The sum of the solutions of the equation $$\left|\sqrt{x}-2\left|+\sqrt{x}\left(\sqrt{x}-4\right)+2=0\right|\right|$$, $$x > 0$$ is equal to:
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The sum of the solutions of the equation $$\left|\sqrt{x}-2\left|+\sqrt{x}\left(\sqrt{x}-4\right)+2=0\right|\right|$$, $$x > 0$$ is equal to:
If $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2} - 2x + 2 = 0$$, then the least value of $$n$$ for which $$\left(\frac{\alpha}{\beta}\right)^{n} = 1$$ is:
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All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occupy even places is:
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The sum of all natural numbers $$n$$ such that $$100 < n < 200$$ and H.C.F. $$(91, n) > 1$$ is:
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The sum of the co-efficient of all even degree terms in $$x$$ in the expansion of $$\left(x + \sqrt{x^{3} - 1}\right)^{6} + \left(x - \sqrt{x^{3} - 1}\right)^{6}$$, $$x \gt 1$$ is equal to:
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The sum of the series $$2 \cdot {}^{20}C_0 + 5 \cdot {}^{20}C_1 + 8 \cdot {}^{20}C_2 + 11 \cdot {}^{20}C_3 + \ldots + 62 \cdot {}^{20}C_{20}$$ is equal to:
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If $$\cos\alpha + \beta = \frac{3}{5}$$, $$\sin(\alpha - \beta) = \frac{5}{13}$$ and $$0 < \alpha, \beta < \frac{\pi}{4}$$, then $$\tan 2\alpha$$ is equal to:
A point on the straight line, $$3x + 5y = 15$$ which is equidistant from the coordinate axes will lie only in:
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The sum of the squares of the lengths of the chords intercepted on the circle, $$x^{2} + y^{2} = 16$$, by the lines, $$x + y = n$$, $$n \in N$$, where N is the set of all natural numbers is:
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Let $$O(0,0)$$ and $$A(0,1)$$ be two fixed points. Then, the locus of a point P such that the perimeter of $$\triangle AOP$$ is 4 is:
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If the tangents on the ellipse $$4x^{2} + y^{2} = 8$$ at the points (1, 2) and (a, b) are perpendicular to each other, then $$a^{2}$$ is equal to:
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$$\lim_{x \to 0} \frac{\sin^{2}x}{\sqrt{2} - \sqrt{1 + \cos x}}$$ equals:
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The contrapositive of the statement "If you are born in India, then you are a citizen of India", is:
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The mean and variance for seven observations are 8 and 16 respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is:
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Let $$A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$$, $$a \in R$$ such that $$A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$. Then, a value of $$\alpha$$ is:
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The greatest value of $$c \in R$$ for which the system of linear equations $$x - cy - cz = 0$$, $$cx - y + cz = 0$$, $$cx + cy - z = 0$$ has a non-trivial solution, is:
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If $$\alpha = \cos^{-1}\frac{3}{5}$$, $$\beta = \tan^{-1}\frac{1}{3}$$, where $$0 < \alpha, \beta < \frac{\pi}{2}$$, then $$\alpha - \beta$$ is equal to:
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If $$f(x) = \log_e\frac{1-x}{1+x}$$, $$|x| < 1$$, then $$f\left(\frac{2x}{1+x^{2}}\right)$$ is equal to:
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If $$2y=\left(\cot^{-1}\left(\frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x}\right)^{ }\right)^2$$, $$\forall x \in \left(0, \frac{\pi}{2}\right)$$, then $$\frac{dy}{dx}$$ is equal to:
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The shortest distance between the line $$y = x$$ and the curve $$y^{2} = x - 2$$ is:
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If $$S_1$$ and $$S_2$$ are respectively the sets of local minimum and local maximum points of the function, $$f(x) = 9x^{4} + 12x^{3} - 36x^{2} + 25$$, $$x \in R$$, then:
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Let $$f: [0, 2] \rightarrow R$$ be a twice differentiable function such that $$f''(x) > 0$$, for all $$x \in [0, 2]$$. If $$\phi(x) = f(x) + f(2 - x)$$, then $$\phi$$ is:
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$$\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}} dx$$ is equal to:
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If $$f(x) = \frac{2 - x\cos x}{2 + x\cos x}$$ and $$g(x) = \log_e x$$, then the value of the integral $$\int_{-\pi/4}^{\pi/4} g(f(x)) \, dx$$ is:
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The area (in sq. units) of the region $$A = \{(x, y) \in R \times R \mid 0 \le x \le 3, 0 \le y \le 4, y \le x^{2} + 3x\}$$ is:
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Let $$y = y(x)$$ be the solution of the differential equation, $$(x^{2} + 1)^{2}\frac{dy}{dx} + 2x(x^{2} + 1)y = 1$$ such that $$y(0) = 0$$. If $$\sqrt{a} \; y(1) = \frac{\pi}{32}$$, then the value of $$a$$ is:
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The magnitude of the projection of the vector $$2\hat{i} + 3\hat{j} + \hat{k}$$ on the vector perpendicular to the plane containing the vectors $$\hat{i} + \hat{j} + \hat{k}$$ and $$\hat{i} + 2\hat{j} + 3\hat{k}$$, is:
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The length of the perpendicular from the point (2, -1, 4) on the straight line $$\frac{x + 3}{10} = \frac{y - 2}{-7} = \frac{z}{1}$$ is:
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The equation of a plane containing the line of intersection of the planes $$2x - y - 4 = 0$$ and $$y + 2z - 4 = 0$$ and passing through the point (1, 1, 0) is:
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Let $$A$$ and $$B$$ be two non-null events such that $$A \subset B$$. Then, which of the following statements is always correct?
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