Mathematics Official 2023-2024 (English Medium) ICSE Class 10 Question Paper Solution

For an Intra-state sale, the CGST paid by a dealer to the Central government is ₹ 120. If the marked price of the article is ₹ 2000, the rate of GST is ______.

6%

10%

12%

16.67%

What must be subtracted from the polynomial x³ + x² – 2x + 1, so that the result is exactly divisible by (x – 3)?

– 31

– 30

30

31

The roots of the quadratic equation px² – qx + r = 0 are real and equal if ______.

p² = 4qr

q² = 4pr

– q² = 4pr

p² > 4pr

If matrix A = `[(2, 2),(0, 2)]` and A² = `[(4, x),(0, 4)]`, then the value of x is ______.

2

4

8

10

The median of the following observations arranged in ascending order is 64. Find the value of x:

27, 31, 46, 52, x, x + 4, 71, 79, 85, 90

60

61

62

66

Points A(x, y), B(3, –2) and C(4, –5) are collinear. The value of y in terms of x is ______.

3x – 11

11 – 3x

3x – 7

7 – 3x

The given table shows the distance covered and the time taken by a train moving at a uniform speed along a straight track:

The values of x and y are:

x = 4, y = 150

x = 3, y = 100

x = 4, y = 100

x = 3, y = 150

The 7^th term of the given Arithmetic Progression (A.P.):

`1/a, (1/a + 1), (1/a + 2)`... is:

`(1/a + 6)`

`(1/a + 7)`

`(1/a + 8)`

`(1/a + 7^7)`

The sum invested to purchase 15 shares of a company of nominal value ₹ 75 available at a discount of 20% is ______.

₹ 60

₹ 90

₹ 1350

₹ 900

The circumcentre of a triangle is the point which is ______.

at equal distance from the three sides of the triangle.

at equal distance from the three vertices of the triangle.

the point of intersection of the three medians.

the point of intersection of the three altitudes of the triangle.

Statement 1: sin²θ + cos²θ = 1

Statement 2: cosec²θ + cot²θ = 1

Which of the following is valid?

Only 1

Only 2

Both 1 and 2

Neither 1 nor 2

In the given diagram, PS and PT are the tangents to the circle. SQ || PT and ∠SPT = 80°. The value of ∠QST is ______.

140°

90°

80°

50°

Assertion (A): A die is thrown once and the probability of getting an even number is `2/3`.

Reason (R): The sample space for even numbers on a die is {2, 4, 6}.

A is true, R is false.

A is false, R is true.

Both A and R are true.

Both A and R are false.

A rectangular sheet of paper of size 11 cm × 7 cm is first rotated about the side 11 cm and then about the side 7 cm to form a cylinder, as shown in the diagram. The ratio of their curved surface areas is ______.

1 : 1

7 : 11

11 : 7

`(11π)/7 : (7π)/11`

In the given diagram, ΔABC ∼ ΔPQR. If AD and PS are bisectors of ∠BAC and ∠QPR respectively then ______.

ΔABC ∼ ΔPQS

ΔABD ∼ ΔPQS

ΔABD ∼ ΔPSR

ΔABC ∼ ΔPSR

A = `[(x, 0),(1, 1)]`, B = `[(4, 0),(y, 1)]` and C = `[(4, 0),(x, 1)]`. Find the values of x and y, if AB = C.

A solid metallic cylinder is cut into two identical halves along its height (as shown in the diagram). The diameter of the cylinder is 7 cm and the height is 10 cm.

Find:

1. The total surface area (both the halves).
2. The total cost of painting the two halves at the rate of ₹ 30 per cm² `("Use"  π = 22/7)`.
   

15, 30, 60, 120.... are in G.P. (Geometric Progression):

1. Find the n^th term of this G.P. in terms of n.
2. How many terms of the above G.P. will give the sum 945?

Factorize: sin³θ + cos³θ

Hence, prove the following identity:

`(sin^3θ + cos^3θ)/(sin θ + cos θ) + sin θ cos θ = 1`

In the given diagram, O is the centre of the circle. PR and PT are two tangents drawn from the external point P and touching the circle at Q and S respectively. MN is a diameter of the circle. Given ∠PQM = 42° and ∠PSM = 25°.

Find:

1. ∠OQM
2. ∠QNS
3. ∠QOS
4. ∠QMS

Use graph sheet for this question. Take 2 cm = 1 unit along the axes.

