KARNATAKA STATE BOARD CLASS 10TH MATHS 2023-1 Solutions &

Q1 mcq 1 mark

The H.C.F. of any two prime numbers is

a.

0

b.

2

c.

1 Correct Answer

d.

– 1

Step-by-Step AI Explanation

Verified Method

By definition, a prime number has exactly two factors: 1 and itself. Since two distinct prime numbers share no common factors other than 1, their Highest Common Factor (H.C.F.) must be 1. Options A, B, and D are incorrect because they do not satisfy the definition of prime factorization.

Q2 mcq 1 mark

The degree of the polynomial $P(x) = 3x^3 - 8x^2 + 6x - 3$ is

a.

3 Correct Answer

b.

2

c.

1

d.

0

Step-by-Step AI Explanation

Verified Method

The degree of a polynomial in one variable is defined as the highest power of the variable present in the expression. In the polynomial $P(x) = 3x^3 - 8x^2 + 6x - 3$, the powers of $x$ are 3, 2, 1, and 0. The highest power is 3; therefore, the degree is 3.

Q3 mcq 1 mark

The coordinates of the midpoint of the line segment joining the points $(3, 4)$ and $(5, 6)$ is

a.

$( - 4, - 5 )$

b.

$( 4, 5 )$ Correct Answer

c.

$( 4, - 5 )$

d.

$( - 4, 5 )$

Step-by-Step AI Explanation

Verified Method

The midpoint formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ is $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$. Substituting the given coordinates $(3, 4)$ and $(5, 6)$, we get $(\frac{3+5}{2}, \frac{4+6}{2}) = (\frac{8}{2}, \frac{10}{2}) = (4, 5)$.

Q4 mcq 1 mark

The probability of winning a game is $\frac{3}{4}$. The probability of losing the same game is

a.

$\frac{1}{2}$

b.

$\frac{3}{4}$

c.

$-\frac{1}{4}$

d.

$\frac{1}{4}$ Correct Answer

Step-by-Step AI Explanation

Verified Method

The sum of the probability of an event occurring, $P(E)$, and the probability of it not occurring, $P(\text{not } E)$, is always 1. Thus, $P(\text{losing}) = 1 - P(\text{winning}) = 1 - \frac{3}{4} = \frac{1}{4}$.

Q5 mcq 1 mark

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

a.

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

b.

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

c.

$\frac{AD}{AE} = \frac{CE}{BD}$

d.

$\frac{DE}{BC} = \frac{AE}{AD}$

Step-by-Step AI Explanation

Verified Method

According to Thales' Theorem (Basic Proportionality Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. For $DE \parallel BC$, we have $\frac{AD}{BD} = \frac{AE}{EC}$.

Q6 mcq 1 mark

The distance between two parallel tangents in a circle of radius 3 cm is

a.

3 cm

b.

1.5 cm

c.

9 cm

d.

6 cm Correct Answer

Step-by-Step AI Explanation

Verified Method

The tangents at the ends of a diameter of a circle are parallel to each other. The distance between these two parallel tangents is equal to the diameter of the circle. Since the radius $r = 3$ cm, the diameter $d = 2r = 2 \times 3 = 6$ cm.

Q7 mcq 1 mark

The formula to find the volume of a solid cylinder having base radius 'r' and height 'h' is

a.

$V = 4 \pi r^2$

b.

$V = \pi r^2 h$ Correct Answer

c.

$V = \pi r l$

d.

$V = \frac{1}{3} \pi r^2 h$

Step-by-Step AI Explanation

Verified Method

The volume of a cylinder is calculated by multiplying the area of the circular base by the height. The area of the base is $\pi r^2$. Therefore, the volume $V = \pi r^2 h$. Option A is surface area related, C is for a cone's slant height, and D is for a cone's volume.

Q8 mcq 1 mark

If the $n^{th}$ term of an arithmetic progression is $a_n = 2n + 1$ then its $(n - 1)^{th}$ term is

a.

$(2n - 2)$

b.

$(2n + 3)$

c.

$(2n - 1)$ Correct Answer

d.

$2n$

Step-by-Step AI Explanation

Verified Method

To find the $(n-1)^{th}$ term, we substitute $(n-1)$ in place of $n$ in the formula $a_n = 2n + 1$. Thus, $a_{n-1} = 2(n-1) + 1 = 2n - 2 + 1 = 2n - 1$.

Q9 short answer 1 mark

According to Euclid’s division lemma, if $13 = 4 \times 3 + r$, then find the value of 'r'.

Step-by-Step AI Explanation

Verified Method

According to Euclid's division lemma, $a = bq + r$ where $0 \leq r < b$. Given $13 = 4 \times 3 + r$, we have $13 = 12 + r$, which implies $r = 13 - 12 = 1$.

Q10 short answer 1 mark

How many solutions do the pair of linear equations $x + 2y - 4 = 0$ and $3x + 2y - 5 = 0$ have ?

Step-by-Step AI Explanation

Verified Method

Comparing the equations with $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, we have $a_1=1, b_1=2$ and $a_2=3, b_2=2$. Since $\frac{a_1}{a_2} = \frac{1}{3}$ and $\frac{b_1}{b_2} = \frac{2}{2} = 1$. Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines intersect at a single point, therefore the pair of linear equations has a unique solution (exactly one solution).

Q11 short answer 1 mark

If $x, 7, 10 ...$ are in arithmetic progression then write the value of $x$.

Correct Answer & Step-by-Step Solution Locked