1
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let $$\alpha x+\beta y+\gamma z=1$$ be the equation of a plane passing through the point $$(3,-2,5)$$ and perpendicular to the line joining the points $$(1,2,3)$$ and $$(-2,3,5)$$. Then the value of $$\alpha \beta y$$ is equal to _____________.

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2
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1
Change Language

If the term without $$x$$ in the expansion of $$\left(x^{\frac{2}{3}}+\frac{\alpha}{x^{3}}\right)^{22}$$ is 7315 , then $$|\alpha|$$ is equal to ___________.

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3
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1
Change Language

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ____________.

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4
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

The line $$x=8$$ is the directrix of the ellipse $$\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ with the corresponding focus $$(2,0)$$. If the tangent to $$\mathrm{E}$$ at the point $$\mathrm{P}$$ in the first quadrant passes through the point $$(0,4\sqrt3)$$ and intersects the $$x$$-axis at $$\mathrm{Q}$$, then $$(3\mathrm{PQ})^{2}$$ is equal to ____________.

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