NTA JEE Main 17th March 2021 Shift 1

If $$(2021)^{3762}$$ is divided by 17, then the remainder is ________.

The minimum distance between any two points $$P_1$$ and $$P_2$$ while considering point $$P_1$$ on one circle and point $$P_2$$ on the other circle for the given circles' equations:
$$x^2 + y^2 - 10x - 10y + 41 = 0$$
$$x^2 + y^2 - 24x - 10y + 160 = 0$$ is ________.

If $$A = \begin{bmatrix} 2 & 3 \\ 0 & -1 \end{bmatrix}$$, then the value of $$\det(A^4) + \det\left(A^{10} - (\text{Adj}(2A))^{10}\right)$$ is equal to ________.

If the function $$f(x) = \frac{\cos(\sin x) - \cos x}{x^4}$$ is continuous at each point in its domain and $$f(0) = \frac{1}{k}$$, then $$k$$ is ________.

If $$f(x) = \sin\left(\cos^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)\right)$$ and its first derivative with respect to $$x$$ is $$-\frac{b}{a}\log_e 2$$ when $$x = 1$$, where $$a$$ and $$b$$ are integers, then the minimum value of $$|a^2 - b^2|$$ is ________.

The maximum value of $$z$$ in the following equation $$z = 6xy + y^2$$, where $$3x + 4y \leq 100$$ and $$4x + 3y \leq 75$$ for $$x \geq 0$$ and $$y \geq 0$$ is ________.

If $$[\cdot]$$ represents the greatest integer function, then the value of $$\left|\int_0^{\sqrt{\frac{\pi}{2}}} \left[\left[x^2\right] - \cos x\right] dx\right|$$ is ________.

If $$\vec{a} = \alpha\hat{i} + \beta\hat{j} + 3\hat{k}$$, $$\vec{b} = -\beta\hat{i} - \alpha\hat{j} - \hat{k}$$ and $$\vec{c} = \hat{i} - 2\hat{j} - \hat{k}$$ such that $$\vec{a} \cdot \vec{b} = 1$$ and $$\vec{b} \cdot \vec{c} = -3$$, then $$\frac{1}{3}\left((\vec{a} \times \vec{b}) \cdot \vec{c}\right)$$ is equal to ________.

If the equation of the plane passing through the line of intersection of the planes $$2x - 7y + 4z - 3 = 0$$, $$3x - 5y + 4z + 11 = 0$$ and the point $$(-2, 1, 3)$$ is $$ax + by + cz - 7 = 0$$, then the value of $$2a + b + c - 7$$ is ________.

Let there be three independent events $$E_1, E_2$$ and $$E_3$$. The probability that only $$E_1$$ occurs is $$\alpha$$, only $$E_2$$ occurs is $$\beta$$ and only $$E_3$$ occurs is $$\gamma$$. Let '$$p$$' denote the probability of none of events occurs that satisfies the equations $$(\alpha - 2\beta)p = \alpha\beta$$ and $$(\beta - 3\gamma)p = 2\beta\gamma$$. All the given probabilities are assumed to lie in the interval $$(0, 1)$$. Then, $$\frac{\text{Probability of occurrence of } E_1}{\text{Probability of occurrence of } E_3}$$ is equal to ________.