1
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1
Change Language

The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________.

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2
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1
Change Language

Let p and p + 2 be prime numbers and let

$$ \Delta=\left|\begin{array}{ccc} \mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\ (\mathrm{p}+1) ! & (\mathrm{p}+2) ! & (\mathrm{p}+3) ! \\ (\mathrm{p}+2) ! & (\mathrm{p}+3) ! & (\mathrm{p}+4) ! \end{array}\right| $$

Then the sum of the maximum values of $$\alpha$$ and $$\beta$$, such that $$\mathrm{p}^{\alpha}$$ and $$(\mathrm{p}+2)^{\beta}$$ divide $$\Delta$$, is __________.

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3
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

If $$\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{100 \times 101 \times 102}=\frac{\mathrm{k}}{101}$$, then 34 k is equal to _________.

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4
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1
Change Language

Let $$S=\{4,6,9\}$$ and $$T=\{9,10,11, \ldots, 1000\}$$. If $$A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$$ $$\epsilon S\}$$, then the sum of all the elements in the set $$T-A$$ is equal to __________.

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