NTA JEE Main 27th July 2021 Shift 1

If the mean and variance of the following data: 6, 10, 7, 13, $$a$$, 12, $$b$$, 12 are 9 and $$\frac{37}{4}$$ respectively, then $$(a - b)^2$$ is equal to:

Let $$A = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}$$. If $$A^{-1} = \alpha I + \beta A$$, $$\alpha, \beta \in R$$, $$I$$ is a $$2 \times 2$$ identity matrix, then $$4(\alpha - \beta)$$ is equal to:

Let $$f : \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow R$$ be defined as,
$$f(x) = \begin{cases} (1 + |\sin x|)^{\frac{3a}{|\sin x|}}, & -\frac{\pi}{4} < x < 0 \\ b, & x = 0 \\ e^{\cot 4x / \cot 2x}, & 0 < x < \frac{\pi}{4} \end{cases}$$
If $$f$$ is continuous at $$x = 0$$ then the value of $$6a + b^2$$ is equal to:

The value of $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} \frac{(2j-1) + 8n}{(2j-1) + 4n}$$ is equal to:

The value of the definite integral $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{(1 + e^{x\cos x})(\sin^4 x + \cos^4 x)}$$ is equal to:

If the area of the bounded region $$R = \{(x, y) : \max\{0, \log_e x\} \leq y \leq 2^x, \frac{1}{2} \leq x \leq 2\}$$ is, $$\alpha(\log_e 2)^{-1} + \beta(\log_e 2) + \gamma$$ then the value of $$(\alpha + \beta - 2\gamma)^2$$ is equal to:

Let $$y = y(x)$$ be solution of the differential equation $$\log_e\left(\frac{dy}{dx}\right) = 3x + 4y$$, with $$y(0) = 0$$. If $$y\left(-\frac{2}{3}\log_e 2\right) = \alpha \log_e 2$$, then the value of $$\alpha$$ is equal to:

Let $$\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$$ and $$\vec{b} = -\hat{i} + 2\hat{j} + 3\hat{k}$$. Then the vector product $$\left(\vec{a} + \vec{b}\right) \times \left(\left(\vec{a} \times \left(\left(\vec{a} - \vec{b}\right) \times \vec{b}\right)\right) \times \vec{b}\right)$$ is equal to:

Let the plane passing through the point $$(-1, 0, -2)$$ and perpendicular to each of the planes $$2x + y - z = 2$$ and $$x - y - z = 3$$ be $$ax + by + cz + 8 = 0$$. Then the value of $$a + b + c$$ is equal to:

The probability that a randomly selected 2-digit number belongs to the set $$\{n \in N : (2^n - 2)$$ is a multiple of 3$$\}$$ is equal to