KCET 2026 Mathematics Solved Paper

Concept:The angle between two lines depends only on their direction ratios, using the cosine formula.Explanation:For the first line, $x = k+1$, $y = 2k-1$, $z = 2k+3$, the direction ratios are the coefficients of $k$: $(1, 2, 2)$.For the second line, $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{1}$, the direction ratios are $(2, 1, 1)$.Let $\theta$ be the angle between the lines.The formula is:$$\cos \theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$$Substituting $a_1=1$, $b_1=2$, $c_1=2$ and $a_2=2$, $b_2=1$, $c_2=1$:$$\cos \theta = \frac{|1 \cdot 2 + 2 \cdot 1 + 2 \cdot 1|}{\sqrt{1^2+2^2+2^2}\sqrt{2^2+1^2+1^2}}$$$$= \frac{6}{\sqrt{9}\sqrt{6}} = \frac{6}{3\sqrt{6}} = \frac{2}{\sqrt{6}} = \sqrt{\frac{2}{3}}$$Therefore, $\theta = \cos^{-1}\left(\sqrt{\frac{2}{3}}\right)$.Answer:The correct option is B: $\cos^{-1}\left(\sqrt{\frac{2}{3}}\right)$.