TS 10th Class Maths Question Paper March 2025

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TS 10th Class Maths Question Paper March 2025

Time: 3 Hours
Maximum Marks: 80

General Instructions:

1. Answer all the questions under Part – A on a separate answer book.
2. Write the answers to the questions under Part – B on the question paper itself and attach it to the answer book of Part – A.

Part – A (60 Marks)
Section – I (6 × 2 = 12 Marks)

Question 1.
Show that the points A(- 6, 10), B(- 4, 6), and C(3, – 8) are collinear.
Solution:

Question 2.
Write a Quadratic equation whose roots are the values of sin 30° and cos 60°.
Solution:
sin 30° = \(\frac{1}{2}\) ; cos 60° = \(\frac{1}{2}\)
Sum of roots = \(\frac{1}{2}\) + \(\frac{1}{2}\) = 1
Product of roots = \(\frac{1}{2}\) × \(\frac{1}{2}\) = \(\frac{1}{4}\)
Quadratic equation
x² – x(sum of roots) + Product of roots = 0
⇒ x² – x(1) + \(\frac{1}{4}\) = 0
∴ 4x² – 4x + \(\frac{1}{4}\) = 0 is the required quadratic
equation.

Question 3.
In an arithmetic progression, first term is T, last term is 20 and the sum of all the terms is 399, then find the number of terms in the progression.
Solution:

Question 4.
“An observer standing at a distance of 50 metre from the foot of a tower observes its top at an angle of elevation of 45°. Draw a suitable diagram for this situation.
Solution:

AB – Tower
C – Position of observer
60° – Angle of elevation

Question 5.
If the pair of linear equations (3k + 1)x + 3y – 2 = 0 and (k² + 1) x + (k – 2) y – 5 = 0 has no solutions, then find the value of k.
Solution:
a₁ = 3k + 1, b₁ = 3, c₁ = -2
a₂ = k² + 1, b₂ = k – 2, c₂ = -5
Given pair of linear equations has no solution

Question 6.
The ratio of radius and slant height of a Right circular cone is 7:25. If its curved surface area is 550 cm², then find its radius.
Solution:

Section – II (6 × 4 = 24 Marks)

Note:

1. Answer ALL the following questions.
2. Each question carries 4 marks.

Question 7.
From the given Venn diagram, find the sets A ∪ B, A ∩ B, A – B and B – A.

Solution:
A = {1, 2, 3, 5, 6, 10, 15, 30}
B = {1, 2, 3, 4, 6, 8, 12, 24}
A∪B = {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 30}
A ∩ B = {1, 2, 3, 6}
A – B = {5, 10, 15, 30}
B – A = {4, 8, 12, 24}

Question 8.
From a well-shuffled deck of cards if a card is selected randomly, then find the probability of getting
i) A red coloured king
ii) A black coloured face card
ii) A diamond card with number 11 on it
iv) Queen of clubs
Solution:
i) P(Getting a red coloured King)
\(=\frac{\text { Number of favourable outcomes }}{\text { Number of total outcomes }}\)
= \(\frac{2}{52}\) = \(\frac{1}{26}\)

ii) P (Getting a black coloured face card)
= \(\frac{6}{52}\) = \(\frac{3}{26}\)

iii) P (Getting a diamond card with number 11)
= \(\frac{0}{52}\) = 0 (∵ Impossible event)

iv) P (Getting Queen of clubs) = \(\frac{1}{52}\)

Question 9.
Show that \(\left[\frac{1+\tan ^2 A}{1+\cot ^2 A}\right]\) = \(\left[\frac{1-\tan A}{1-\cot A}\right]^2\) = tan²A
Solution:

Question 10.
In ΔABC, DE || BC
If AD = x – 2, DB = 5, AE = 2x – 1, EC = 2x + 5, then find the value of x.

Solution:

⇒ 2x² + 5x – 4x – 10 = 10x – 5
⇒ 2x² – 9x – 5 = 0
⇒ 2x² – 10x + x – 5 = 0
⇒ 2x(x – 5) + 1 (x – 5) = 0
⇒ (x – 5)(2x + 1) = 0
⇒ x = 5 or x = –\(\frac{1}{2}\)
But x value can’t be –\(\frac{1}{2}\) as length can’t be negative.
So, x – 5 is the required solution.

