The number of solutions, of the equation $$e^{\sin x} - 2e^{-\sin x} = 2$$ is
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The number of solutions, of the equation $$e^{\sin x} - 2e^{-\sin x} = 2$$ is
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Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1 + z_2 = 5$$ and $$z_1^3 + z_2^3 = 20 + 15i$$. Then $$|z_1^4 + z_2^4|$$ equals
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The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
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Let 2$$^{nd}$$, 8$$^{th}$$ and 44$$^{th}$$, terms of a non-constant A.P. be respectively the 1$$^{st}$$, 2$$^{nd}$$ and 3$$^{rd}$$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to
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If for some n; $${}^{6}C_{m}+2^{6}C_{m+1}+{}^{6}C_{m+2}>{}^{8}C_{3}$$ and $$^{n-1}P_3 : ^nP_4 = 1:8$$, then $$^nP_{m+1} + ^{n+1}C_m$$ is equal to
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Let $$A(a, b)$$, $$B(3, 4)$$ and $$(-6, -8)$$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $$P(2a + 3, 7b + 5)$$ from the line $$2x + 3y - 4 = 0$$ measured parallel to the line $$x - 2y - 1 = 0$$ is
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Let a variable line passing through the centre of the circle $$x^2 + y^2 - 16x - 4y = 0$$, meet the positive coordinate axes at the point $$A$$ and $$B$$. Then the minimum value of $$OA + OB$$, where $$O$$ is the origin, is equal to
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Let $$P$$ be a parabola with vertex $$(2, 3)$$ and directrix $$2x + y = 6$$. Let an ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$ of eccentricity $$\frac{1}{\sqrt{2}}$$ pass through the focus of the parabola $$P$$. Then the square of the length of the latus rectum of $$E$$, is
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Let $$f: \mathbb{R} \rightarrow (0, \infty)$$ be strictly increasing function such that $$\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1$$. Then, the value of $$\lim_{x \to \infty} \left[\frac{f(5x)}{f(x)} - 1\right]$$ is equal to
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Let the mean and the variance of 6 observations $$a, b, 68, 44, 48, 60$$ be 55 and 194, respectively. If $$a > b$$, then $$a + 3b$$ is
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Let A be a $$3 \times 3$$ real matrix such that $$A\begin{pmatrix}1\\0\\1\end{pmatrix} = 2\begin{pmatrix}1\\0\\1\end{pmatrix}$$, $$A\begin{pmatrix}-1\\0\\1\end{pmatrix} = 4\begin{pmatrix}-1\\0\\1\end{pmatrix}$$, $$A\begin{pmatrix}0\\1\\0\end{pmatrix} = 2\begin{pmatrix}0\\1\\0\end{pmatrix}$$. Then, the system $$(A - 3I)\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}$$ has
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If $$a = \sin^{-1}(\sin 5)$$ and $$b = \cos^{-1}(\cos 5)$$, then $$a^2 + b^2$$ is equal to
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If the function $$f: (-\infty, -1] \rightarrow [a, b]$$ defined by $$f(x) = e^{x^3 - 3x + 1}$$ is one-one and onto, then the distance of the point $$P(2b + 4, a + 2)$$ from the line $$x + e^{-3}y = 4$$ is:
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Consider the function $$f: (0, \infty) \rightarrow R$$ defined by $$f(x) = e^{-|\log_e x|}$$. If $$m$$ and $$n$$ be respectively the number of points at which $$f$$ is not continuous and $$f$$ is not differentiable, then $$m + n$$ is
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Let $$f, g: [0, \infty) \rightarrow R$$ be two functions defined by $$f(x) = \int_{-x}^{x}(|t| - t^2)e^{-t^2}dt$$ and $$g(x) = \int_{0}^{x^2}t^{1/2}e^{-t}dt$$. Then the value of $$9(f(\sqrt{\log_e 9}) + g(\sqrt{\log_e 9}))$$ is equal to
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The area of the region enclosed by the parabola $$y = 4x - x^2$$ and $$3y = (x - 4)^2$$ is equal to
