NTA JEE Main 25th June 2022 Shift 2

The system of equations
$$-kx + 3y - 14z = 25$$
$$-15x + 4y - kz = 3$$
$$-4x + y + 3z = 4$$
is consistent for all $$k$$ in the set

The value of $$\tan^{-1}\left(\frac{\cos\frac{15\pi}{4} - 1}{\sin\frac{\pi}{4}}\right)$$ is equal to

Water is being filled at the rate of $$1$$ cm$$^3$$ sec$$^{-1}$$ in a right circular conical vessel (vertex downwards) of height $$35$$ cm and diameter $$14$$ cm. When the height of the water level is $$10$$ cm, the rate (in cm$$^2$$ sec$$^{-1}$$) at which the wet conical surface area of the vessel increases is

If the line $$y = 4 + kx, k > 0$$, is the tangent to the parabola $$y = x - x^2$$ at the point $$P$$ and $$V$$ is the vertex of the parabola, then the slope of the line through $$P$$ and $$V$$ is

If the angle made by the tangent at the point $$(x_0, y_0)$$ on the curve $$x = 12(t + \sin t \cos t), y = 12(1 + \sin t)^2, 0 < t < \frac{\pi}{2}$$, with the positive $$x$$-axis is $$\frac{\pi}{3}$$, then $$y_0$$ is equal to

If $$b_n = \int_0^{\pi/2} \frac{\cos^2(nx)}{\sin x} dx, n \in \mathbb{N}$$, then

The area of the region enclosed between the parabolas $$y^2 = 2x - 1$$ and $$y^2 = 4x - 3$$ is

If $$y = yx$$ is the solution of the differential equation $$2x^2\frac{dy}{dx} - 2xy + 3y^2 = 0$$ such that $$y(e) = \frac{e}{3}$$, then $$y(1)$$ is equal to

Let $$P$$ be the plane passing through the intersection of the planes $$\vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 5$$ and $$\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 3$$, and the point $$(2, 1, -2)$$. Let the position vectors of the points $$X$$ and $$Y$$ be $$\hat{i} - 2\hat{j} + 4\hat{k}$$ and $$5\hat{i} - \hat{j} + 2\hat{k}$$ respectively. Then the points

A biased die is marked with numbers $$2, 4, 8, 16, 32, 32$$ on its faces and the probability of getting a face with mark $$n$$ is $$\frac{1}{n}$$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is $$48$$, is