If for $$x \in \left(0, \frac{\pi}{2}\right)$$, $$\log_{10} \sin x + \log_{10} \cos x = -1$$ and $$\log_{10}(\sin x + \cos x) = \frac{1}{2}(\log_{10} n - 1)$$, $$n > 0$$, then the value of $$n$$ is equal to:
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If for $$x \in \left(0, \frac{\pi}{2}\right)$$, $$\log_{10} \sin x + \log_{10} \cos x = -1$$ and $$\log_{10}(\sin x + \cos x) = \frac{1}{2}(\log_{10} n - 1)$$, $$n > 0$$, then the value of $$n$$ is equal to:
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Let a complex number $$z$$, $$|z| \neq 1$$, satisfy $$\log_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2$$. Then, the largest value of $$|z|$$ is equal to:
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If $$n$$ is the number of irrational terms in the expansion of $$\left(3^{1/4} + 5^{1/8}\right)^{60}$$, then $$(n-1)$$ is divisible by:
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Let $$[x]$$ denote greatest integer less than or equal to $$x$$. If for $$n \in N$$, $$\left(1 - x + x^3\right)^n = \sum_{j=0}^{3n} a_j x^j$$, then $$\sum_{j=0}^{\left[\frac{3n}{2}\right]} a_{2j} + 4\sum_{j=0}^{\left[\frac{3n-1}{2}\right]} a_{2j+1}$$ is equal to:
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The number of roots of the equation, $$(81)^{\sin^2 x} + (81)^{\cos^2 x} = 30$$ in the interval $$[0, \pi]$$ is equal to:
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If the three normals drawn to the parabola, $$y^2 = 2x$$ pass through the point $$(a, 0)$$, $$a \neq 0$$, then $$a$$ must be greater than:
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The locus of the midpoints of the chord of the circle, $$x^2 + y^2 = 25$$ which is tangent to the hyperbola, $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$ is:
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Which of the following Boolean expression is a tautology?
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Consider three observations $$a, b$$ and $$c$$ such that $$b = a + c$$. If the standard deviation of $$a+2, b+2, c+2$$ is $$d$$, then which of the following is true?
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Let $$A = \begin{bmatrix} i & -i \\ -i & i \end{bmatrix}$$, $$i = \sqrt{-1}$$. Then, the system of linear equations $$A^8 \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 64 \end{bmatrix}$$ has:
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Let $$S_k = \sum_{r=1}^{k} \tan^{-1}\left(\frac{6^r}{2^{2r+1} + 3^{2r+1}}\right)$$, then $$\lim_{k \to \infty} S_k$$ is equal to:
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The number of elements in the set $$\{x \in R : (|x| - 3)|x + 4| = 6\}$$ is equal to:
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Let the functions $$f : R \to R$$ and $$g : R \to R$$ be defined as:
$$f(x) = \begin{cases} x+2, & x < 0 \\ x^2, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x^3, & x < 1 \\ 3x-2, & x \geq 1 \end{cases}$$
Then, the number of points in $$R$$ where $$(f \circ g)(x)$$ is NOT differentiable is equal to:
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The range of $$a \in R$$ for which the function $$f(x) = (4a-3)(x + \log_e 5) + 2(a-7)\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right)$$, $$x \neq 2n\pi$$, $$n \in N$$, has critical points, is:
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If $$y = y(x)$$ is the solution of the differential equation, $$\frac{dy}{dx} + 2y\tan x = \sin x$$, $$y\left(\frac{\pi}{3}\right) = 0$$, then the maximum value of the function $$y(x)$$ over $$R$$ is equal to:
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Let a vector $$\alpha\hat{i} + \beta\hat{j}$$ be obtained by rotating the vector $$\sqrt{3}\hat{i} + \hat{j}$$ by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $$(\alpha, \beta)$$, $$(0, \beta)$$ and $$(0, 0)$$ is equal to:
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If for $$a > 0$$, the feet of perpendiculars from the points $$A(a, -2a, 3)$$ and $$B(0, 4, 5)$$ on the plane $$lx + my + nz = 0$$ are points $$C(0, -a, -1)$$ and $$D$$ respectively, then the length of line segment $$CD$$ is equal to:
