1
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1
Change Language

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____________.

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2
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1
Change Language

If the maximum value of the term independent of $$t$$ in the expansion of $$\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}, x \geqslant 0$$, is $$\mathrm{K}$$, then $$8 \mathrm{~K}$$ is equal to ____________.

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3
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1
Change Language

Let $$a, b$$ be two non-zero real numbers. If $$p$$ and $$r$$ are the roots of the equation $$x^{2}-8 \mathrm{a} x+2 \mathrm{a}=0$$ and $$\mathrm{q}$$ and s are the roots of the equation $$x^{2}+12 \mathrm{~b} x+6 \mathrm{~b}=0$$, such that $$\frac{1}{\mathrm{p}}, \frac{1}{\mathrm{q}}, \frac{1}{\mathrm{r}}, \frac{1}{\mathrm{~s}}$$ are in A.P., then $$\mathrm{a}^{-1}-\mathrm{b}^{-1}$$ is equal to _____________.

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4
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let $$a_{1}=b_{1}=1, a_{n}=a_{n-1}+2$$ and $$b_{n}=a_{n}+b_{n-1}$$ for every

natural number $$n \geqslant 2$$. Then $$\sum\limits_{n = 1}^{15} {{a_n}.{b_n}} $$ is equal to ___________.

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