1
JEE Main 2023 (Online) 31st January Morning Shift
Numerical
+4
-1
Out of Syllabus

Let $$\theta$$ be the angle between the planes $$P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$$ and $$P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$$. Let $$\mathrm{L}$$ be the line that meets $$P_{2}$$ at the point $$(4,-2,5)$$ and makes an angle $$\theta$$ with the normal of $$P_{2}$$. If $$\alpha$$ is the angle between $$\mathrm{L}$$ and $$P_{2}$$, then $$\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$$ is equal to ____________.

2
JEE Main 2023 (Online) 30th January Evening Shift
Numerical
+4
-1
Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3 \alpha^2$ is equal to :
3
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Out of Syllabus

If the equation of the plane passing through the point $$(1,1,2)$$ and perpendicular to the line $$x-3 y+ 2 z-1=0=4 x-y+z$$ is $$\mathrm{A} x+\mathrm{B} y+\mathrm{C} z=1$$, then $$140(\mathrm{C}-\mathrm{B}+\mathrm{A})$$ is equal to ___________.

4
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Out of Syllabus

If $$\lambda_{1} < \lambda_{2}$$ are two values of $$\lambda$$ such that the angle between the planes $$P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7$$ and $$P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9$$ is $$\sin ^{-1}\left(\frac{2 \sqrt{6}}{5}\right)$$, then the square of the length of perpendicular from the point $$\left(38 \lambda_{1}, 10 \lambda_{2}, 2\right)$$ to the plane $$P_{1}$$ is ______________.

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