NTA JEE Main 29th June 2022 Shift 1

Let $$A = [a_{ij}]$$ be a square matrix of order 3 such that $$a_{ij} = 2^{j-i}$$, for all $$i, j = 1, 2, 3$$. Then, the matrix $$A^2 + A^3 + \ldots + A^{10}$$ is equal to

If the system of linear equations
$$2x + y - z = 7$$
$$x - 3y + 2z = 1$$
$$x + 4y + \delta z = k$$, where $$\delta, k \in R$$
has infinitely many solutions, then $$\delta + k$$ is equal to

The domain of the function $$\cos^{-1}\left(\frac{2\sin^{-1}\left(\frac{1}{4x^2-1}\right)}{\pi}\right)$$ is

Let $$f : R \to R$$ be a function defined by:
$$f(x) = \begin{cases} \max\{t^3 - 3t\}; & t \leq x, \quad x \leq 2 \\ x^2 + 2x - 6; & 2 < x < 3 \\ [x-3] + 9; & 3 \leq x \leq 5 \\ 2x + 1; & x > 5 \end{cases}$$
Where $$[t]$$ is the greatest integer less than or equal to $$t$$. Let $$m$$ be the number of points where $$f$$ is not differentiable and $$I = \int_{-2}^{2} f(x) dx$$. Then the ordered pair $$(m, I)$$ is equal to

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is

$$\int_0^5 \cos\left(\pi\left(x - \left[\frac{x}{2}\right]\right)\right)dx$$, where $$[t]$$ denotes greatest integer less than or equal to $$t$$, is equal to

The area enclosed by $$y^2 = 8x$$ and $$y = \sqrt{2}x$$ that lies outside the triangle formed by $$y = \sqrt{2}x$$, $$x = 1$$, $$y = 2\sqrt{2}$$, is equal to

Let the solution curve of the differential equation $$x\frac{dy}{dx} - y = \sqrt{y^2 + 16x^2}$$, $$y(1) = 3$$ be $$y = y(x)$$. Then $$y(2)$$ is equal to

Let $$\vec{a} = \alpha \hat{i} + 3\hat{j} - \hat{k}$$, $$\vec{b} = 3\hat{i} - \beta \hat{j} + 4\hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 2\hat{k}$$ where $$\alpha, \beta \in R$$. If the projection of $$\vec{a}$$ on $$\vec{c}$$ is $$\frac{10}{3}$$ and $$\vec{b} \times \vec{c} = -6\hat{i} + 10\hat{j} + 7\hat{k}$$, then the value of $$\alpha + \beta$$ equal to

If the mirror image of the point $$(2, 4, 7)$$ in the plane $$3x - y + 4z = 2$$ is $$(a, b, c)$$, the $$2a + b + 2c$$ is equal to