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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
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$$16\sin(20°)\sin(40°)\sin(80°)$$ is equal to
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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
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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
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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
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$$\lim_{x \to 0} \frac{\cos(\sin x) - \cos x}{x^4}$$ is equal to
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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
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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
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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
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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
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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:
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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
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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
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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
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The area of the region bounded by $$y^2 = 8x$$ and $$y^2 = 16(3 - x)$$ is equal to
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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
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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
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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
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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
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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
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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 ______
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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 ______
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The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______
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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 ______
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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 ______
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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 ______
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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 ______
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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 ______
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The integral $$\frac{24}{\pi}\int_0^{\sqrt{2}} \frac{(2-x^2)dx}{(2+x^2)\sqrt{4+x^4}}$$ is equal to ______
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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 ______
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