1
JEE Main 2023 (Online) 25th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

A
120
B
6
C
72
D
12
2
JEE Main 2023 (Online) 25th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$y=y(t)$$ be a solution of the differential equation $${{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}$$ where, $$\alpha > 0,\beta > 0$$ and $$\gamma > 0$$. Then $$\mathop {\lim }\limits_{t \to \infty } y(t)$$

A
is 0
B
is 1
C
is $$-1$$
D
does not exist
3
JEE Main 2023 (Online) 25th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - i} \cr 0 & 1 \cr } } \right]$$, where $$i = \sqrt { - 1} $$. If $$\mathrm{M=A^T B A}$$, then the inverse of the matrix $$\mathrm{AM^{2023}A^T}$$ is

A
$$\left[ {\matrix{ 1 & { - 2023i} \cr 0 & 1 \cr } } \right]$$
B
$$\left[ {\matrix{ 1 & 0 \cr {2023i} & 1 \cr } } \right]$$
C
$$\left[ {\matrix{ 1 & {2023i} \cr 0 & 1 \cr } } \right]$$
D
$$\left[ {\matrix{ 1 & 0 \cr { - 2023i} & 1 \cr } } \right]$$
4
JEE Main 2023 (Online) 25th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

$$\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}} $$ is equal to :

A
$$\mathrm{{}^{51}{C_4} - {}^{45}{C_4}}$$
B
$$\mathrm{{}^{51}{C_3} - {}^{45}{C_3}}$$
C
$$\mathrm{{}^{52}{C_3} - {}^{45}{C_3}}$$
D
$$\mathrm{{}^{52}{C_4} - {}^{45}{C_4}}$$
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