The functions $f$ and $g$ are twice differentiable 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?

1

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)

2

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)

3

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)

4

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)

5

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)

6

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)

7

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)

8

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)

9

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)

10

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)