the foot of the tower, the angle of elevation of the top and bottom of the flagstaff are 54° and 42° respectively. Find the height of the flagstaff. Give your answer correct to nearest metre.

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.

In a TV show, a contestant opts for video call a friend life line to get an answer from three of his friends, named Amar, Akbar and Anthony. The question which he asks from one of his friends has four options.

Find the probability that:

1. Akbar is chosen for the call.
2. Akbar couldn't give the correct answer.

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 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`)

The table given below shows the runs scored by a cricket team during the overs of a match.

Use graph sheet for this question.

Take 2 cm = 10 overs along one axis and 2 cm = 10 runs along the other axis.

1. Draw a histogram representing the above distribution.
2. Estimate the modal runs scored.

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, that is, p, q and r are in A.P.

A car travels a distance of 72 km at a certain average speed of x km per hour and then travels a distance of 81 km at an average speed of 6 km per hour more than its original average speed. If it takes 3 hours to complete the total journey then form a quadratic equation and solve it to find its original average speed.

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

Use ruler and compasses for the following question taking a scale of 10 m = 1 cm. A park in a city is bounded by straight fences AB, BC, CD and DA. Given that AB = 50 m, BC = 63 m, ∠ABC = 75°. D is a point equidistant from the fences AB and BC. If ∠BAD = 90°, construct the outline of the park ABCD. Also locate a point P on the line BD for the flag post which is equidistant from the corners of the park A and B.