1
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

If $$1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})$$ is equal to $$2^{\mathrm{n}} \cdot \mathrm{m}$$, where $$\mathrm{m}$$ is odd, then $$\mathrm{n}+\mathrm{m}$$ is equal to __________.

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2
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Two tangent lines $$l_{1}$$ and $$l_{2}$$ are drawn from the point $$(2,0)$$ to the parabola $$2 \mathrm{y}^{2}=-x$$. If the lines $$l_{1}$$ and $$l_{2}$$ are also tangent to the circle $$(x-5)^{2}+y^{2}=r$$, then 17r is equal to ___________.

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3
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language
$${6 \over {{3^{12}}}} + {{10} \over {{3^{11}}}} + {{20} \over {{3^{10}}}} + {{40} \over {{3^9}}} + \,\,...\,\, + \,\,{{10240} \over 3} = {2^n}\,.\,m$$, where m is odd, then m . n is equal to ____________.
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4
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let $$S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$. Then the number of elements in the set $$\mid A=\{\theta \in S: \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (2 \theta)\}$$ is __________.

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