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Let the position vectors of two points $$P$$ and $$Q$$ be $$3\hat{i} - \hat{j} + 2\hat{k}$$ and $$\hat{i} + 2\hat{j} - 4\hat{k}$$, respectively. Let $$R$$ and $$S$$ be two points such that the direction ratios of lines $$PR$$ and $$QS$$ are $$(4, -1, 2)$$ and $$(-2, 1, -2)$$, respectively. Let lines $$PR$$ and $$QS$$ intersect at $$T$$. If the vector $$\vec{TA}$$ is perpendicular to both $$\vec{PR}$$ and $$\vec{QS}$$ and the length of vector $$\vec{TA}$$ is $$\sqrt{5}$$ units, then the modulus of a position vector of $$A$$ is:
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Let $$P$$ be a plane $$lx + my + nz = 0$$ containing the line, $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$. If plane $$P$$ divides the line segment $$AB$$ joining points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ in ratio $$k:1$$ then the value of $$k$$ is equal to:
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A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is:
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Let $$z$$ and $$w$$ be two complex numbers such that $$w = z\bar{z} - 2z + 2$$, $$\left|\frac{z+i}{z-3i}\right| = 1$$ and $$\text{Re}(w)$$ has minimum value. Then, the minimum value of $$n \in N$$ for which $$w^n$$ is real, is equal to ________.
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Consider an arithmetic series and a geometric series having four initial terms from the set $$\{11, 8, 21, 16, 26, 32, 4\}$$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ________.
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Let $$ABCD$$ be a square of side of unit length. Let a circle $$C_1$$ centered at $$A$$ with unit radius is drawn. Another circle $$C_2$$ which touches $$C_1$$ and the lines $$AD$$ and $$AB$$ are tangent to it, is also drawn. Let a tangent line from the point $$C$$ to the circle $$C_2$$ meet the side $$AB$$ at $$E$$. If the length of $$EB$$ is $$\alpha + \sqrt{3}\beta$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to ________.
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If $$\lim_{x \to 0} \frac{ae^x - b\cos x + ce^{-x}}{x \sin x} = 2$$, then $$a + b + c$$ is equal to ________.
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Let $$P = \begin{bmatrix} -30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}$$ and $$A = \begin{bmatrix} 2 & 7 & \omega^2 \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1 \end{bmatrix}$$ where $$\omega = \frac{-1+i\sqrt{3}}{2}$$, and $$I_3$$ be the identity matrix of order 3. If the determinant of the matrix $$\left(P^{-1}AP - I_3\right)^2$$ is $$\alpha\omega^2$$, then the value of $$\alpha$$ is equal to ________.
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The total number of $$3 \times 3$$ matrices $$A$$ having entries from the set $$\{0, 1, 2, 3\}$$ such that the sum of all the diagonal entries of $$AA^T$$ is 9, is equal to ________.
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Let $$f : (0, 2) \to R$$ be defined as $$f(x) = \log_2\left(1 + \tan\left(\frac{\pi x}{4}\right)\right)$$. Then, $$\lim_{n \to \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \ldots + f(1)\right)$$ is equal to ________.
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If the normal to the curve $$y(x) = \int_0^x (2t^2 - 15t + 10)\,dt$$ at a point $$(a, b)$$ is parallel to the line $$x + 3y = -5, a > 1$$, then the value of $$|a + 6b|$$ is equal to ________.
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Let $$f : R \to R$$ be a continuous function such that $$f(x) + f(x+1) = 2$$ for all $$x \in R$$. If $$I_1 = \int_0^8 f(x)\,dx$$ and $$I_2 = \int_{-1}^3 f(x)\,dx$$, then the value of $$I_1 + 2I_2$$ is equal to ________.
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Let the curve $$y = y(x)$$ be the solution of the differential equation, $$\frac{dy}{dx} = 2(x+1)$$. If the numerical value of area bounded by the curve $$y = y(x)$$ and $$x$$-axis is $$\frac{4\sqrt{8}}{3}$$, then the value of $$y(1)$$ is equal to ________.
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