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The temperature $$T(t)$$ of a body at time $$t = 0$$ is $$160°F$$ and it decreases continuously as per the differential equation $$\frac{dT}{dt} = -K(T - 80)$$, where $$K$$ is positive constant. If $$T(15) = 120°F$$, then $$T(45)$$ is equal to
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Let $$(\alpha, \beta, \gamma)$$ be mirror image of the point $$(2, 3, 5)$$ in the line $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$. Then $$2\alpha + 3\beta + 4\gamma$$ is equal to
The shortest distance between lines $$L_1$$ and $$L_2$$, where $$L_1: \frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+4}{2}$$ and $$L_2$$ is the line passing through the points $$A(-4, 4, 3)$$, $$B(-1, 6, 3)$$ and perpendicular to the line $$\frac{x-3}{-2} = \frac{y}{3} = \frac{z-1}{1}$$, is
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A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is
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Let $$a, b, c$$ be the length of three sides of a triangle satisfying the condition $$(a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0$$. If the set of all possible values of $$x$$ is in the interval $$(\alpha, \beta)$$, then $$12(\alpha^2 + \beta^2)$$ is equal to
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Let the coefficient of $$x^r$$ in the expansion of $$(x+3)^{n-1} + (x+3)^{n-2}(x+2) + (x+3)^{n-3}(x+2)^2 + \ldots + (x+2)^{n-1}$$ be $$\alpha_r$$. If $$\sum_{r=0}^{n}\alpha_r = \beta^n - \gamma^n$$, $$\beta, \gamma \in \mathbb{N}$$, then the value of $$\beta^2 + \gamma^2$$ equals
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Let $$A(-2, -1)$$, $$B(1, 0)$$, $$C(\alpha, \beta)$$ and $$D(\gamma, \delta)$$ be the vertices of a parallelogram $$ABCD$$. If the point $$C$$ lies on $$2x - y = 5$$ and the point $$D$$ lies on $$3x - 2y = 6$$, then the value of $$|\alpha + \beta + \gamma + \delta|$$ is equal to
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If $$\lim_{x \to 0} \frac{ax^2e^x - b\log_e(1+x) + cxe^{-x}}{x^2\sin x} = 1$$, then $$16(a^2 + b^2 + c^2)$$ is equal to
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Let $$A = \{1, 2, 3, \ldots, 100\}$$. Let $$R$$ be a relation on $$A$$ defined by $$(x, y) \in R$$ if and only if $$2x = 3y$$. Let $$R_1$$ be a symmetric relation on $$A$$ such that $$R \subset R_1$$ and the number of elements in $$R_1$$ is $$n$$. Then the minimum value of $$n$$ is
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Let $$A$$ be a $$3 \times 3$$ matrix and $$\det(A) = 2$$. If $$n = \det(\underbrace{adj(adj(\ldots adj(A)))}_{\text{2024 times}})$$, then the remainder when $$n$$ is divided by 9 is equal to
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$$\frac{120}{\pi^3}\int_{0}^{\pi}\frac{x^2\sin x\cos x}{\sin^4 x + \cos^4 x}dx$$ is equal to
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Let $$y = y(x)$$ be the solution of the differential equation $$\sec^2 x \, dx + e^{2y}(\tan^2 x + \tan x) \, dy = 0$$, $$0 \lt x \lt \frac{\pi}{2}$$, $$y\left(\frac{\pi}{4}\right) = 0$$. If $$y\left(\frac{\pi}{6}\right) = \alpha$$, then $$e^{8\alpha}$$ is equal to
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Let $$\vec{a} = 3\hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + \vec{b}) \times \vec{c} = 2(\vec{a} \times \vec{b}) + 24\hat{j} - 6\hat{k}$$ and $$(\vec{a} - \vec{b} + \hat{i}) \cdot \vec{c} = -3$$. Then $$|\vec{c}|^2$$ is equal to
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A line passes through $$A(4, -6, -2)$$ and $$B(16, -2, 4)$$. The point $$P(a, b, c)$$ where $$a, b, c$$ are non-negative integers, on the line $$AB$$ lies at a distance of 21 units, from the point $$A$$. The distance between the points $$P(a, b, c)$$ and $$Q(4, -12, 3)$$ is equal to
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