KARNATAKA STATE BOARD CLASS 10TH MATHS 2023-2 Solutions &

Q1 mcq 1 mark

The number of zeroes of the polynomial $y = p(x)$ in the given graph is

a.

3

b.

2 Correct Answer

c.

1

d.

4

Step-by-Step AI Explanation

Verified Method

The number of zeroes of a polynomial $p(x)$ is given by the number of points where the graph intersects the x-axis. Observing the provided graph, the curve crosses the x-axis at exactly 2 distinct points. Therefore, the number of zeroes is 2.

Q2 mcq 1 mark

For an event 'E', if $P(E) = 0\cdot75$, then $P(\bar{E})$ is

a.

2\cdot5

b.

0\cdot25 Correct Answer

c.

0\cdot025

d.

1\cdot25

Step-by-Step AI Explanation

Verified Method

The sum of the probability of an event $E$ and its complement $\bar{E}$ is 1, i.e., $P(E) + P(\bar{E}) = 1$. Given $P(E) = 0.75$, then $P(\bar{E}) = 1 - 0.75 = 0.25$.

Q3 mcq 1 mark

The total surface area of a right circular cylinder having radius 'r' and height 'h' is

a.

$\pi r(r + h)$

b.

$2\pi rh$

c.

$2\pi r(r - h)$

d.

$2\pi r(r + h)$ Correct Answer

Step-by-Step AI Explanation

Verified Method

The total surface area of a right circular cylinder is the sum of the curved surface area ($2\pi rh$) and the areas of the two circular bases ($2 \times \pi r^2$). Factoring this expression gives $2\pi r^2 + 2\pi rh = 2\pi r(r + h)$.

Q4 mcq 1 mark

The number that represents the remainder when $19 = 6 \times 3 + 1$ is compared with Euclid's division lemma $a = bq + r$ is

a.

3

b.

6

c.

1 Correct Answer

d.

19

Step-by-Step AI Explanation

Verified Method

Euclid's division lemma states $a = bq + r$, where $0 \le r < b$. Comparing $19 = 6 \times 3 + 1$ with $a = bq + r$, we identify $a=19$, $b=6$, $q=3$, and $r=1$.

Q5 mcq 1 mark

In the given figure, $PB$ is a tangent drawn at the point $A$ to the circle with centre 'O'. If $\angle AOP = 45^\circ$, then the measure of $\angle OPA$ is

a.

45^\circ Correct Answer

b.

90^\circ

c.

35^\circ

d.

65^\circ

Step-by-Step AI Explanation

Verified Method

A tangent at any point of a circle is perpendicular to the radius at the point of contact. Thus, $\angle OAP = 90^\circ$. In $\triangle OAP$, the sum of angles is $180^\circ$. So, $\angle OPA = 180^\circ - (90^\circ + 45^\circ) = 180^\circ - 135^\circ = 45^\circ$.

Q6 mcq 1 mark

In the figure, if $DE \parallel BC$, then the correct relation among the following is

a.

$\frac{EC}{AE} = \frac{AB}{AD}$

b.

$\frac{AE}{EC} = \frac{DB}{AD}$

c.

$\frac{EC}{AE} = \frac{DB}{AD}$ Correct Answer

d.

$\frac{EC}{AE} = \frac{AD}{DB}$

Step-by-Step AI Explanation

Verified Method

By Thales' Theorem (Basic Proportionality Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides the two sides in the same ratio. Thus, $\frac{AD}{DB} = \frac{AE}{EC}$, which can be rearranged to $\frac{EC}{AE} = \frac{DB}{AD}$.

Q7 mcq 1 mark

The lines represented by the equations $4x + 5y - 10 = 0$ and $8x + 10y + 20 = 0$ are

a.

intersecting lines

b.

perpendicular lines to each other

c.

coincident lines

d.

parallel lines Correct Answer

Step-by-Step AI Explanation

Verified Method

For equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, the ratio $\frac{a_1}{a_2} = \frac{4}{8} = \frac{1}{2}$ and $\frac{b_1}{b_2} = \frac{5}{10} = \frac{1}{2}$, but $\frac{c_1}{c_2} = \frac{-10}{20} = -\frac{1}{2}$. Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel.

Q8 mcq 1 mark

The distance of the point $(-8, 3)$ from the x-axis is

a.

-8 units

b.

3 units Correct Answer

c.

-3 units

d.

8 units

Step-by-Step AI Explanation

Verified Method

The distance of a point $(x, y)$ from the x-axis is given by the absolute value of its y-coordinate, i.e., $|y|$. For the point $(-8, 3)$, the distance is $|3| = 3$ units.

Q9 short answer 1 mark

Express the denominator of $\frac{7}{80}$ in the form of $2^n \times 5^m$.

Step-by-Step AI Explanation

Verified Method

The denominator is 80. Prime factorization of 80 is: $80 = 8 \times 10 = 2^3 \times (2 \times 5) = 2^4 \times 5^1$. Thus, it is expressed as $2^4 \times 5^1$.

Q10 short answer 1 mark

If the pair of lines represented by the linear equations $x + 2y - 4 = 0$ and $ax + by - 12 = 0$ are coincident lines, then find the values of 'a' and 'b'.

Step-by-Step AI Explanation

Verified Method

For coincident lines, the condition is $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. Here, $\frac{1}{a} = \frac{2}{b} = \frac{-4}{-12}$. This simplifies to $\frac{1}{a} = \frac{2}{b} = \frac{1}{3}$. Therefore, $\frac{1}{a} = \frac{1}{3} \implies a = 3$, and $\frac{2}{b} = \frac{1}{3} \implies b = 6$.

Q11 short answer 1 mark

$\Delta ABC \sim \Delta PQR$. Area of the $\Delta ABC$ is $64\text{ cm}^2$ and the area of the $\Delta PQR$ is $100\text{ cm}^2$. If $AB = 8\text{ cm}$, then find the length of $PQ$.

Correct Answer & Step-by-Step Solution Locked