1. Plot A(0, 3), B(2, 1) and C(4, –1).
2. Reflect point B and C in y-axis and name their images as B' and C' respectively. Plot and write coordinates of the points B' and C'.
3. Reflect point A in the line BB' and name its images as A'.
4. Plot and write coordinates of point A'.
5. Join the points ABA'B' and give the geometrical name of the closed figure so formed.

Suresh has a recurring deposit account in a bank. He deposits ₹ 2000 per month and the bank pays interest at the rate of 8% per annum. If he gets ₹ 1040 as interest at the time of maturity, find in years total time for which the account was held.

The following table gives the duration of movies in minutes:

Using step-deviation method, find the mean duration of the movies.

If `(a + b)^3/(a - b)^3 = 64/27`.

1. Find `(a + b)/(a - b)`
2. Hence using properties of proportion, find a : b.

The given graph with a histogram represents the number of plants of different heights grown in a school campus. Study the graph carefully and answer the following questions:

1. Make a frequency table with respect to the class boundaries and their corresponding frequencies.
2. State the modal class.
3. Identify and note down the mode of the distribution.
4. Find the number of plants whose height range is between 80 cm to 90 cm.

The angle of elevation of the top of a 100 m high tree from two points A and B on the opposite side of the tree are 52° and 45° respectively. Find the distance AB, to the nearest metre.

Solve the following equation for x and give, in the following case, your answer correct to 2 decimal places:

2x² – 10x + 5 = 0

The n^th term of an Arithmetic Progression (A.P.) is given by the relation T_n = 6(7 – n)..

Find:

1. its first term and common difference
2. sum of its first 25 terms

In the given diagram ΔADB and ΔACB are two right angled triangles with ∠ADB = ∠BCA = 90°. If AB = 10 cm, AD = 6 cm, BC = 2.4 cm and DP = 4.5 cm.

1. Prove that ΔAPD ∼ ΔBPC
2. Find the length of BD and PB
3. Hence, find the length of PA
4. Find area ΔAPD : area ΔBPC.

In the given diagram an isosceles ΔABC is inscribed in a circle with centre O. PQ is a tangent to the circle at C. OM is perpendicular to chord AC and ∠COM = 65°.

Find:

1. ∠ABC
2. ∠BAC
3. ∠BCQ

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

`-3 + x ≤ (7x)/2 + 2 < 8 + 2x, x ∈ I`

In the given diagram, ABC is a triangle, where B(4, – 4) and C(– 4, –2). D is a point on AC.

1. Write down the coordinates of A and D.
2. Find the coordinates of the centroid of ΔABC.
3. If D divides AC in the ratio k : 1, find the value of k.
4. Find the equation of the line BD.

The polynomial 3x³ + 8x² – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

The following letters A, D, M, N, O, S, U, Y of the English alphabet are written on separate cards and put in a box. The cards are well shuffled and one card is drawn at random. What is the probability that the card drawn is a letter of the word,

1. MONDAY?
2. Which does not appear in MONDAY?
3. Which appears both in SUNDAY and MONDAY?

Oil is stored in a spherical vessel occupying `3/4` of its full capacity. Radius of this spherical vessel is 28 cm. This oil is then poured into a cylindrical vessel with a radius of 21 cm. Find the height of the oil in the cylindrical vessel (correct to the nearest cm). Take π = `22/7`

The figure shows a circle of radius 9 cm with 0 as the centre. The diameter AB produced meets the tangent PQ at P. If PA = 24 cm, find the length of tangent PQ:

Mr. Gupta invested ₹ 33000 in buying ₹ 100 shares of a company at 10% premium. The dividend declared by the company is 12%.

Find:

1. the number of shares purchased by him
2. his annual dividend.

A life insurance agent found the following data for distribution of ages of 100 policy holders.

On a graph sheet draw an ogive using the given data. Take 2 cm = 5 years along one axis and 2 cm = 10 policy holders along the other axis.

Use your graph to find:

1. The median age.
2. Number of policy holders whose age is above 52 years.

Rohan bought the following eatables for his friends:

Calculate:

1. Total GST paid.
2. Total bill amount including GST.

If the lines kx – y + 4 = 0 and 2y = 6x + 7 are perpendicular to each other, find the value of k.

Find the equation of a line parallel to 2y = 6x + 7 and passing through (–1, 1).

Use ruler and compass to answer this question. Construct ∠ABC = 90°, where AB = 6 cm, BC = 8 cm.

1. Construct the locus of points equidistant from B and C.
2. Construct the locus of points equidistant from A and B.
3. Mark the point which satisfies both the conditions (a) and (b) as 0. Construct the locus of points keeping a fixed distance OA from the fixed point 0.
4. Construct the locus of points which are equidistant from BA and BC.