NTA JEE Main 17th March 2021 Shift 2

If $$x, y, z$$ are in arithmetic progression with common difference $$d$$, $$x \neq 3d$$, and the determinant of the matrix $$\begin{bmatrix} 3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z \end{bmatrix}$$ is zero, then the value of $$k^2$$ is:

The number of solutions of the equation $$\sin^{-1}\left[x^2 + \frac{1}{3}\right] + \cos^{-1}\left[x^2 - \frac{2}{3}\right] = x^2$$ for $$x \in [-1, 1]$$, and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is:

Consider the function $$f : R \to R$$ defined by $$f(x) = \begin{cases} \left(2 - \sin\left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$$. Then $$f$$ is:

Let $$f : R \to R$$ be defined as $$f(x) = e^{-x}\sin x$$. If $$F : [0, 1] \to R$$ is a differentiable function such that $$F(x) = \int_0^x f(t)dt$$, then the value of $$\int_0^1 (F'(x) + f(x))e^x dx$$ lies in the interval:

If the integral $$\int_0^{10} \frac{|\sin 2\pi x|}{e^{[x]}}dx = \alpha e^{-1} + \beta e^{-\frac{1}{2}} + \gamma$$, where $$\alpha, \beta, \gamma$$ are integers and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, then the value of $$\alpha + \beta + \gamma$$ is equal to:

Let $$y = y(x)$$ be the solution of the differential equation $$\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx$$, $$0 \leq x \leq \frac{\pi}{2}$$, $$y(0) = 0$$. Then, $$y\left(\frac{\pi}{3}\right)$$ is equal to:

If the curve $$y = y(x)$$ is the solution of the differential equation $$2(x^2 + x^{5/4})dy - y(x + x^{1/4})dx = 2x^{9/4}dx$$, $$x > 0$$ which passes through the point $$\left(1, 1 - \frac{4}{3}\log_e 2\right)$$, then the value of $$y(16)$$ is equal to:

Let $$O$$ be the origin. Let $$\vec{OP} = x\hat{i} + y\hat{j} - \hat{k}$$ and $$\vec{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$$, $$x, y \in R$$, $$x > 0$$, be such that $$|\vec{PQ}| = \sqrt{20}$$ and the vector $$\vec{OP}$$ is perpendicular to $$\vec{OQ}$$. If $$\vec{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$$, $$z \in R$$, is coplanar with $$\vec{OP}$$ and $$\vec{OQ}$$, then the value of $$x^2 + y^2 + z^2$$ is equal to:

If the equation of plane passing through the mirror image of a point $$(2, 3, 1)$$ with respect to line $$\frac{x+1}{2} = \frac{y-3}{1} = \frac{z+2}{-1}$$ and containing the line $$\frac{x-2}{3} = \frac{1-y}{2} = \frac{z+1}{1}$$ is $$\alpha x + \beta y + \gamma z = 24$$ then $$\alpha + \beta + \gamma$$ is equal to:

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $$\frac{1}{2}$$ and probability of occurrence of 0 at the odd place be $$\frac{1}{3}$$. Then the probability that 10 is followed by 01 is equal to: