NTA JEE Main 13th April 2023 Shift 1

$$\max_{0 \leq x \leq \pi} \left\{x - 2\sin x \cos x + \frac{1}{3}\sin 3x\right\} =$$

The set of all $$a \in \mathbb{R}$$ for which the equation $$x|x-1| + |x+2| + a = 0$$ has exactly one real root, is

$$\int_0^{\infty} \frac{6}{e^{3x} + 6e^{2x} + 11e^x + 6} dx =$$

Among
$$(S1) : \lim_{n \to \infty} \frac{1}{n^2}(2 + 4 + 6 + \ldots + 2n) = 1$$
$$(S2) : \lim_{n \to \infty} \frac{1}{n^{16}}(1^{15} + 2^{15} + 3^{15} + \ldots + n^{15}) = \frac{1}{16}$$

The area of the region enclosed by the curve $$f(x) = \max\{\sin x, \cos x\}$$, $$-\pi \leq x \leq \pi$$ and the $$x-$$axis is

Let $$y = y_1(x)$$ and $$y = y_2(x)$$ be the solution curves the differential equation $$\frac{dy}{dx} = y + 7$$ with initial conditions $$y_1(0) = 0$$ and $$y_2(0) = 1$$ respectively. Then the curves $$y = y_1(x)$$ and $$y = y_2(x)$$ intersect at

Let $$\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$$, $$\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$$ and $$\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$$. If a vector $$\vec{d}$$ satisfies $$\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{d} \cdot \vec{a} = 24$$, then $$|\vec{d}|^2$$ is equal to

Let the equation of plane passing through the line of intersection of the planes $$x + 2y + az = 2$$ and $$x - y + z = 3$$ be $$5x - 11y + bz = 6a - 1$$. For $$c \in \mathbb{Z}$$, if the distance of this plane from the point $$(a, -c, c)$$ is $$\frac{2}{\sqrt{a}}$$, then $$\frac{a+b}{c}$$ is equal to

The distance of the point $$(-1, 2, 3)$$ from the plane $$\vec{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 10$$ parallel to the line of the shortest distance between the lines $$\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{k})$$ and $$\vec{r} = (2\hat{i} - \hat{j}) + \mu(\hat{i} - \hat{j} + \hat{k})$$ is

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $$X$$ denotes the number of tosses of the coin, then the mean of $$X$$ is