1
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
$$\left[ {\matrix{ 1 & 0 \cr { - 2023i} & 1 \cr } } \right]$$
2
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
$$\mathrm{{}^{52}{C_4} - {}^{45}{C_4}}$$
3
JEE Main 2023 (Online) 25th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N$$, and $$f(4) = 133,f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) - f(2))$$ is

A
60
B
58
C
61
D
59
4
JEE Main 2023 (Online) 25th January Evening Shift
Numerical
+4
-1
Change Language

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $$\frac{k}{10}%$$. Then the value of k is __________.

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