NTA JEE Main 25th June 2022 Shift 1

Let $$f : N \to R$$ be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers $$x$$ and $$y$$. If $$f(1) = 2$$, then the value of $$\alpha$$ for which $$\sum_{k=1}^{10} f(\alpha + k) = \frac{512}{3}2^{20} - 1$$ holds, is

Let $$f : R \to R$$ be defined as $$f(x) = x^3 + x - 5$$. If $$g(x)$$ is a function such that $$f(g(x)) = x, \forall x \in R$$, then $$g'(63)$$ is equal to

Let $$f : R \to R$$ and $$g : R \to R$$ be two functions defined by $$f(x) = \log_e(x^2 + 1) - e^{-x} + 1$$ and $$g(x) = \frac{1 - 2e^{2x}}{e^x}$$. Then, for which of the following range of $$\alpha$$, the inequality $$f\left(g\left(\frac{\alpha - 1}{3}\right)\right) > f\left(g\left(\alpha - \frac{5}{3}\right)\right)$$ holds?

Let $$g : (0, \infty) \to R$$ be a differentiable function such that $$\int \frac{x\cos x - \sin x}{e^x + 1} + \frac{g(x)e^x + 1 - xe^x}{(e^x + 1)^2} dx = \frac{xg(x)}{e^x + 1} + C$$, for all $$x > 0$$, where $$C$$ is an arbitrary constant. Then

The value of $$\int_0^{\pi} \frac{e^{\cos x} \sin x}{1 + \cos^2 x \cdot e^{\cos x} + e^{-\cos x}} dx$$ is equal to

Let $$y = y(x)$$ be the solution of the differential equation $$(x + 1)y' - y = e^{3x}(x + 1)^2$$, with $$y(0) = \frac{1}{3}$$. Then, the point $$x = -\frac{4}{3}$$ for the curve $$y = y(x)$$ is

If the solution curve $$y = y(x)$$ of the differential equation $$y^2 dx + (x^2 - xy + y^2)dy = 0$$, which passes through the point $$(1, 1)$$ and intersects the line $$y = \sqrt{3}x$$ at the point $$(\alpha, \sqrt{3}\alpha)$$, then value of $$\log_e \sqrt{3}\alpha$$ is equal to

Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}, a_i > 0, i = 1, 2, 3$$ be a vector which makes equal angles with the coordinate axes $$OX, OY$$ and $$OZ$$. Also, let the projection of $$\vec{a}$$ on the vector $$3\hat{i} + 4\hat{j}$$ be $$7$$. Let $$\vec{b}$$ be a vector obtained by rotating $$\vec{a}$$ with $$90°$$. If $$\vec{a}, \vec{b}$$ and x-axis are coplanar, then projection of a vector $$\vec{b}$$ on $$3\hat{i} + 4\hat{j}$$ is equal to

Let $$Q$$ be the mirror image of the point $$P(1, 0, 1)$$ with respect to the plane $$S : x + y + z = 5$$. If a line $$L$$ passing through $$(1, -1, -1)$$, parallel to the line $$PQ$$ meets the plane $$S$$ at $$R$$, then $$QR^2$$ is equal to

Let $$E_1$$ and $$E_2$$ be two events such that the conditional probabilities $$P(E_1 | E_2) = \frac{1}{2}$$, $$P(E_2 | E_1) = \frac{3}{4}$$ and $$P(E_1 \cap E_2) = \frac{1}{8}$$. Then