NTA JEE Main 16th March 2021 Shift 1

Let $$S_k = \sum_{r=1}^{k} \tan^{-1}\left(\frac{6^r}{2^{2r+1} + 3^{2r+1}}\right)$$, then $$\lim_{k \to \infty} S_k$$ is equal to:

The number of elements in the set $$\{x \in R : (|x| - 3)|x + 4| = 6\}$$ is equal to:

Let the functions $$f : R \to R$$ and $$g : R \to R$$ be defined as:
$$f(x) = \begin{cases} x+2, & x < 0 \\ x^2, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x^3, & x < 1 \\ 3x-2, & x \geq 1 \end{cases}$$
Then, the number of points in $$R$$ where $$(f \circ g)(x)$$ is NOT differentiable is equal to:

The range of $$a \in R$$ for which the function $$f(x) = (4a-3)(x + \log_e 5) + 2(a-7)\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right)$$, $$x \neq 2n\pi$$, $$n \in N$$, has critical points, is:

If $$y = y(x)$$ is the solution of the differential equation, $$\frac{dy}{dx} + 2y\tan x = \sin x$$, $$y\left(\frac{\pi}{3}\right) = 0$$, then the maximum value of the function $$y(x)$$ over $$R$$ is equal to:

Let a vector $$\alpha\hat{i} + \beta\hat{j}$$ be obtained by rotating the vector $$\sqrt{3}\hat{i} + \hat{j}$$ by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $$(\alpha, \beta)$$, $$(0, \beta)$$ and $$(0, 0)$$ is equal to:

If for $$a > 0$$, the feet of perpendiculars from the points $$A(a, -2a, 3)$$ and $$B(0, 4, 5)$$ on the plane $$lx + my + nz = 0$$ are points $$C(0, -a, -1)$$ and $$D$$ respectively, then the length of line segment $$CD$$ is equal to:

Let the position vectors of two points $$P$$ and $$Q$$ be $$3\hat{i} - \hat{j} + 2\hat{k}$$ and $$\hat{i} + 2\hat{j} - 4\hat{k}$$, respectively. Let $$R$$ and $$S$$ be two points such that the direction ratios of lines $$PR$$ and $$QS$$ are $$(4, -1, 2)$$ and $$(-2, 1, -2)$$, respectively. Let lines $$PR$$ and $$QS$$ intersect at $$T$$. If the vector $$\vec{TA}$$ is perpendicular to both $$\vec{PR}$$ and $$\vec{QS}$$ and the length of vector $$\vec{TA}$$ is $$\sqrt{5}$$ units, then the modulus of a position vector of $$A$$ is:

Let $$P$$ be a plane $$lx + my + nz = 0$$ containing the line, $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$. If plane $$P$$ divides the line segment $$AB$$ joining points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ in ratio $$k:1$$ then the value of $$k$$ is equal to:

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is: