NTA JEE Main 25th July 2022 Shift 1

Let $$a, b$$ be two non-zero real numbers. If $$p$$ and $$r$$ are the roots of the equation $$x^2 - 8ax + 2a = 0$$ and $$q$$ and $$s$$ are the roots of the equation $$x^2 + 12bx + 6b = 0$$, such that $$\dfrac{1}{p}, \dfrac{1}{q}, \dfrac{1}{r}, \dfrac{1}{s}$$ are in A.P., then $$a^{-1} - b^{-1}$$ is equal to ______.

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.

Let $$a_1 = b_1 = 1$$, $$a_n = a_{n-1} + 2$$ and $$b_n = a_n + b_{n-1}$$ for every natural number $$n \ge 2$$. Then $$\displaystyle\sum_{n=1}^{15} a_n \cdot b_n$$ is equal to ______.

If the maximum value of the term independent of $$t$$ in the expansion of $$\left(t^2 x^{1/5} + \dfrac{(1-x)^{1/10}}{t}\right)^{15}$$, $$x \ge 0$$, is $$K$$, then $$8K$$ is equal to ______.

The sum of diameters of the circles that touch (i) the parabola $$75x^2 = 64(5y - 3)$$ at the point $$\left(\dfrac{8}{5}, \dfrac{6}{5}\right)$$ and (ii) the $$y$$-axis, is equal to ______.

Let the equation of two diameters of a circle $$x^2 + y^2 - 2x + 2fy + 1 = 0$$ be $$2px - y = 1$$ and $$2x + py = 4p$$. Then the slope $$m \in (0, \infty)$$ of the tangent to the hyperbola $$3x^2 - y^2 = 3$$ passing through the centre of the circle is equal to ______.

Let $$A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$$ and $$B = A - I$$. If $$\omega = \dfrac{\sqrt{3}i - 1}{2}$$, then the number of elements in the set $$\{n \in \{1, 2, \ldots, 100\} : A^n + (\omega B)^n = A + B\}$$ is equal to ______.

Let $$f(x) = \begin{cases} \{4x^2 - 8x + 5\}, & \text{if } 8x^2 - 6x + 1 \ge 0 \\ [4x^2 - 8x + 5], & \text{if } 8x^2 - 6x + 1 < 0 \end{cases}$$, where $$[\alpha]$$ denotes the greatest integer less than or equal to $$\alpha$$ . Then the number of points in $$\mathbb{R}$$ where $$f$$ is not differentiable is ______.

If $$\displaystyle\lim_{n \to \infty} \dfrac{(n+1)^{k-1}}{n^{k+1}} \left[(nk+1) + (nk+2) + \ldots + (nk+n)\right] = 33 \cdot \lim_{n \to \infty} \dfrac{1}{n^{k+1}} \left(1^k + 2^k + 3^k + \ldots + n^k\right)$$, then the integral value of $$k$$ is equal to ______.

The line of shortest distance between the lines $$\dfrac{x-2}{0} = \dfrac{y-1}{1} = \dfrac{z}{1}$$ and $$\dfrac{x-3}{2} = \dfrac{y-5}{2} = \dfrac{z-1}{1}$$ makes an angle of $$\sin^{-1}\sqrt{\dfrac{2}{27}}$$ with the plane $$P: ax - y - z = 0$$, $$a > 0$$. If the image of the point $$(1, 1, -5)$$ in the plane $$P$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta - \gamma$$ is equal to ______.