Karnataka SSLC 10th Maths Question Paper 2025

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.

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 $3$.

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 $d=1$.

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.

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 $\cos A = \frac{1}{2/\sqrt{3}} = \frac{\sqrt{3}}{2}$.