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Let $$S$$ be the set of positive integral values of $$a$$ for which $$\frac{ax^2 + 2(a+1)x + 9a + 4}{x^2 - 8x + 32} < 0, \forall x \in \mathbb{R}$$. Then, the number of elements in $$S$$ is:
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For $$0 \lt c \lt b \lt a$$, let $$(a + b - 2c)x^2 + (b + c - 2a)x + (c + a - 2b) = 0$$ and $$\alpha \neq 1$$ be one of its root. Then, among the two statements (I) If $$\alpha \in (-1, 0)$$, then $$b$$ cannot be the geometric mean of $$a$$ and $$c$$. (II) If $$\alpha \in (0, 1)$$, then $$b$$ may be the geometric mean of $$a$$ and $$c$$.
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The sum of the series $$\frac{1}{1 - 3 \cdot 1^2 + 1^4} + \frac{2}{1 - 3 \cdot 2^2 + 2^4} + \frac{3}{1 - 3 \cdot 3^2 + 3^4} + \ldots$$ up to 10 terms is
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Let $$\alpha, \beta, \gamma, \delta \in \mathbb{Z}$$ and let $$A(\alpha, \beta), B(1, 0), C(\gamma, \delta)$$ and $$D(1, 2)$$ be the vertices of a parallelogram $$ABCD$$. If $$AB = \sqrt{10}$$ and the points $$A$$ and $$C$$ lie on the line $$3y = 2x + 1$$, then $$2(\alpha + \beta + \gamma + \delta)$$ is equal to
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If one of the diameters of the circle $$x^2 + y^2 - 10x + 4y + 13 = 0$$ is a chord of another circle $$C$$, whose center is the point of intersection of the lines $$2x + 3y = 12$$ and $$3x - 2y = 5$$, then the radius of the circle $$C$$ is
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If the foci of a hyperbola are same as that of the ellipse $$\frac{x^2}{9} + \frac{y^2}{25} = 1$$ and the eccentricity of the hyperbola is $$\frac{15}{8}$$ times the eccentricity of the ellipse, then the smaller focal distance of the point $$\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$$ on the hyperbola, is equal to
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$$\lim_{x \to 0} \frac{e^{2\sin x} - 2\sin x - 1}{x^2}$$
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Let $$a$$ be the sum of all coefficients in the expansion of $$(1 - 2x + 2x^2)^{2023}(3 - 4x^2 + 2x^3)^{2024}$$ and $$b = \lim_{x \to 0} \frac{\int_0^x \frac{\log(1+t)}{t^{2024}+1}dt}{x^2}$$. If the equations $$cx^2 + dx + e = 0$$ and $$2bx^2 + ax + 4 = 0$$ have a common root, where $$c, d, e \in \mathbb{R}$$, then $$d : c : e$$ equals
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If $$f(x) = \begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$$ for all $$x \in \mathbb{R}$$, then $$2f(0) + f'(0)$$ is equal to
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If the system of linear equations $$x - 2y + z = -4$$, $$2x + \alpha y + 3z = 5$$, $$3x - y + \beta z = 3$$ has infinitely many solutions, then $$12\alpha + 13\beta$$ is equal to
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For $$\alpha, \beta, \gamma \neq 0$$. If $$\sin^{-1}\alpha + \sin^{-1}\beta + \sin^{-1}\gamma = \pi$$ and $$(\alpha + \beta + \gamma)(\alpha - \gamma + \beta) = 3\alpha\beta$$, then $$\gamma$$ equal to
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If $$f(x) = \frac{4x+3}{6x-4}, x \neq \frac{2}{3}$$ and $$(f \circ f)(x) = g(x)$$, where $$g: \mathbb{R} - \left\{\frac{2}{3}\right\} \to \mathbb{R} - \left\{\frac{2}{3}\right\}$$, then $$(g \circ g \circ g)(4)$$ is equal to
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Let $$g(x)$$ be a linear function and $$f(x) = \begin{cases} g(x), & x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{1/x}, & x > 0 \end{cases}$$, is continuous at $$x = 0$$. If $$f'(1) = f(-1)$$, then the value of $$g(3)$$ is
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The area of the region $$\left\{(x, y): y^2 \leq 4x, x < 4, \frac{xy(x-1)(x-2)}{(x-3)(x-4)} > 0, x \neq 3\right\}$$ is
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The solution curve of the differential equation $$y\frac{dx}{dy} = x(\log_e x - \log_e y + 1), x > 0, y > 0$$ passing through the point $$(e, 1)$$ is
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = \frac{\tan x + y}{\sin x(\sec x - \sin x \tan x)}, x \in \left(0, \frac{\pi}{2}\right)$$ satisfying the condition $$y\left(\frac{\pi}{4}\right) = 2$$. Then, $$y\left(\frac{\pi}{3}\right)$$ is
