NTA JEE Main 27th June 2022 Shift 2

Let $$f(x) = \begin{vmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{vmatrix}, a \in R$$. Then the sum of the squares of all the values of $$a$$ for
$$2f'(10) - f'(5) + 100 = 0$$ is

The value of $$\cot\left(\sum_{n=1}^{50} \tan^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$$ is

If $$m$$ and $$n$$ respectively are the number of local maximum and local minimum points of the function $$f(x) = \int_0^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} dt$$, then the ordered pair $$(m, n)$$ is equal to

Let $$f$$ be a differentiable function in $$\left(0, \frac{\pi}{2}\right)$$. If $$\int_{\cos x}^{1} t^2 f(t) dt = \sin^3 x + \cos x$$, then $$\frac{1}{\sqrt{3}} f'\left(\frac{1}{\sqrt{3}}\right)$$ is equal to

The value of the integral $$\int_0^1 \frac{1}{7^{[\frac{1}{x}]}} dx$$, where $$[\cdot]$$ denotes the greatest integer function, is equal to

If the solution curve of the differential equation $$((\tan^{-1}y) - x)dy = (1 + y^2)dx$$ passes through the point $$(1, 0)$$ then the abscissa of the point on the curve whose ordinate is $$\tan(1)$$ is

Let $$\vec{a}$$ and $$\vec{b}$$ be the vectors along the diagonal of a parallelogram having area $$2\sqrt{2}$$. Let the angle between $$\vec{a}$$ and $$\vec{b}$$ be acute. $$|\vec{a}| = 1$$ and $$|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$$. If $$\vec{c} = 2\sqrt{2}(\vec{a} \times \vec{b}) - 2\vec{b}$$, then an angle between $$\vec{b}$$ and $$\vec{c}$$ is

Let the foot of the perpendicular from the point $$(1, 2, 4)$$ on the line $$\frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3}$$ be $$P$$. Then the distance of $$P$$ from the plane $$3x + 4y + 12z + 23 = 0$$ is

The shortest distance between the lines $$\frac{x-3}{2} = \frac{y-2}{3} = \frac{z-1}{-1}$$ and $$\frac{x+3}{2} = \frac{y-6}{1} = \frac{z-5}{3}$$ is

If a point $$A(x, y)$$ lies in the region bounded by the y-axis, straight lines $$2y + x = 6$$ and $$5x - 6y = 30$$, then the probability that $$y < 1$$ is