NTA JEE Main 26th June 2022 Shift 2 - Mathematics

If $$A = \sum_{n=1}^{\infty} \frac{1}{(3+(- 1)^n)^n}$$ and $$B = \sum_{n=1}^{\infty} \frac{(-1)^n}{(3+(-1)^n)^n}$$, then $$\frac{A}{B}$$ is equal to

$$16\sin(20°)\sin(40°)\sin(80°)$$ is equal to

If $$m$$ is the slope of a common tangent to the curves $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$ and $$x^2 + y^2 = 12$$, then $$12m^2$$ is equal to

The locus of the mid-point of the line segment joining the point $$(4, 3)$$ and the points on the ellipse $$x^2 + 2y^2 = 4$$ is an ellipse with eccentricity

The normal to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{9} = 1$$ at the point $$(8, 3\sqrt{3})$$ on it passes through the point

$$\lim_{x \to 0} \frac{\cos(\sin x) - \cos x}{x^4}$$ is equal to

Let $$r \in (P, q, \sim p, \sim q)$$ be such that the logical statement $$r \vee (\sim p) \Rightarrow (p \wedge q) \vee r$$ is a tautology. Then $$r$$ is equal to

Let the mean of 50 observations is 15 and the standard deviation is 2. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is 70. If the mean of the correct set of observations is 16, then the variance of the correct set is equal to

If the system of equations $$\alpha x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = \beta$$. Has infinitely many solutions, then the ordered pair $$(\alpha, \beta)$$ is equal to

If the inverse trigonometric functions take principal values, then
$$\cos^{-1}\left(\frac{3}{10}\cos\left(\tan^{-1}\left(\frac{4}{3}\right)\right) + \frac{2}{5}\sin\left(\tan^{-1}\left(\frac{4}{3}\right)\right)\right)$$ is equal to

Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = x - 1$$ and $$g : R \to \{1, -1\} \to \mathbb{R}$$ be defined as $$g(x) = \frac{x^2}{x^2 - 1}$$. Then the function $$fog$$ is:

Let $$f(x) = \min\{1, 1 + x\sin x\}, 0 \leq x \leq 2\pi$$. If $$m$$ is the number of points, where $$f$$ is not differentiable and $$n$$ is the number of points, where $$f$$ is not continuous, then the ordered pair $$(m, n)$$ is equal to

Consider a cuboid of sides $$2x, 4x$$ and $$5x$$ and a closed hemisphere of radius $$r$$. If the sum of their surface areas is constant $$k$$, then the ratio $$x : r$$, for which the sum of their volumes is maximum, is

If $$\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}} dx = g(x) + c, g(1) = 0$$, then $$g\left(\frac{1}{2}\right)$$ is equal to

The area of the region bounded by $$y^2 = 8x$$ and $$y^2 = 16(3 - x)$$ is equal to

If $$y = y(x)$$ is the solution of the differential equation $$x\frac{dy}{dx} + 2y = xe^x, y(1) = 0$$ then the local maximum value of the function $$z(x) = x^2y(x) - e^x, x \in R$$ is

If $$\frac{dy}{dx} + e^x(x^2 - 2)y = (x^2 - 2x)(x^2 - 2)e^{2x}$$ and $$y(0) = 0$$, then the value of $$y(2)$$ is

Let $$\vec{a} = \hat{i} + \hat{j} + 2\hat{k}, \vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + \hat{k}$$ be the three given vectors. Let $$\vec{v}$$ be a vector in the plane of $$\vec{a}$$ and $$\vec{b}$$ whose projection on $$\vec{c}$$ is $$\frac{2}{\sqrt{3}}$$. If $$\vec{v} \cdot \hat{j} = 7$$, then $$\vec{v} \cdot (\hat{i} + \hat{k})$$ is equal to

If the plane $$2x + y - 5z = 0$$ is rotated about its line of intersection with the plane $$3x - y + 4z - 7 = 0$$ by an angle of $$\frac{\pi}{2}$$, then the plane after the rotation passes through the point

If the lines $$\vec{r} = (\hat{i} - \hat{j} + \hat{k}) + \lambda(3\hat{j} - \hat{k})$$ and $$\vec{r} = (\alpha\hat{i} - \hat{j}) + \mu(2\hat{i} - 3\hat{k})$$ are co-planar, then the distance of the plane containing these two lines from the point $$(\alpha, 0, 0)$$ is

If $$p$$ and $$q$$ are real number such that $$p + q = 3, p^4 + q^4 = 369$$, then the value of $$\left(\frac{1}{p} + \frac{1}{q}\right)^{-2}$$ is equal to ______

If $$z^2 + z + 1 = 0, z \in C$$, then $$\left|\sum_{n=1}^{15}\left(z^n + (-1)^n \frac{1}{z^n}\right)^2\right|$$ is equal to ______

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______

If $$a_1(> 0), a_2, a_3, a_4, a_5$$ are in a G.P., $$a_2 + a_4 = 2a_3 + 1$$ and $$3a_2 + a_3 = 2a_4$$, then $$a_2 + a_4 + 2a_5$$ is equal to ______

If $$^{40}C_0 + ^{41}C_1 + ^{42}C_2 + \cdots + ^{60}C_{20} = \frac{m}{n} \times ^{60}C_{20}$$ where $$m$$ & $$n$$ are co-prime, then $$m + n$$ is equal to ______

Let a line $$L_1$$ be tangent to the hyperbola $$\frac{x^2}{16} - \frac{y^2}{4} = 1$$ and let $$L_2$$ be the line passing through the origin and perpendicular to $$L_1$$. If the locus of the point of intersection of $$L_1$$ and $$L_2$$ is $$(x^2 + y^2)^2 = \alpha x^2 + \beta y^2$$, then $$\alpha + \beta$$ is equal to ______

Let $$X = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$, $$Y = \alpha I + \beta X + \gamma X^2$$ and $$Z = \alpha^2 I - \alpha\beta X + (\beta^2 - \alpha\gamma)X^2, \alpha, \beta, \gamma \in \mathbb{R}$$.
If $$Y^{-1} = \begin{bmatrix} \frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{-2}{5} \\ 0 & 0 & \frac{1}{5} \end{bmatrix}$$, then $$(\alpha - \beta + \gamma)^2$$ is equal to ______

Let $$f : \mathbb{R} \to \mathbb{R}$$ satisfy $$f(x + y) = 2^x f(y) + 4^y f(x), \forall x, y \in \mathbb{R}$$. If $$f(2) = 3$$, then $$14 \cdot \frac{f'(4)}{f'(2)}$$ is equal to ______

The integral $$\frac{24}{\pi}\int_0^{\sqrt{2}} \frac{(2-x^2)dx}{(2+x^2)\sqrt{4+x^4}}$$ is equal to ______

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is $$p$$, then $$96p$$ is equal to ______