KARNATAKA STATE BOARD CLASS 10TH MATHS 2026-2 Solutions &

Q1 mcq 1 mark

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

a.

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

b.

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

c.

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

d.

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

Step-by-Step AI Explanation

Verified Method

Two lines are coincident if they have infinitely many solutions, which occurs when their coefficients are in the same proportion: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. Option (c) represents parallel lines, (d) represents intersecting lines, and (b) is not a standard condition.

Q2 mcq 1 mark

The coordinates of the point ‘P’ which divides the line segment joining the points $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $1:1$ is,

a.

$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ Correct Answer

b.

$(x_1 + x_2, y_1 + y_2)$

c.

$(x_1 - x_2, y_1 - y_2)$

d.

$\left(\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2}\right)$

Step-by-Step AI Explanation

Verified Method

The section formula for a point dividing the line in ratio $m:n$ is $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$. For a midpoint ($1:1$), this simplifies to $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

Q3 mcq 1 mark

If $\sin \theta = \frac{5}{13}$, then the value of $\text{cosec } \theta$ is,

a.

$\frac{13}{12}$

b.

$\frac{5}{12}$

c.

$\frac{13}{5}$ Correct Answer

d.

$\frac{12}{13}$

Step-by-Step AI Explanation

Verified Method

By definition, $\text{cosec } \theta$ is the reciprocal of $\sin \theta$. Since $\sin \theta = \frac{5}{13}$, then $\text{cosec } \theta = \frac{1}{\sin \theta} = \frac{13}{5}$.

Q4 mcq 1 mark

The surface area of a sphere of radius $7\text{ cm}$ is

a.

$77\text{ cm}^2$

b.

$154\text{ cm}^2$

c.

$308\text{ cm}^2$

d.

$616\text{ cm}^2$ Correct Answer

Step-by-Step AI Explanation

Verified Method

The surface area of a sphere is $A = 4\pi r^2$. Substituting $r = 7\text{ cm}$ and $\pi \approx \frac{22}{7}$: $A = 4 \times \frac{22}{7} \times 7^2 = 4 \times 22 \times 7 = 616\text{ cm}^2$.

Q5 mcq 1 mark

If the $n^{\text{th}}$ term of an arithmetic progression $a_n = 3n - 2$, then its $9^{\text{th}}$ term is

a.

$-5$

b.

$25$ Correct Answer

c.

$5$

d.

$-25$

Step-by-Step AI Explanation

Verified Method

Given $a_n = 3n - 2$. For the $9^{\text{th}}$ term, substitute $n = 9$: $a_9 = 3(9) - 2 = 27 - 2 = 25$.

Q6 mcq 1 mark

Which of the following can be the probability of an event?

a.

$\frac{3}{2}$

b.

$-1.5$

c.

$105\%$

d.

$0.7$ Correct Answer

Step-by-Step AI Explanation

Verified Method

The probability of an event must lie between $0$ and $1$ (inclusive). Option (a) $>1$, (b) is negative, and (c) $=1.05 > 1$. Thus, only $0.7$ is a valid probability.

Q7 mcq 1 mark

In the figure, the value of ‘x’ is

a.

$60\text{ cm}$ Correct Answer

b.

$25\text{ cm}$

c.

$64\text{ cm}$

d.

$53\text{ cm}$

Step-by-Step AI Explanation

Verified Method

Since $CD \parallel AB$, $\triangle ODC \sim \triangle OAB$ by AA similarity. Therefore, the ratio of corresponding sides is equal: $\frac{AB}{CD} = \frac{OA}{OD}$. Thus, $\frac{x}{24} = \frac{40}{15}$. Simplifying $\frac{40}{15} = \frac{8}{3}$. So, $x = 24 \times \frac{8}{3} = 8 \times 8 = 64\text{ cm}$. Wait, checking calculation: $\frac{x}{24} = \frac{40}{15} \implies x = 24 \times \frac{8}{3} = 64$. However, looking at the choices, $64$ is (C). Let me re-read: $\frac{x}{24} = \frac{40}{15} = 2.66$. $x = 64$. The correct answer is (C).

Q8 mcq 1 mark

If the sum of the zeroes of the polynomial $P(x) = kx^2 + 2x + 3k$ is equal to the product of its zeroes, then the value of ‘k’ is

a.

$\frac{2}{3}$

b.

$\frac{3}{4}$

c.

$-\frac{2}{3}$

d.

$-\frac{3}{4}$ Correct Answer

Step-by-Step AI Explanation

Verified Method

For $ax^2+bx+c$, sum of zeros $= -b/a = -2/k$ and product of zeros $= c/a = 3k/k = 3$. Setting them equal: $-2/k = 3 \implies 3k = -2 \implies k = -2/3$. Wait, checking the product term: $c=3k$, $a=k$, product is $3k/k=3$. $-2/k = 3 o k = -2/3$. Looking at the provided options, if the polynomial was $kx^2+2x+3k$, product is $3k/k=3$. If the question meant $P(x)=kx^2+2x+3$, product would be $3/k$. $-2/k = 3/k o -2=3$ (impossible). Re-evaluating: sum of zeros $\alpha+\beta = -2/k$. Product $\alpha\beta = 3k/k = 3$. $-2/k = 3 \to k = -2/3$. None of the options match $-2/3$. If the polynomial was $kx^2+2x+3$, sum $-2/k$, product $3/k$, $-2/k=3/k$ is impossible. If the equation was $3x^2+2x+k$, sum $-2/3$, product $k/3$. $-2/3 = k/3 \to k=-2$. Checking math again: Sum of zeros is $-b/a = -2/k$. Product is $c/a = 3k/k = 3$. Set $-2/k = 3 \Rightarrow k = -2/3$. Option C is -2/3.

Q9 short answer 1 mark

Write the degree of a cubic polynomial.

Step-by-Step AI Explanation

Verified Method

The degree of a cubic polynomial is 3.

Q10 short answer 1 mark

How many solutions does the pair of linear equations in two variables have if they are inconsistent ?

Step-by-Step AI Explanation

Verified Method

A pair of linear equations that are inconsistent has zero (no) solutions.

Q11 short answer 1 mark

State Basic Proportionality theorem ( Thales theorem ).