Karnataka SSLC 10th Maths Question Paper 2021

The nth term is given by $a_n = 4n + 5$. To find the 5th term, substitute $n = 5$: $a_5 = 4(5) + 5 = 20 + 5 = 25$. Thus, option (c) is correct.

Given the equation $5x^2 = 2(2x + 3)$, expand the right side: $5x^2 = 4x + 6$. Moving all terms to the left gives the standard form: $5x^2 - 4x - 6 = 0$. The constant term is $-6$. Thus, option (d) is correct.

For equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, we compare the ratios: $\frac{a_1}{a_2} = \frac{1}{3}$ and $\frac{b_1}{b_2} = \frac{-2}{4} = -\frac{1}{2}$. Since $\frac{1}{3} \neq -\frac{1}{2}$, the lines are intersecting. Thus, option (a) is correct.

From the graph, Line 1 passes through $(1, 0)$ and $(0, -2)$. Using the intercept form $\frac{x}{a} + \frac{y}{b} = 1$, we get $\frac{x}{1} + \frac{y}{-2} = 1$, which simplifies to $2x - y = 2$. Line 2 passes through $(1, 0)$ and $(2, 4)$. Its slope is $\frac{4-0}{2-1} = 4$. Using $y - y_1 = m(x - x_1)$, $y - 0 = 4(x - 1) \implies y = 4x - 4 \implies 4x - y = 4$. Thus, option (c) is correct.

A pair of linear equations represents parallel lines if and only if the ratios of the coefficients of $x$ and $y$ are equal, but the ratio of the constants is different, i.e., $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$. Thus, option (a) is correct.