Question 11.
Write the formula for the sum of first ‘n’ terms of an arithmetic progression and explain each term in it.
Solution:
Sum of first ‘n’ terms of an arithmetic progression
S_n = \(\frac{n}{2}\)[2a + (n – 1)d] S_n = \(\frac{n}{2}\)[a + l]
n = number of terms
a = first term
d = common difference
S_n = sum of first
(or)
n = number of terms
a = first term ‘n’ terms
l = last term
S_n = sum of first ‘n’ terms

Question 12.
Show that the triangle with vertices
A(- 4, 2), B(2, – 4), C(12, 6) forms a Right – angled triangle.
Solution:
A(-4, 2) B(2, -4), C(12, 6)
If (x₁, y₁) (x₂, y₂) are any two points, then the distance between those two points.

Section – III (4 × 6 = 24 Marks)

Note :

1. Answer any 4 questions from the given six following questions.
2. Each question carries 6 marks.

Question 13.
Draw the graph of the polynomial p(x) = x² + 2x – 3 and find zeroes of the polynomial from the graph.
Solution:

Question 14.
Find the mode of the following data :

Solution:

Since the maximum frequency is 20.
the modal class is 40 – 50.
Lower boundary of the modal class ‘l’ = 40
Frequency of the modal class, f₁ = 20
Frequency of the class preceding the modal class, f₀ = 12
Frequency of the class succeeding the modal class, f₂ = 11

Question 15.
Draw a circle of radius 3 cm. Construct a pair of tangents to the circle from an external point which is at a distance of 8 cm from the centre of the circle.
Solution:

Steps of construction

1. Draw a circle from a point ‘O’ with a radius of 3 cm.
2. Now trace a point ‘P’ at a distance of 8 cm from the centre ‘O’, then \(\overline{\mathrm{OP}}\) = 8 cm.
3. Construct perpendicular bisector \(\overline{\mathrm{KL}}\) which meets \(\overline{\mathrm{OP}}\) at ‘M’.
4. Now construct a circle from the point ‘M’ with a radius \(\overline{\mathrm{OM}}\) or \(\overline{\mathrm{MP}}\), which meets the previous circle at the points A, B.
5. Then join \(\overline{\mathrm{PA}}\) and \(\overline{\mathrm{PB}}\) which are required tangents.

Question 16.
If x² + y² = 27xy, then show that 2 log (x – y) = 2 log 5 + log x + log y.
Solution:
x² + y² = 27xy (Given)
Subtracting 2xy on both sides,
x² + y² – 2xy = 27xy – 2xy
(x – y)² = 25xy
Applying ‘log’ on both sides
log (x – y)² = log (25xy)
= log (5² × x × y)
2 log (x + y) = log 5² + log x + log y
= 2 log 5 + log x + log y
Alternative method:
LHS = 2 log (x – y)
= log (x – y)²
= log (x² + y² – 2xy)
= log (27xy – 2xy)
[∵ x² + y² = 27xy (given)]
= log (25 xy)
= log 25 + log x + log y
= log 5² + log x + log y
= 2 log 5 + log x + log y
= RHS.

Question 17.
Find the coordinates of the points which divide the line segment joining the points A(-2, 2) and B(2, 8) into four equal parts.
Solution:
Given, A (-2, 2) and B (2, 8).
Let P, Q and R be the points which divide AB into four equal parts.

Question 18.
A metallic vessel is in the shape of a cylinder surmounted over a hemisphere. The radii of cylinder and hemisphere are same and the height of the cylindrical part is 10 cm. If the outer surface area of the vessel is 748 cm², then find their radii.
Solution:

Part – B (20 Marks)

Note :

1. ALL questions are to be answered.
2. Each question carries 1 mark.
3. Answers are to be written in the Question paper only.
4. Marks will not be given for over – writing, rewriting or erased answers.

I. Write the capital (A, B, C, D) of the correct answer in the brackets provided against each question, 20 × 1 = 20

Question 1.
Set-builder form of the set {1, 2, 3, 4, 5} [ ]
A) {x : x ∈ W and 0 < x < 5}
B) {x : x ∈ W and 0 < x ≤ 5}
C) {X : X ∈ W and 0 ≤ x ≤ 5}
D) {x : x ∈ W and 0 ≤ x ≤ 5}
Solution:
B) {x : x ∈ W and 0 < x ≤ 5}

Question 2.
The \(\frac{\mathrm{p}}{\mathrm{q}}\) (q ≠ 0) form of 0.125 is
A) \(\frac{125}{10}\)
B) \(\frac{125}{100}\)
C) \(\frac{125}{1000}\)
D) \(\frac{0.125}{100}\)
Solution:
C) \(\frac{125}{1000}\)

