NTA JEE Main 27th July 2022 Shift 1

Let $$A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^2 + \beta A = 2I$$. Then $$\alpha + \beta$$ is equal to

Let $$f, g : \mathbb{N} - \{1\} \to \mathbb{N}$$ be functions defined by $$f(a) = \alpha$$, where $$\alpha$$ is the maximum of the powers of those primes $$p$$ such that $$p^\alpha$$ divides $$a$$, and $$g(a) = a + 1$$, for all $$a \in \mathbb{N} - \{1\}$$. Then, the function $$f + g$$ is

Let a function $$f : \mathbb{R} \to \mathbb{R}$$ be defined as:

$$f(x) = \begin{cases} \int_0^x (5 - |t - 3|) \, dt, & x > 4 \\ x^2 + bx, & x \leq 4 \end{cases}$$

where $$b \in \mathbb{R}$$. If $$f$$ is continuous at $$x = 4$$, then which of the following statements is NOT true?

$$I = \int_{\pi/4}^{\pi/3} \left(\frac{8 \sin x - \sin 2x}{x}\right) dx$$. Then

The area of the smaller region enclosed by the curves $$y^2 = 8x + 4$$ and $$x^2 + y^2 + 4\sqrt{3}x - 4 = 0$$ is equal to

Let $$y = y_1(x)$$ and $$y = y_2(x)$$ be two distinct solutions of the differential equation $$\frac{dy}{dx} = x + y$$, with $$y_1(0) = 0$$ and $$y_2(0) = 1$$ respectively. Then, the number of points of intersection of $$y = y_1(x)$$ and $$y = y_2(x)$$ is

Let $$\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}$$ and $$\vec{b} = 3\hat{i} - 5\hat{j} + 4\hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12\hat{k}$$. Then the projection of $$\vec{b} - 2\vec{a}$$ on $$\vec{b} + \vec{a}$$ is equal to

Let $$\vec{a} = 2\hat{i} - \hat{j} + 5\hat{k}$$ and $$\vec{b} = \alpha \hat{i} + \beta \hat{j} + 2\hat{k}$$. If $$\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right) \cdot \hat{k} = \frac{23}{2}$$, then $$\left|\vec{b} \times 2\hat{j}\right|$$ is equal to

If the plane $$P$$ passes through the intersection of two mutually perpendicular planes $$2x + ky - 5z = 1$$ and $$3kx - ky + z = 5$$, $$k < 3$$ and intercepts a unit length on positive $$x$$-axis, then the intercept made by the plane $$P$$ on the $$y$$-axis is

Let $$S$$ be the sample space of all five digit numbers. If $$p$$ is the probability that a randomly selected number from $$S$$, is a multiple of $$7$$ but not divisible by $$5$$, then $$9p$$ is equal to