NTA JEE Main 25th June 2022 Shift 2 - Mathematics

Let $$A = \{x \in R : |x + 1| < 2\}$$ and $$B = \{x \in R : |x - 1| \geq 2\}$$. Then which one the following statements is NOT true?

Let $$a, b \in R$$ be such that the equation $$ax^2 - 2bx + 15 = 0$$ has repeated root $$\alpha$$ and if $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 - 2bx + 21 = 0$$, then $$\alpha^2 + \beta^2$$ is equal to:

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$\bar{z}_1 = iz_2$$ and $$\arg\frac{z_1}{z_2} = \pi$$, then the argument of $$z_1$$ is

The sum $$1 + 2 \cdot 3 + 3 \cdot 3^2 + \ldots + 10 \cdot 3^9$$ is equal to

The coefficient of $$x^{101}$$ in the expression
$$5 + x^{500} + x5 + x^{499} + x^2(5 + x)^{498} + \ldots + x^{500}, x > 0$$ is

The value of $$2\sin 12° - \sin 72°$$ is

A circle touches both the $$y$$-axis and the line $$x + y = 0$$. Then the locus of its center is

The line $$y = x + 1$$ meets the ellipse $$\frac{x^2}{4} + \frac{y^2}{2} = 1$$ at two points $$P$$ and $$Q$$. If $$r$$ is the radius of the circle with $$PQ$$ as diameter then $$(3r)^2$$ is equal to

$$\lim_{x \to \frac{\pi}{2}} {\tan^2 x(2\sin^2 x + 3\sin x + 4)^{\frac{1}{2}} - (\sin^2 x + 6\sin x + 2)^{\frac{1}{2}}}{2}$$ is equal to

The negation of the Boolean expression $$\sim q \wedge p \Rightarrow \sim p \vee q$$ is logically equivalent to

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

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 ______.