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

The given quadratic equation `3x^2 + sqrt7x + 2 = 0` has ______.

two equal real roots.

two distinct real roots.

more than two real roots.

no real roots.

Mr. Anuj deposits ₹500 per month for 18 months in a recurring deposit account at a certain rate. If he earns ₹570 as interest at the time of maturity, then his matured amount is ______.

₹(500 × 18 + 570)

₹(500 × 19 + 570)

₹(500 × 18 × 19 + 570)

₹(500 × 9 × 19 + 570)

Which of the following cannot be the probability of any event?

`5/4`

0.25

`1/33`

67%

The equation of the line passing through origin and parallel to the line 3x + 4y + 7 = 0 is ______.

3x + 4y + 5 = 0

4x − 3y − 5 = 0

4x − 3y = 0

3x + 4y = 0

If A = `[(0,1),(1,0)]`, then A² is equal to ______.

`[(1,1),(0,0)]`

`[(0,0),(1,1)]`

`[(1,0),(0,1)]`

`[(0,1),(1,0)]`

In the given diagram, chords AC and BC are equal. If ∠ACD = 120°, then ∠AEC is ______.

30°

60°

90°

120°

The factor common to the two polynomials x² − 4 and x³ − x² − 4x + 4 is ______.

(x + 1)

(x − 1)

(x − 2)

(x − 4)

A man invested in a company paying 12% dividend on its share. If the percentage return on his investment is 10%, then the shares are ______.

at par

below par

above par

cannot be determined

Statement 1: The point which is equidistant from three non-collinear points D, E and F is the circumcentre of the ΔDEF.

Statement 2: The incentre of a triangle is the point where the bisector of the angles intersects.

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.

Assertion(A): If sin² A + Sin A = 1 then cos⁴A + cos² A = 1

Reason(R): 1 − Sin²A = Cos²A

(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 the incorrect reason for (A).

In the given diagram ΔABC ~ ΔEFG. If ∠ABC = 2 EFG = 60°, then the length of the side FG is ______.

15 cm

20 cm

25 cm

30 cm

If the volume of two spheres is in the ratio 27 ∶ 64, then the ratio of their radii is ______.

3 : 4

4 : 3

9 : 16

16 : 9

The marked price of an article is ₹1375. If the CGST is charged at a rate of 4%, then the price of the article including GST is ______.

₹55

₹110

₹1430

₹1485

The solution set for `0 < − x/3 < 2, x ∈ z` is ______.

{−5, −4, −3, −2, −1}

{−6, −5, −4, −3, −2, 1}

{−5, −4, −3, −2, −1, −0}

{−6, −5, −4, −3, −2, −1, 0}

Assertion (A): The mean of first 9 natural numbers is 4.5.

Reason (R): Mean = `"Sum of all the observations"/"Total number of observations"`

(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 the incorrect reason for (A).

Solve the following quadratic equation 2x² − 5x − 4 = 0.

Mrs. Rao deposited ₹250 per month in a recurring deposit account for a period of 3 years. She received ₹10,110 at the time of maturity. Find:

1. the rate of interest. 
2. how much more interest Mrs. Rao will receive if she had deposited ₹50 more per month at the same rate of interest and for the same time.

In ΔABC, ∠ABC = 90°, AB = 20 cm, AC = 25 cm, DE is perpendicular to AC such that ∠DEA = 90° and DE = 3 cm as shown in the given figure.

1. Prove that ΔABC ~ ΔAED.
2. Find the lengths of BC, AD, and AE.
3. If BCED represents a plot of land on a map whose actual area on the ground is 576 m², then find the scale factor of the map.

Use ruler and compass for the following construction. Construct a ΔABC, where AB = 6 cm, AC = 4.5 cm and ∠BAC = 120°. Construct a circle circumscribing the ΔABC. Measure and write down the length of the radius of the circle.

If A = `[(1,2),(3,4)], B = [(2,1),(4,2)] and C = [(−5,1),(7,−4)]` Find:

1. A + C
2. B(A+C)
3. 5B
4. B(A+C) – 5B

In the given graph ABCD is a parallelogram.

Using the graph, answer the following:

1. write down the coordinates of A, B, C and D.
2. calculate the coordinates of ‘P’, the point of intersection of the diagonals AC and BD.
3. find the slope of sides CB and DA and verify that they represent parallel lines.
4. find the equation of the diagonal AC.