Question 3.
In a random experiment E and \(\overline{\mathbf{E}}\) are complementary events. If P(E) = 0.43, then the value of P(\(\overline{\mathbf{E}}\)) is [ ]
A) 0.53
B) 0.47
C) 0.43
D) 0.57
Solution:
D) 0.57

Question 4.
A = {x : x is a letter of the word “RAMANUJAN”}, then n(A) = ? [ ]
A) 9
B) 8
C) 7
D) 6
Solution:
D) 6

Question 5.
The quadratic equation, whose sum of the roots is 5 and product of the roots is 6, is [ ]
A) x² – 5x + 6 = 0
B) x² + 5x + 6 = 0
C) x² – 6x + 5 = 0
D) x² + 6x + 5 = 0
Solution:
A) x² – 5x + 6 = 0

Question 6.
Centroid of the triangle with vertices A(0, 4), B(4, -2) and C(5, 10) is [ ]
A) (3, 4)
B) (3, 3)
C) (4, 3)
D) (4, 4)
Solution:
A) (3, 4)

Question 7.
If one root of a quadratic equation x² – 5x + k = 0 is 2, then the value of k is [ ]
A) 2
B) 4
C) 6
D) – 6
Solution:
C) 6

Question 8.
The lengths of tangents from an external point P to a circle with centre ‘O’ are 2x² – 3 and x² + 22, then the value of x is

A) 3
B) 5
C) 22
D) 19
Solution:
B) 5

Question 9.
Sum of first 10 terms of arithmetic progression log 3, log 9, log 27, ………… is [ ]
A) 45 log 3
B) 90 log 3
C) 10 log 3
D) 55 log 3
Solution:
D) 55 log 3

Question 10.
Nature of roots of a quadratic equation x² – 4x + 4 = 0 is
A) Real and distinct
B) Real and equal
C) Not Real and distinct
D) Not Real and equal
Solution:
B) Real and equal

Question 11.
The median of factors of 36 is
A) 4
B) 9
C) 6
D) 12
Solution:
C) 6

Question 12.
Among the following, a pair of inconsistent equation is
A) x + 2y = 5, 2x + 4y = 10
B) x + 2y = 5, 2x + 4y = 7
C) x + 2y = 5, 4x + 2y = 8
D) 2x + y = 4, 2x + 4y = 10
Solution:
B) x + 2y = 5, 2x + 4y = 7

Question 13.
If sin α = \(\frac{\sqrt{3}}{2}\) and cos β = \(\frac{\sqrt{3}}{2}\) (0° < α, β < 90°), then the value of tan(α – β) is
A) \(\sqrt{3}\)
B) 0
C) 1
D) \(\frac{1}{\sqrt{3}}\)
Solution:
D) \(\frac{1}{\sqrt{3}}\)

Question 14.
A ladder of length 10m is leaning against a wall. If it touches the wall at height of 5m, then the angle made by the ladder with the ground is
A) 30°
B) 60°
C) 45°
D) 90°
Solution:
A) 30°

Question 15.
The area of one face of a cube is 25cm², then its volume is (in cm³)
A) 150
B) 100
C) 125
D) 200
Solution:
C) 125

Question 16.
If \(\frac{1}{x}\) + \(\frac{1}{x}\) = \(\frac{5}{6}\), and \(\frac{1}{x}\) – \(\frac{1}{y}\) = \(\frac{1}{6}\), then the values of x and y respectively are
A) \(\frac{1}{2}\), \(\frac{1}{3}\)
B) 2, 3
C) \(\frac{1}{3}\), \(\frac{1}{2}\)
D) 3, 2
Solution:
B) 2, 3

Question 17.
The radii of two concentric circles with centre ‘O’ are 8 cm and 10 cm. If the chord AB ol larger circle is a tangent to the smaller circle at P, then the length of chord AB is

A) 12 cm
B) 6 cm
C) 18 cm
D) 9 cm
Solution:
A) 12 cm

Question 18.
Among the following, which is an example for a certain (sure) event ?
A) Getting a black ball when a ball is selected randomly from a bag containing black balls.
B) Getting a white ball when a ball is selected randomly from a bag containing black balls.
C) Getting a black ball when a ball is selected randomly from a bag containing 3 black and 4 white balls.
D) Not getting a white ball when a ball is selected randomly from a bag containing 3 black and 4 white balls.
Solution:
A) Getting a black ball when a ball is selected randomly from a bag containing black balls.

Question 19.
If p(x) = 2x² – 5x + 6, then p(1) + p(-1) = ______.
A) 3
B) 13
C) 16
D) 10
Solution:
C) 16

Question 20.
Among the following, which pair of triangles are not similar ?

Solution: