NTA JEE Main 6th April 2023 Shift 1

If the system of equations
$$x + y + az = b$$
$$2x + 5y + 2z = 6$$
$$x + 2y + 3z = 3$$
has infinitely many solutions, then $$2a + 3b$$ is equal to

Let $$5f(x) + 4f\left(\dfrac{1}{x}\right) = \dfrac{1}{x} + 3$$, $$x > 0$$. Then $$18\int_1^2 f(x)\,dx$$ is equal to

Let $$A = \left\{x \in \mathbb{R}: |x+3| + |x+4| \le 3\right\}$$, $$B = \left\{x \in \mathbb{R}: 3^x \sum_{r=1}^{\infty} \dfrac{3^{x-3}}{10^r} < 3^{-3x}\right\}$$, where $$[t]$$ denotes greatest integer function. Then,

If $$2x^y + 3y^x = 20$$, then $$\dfrac{dy}{dx}$$ at (2, 2) is equal to:

Let $$Ix = \int \dfrac{x^x \sec^2 + \tan x}{(x \tan x + 1)^2} dx$$. If $$I(0) = 0$$, then $$I\left(\dfrac{\pi}{4}\right)$$ is equal to

Let the position vectors of the points A, B, C and D be $$5\hat{i} + 5\hat{j} + 2\lambda\hat{k}$$, $$\hat{i} + 2\hat{j} + 3\hat{k}$$, $$-2\hat{i} + \lambda\hat{j} + 4\hat{k}$$ and $$-\hat{i} + 5\hat{j} + 6\hat{k}$$. Let the set $$S = \{\lambda \in \mathbb{R}$$: the points A, B, C and D are coplanar$$\}$$. The $$\sum_{\lambda \in S} (\lambda + 2)^2$$ is equal to

Let $$\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$, $$\vec{b} = \hat{i} - 2\hat{j} - 2\hat{k}$$ and $$\vec{c} = -\hat{i} + 4\hat{j} + 3\hat{k}$$. If $$\vec{d}$$ is a vector perpendicular to both $$\vec{b}$$ and $$\vec{c}$$, and $$\vec{a} \cdot \vec{d} = 18$$, then $$|\vec{a} \times \vec{d}|^2$$ is equal to

One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is

If the equation of the plane passing through the line of intersection of the planes $$2x - y + z = 3$$, $$4x - 3y + 5z + 9 = 0$$ and parallel to the line $$\dfrac{x+1}{-2} = \dfrac{y+3}{4} = \dfrac{z-2}{5}$$ is $$ax + by + cz + 6 = 0$$, then $$a + b + c$$ is equal to

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $$\dfrac{k}{3^{11}}$$, then $$k$$ is equal to