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Let $$\vec{a} = 3\hat{i} + \hat{j} - 2\hat{k}$$, $$\vec{b} = 4\hat{i} + \hat{j} + 7\hat{k}$$ and $$\vec{c} = \hat{i} - 3\hat{j} + 4\hat{k}$$ be three vectors. If a vector $$\vec{p}$$ satisfies $$\vec{p} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{p} \cdot \vec{a} = 0$$, then $$\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k})$$ is equal to
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The distance of the point $$Q(0, 2, -2)$$ from the line passing through the point $$P(5, -4, 3)$$ and perpendicular to the lines $$\vec{r} = -3\hat{i} + 2\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 5\hat{k}), \lambda \in \mathbb{R}$$ and $$\vec{r} = \hat{i} - 2\hat{j} + \hat{k} + \mu(-\hat{i} + 3\hat{j} + 2\hat{k}), \mu \in \mathbb{R}$$ is
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Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is
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Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $$x$$ to be the number of rotten apples in a draw of two apples, the variance of $$x$$ is
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If $$\alpha$$ denotes the number of solutions of $$|1 - i|^x = 2^x$$ and $$\beta = \frac{|z|}{\arg(z)}$$, where $$z = \frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi}\cdot i}{\sqrt{\pi}+i} + \frac{\sqrt{\pi}-i}{1+\sqrt{\pi}\cdot i}\right)$$, $$i = \sqrt{-1}$$, then the distance of the point $$(\alpha, \beta)$$ from the line $$4x - 3y = 7$$ is ______
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The total number of words (with or without meaning) that can be formed out of the letters of the word "DISTRIBUTION" taken four at a time, is equal to ______.
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In the expansion of $$(1+x)(1-x)^2\left(1 + \frac{3}{x} + \frac{3}{x^2} + \frac{1}{x^3}\right)^5, x \neq 0$$, the sum of the coefficient of $$x^3$$ and $$x^{-13}$$ is equal to ______
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Let the foci and length of the latus rectum of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$$ be $$(\pm 5, 0)$$ and $$\sqrt{50}$$, respectively. Then, the square of the eccentricity of the hyperbola $$\frac{x^2}{b^2} - \frac{y^2}{a^2 b^2} = 1$$ equals
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Let $$A = \{1, 2, 3, 4\}$$ and $$R = \{(1,2), (2,3), (1,4)\}$$ be a relation on $$A$$. Let $$S$$ be the equivalence relation on $$A$$ such that $$R \subset S$$ and the number of elements in $$S$$ is $$n$$. Then, the minimum value of $$n$$ is _______
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Let $$f: \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = \frac{4^x}{4^x + 2}$$ and $$M = \int_{f(a)}^{f(1-a)} x\sin^4(x(1-x))dx$$, $$N = \int_{f(a)}^{f(1-a)} \sin^4(x(1-x))dx; a \neq \frac{1}{2}$$. If $$\alpha M = \beta N, \alpha, \beta \in \mathbb{N}$$, then the least value of $$\alpha^2 + \beta^2$$ is equal to ______
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Let $$S = [-1, \infty)$$ and $$f: S \to \mathbb{R}$$ be defined as $$f(x) = \int_{-1}^{x} (e^t - 1)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61}dt$$. Let $$p$$ = Sum of square of the values of $$x$$, where $$f(x)$$ attains local maxima on $$S$$, and $$q$$ = Sum of the values of $$x$$, where $$f(x)$$ attains local minima on $$S$$. Then, the value of $$p^2 + 2q$$ is ________
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If the integral $$525\int_0^{\pi/2} \sin(2x) \cos^{11/2}(x)(1 + \cos^{5/2}(x))^{1/2}dx$$ is equal to $$n\sqrt{2} - 64$$, then $$n$$ is equal to ________
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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}| = 1, |\vec{b}| = 4$$ and $$\vec{a} \cdot \vec{b} = 2$$. If $$\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$$ and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\alpha$$, then $$192\sin^2\alpha$$ is equal to _________
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Let $$Q$$ and $$R$$ be the feet of perpendiculars from the point $$P(a, a, a)$$ on the lines $$x = y, z = 1$$ and $$x = -y, z = -1$$ respectively. If $$\angle QPR$$ is a right angle, then $$12a^2$$ is equal to ________
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