NTA JEE Main 26th July 2022 Shift 1

If the function $$f(x) = \begin{cases} \dfrac{\log_e(1-x+x^2) + \log_e(1+x+x^2)}{\sec x - \cos x}, & x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) - \{0\} \\ k, & x = 0 \end{cases}$$ is continuous at $$x = 0$$, then $$k$$ is equal to:

If $$f(x) = \begin{cases} x + a, & x \le 0 \\ |x - 4|, & x > 0 \end{cases}$$ and $$g(x) = \begin{cases} x + 1, & x < 0 \\ (x-4)^2 + b, & x \ge 0 \end{cases}$$ are continuous on $$\mathbb{R}$$, then $$(gof)(2) + (fog)(-2)$$ is equal to:

Let $$f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \le 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$$. Then the set of all values of $$b$$, for which $$f(x)$$ has maximum value at $$x = 1$$, is:

If $$a = \displaystyle\lim_{n \to \infty} \sum_{k=1}^{n} \dfrac{2n}{n^2 + k^2}$$ and $$f(x) = \sqrt{\dfrac{1-\cos x}{1+\cos x}}$$, $$x \in (0, 1)$$, then:

The odd natural number $$a$$, such that the area of the region bounded by $$y = 1$$, $$y = 3$$, $$x = 0$$, $$x = y^a$$ is $$\dfrac{364}{3}$$, is equal to:

If $$\dfrac{dy}{dx} + 2y \tan x = \sin x$$, $$0 < x < \dfrac{\pi}{2}$$ and $$y\left(\dfrac{\pi}{3}\right) = 0$$, then the maximum value of $$y(x)$$ is

Let $$\vec{a} = \alpha\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}$$, $$\alpha > 0$$. If the projection of $$\vec{a} \times \vec{b}$$ on the vector $$-\hat{i} + 2\hat{j} - 2\hat{k}$$ is $$30$$, then $$\alpha$$ is equal to

The length of the perpendicular from the point $$(1, -2, 5)$$ on the line passing through $$(1, 2, 4)$$ and parallel to the line $$x + y - z = 0 = x - 2y + 3z - 5$$ is:

The mean and variance of a binomial distribution are $$\alpha$$ and $$\dfrac{\alpha}{3}$$ respectively. If $$P(X = 1) = \dfrac{4}{243}$$, then $$P(X = 4 \text{ or } 5)$$ is equal to:

Let $$E_1, E_2, E_3$$ be three mutually exclusive events such that $$P(E_1) = \dfrac{2+3p}{6}$$, $$P(E_2) = \dfrac{2-p}{8}$$ and $$P(E_3) = \dfrac{1-p}{2}$$. If the maximum and minimum values of $$p$$ are $$p_1$$ and $$p_2$$ then $$(p_1 + p_2)$$ is equal to: