JEE Main 2024 — Definite Integration Question with Solution

$$L_1:\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$$

$$\begin{array}{rl}& {\vec{b}}_1=2\hat{i}+3\hat{j}+4\hat{k}\\ & {\vec{a}}_1=-2\hat{i}-3\hat{j}+5\hat{k}\end{array}$$

$$L_2=\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$$

$$\begin{array}{rl}& {\vec{a}}_2=3\hat{i}+2\hat{j}-4\hat{k}\\ & {\vec{b}}_2=1\hat{i}-3\hat{j}+2\hat{k}\end{array}$$

$$d=\left|\frac{\left({\vec{a}}_2-{\vec{a}}_1\right)\cdot \left({\vec{b}}_1\times {\vec{b}}_2\right)}{\left|{\vec{b}}_1\times {\vec{b}}_2\right|}\right|$$

$$\begin{array}{c}d=\left|\frac{(5\hat{i}+5\hat{j}-9\hat{k})\cdot (18\hat{i}-9\hat{k})}{\sqrt{324+81}}\right|\\ \left|{\vec{b}}_1\times \vec{b}\right|\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 4\\ 1& -3& 2\end{array}\right|\\ \Rightarrow \hat{i}(6+12)-\hat{j}(4-4)+\hat{k}(-6-3)\\ \Rightarrow (18\hat{i}-9\hat{k})\end{array}$$

$$\begin{array}{rl}& d=\left|\frac{90+81}{9\sqrt{5}}\right|\\ & d=\frac{171}{9\sqrt{5}}\\ & \frac{38}{3\sqrt{5}}k=\frac{171}{9\sqrt{5}}\\ & \frac{38}{3\sqrt{5}}k=\frac{57}{3\sqrt{5}}\end{array}$$

\begin{aligned} & {k=\frac{57}{38}}=\frac{3}{2} \\ & \int_\limits0^{\frac{3}{2}}\left[x^2\right] d x \\ & \int_\limits0^1 0 d x+\int_\limits0^{\sqrt{ }} 1 d x+\int_\limits{\sqrt{2}}^{\frac{3}{2}} 2 d x \\ & 0+(\sqrt{2}-1)+2\left(\frac{3}{2}-\sqrt{2}\right) \\ & \sqrt{2}-1+3-2 \sqrt{2} \\ & 2-\sqrt{2} \end{aligned}

$$\begin{array}{rl}& \alpha =2\\ & 6{\alpha }^3=6(2{)}^3=48\end{array}$$