1
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let K be the sum of the coefficients of the odd powers of $$x$$ in the expansion of $$(1+x)^{99}$$. Let $$a$$ be the middle term in the expansion of $${\left( {2 + {1 \over {\sqrt 2 }}} \right)^{200}}$$. If $${{{}^{200}{C_{99}}K} \over a} = {{{2^l}m} \over n}$$, where m and n are odd numbers, then the ordered pair $$(l,\mathrm{n})$$ is equal to

A
(50, 101)
B
(50, 51)
C
(51, 101)
D
(51, 99)
2
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of the integral $$\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx} $$ is equal to :

A
$${\pi \over 2}{\log _e}2$$
B
$${\pi \over 4}{\log _e}2$$
C
$${1 \over 2}{\log _e}2$$
D
$$\pi {\log _e}2$$
3
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The shortest distance between the lines $${{x - 1} \over 2} = {{y + 8} \over -7} = {{z - 4} \over 5}$$ and $${{x - 1} \over 2} = {{y - 2} \over 1} = {{z - 6} \over { - 3}}$$ is :

A
$$2\sqrt3$$
B
$$3\sqrt3$$
C
$$4\sqrt3$$
D
$$5\sqrt3$$
4
JEE Main 2023 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f$$ and $$g$$ be the twice differentiable functions on $$\mathbb{R}$$ such that

$$f''(x)=g''(x)+6x$$

$$f'(1)=4g'(1)-3=9$$

$$f(2)=3g(2)=12$$.

Then which of the following is NOT true?

A
$$g(-2)-f(-2)=20$$
B
There exists $$x_0\in(1,3/2)$$ such that $$f(x_0)=g(x_0)$$
C
$$|f'(x)-g'(x)| < 6\Rightarrow -1 < x < 1$$
D
If $$-1 < x < 2$$, then $$|f(x)-g(x)| < 8$$
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