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

`2x − 5/3 < "3x"/5 + 10 ≤ "4x"/5 + 11; x ∈ R`

The first term of an Arithmetic Progression (A.P.) is 5, the last term is 50 and their sum is 440. Find:

1. the number of terms 
2. common difference

Prove that:

`(cotA + tanA − 1) (sinA + cosA)/sin^3A + cos^3A = secA × cosec A`

Using properties of proportion, find the value of ‘x’:

`(6x^2 + 3x − 5)/(3x − 5) = (9x^2 + 2x + 5)/(2x + 5); x ≠ 0`

It is given that (x − 2) is a factor of polynomial 2x3 − 7x2+ kx − 2.

Find:

1. the value of ‘k’. 
2. Hence, factorise the resulting polynomial completely.

A solid wooden capsule is shown in Figure 1. The capsule is formed of a cylindrical block and two hemispheres.

Find the sum of total surface area of the three parts as shown in Figure 2. Given, the radius of the capsule is 3.5 cm and the length of the cylindrical block is 14 cm.

(use π = `22/7`)

Use a graph paper for this question taking 2 cm = 1 unit along both axes.

1. Plot A(1, 3), B(1, 2) and C(3, 0).
2. Reflect A and B on the x-axis and name their images as E and D respectively. Write down their coordinates.
3. Reflect A and B through the origin and name their images as F and G respectively.
4. Reflect A, B and C on the y-axis and name their images as J, I and H respectively.
5. Join all the points A, B, C, D, E, F, G, H, I and J in order and name the closed figure so formed.

In the given diagram, AB is a vertical tower 100 m away from the foot of a 30 storied building CD. The angles of depression from the point C and E, (E being the mid-point of CD), are 35° and 14° respectively.

(Use a mathematical table for the required values rounded off correct to two places of decimals only)

Find the height of the:

1. tower AB
2. building CD

Draw a Histogram for the following distribution which gives the marks obtained by 164 students in a particular class and hence find the Mode.

In the given graph, P and Q are points such that PQ cuts off intercepts of 5 units and 3 units along the x-axis and y-axis respectively. Line RS is perpendicular to PQ and passes through the origin. Find the:

1. coordinates of P and Q 
2. equation of line RS

Refer to the given bill.

A customer paid ₹2000 (rounded off to the nearest ₹10) to clear the bill.

Note: 5% discount is applicable on an article if 10 or more such articles are purchased.

Check whether the total amount paid by the customer is correct or not. Justify your answer with the necessary working.

A man bought ₹200 shares of a company at 25% premium. If he received a return of 5% on his investment. Find the:

1. market value 
2. dividend percent declared 
3. number of shares purchased, if annual dividend is ₹1000.

For the given frequency distribution, find the:

1. mean, to the nearest whole number
2. median

Mr. and Mrs. Das were travelling by car from Delhi to Kasauli for a holiday. Distance between Delhi and Kasauli is approximately 350 km (via NH 152D). Due to heavy rain they had to slow down. The average speed of the car was reduced by 20 km/hr and time of the journey increased by 2 hours. Find:

1. the original speed of the car. 
2. with the reduced speed, the number of hours they took to reach their destination.

A hollow sphere of external diameter 10 cm and internal diameter 6 cm is melted and made into a solid right circular cone of height 8 cm. Find the radius of the cone so formed.

[Use π = `22/7`]

Ms. Sushmita went to a fair and participated in a game. The game consisted of a box having number cards with numbers from 01 to 30. The three prizes were as per the given table:

Find the probability of winning a:

1. Wall Clock
2. Water Bottle
3. Purse

X, Y, Z and C are the points on the circumference of a circle with centre ‘O’. AB is a tangent to the circle at ‘X’ and ZY = XY.

Given ∠OBX = 32° and ∠AXZ = 66°. Find:

1. ∠BOX
2. ∠CYZ
3. ∠ZYX
4. ∠OXY

If 1701 is the n^th term of the Geometric Progression (G.P.) 7, 21, 63 ……., find:

1. the value of ‘n’
2. hence find the sum of the ‘n’ terms of the G.P

In the given diagram ‘O’ is the centre of the circle. Chord SR produced meets the tangent XTP at P.

1. Prove that ΔPTR ~ ΔPST
2. Prove that PT² = PR × PS
3. If PR = 4 cm and PS = 16 cm, find the length of the tangent PT.

The given graph represents the monthly salaries (in ₹) of workers of a factory.

Using graph answer the following:

1. the total number of workers. 
2. the median class. 
3. the lower-quartile class. 
4. number of workers having monthly salary more than or equal to ₹6,000 but less than ₹10,000.