KARNATAKA STATE BOARD CLASS 10TH MATHS 2025-3 Solutions &

Q1 mcq 1 mark

The degree of the polynomial $p(x) = x^{3} - 7x^{2} + 5x + 1$ is

a.

1

b.

2

c.

3 Correct Answer

d.

0

Step-by-Step AI Explanation

Verified Method

The degree of a polynomial is the highest power of the variable present in the expression. In $p(x) = x^{3} - 7x^{2} + 5x + 1$, the highest power is 3. Thus, the degree is 3.

Q2 mcq 1 mark

The roots of the equation $(x + 2)(x - 3) = 0$ are

a.

$-2$ and $-3$

b.

$-2$ and $3$ Correct Answer

c.

$2$ and $3$

d.

$2$ and $-3$

Step-by-Step AI Explanation

Verified Method

For the equation $(x + 2)(x - 3) = 0$, the roots are found by setting each factor equal to zero: $x + 2 = 0 \implies x = -2$ and $x - 3 = 0 \implies x = 3$. Therefore, the roots are $-2$ and $3$.

Q3 mcq 1 mark

The formula to find the sum of first 'n' natural numbers is

a.

$S_{n} = \frac{n}{2}$

b.

$S_{n} = n(n + 1)$

c.

$S_{n} = n^{2}$

d.

$S_{n} = \frac{n(n + 1)}{2}$ Correct Answer

Step-by-Step AI Explanation

Verified Method

The sum of the first 'n' natural numbers is given by the arithmetic series formula $S_{n} = \frac{n(n + 1)}{2}$. This is a standard formula derived from the sum of an arithmetic progression with first term $a=1$ and common difference $d=1$.

Q4 mcq 1 mark

In the figure, $\Delta ABC \sim \Delta DEF$. If $AB = 5$ cm, $BC = 10$ cm, $DE = 3$ cm then value of $EF$ is

a.

6 cm Correct Answer

b.

4 cm

c.

7 cm

d.

5 cm

Step-by-Step AI Explanation

Verified Method

Since $\Delta ABC \sim \Delta DEF$, the ratio of their corresponding sides must be equal: $\frac{AB}{DE} = \frac{BC}{EF}$. Substituting the given values: $\frac{5}{3} = \frac{10}{EF}$. Solving for $EF$: $5 \times EF = 10 \times 3 \implies 5 \times EF = 30 \implies EF = \frac{30}{5} = 6$ cm.

Q5 mcq 1 mark

If $\sec A = \frac{2}{\sqrt{3}}$ then the value of $\cos A$ is

a.

2

b.

$\frac{2}{\sqrt{3}}$

c.

$\sqrt{3}$

d.

$\frac{\sqrt{3}}{2}$ Correct Answer

Step-by-Step AI Explanation

Verified Method

Since $\cos A$ is the reciprocal of $\sec A$, we have $\cos A = \frac{1}{\sec A}$. Given $\sec A = \frac{2}{\sqrt{3}}$, therefore $\cos A = \frac{1}{2/\sqrt{3}} = \frac{\sqrt{3}}{2}$.

Q6 mcq 1 mark

The formula to find the total surface area of a solid hemisphere of radius 'r' is

a.

$\pi r^{2}$

b.

$2\pi r^{2}$

c.

$3\pi r^{2}$ Correct Answer

d.

$4\pi r^{2}$

Step-by-Step AI Explanation

Verified Method

The total surface area of a solid hemisphere includes the curved surface area ($2\pi r^{2}$) plus the area of the circular base ($\pi r^{2}$). Thus, the total surface area is $2\pi r^{2} + \pi r^{2} = 3\pi r^{2}$.

Q7 mcq 1 mark

Diagonals of a parallelogram bisect each other at the point $(2, -3)$. If the coordinates of the point of one end of a diagonal are $(1, 4)$ then the coordinates of the point of the other end of the same diagonal are

a.

$(3, 10)$

b.

$(3, -10)$ Correct Answer

c.

$(-3, 10)$

d.

$(-3, -10)$

Step-by-Step AI Explanation

Verified Method

Let the unknown coordinates be $(x, y)$. Since the diagonals bisect each other, the point $(2, -3)$ is the midpoint of the segment connecting $(1, 4)$ and $(x, y)$. Using the midpoint formula: $\frac{1 + x}{2} = 2 \implies 1 + x = 4 \implies x = 3$; and $\frac{4 + y}{2} = -3 \implies 4 + y = -6 \implies y = -10$. Thus, the other end is $(3, -10)$.

Q8 mcq 1 mark

If $P(E) = 0.05$, then the probability of 'not E' is

a.

0.95 Correct Answer

b.

0.05

c.

1

d.

0.9

Step-by-Step AI Explanation

Verified Method

The sum of the probability of an event $P(E)$ and the probability of 'not E' (complementary event) is always 1. So, $P(\text{not } E) = 1 - P(E) = 1 - 0.05 = 0.95$.

Q9 short answer 1 mark

How many solutions does the pair of linear equations $3x + 5y + 8 = 0$ and $6x + 10y + 16 = 0$ have?

Step-by-Step AI Explanation

Verified Method

The given equations are $3x + 5y + 8 = 0$ and $6x + 10y + 16 = 0$. Comparing these with $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, we have $a_1=3, b_1=5, c_1=8$ and $a_2=6, b_2=10, c_2=16$. Calculating the ratios: $\frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{5}{10} = \frac{1}{2}$, and $\frac{c_1}{c_2} = \frac{8}{16} = \frac{1}{2}$. Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident. Therefore, the pair of linear equations has infinitely many solutions.

Q10 short answer 1 mark

Find the zeroes of the polynomial $p(x) = x^{2} - 5$.

Step-by-Step AI Explanation

Verified Method

To find the zeroes, set $p(x) = 0$. So, $x^2 - 5 = 0$. This implies $x^2 = 5$. Therefore, $x = \pm \sqrt{5}$. The zeroes are $\sqrt{5}$ and $-\sqrt{5}$.

Q11 short answer 1 mark

Write the quadratic equation $ax^{2} + c = -bx$ in the standard form.

Correct Answer & Step-by-Step Solution Locked