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 \(\left(2+\frac{1}{\sqrt{2}}\right)^{200}\). If \(\displaystyle \frac{\binom{200}{99} \, K}{a} = \frac{2^{\,\ell} \, m}{n}\), where \(m\) and \(n\) are odd numbers, then the ordered pair \((\ell,n)\) is:
Previous 10 Questions — JEE Main 2023 (29 January Evening Shift)
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The value of the integral \(\displaystyle \int_{1}^{2} \left(\frac{t^{4}+1}{t^{6}+1}\right) dt\) is:
Topic: JEE Main 2023 (29 January Evening Shift)
Let \(\vec{a} = 4\hat{i} + 3\hat{j}\) and \(\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}\).
If \(\vec{c}\) is a vector such…
Topic: JEE Main 2023 (29 January Evening Shift)
The area of the region
\[
A=\{(x,y): |\,\cos x - \sin x\,| \le y \le \sin x,\; 0 \le x \le \tfrac{\pi}{2}\}
\]
is:
Topic: JEE Main 2023 (29 January Evening Shift)
The set of all values of \(\lambda\) for which the equation
\[
\cos^{2}(2x)-2\sin^{4}x-2\cos^{2}x=\lambda
\]
has a real…
Topic: JEE Main 2023 (29 January Evening Shift)
Let \( S=\{w_1,w_2,\ldots\} \) be the sample space of a random experiment.
Let the probabilities satisfy
\[
P(w_n)=\f…
Topic: JEE Main 2023 (29 January Evening Shift)
Consider a function $f:\mathbb{N}\to\mathbb{R}$ satisfying
\[
f(1)+2f(2)+3f(3)+\cdots+xf(x)=x(x+1)f(x),\quad x\ge 2,
…
Topic: JEE Main 2023 (29 January Evening Shift)
The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :
Topic: JEE Main 2023 (29 January Evening Shift)
If $\vec a=\hat i+2\hat k$, $\vec b=\hat i+\hat j+\hat k$, $\vec c=7\hat i-3\hat j+4\hat k$,
$\ \ \vec r\times\vec b+\…
Topic: JEE Main 2023 (29 January Evening Shift)
The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a se…
Topic: JEE Main 2023 (29 January Evening Shift)
Let \(R\) be a relation defined on \(\mathbb{N}\) as \(aRb\) iff \(2a+3b\) is a multiple of \(5\), \(a,b\in\mathbb{N}\)…
Topic: JEE Main 2023 (29 January Evening Shift)
Next 10 Questions — JEE Main 2023 (29 January Evening Shift)
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The value of the integral \(\displaystyle \int_{1/2}^{2} \frac{\tan^{-1}x}{x}\,dx\) is equal to:
Topic: JEE Main 2023 (29 January Evening Shift)
The shortest distance between the lines
\[
\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}
\quad\text{and}\quad
\frac{x-1}{…
Topic: JEE Main 2023 (29 January Evening Shift)
The functions $f$ and $g$ are twice differentiable on $\mathbb{R}$ such that
$f''(x) = g''(x) + 6x$
$f'(1) = 4g'(…
Topic: JEE Main 2023 (29 January Evening Shift)
