NTA JEE Main 25th June 2022 Shift 2

The total number of three-digit numbers, with one digit repeated exactly two times, is ______.

If the sum of the co-efficients of all the positive even powers of $$x$$ in the binomial expansion of $$\left(2x^3 + \frac{3}{x}\right)^{10}$$ is $$5^{10} - \beta \cdot 3^9$$, then $$\beta$$ is equal to ______.

Let the eccentricity of the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$\frac{5}{4}$$. If the equation of the normal at the point $$\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)$$ on the hyperbola is $$8\sqrt{5}x + \beta y = \lambda$$, then $$\lambda - \beta$$ is equal to ______.

If the mean deviation about the mean of the numbers $$1, 2, 3, \ldots, n$$, where $$n$$ is odd, is $$\frac{5(n+1)}{n}$$, then $$n$$ is equal to ______.

Let $$A = \begin{pmatrix} 2 & -2 \\ 1 & -1 \end{pmatrix}$$ and $$B = \begin{pmatrix} -1 & 2 \\ -1 & 2 \end{pmatrix}$$. Then the number of elements in the set $$\{(n, m) : n, m \in \{1, 2, \ldots, 10\}$$ and $$nA^n + mB^m = I\}$$ is ______.

Let $$f(x) = [2x^2 + 1]$$ and $$g(x) = \begin{cases} 2x - 3, & x < 0 \\ 2x + 3, & x \geq 0 \end{cases}$$, where $$[t]$$ is the greatest integer $$\leq t$$. Then, in the open interval $$(-1, 1)$$, the number of points where $$f \circ g$$ is discontinuous is equal to ______.

Let $$f(x) = x|x^2 - 1| - 2|x - 3| + x - 3, x \in \mathbb{R}$$. If $$m$$ and $$M$$ are respectively the number of points of local minimum and local maximum of $$f$$ in the interval $$(0, 4)$$, then $$m + M$$ is equal to ______.

The value of $$b > 3$$ for which $$12\int_3^b \frac{1}{(x^2 - 1)(x^2 - 4)} dx = \log_e\frac{49}{40}$$, is equal to ______.

Let $$\vec{b} = \hat{i} + \hat{j} + \lambda\hat{k}, \lambda \in \mathbb{R}$$. If $$\vec{a}$$ is a vector such that $$\vec{a} \times \vec{b} = 13\hat{i} - \hat{j} - 4\hat{k}$$ and $$\vec{a} \cdot \vec{b} + 21 = 0$$, then
$$\vec{b} - \vec{a} \cdot \hat{k} - \hat{j} + \vec{b} + \vec{a} \cdot \hat{i} - \hat{k}$$ is equal to ______

Let $$l_1$$ be the line in $$xy$$-plane with $$x$$ and $$y$$ intercepts $$\frac{1}{8}$$ and $$\frac{1}{4\sqrt{2}}$$ respectively, and $$l_2$$ be the line in $$zx$$-plane with $$x$$ and $$z$$ intercepts $$-\frac{1}{8}$$ and $$-\frac{1}{6\sqrt{3}}$$ respectively. If $$d$$ is the shortest distance between the line $$l_1$$ and $$l_2$$, then $$d^{-2}$$ is equal to ______.