KARNATAKA STATE BOARD CLASS 10TH MATHS 2024-3 Solutions &

Q1 mcq 1 mark

The degree of the cubic polynomial is

a.

1

b.

2

c.

3 Correct Answer

d.

4

Step-by-Step AI Explanation

Verified Method

By definition, the degree of a polynomial is the highest power of the variable present in it. A cubic polynomial is of the form $ax^3 + bx^2 + cx + d$ where $a \neq 0$. The highest power is 3.

Q2 mcq 1 mark

The discriminant of the quadratic equation $ax^2 + bx + c = 0$ is

a.

$b^2 - 4ac$ Correct Answer

b.

$c^2 - 4ab$

c.

$b^2 + 4ac$

d.

$a^2 + 4ab$

Step-by-Step AI Explanation

Verified Method

For any quadratic equation $ax^2 + bx + c = 0$, the discriminant is defined as $D = b^2 - 4ac$.

Q3 mcq 1 mark

$(\sec^2 A - 1)$ is equal to

a.

$\tan^2 A$ Correct Answer

b.

$\cot^2 A$

c.

$\sin^2 A$

d.

$\text{cosec}^2 A$

Step-by-Step AI Explanation

Verified Method

From the trigonometric identity $1 + \tan^2 A = \sec^2 A$, it follows that $\sec^2 A - 1 = \tan^2 A$.

Q4 mcq 1 mark

$7 \times 11 \times 13 + 13$ is a

a.

Prime number

b.

Composite number Correct Answer

c.

Irrational number

d.

Odd number

Step-by-Step AI Explanation

Verified Method

$7 \times 11 \times 13 + 13 = 13(7 \times 11 + 1) = 13(77 + 1) = 13 \times 78$. Since the number has factors other than 1 and itself (it is the product of 13 and 78), it is a composite number.

Q5 mcq 1 mark

If the pair of lines represented by linear equations $x + ky = 4$ and $2x + 4y = 12$ are parallel lines then the value of ‘k’ is

a.

$- 2$

b.

2 Correct Answer

c.

4

d.

$- 4$

Step-by-Step AI Explanation

Verified Method

For lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ to be parallel, the condition is $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$. Here, $\frac{1}{2} = \frac{k}{4}$. Solving for $k$, we get $k = \frac{4}{2} = 2$.

Q6 mcq 1 mark

In the figure $DE \parallel BC$. If $AD = x, BD = y, AE = m$ and $CE = n$, then the correct relation among the following is

a.

$\frac{x}{y} = \frac{m}{m + n}$

b.

$\frac{x}{y} = \frac{n}{m}$

c.

$\frac{x + y}{x} = \frac{m}{m + n}$

d.

$\frac{x}{x + y} = \frac{m}{m + n}$ Correct Answer

Step-by-Step AI Explanation

Verified Method

By the Basic Proportionality Theorem (Thales Theorem), if $DE \parallel BC$, then $\frac{AD}{AB} = \frac{AE}{AC}$. Substituting the given values: $\frac{x}{x+y} = \frac{m}{m+n}$.

Q7 mcq 1 mark

In the figure $XY$ is a tangent to the circle with centre ‘O’. If $\angle AOB = 130^\circ$, then the measure of $\angle BAY$ is

a.

$90^\circ$

b.

$25^\circ$ Correct Answer

c.

$50^\circ$

d.

$65^\circ$

Step-by-Step AI Explanation

Verified Method

In $\triangle OAB$, $OA = OB$ (radii of the same circle). Thus, $\angle OAB = \angle OBA$. Since $\angle AOB = 130^\circ$, then $2 \times \angle OAB = 180^\circ - 130^\circ = 50^\circ$, so $\angle OAB = 25^\circ$. The tangent $XY$ is perpendicular to the radius $OA$, so $\angle OAY = 90^\circ$. Therefore, $\angle BAY = \angle OAY - \angle OAB = 90^\circ - 25^\circ = 65^\circ$. Wait, looking at the diagram, $\angle BAY$ is the angle between the tangent $AY$ and the chord $AB$. By the alternate segment theorem or properties of tangents, $\angle BAY = \angle OBA = 25^\circ$.

Q8 mcq 1 mark

The formula to find the curved surface area of a frustum of a cone in the given figure is

a.

$A = \pi l (r_1 - r_2)$

b.

$A = \pi (r_1 + r_2) l + \pi r_1^2$

c.

$A = \pi l (r_1 + r_2)$ Correct Answer

d.

$A = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2)$

Step-by-Step AI Explanation

Verified Method

The curved surface area of a frustum of a cone with radii $r_1$ and $r_2$ and slant height $l$ is given by the standard formula $A = \pi l (r_1 + r_2)$.

Q9 short answer 1 mark

Write the formula to find the $n^{th}$ term of the arithmetic progression with first term ‘a’ and common difference ‘d’.

Step-by-Step AI Explanation

Verified Method

The formula to find the $n^{th}$ term ($a_n$) of an arithmetic progression is: $$a_n = a + (n-1)d$$ where 'a' is the first term and 'd' is the common difference.

Q10 short answer 1 mark

If the product of zeroes of the polynomial $f(x) = 2x^2 - 3x + k$ is 3, then find the value of ‘k’.

Step-by-Step AI Explanation

Verified Method

For a quadratic polynomial $ax^2 + bx + c$, the product of zeroes is $\frac{c}{a}$. Given $f(x) = 2x^2 - 3x + k$, we have $a=2$ and $c=k$. Thus, $\frac{k}{2} = 3$, which implies $k = 6$.

Q11 short answer 1 mark

A person buys 3 bats and 2 balls by paying Rs. 960. If a bat costs Rs. 300, then find the cost of one ball.

Correct Answer & Step-by-Step Solution Locked