1
JEE Main 2023 (Online) 12th April Morning Shift
Numerical
+4
-1
Out of Syllabus

Let the plane $$x+3 y-2 z+6=0$$ meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle $$\mathrm{ABC}$$ is $$\left(\alpha, \beta, \frac{6}{7}\right)$$, then $$98(\alpha+\beta)^{2}$$ is equal to ___________.

2
JEE Main 2023 (Online) 11th April Evening Shift
Numerical
+4
-1
Out of Syllabus

Let the line $$l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}$$ meet the plane $$P: x+2 y+3 z=4$$ at the point $$(\alpha, \beta, \gamma)$$. If the angle between the line $$l$$ and the plane $$P$$ is $$\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)$$, then $$\alpha+2 \beta+6 \gamma$$ is equal to ___________.

3
JEE Main 2023 (Online) 11th April Morning Shift
Numerical
+4
-1

Let a line $$l$$ pass through the origin and be perpendicular to the lines

$$l_{1}: \vec{r}=(\hat{\imath}-11 \hat{\jmath}-7 \hat{k})+\lambda(\hat{i}+2 \hat{\jmath}+3 \hat{k}), \lambda \in \mathbb{R}$$ and

$$l_{2}: \vec{r}=(-\hat{\imath}+\hat{\mathrm{k}})+\mu(2 \hat{\imath}+2 \hat{\jmath}+\hat{\mathrm{k}}), \mu \in \mathbb{R}$$.

If $$\mathrm{P}$$ is the point of intersection of $$l$$ and $$l_{1}$$, and $$\mathrm{Q}(\propto, \beta, \gamma)$$ is the foot of perpendicular from P on $$l_{2}$$, then $$9(\alpha+\beta+\gamma)$$ is equal to _____________.

4
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1
Out of Syllabus

Let the foot of perpendicular from the point $$\mathrm{A}(4,3,1)$$ on the plane $$\mathrm{P}: x-y+2 z+3=0$$ be N. If B$$(5, \alpha, \beta), \alpha, \beta \in \mathbb{Z}$$ is a point on plane P such that the area of the triangle ABN is $$3 \sqrt{2}$$, then $$\alpha^{2}+\beta^{2}+\alpha \beta$$ is equal to ___________.