KARNATAKA STATE BOARD CLASS 10TH MATHS 2025-1 Solutions &

Q1 mcq 1 mark

LCM of 2 and 3 is

a.

2

b.

3

c.

5

d.

6 Correct Answer

Step-by-Step AI Explanation

Verified Method

The least common multiple (LCM) of two prime numbers is their product. Since 2 and 3 are prime numbers, their LCM is $2 \times 3 = 6$. Therefore, option (D) is correct.

Q2 mcq 1 mark

If the lines represented by the equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are coincident, then the correct relation is

a.

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ Correct Answer

b.

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

c.

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

d.

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Step-by-Step AI Explanation

Verified Method

For two linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, the lines are coincident if they represent the same line, which occurs when their coefficients are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. Thus, (A) is the correct relation.

Q3 mcq 1 mark

The quadratic equation in the following is

a.

$x^3 - 6x$

b.

$p(x) = x^2 + 7x$

c.

$3x = 9$

d.

$x^2 + 3x + 4 = 0$ Correct Answer

Step-by-Step AI Explanation

Verified Method

A quadratic equation must be of the form $ax^2 + bx + c = 0$ where $a \neq 0$. Option (D) fits this standard form. Option (A) is a cubic expression, (B) is a quadratic polynomial but not an equation, and (C) is a linear equation.

Q4 mcq 1 mark

In the following, the shapes which are always similar, are

a.

any two equilateral triangles Correct Answer

b.

square and rectangle

c.

square and rhombus

d.

any two trapeziums

Step-by-Step AI Explanation

Verified Method

Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional. Any two equilateral triangles have all angles equal to $60^\circ$ and sides in a fixed ratio, making them always similar. Other shapes listed do not share constant similarity.

Q5 mcq 1 mark

The volume of a sphere of radius ‘r’ units is

a.

$\frac{2}{3}\pi r^3$ cubic units

b.

$\frac{4}{3}\pi r^3$ cubic units Correct Answer

c.

$\frac{1}{3}\pi r^3$ cubic units

d.

$\frac{3}{2}\pi r^3$ cubic units

Step-by-Step AI Explanation

Verified Method

The standard formula for the volume $V$ of a sphere with radius $r$ is derived via calculus as $V = \frac{4}{3}\pi r^3$. Option (B) correctly represents this formula.

Q6 mcq 1 mark

The distance of a point P (x, y) from the origin is

a.

$\sqrt{x^2 - y^2}$

b.

$\sqrt{x + y}$

c.

$\sqrt{x^2 + y^2}$ Correct Answer

d.

$\sqrt{x - y}$

Step-by-Step AI Explanation

Verified Method

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For a point $(x, y)$ and the origin $(0, 0)$, this becomes $\sqrt{(x-0)^2 + (y-0)^2} = \sqrt{x^2+y^2}$.

Q7 mcq 1 mark

The common difference of the arithmetic progression -1, -3, -5 ... is

a.

-1

b.

2

c.

-2 Correct Answer

d.

3

Step-by-Step AI Explanation

Verified Method

The common difference $d$ of an AP is found by subtracting a term from its successor: $d = a_2 - a_1$. For the sequence $-1, -3, -5, \dots$, $d = -3 - (-1) = -3 + 1 = -2$.

Q8 mcq 1 mark

In the given figure ‘O’ is the centre of the circle and the length of the arc APB is $4\pi$ cm. If OB = 9 cm, then the measure of angle $\theta$ is

a.

60°

b.

80° Correct Answer

c.

85°

d.

70°

Step-by-Step AI Explanation

Verified Method

The length of an arc $l$ is given by $l = \frac{\theta}{360^\circ} \times 2 \pi r$. Given $l = 4\pi$ and $r = 9$, we have $4\pi = \frac{\theta}{360^\circ} \times 2 \times \pi \times 9$. This simplifies to $4\pi = \frac{\theta \times 18\pi}{360^\circ}$, so $4 = \frac{\theta}{20}$, which gives $\theta = 80^\circ$.

Q9 short answer 1 mark

Write the degree of a linear polynomial.

Step-by-Step AI Explanation

Verified Method

The degree of a linear polynomial is 1.

Q10 short answer 1 mark

Write the formula to find the total surface area of a cube of edge ‘a’ units.

Step-by-Step AI Explanation

Verified Method

The total surface area of a cube with edge length 'a' is $6a^2$ square units.

Q11 short answer 1 mark

In the given frequency distribution table, write the modal class: Class-interval: 1-3, 3-5, 5-7, 7-9 Frequency: 4, 8, 2, 2

Correct Answer & Step-by-Step Solution Locked