Consider a function $f:\mathbb{N}\to\mathbb{R}$ satisfying \[ f(1)+2f(2)+3f(3)+\cdots+xf(x)=x(x+1)f(x),\quad x\ge 2, \] with $f(1)=1$. Then \[ \frac{1}{f(2022)}+\frac{1}{f(2028)} \] is equal to:

1

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)

2

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)

3

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)

4

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)

5

Let $y=y(x)$ be the solution of the differential equation $$x\log_e x \,\frac{dy}{dx}+y=x^2\log_e x,\quad (x>1).$$ If…

Topic: JEE Main 2023 (29 January Evening Shift)

1

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)

2

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)

3

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)

4

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)

5

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)

6

Let \(K\) be the sum of the coefficients of the odd powers of \(x\) in the expansion of \((1+x)^{99}\). Let \(a\) be th…

Topic: JEE Main 2023 (29 January Evening Shift)

7

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)

8

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)

9

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)