Karnataka SSLC 10th Maths Question Paper 2023

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.

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.

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 $(\frac{3+5}{2}, \frac{4+6}{2}) = (\frac{8}{2}, \frac{10}{2}) = (4, 5)$.

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, $P(\text{losing}) = 1 - P(\text{winning}) = 1 - \frac{3}{4} = \frac{1}{4}$.

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 $\frac{AD}{BD} = \frac{AE}{EC}$.