1
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let $$\mathrm{P}_{1}$$ be the plane $$3 x-y-7 z=11$$ and $$\mathrm{P}_{2}$$ be the plane passing through the points $$(2,-1,0),(2,0,-1)$$, and $$(5,1,1)$$. If the foot of the perpendicular drawn from the point $$(7,4,-1)$$ on the line of intersection of the planes $$P_{1}$$ and $$P_{2}$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha+\beta+\gamma$$ is equal to ___________.

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2
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1
Change Language

Let $$\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$$ and $$\mathrm{S}=\{1,2,3,4\}$$. Total number of onto functions $$f: \mathrm{R} \rightarrow \mathrm{S}$$ such that $$f(\mathrm{a}) \neq 1$$, is equal to ______________.

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3
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1
Change Language

Let $$\mathrm{k}$$ and $$\mathrm{m}$$ be positive real numbers such that the function $$f(x)=\left\{\begin{array}{cc}3 x^{2}+k \sqrt{x+1}, & 0 < x < 1 \\ m x^{2}+k^{2}, & x \geq 1\end{array}\right.$$ is differentiable for all $$x > 0$$. Then $$\frac{8 f^{\prime}(8)}{f^{\prime}\left(\frac{1}{8}\right)}$$ is equal to ____________.

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4
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R}(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$\mathrm{P}$$ and $$\mathrm{Q}$$, then $$\frac{\beta^{2}}{\alpha}$$ is equal to _______________.

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