Let $g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$, where $f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R$. Then which one of the following is correct?

1

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and ${{20} \over 3}$, respectively, then the va…

Topic: JEE Main 2021 (20 July Evening Shift)

2

Let y = y(x) satisfies the equation ${{dy} \over {dx}} - |A| = 0$, for all x > 0, where $A = \left[ {\matrix{ y &a…

Topic: JEE Main 2021 (20 July Evening Shift)

3

The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3…

Topic: JEE Main 2021 (20 July Evening Shift)

4

Let A, B and C be three events such that the probability that exactly one of A and B occurs is (1 $-$ k), the probabili…

Topic: JEE Main 2021 (20 July Evening Shift)

5

If $f:R \to R$ is given by $f(x) = x + 1$, then the value of $\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ …

Topic: JEE Main 2021 (20 July Evening Shift)

6

Let $f:R - \left\{ {{\alpha \over 6}} \right\} \to R$ be defined by $f(x) = {{5x + 3} \over {6x - \alpha }}$. Then the…

Topic: JEE Main 2021 (20 July Evening Shift)

7

If the real part of the complex number ${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$ is ${1 \over 5}$ for $\theta \in…

Topic: JEE Main 2021 (20 July Evening Shift)

8

If [x] denotes the greatest integer less than or equal to x, then the value of the integral $\int_{ - \pi /2}^{\pi /2} …

Topic: JEE Main 2021 (20 July Evening Shift)

9

The lines x = ay $-$ 1 = z $-$ 2 and x = 3y $-$ 2 = bz $-$ 2, (ab $\ne$ 0) are coplanar, if :

Topic: JEE Main 2021 (20 July Evening Shift)

10

The value of $\tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \rig…

Topic: JEE Main 2021 (20 July Evening Shift)

1

Let P be a variable point on the parabola $y = 4{x^2} + 1$. Then, the locus of the mid-point of the point P and the foo…

Topic: JEE Main 2021 (20 July Evening Shift)

2

The value of k $\in$R, for which the following system of linear equations. 3x $-$ y + 4z = 3,x + 2y $-$ 3z = $-$2, 6x +…

Topic: JEE Main 2021 (20 July Evening Shift)

3

In a triangle ABC, if $\left| {\overrightarrow {BC} } \right| = 3$, $\left| {\overrightarrow {CA} } \right| = 5$ and $\…

Topic: JEE Main 2021 (20 July Evening Shift)