NTA JEE Main 12th April 2023 Shift 1

Let $$A = \begin{bmatrix} 1 & \frac{1}{51} \\ 0 & 1 \end{bmatrix}$$. If $$B = \begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$$, then the sum of all the elements of the matrix $$\sum_{n=1}^{50} B^n$$ is equal to

Let $$D$$ be the domain of the function $$f(x) = \sin^{-1}\left(\log_{3x}\left(\frac{6 + 2\log_{3}x}{-5x}\right)\right)$$. If the range of the function $$g : D \to \mathbb{R}$$ defined by $$g(x) = x - [x]$$, ($$[x]$$ is the greatest integer function), is $$(\alpha, \beta)$$, then $$\alpha^2 + \frac{5}{\beta}$$ is equal to

If the total maximum value of the function $$f(x) = \left(\frac{\sqrt{3e}}{2\sin x}\right)^{\sin^2 x}$$, $$x \in \left(0, \frac{\pi}{2}\right)$$, is $$\frac{k}{e}$$, then $$\left(\frac{k}{e}\right)^8 + \frac{k^8}{e^5} + k^8$$ is equal to

The area of the region enclosed by the curve $$y = x^3$$ and its tangent at the point $$(-1, -1)$$ is

Let $$y = y(x)$$, $$y > 0$$, be a solution curve of the differential equation $$(1 + x^2)dy = y(x - y)dx$$. If $$y(0) = 1$$ and $$y(2\sqrt{2}) = \beta$$, then

Let a, b, c be three distinct real numbers, none equal to one. If the vectors $$a\hat{i} + \hat{j} + \hat{k}$$, $$\hat{i} + b\hat{j} + \hat{k}$$ and $$\hat{i} + \hat{j} + c\hat{k}$$ are coplanar, then $$\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}$$ is equal to

Let $$\lambda \in \mathbb{Z}$$, $$\vec{a} = \lambda \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$$. Let $$\vec{c}$$ be a vector such that $$(\vec{a} + \vec{b}) \times \vec{c} = 0$$, $$\vec{a} \cdot \vec{c} = -17$$ and $$\vec{b} \cdot \vec{c} = -20$$. Then $$|\vec{c} \times (\lambda\hat{i} + \hat{j} + \hat{k})|^2$$ is equal to

Let the lines $$L_1 : \frac{x+5}{3} = \frac{y+4}{1} = \frac{z-\alpha}{-2}$$ and $$L_2 : 3x + 2y + z - 2 = 0 = x - 3y + 2z - 13$$ be coplanar. If the point $$P(a, b, c)$$ on $$L_1$$ is nearest to the point $$Q(-4, -3, 2)$$, then $$|a| + |b| + |c|$$ is equal to

Let the plane $$P : 4x - y + z = 10$$ be rotated by an angle $$\frac{\pi}{2}$$ about its line of intersection with the plane $$x + y - z = 4$$. If $$\alpha$$ is the distance of the point $$(2, 3, -4)$$ from the new position of the plane $$P$$, then $$35\alpha$$ is equal to

Two dice $$A$$ and $$B$$ are rolled. Let the numbers obtained on $$A$$ and $$B$$ be $$\alpha$$ and $$\beta$$ respectively. If the variance of $$\alpha - \beta$$ is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then the sum of the positive divisors of $$p$$ is equal to