KARNATAKA STATE BOARD CLASS 10TH MATHS 2022-2 Solutions &

Q1 mcq 1 mark

The graphical representation of the pair of lines $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ is

a.

intersecting lines

b.

parallel lines Correct Answer

c.

coincident lines

d.

perpendicular lines.

Step-by-Step AI Explanation

Verified Method

For the equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$, we check the ratios of the coefficients: $a_1/a_2 = 1/2$, $b_1/b_2 = 2/4 = 1/2$, and $c_1/c_2 = -4/-12 = 1/3$. Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, the lines are parallel.

Q2 mcq 1 mark

The common difference of the Arithmetic progression 8, 5, 2, - 1, ... is

a.

- 3 Correct Answer

b.

- 2

c.

3

d.

8.

Step-by-Step AI Explanation

Verified Method

The common difference $d$ is calculated by subtracting a term from its succeeding term: $d = a_2 - a_1 = 5 - 8 = -3$.

Q3 mcq 1 mark

The standard form of $2x^{2} = x - 7$ is

a.

$2x^{2} - x = -7$

b.

$2x^{2} + x - 7 = 0$

c.

$2x^{2} - x + 7 = 0$ Correct Answer

d.

$2x^{2} + x + 7 = 0$.

Step-by-Step AI Explanation

Verified Method

The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Starting from $2x^2 = x - 7$, we subtract $x$ and add $7$ to both sides to get $2x^2 - x + 7 = 0$.

Q4 mcq 1 mark

The value of $\cos ( 90^{\circ} - 30^{\circ} )$ is

a.

- 1

b.

$\frac{1}{2}$ Correct Answer

c.

0

d.

1.

Step-by-Step AI Explanation

Verified Method

$\cos(90^{\circ} - 30^{\circ}) = \cos(60^{\circ})$. The value of $\cos(60^{\circ})$ is $1/2$.

Q5 mcq 1 mark

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

a.

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

b.

$x^{2} + y^{2}$

c.

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

d.

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

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 the origin $(0,0)$ and $P(x,y)$, this becomes $\sqrt{x^2 + y^2}$.

Q6 mcq 1 mark

In a circle, the angle between the tangent and the radius at the point of contact is

a.

$30^{\circ}$

b.

$60^{\circ}$

c.

$90^{\circ}$ Correct Answer

d.

$180^{\circ}$.

Step-by-Step AI Explanation

Verified Method

According to the tangent theorem, the radius of a circle is always perpendicular to the tangent at the point of contact, meaning the angle between them is always $90^{\circ}$.

Q7 mcq 1 mark

In the given figure, the volume of the frustum of a cone is

a.

$\pi (r_{1} + r_{2})l$

b.

$\pi (r_{1} - r_{2})l$

c.

$\frac{1}{3} \pi h (r_{1}^{2} - r_{2}^{2} - r_{1}r_{2})$

d.

$\frac{1}{3} \pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})$ Correct Answer

Step-by-Step AI Explanation

Verified Method

The formula for the volume $V$ of a frustum of a cone with height $h$ and radii $r_1$ and $r_2$ is given by $V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1r_2)$.

Q8 mcq 1 mark

Surface area of a sphere of radius ‘r’ unit is

a.

$\pi r^{2}$ sq.units

b.

$2\pi r^{2}$ sq.units

c.

$3\pi r^{2}$ sq.units

d.

$4\pi r^{2}$ sq.units. Correct Answer

Step-by-Step AI Explanation

Verified Method

The surface area of a sphere of radius $r$ is a standard geometric formula given by $4\pi r^2$ square units.

Q9 short answer 1 mark

If the pair of linear equations in two variables are inconsistent, then how many solutions do they have ?

Step-by-Step AI Explanation

Verified Method

If a pair of linear equations is inconsistent, they represent parallel lines, meaning they have zero or no solutions.

Q10 short answer 1 mark

In an Arithmetic progression if ‘a’ is the first term and ‘d’ is the common difference, then write its $n^{th}$ term.

Step-by-Step AI Explanation

Verified Method

The $n^{th}$ term of an Arithmetic progression is given by the formula $a_n = a + (n - 1)d$.

Q11 short answer 1 mark

Write the standard form of quadratic equation.

Correct Answer & Step-by-Step Solution Locked