NTA JEE Main 29th January 2023 Shift 1

The domain of $$f(x) = \frac{\log_{(x+1)}(x-2)}{e^{2\log_e x^2 - (2x+3)}}$$, $$x \in R$$ is

Let $$f : R \to R$$ be a function such that $$f(x) = \frac{x^2+2x+1}{x^2+1}$$. Then

Let $$f(x) = x + \frac{a}{\pi^2-4}\sin x + \frac{b}{\pi^2-4}\cos x$$, $$x \in \mathbb{R}$$ be a function which satisfies $$f(x) = x + \int_0^{\pi/2} \sin(x+y)f(y)dy$$. Then $$(a+b)$$ is equal to

Let $$[x]$$ denote the greatest integer $$\leq x$$. Consider the function $$f(x) = \max\{x^2, 1 + [x]\}$$. Then the value of the integral $$\int_0^2 f(x) dx$$ is:

Let $$A = \left\{(x,y) \in \mathbb{R}^2 : y \geq 0, 2x \leq y \leq \sqrt{4-(x-1)^2}\right\}$$ and
$$B = \left\{(x,y) \in \mathbb{R} \times \mathbb{R} : 0 \leq y \leq \min\left\{2x, \sqrt{4-(x-1)^2}\right\}\right\}$$. Then the ratio of the area of $$A$$ to the area of $$B$$ is

Let $$\Delta$$ be the area of the region $$\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 21, y^2 \leq 4x, x \geq 1\}$$. Then $$\frac{1}{2}\left(\Delta - 21\sin^{-1}\frac{2}{\sqrt{7}}\right)$$ is equal to

Let $$y = f(x)$$ be the solution of the differential equation $$y(x+1)dx - x^2 dy = 0$$, $$y(1) = e$$. Then $$\lim_{x \to 0^+} f(x)$$ is equal to

If the vectors $$\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$$, $$\vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$$ are coplanar and the projection of $$\vec{a}$$ on the vector $$\vec{b}$$ is $$\sqrt{54}$$ units, then the sum of all possible values of $$\lambda + \mu$$ is equal to

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $$\mu$$ and $$\sigma^2$$ represent mean and variance of X, respectively, then $$10(\mu^2 + \sigma^2)$$ is equal to