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Let $$\alpha$$ and $$\beta$$ be two real roots of the equation $$(k + 1)\tan^2 x - \sqrt{2} \cdot \lambda \tan x = (1 - k)$$, where $$k(\neq -1)$$ and $$\lambda$$ are real numbers. If $$\tan^2(\alpha + \beta) = 50$$, then a value of $$\lambda$$ is
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If $$\text{Re}\left(\frac{z-1}{2z+i}\right) = 1$$, where $$z = x + iy$$, then the point $$(x, y)$$ lies on a
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Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appears, is
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Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $$-\frac{1}{2}$$, then the greatest number amongst them is
The greatest positive integer $$k$$, for which $$49^k + 1$$ is a factor of the sum $$49^{125} + 49^{124} + \ldots + 49^2 + 49 + 1$$, is
If $$y = mx + 4$$ is a tangent to both the parabolas, $$y^2 = 4x$$ and $$x^2 = 2by$$, then $$b$$ is equal to
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If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is
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For two statements $$p$$ and $$q$$, the logical statement $$(p \rightarrow q) \wedge (q \rightarrow \sim p)$$ is equivalent to
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Let $$\alpha$$ be a root of the equation $$x^2 + x + 1 = 0$$ and the matrix $$A = \frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha^4 \end{bmatrix}$$, then the matrix $$A^{31}$$ is equal to
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If the system of linear equations
$$2x + 2ay + az = 0$$
$$2x + 3by + bz = 0$$
$$2x + 4cy + cz = 0$$,
where $$a, b, c \in R$$ are non-zero and distinct; has a non-zero solution, then
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If $$g(x) = x^2 + x - 1$$ and $$(g \circ f)(x) = 4x^2 - 10x + 5$$, then $$f\left(\frac{5}{4}\right)$$ is equal to
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If $$y(\alpha) = \sqrt{2\left(\frac{\tan\alpha + \cot\alpha}{1+\tan^2\alpha}\right) + \frac{1}{\sin^2\alpha}}$$, $$\alpha \in \left(\frac{3\pi}{4}, \pi\right)$$, then $$\frac{dy}{d\alpha}$$ at $$\alpha = \frac{5\pi}{6}$$ is
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Let $$x^k + y^k = a^k$$, $$(a, k > 0)$$ and $$\frac{dy}{dx} + \left(\frac{y}{x}\right)^{\frac{1}{3}} = 0$$, then $$k$$ is
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Let the function $$f : [-7, 0] \rightarrow R$$ be continuous on $$[-7, 0]$$ and differentiable on $$(-7, 0)$$. If $$f(-7) = -3$$ and $$f'(x) \le 2$$ for all $$x \in (-7, 0)$$, then for all such functions $$f$$, $$f(-1) + f(0)$$ lies in the interval
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If $$f(a + b + 1 - x) = f(x)$$, for all $$x$$, where $$a$$ and $$b$$ are fixed positive real numbers, then $$\frac{1}{a+b}\int_a^b x(f(x) + f(x + 1))dx$$ is equal to
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The area of the region (in sq. units), enclosed by the circle $$x^2 + y^2 = 2$$ which is not common to the region bounded by the parabola $$y^2 = x$$ and the straight line $$y = x$$, is
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If $$y = y(x)$$ is the solution of the differential equation, $$e^y\left(\frac{dy}{dx} - 1\right) = e^x$$ such that $$y(0) = 0$$, then $$y(1)$$ is equal to
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A vector $$\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k}$$ $$(\alpha, \beta \in R)$$ lies in the plane of the vectors, $$\vec{b} = \hat{i} + \hat{j}$$ and $$\vec{c} = \hat{i} - \hat{j} + 4\hat{k}$$. If $$\vec{a}$$ bisects the angle between $$\vec{b}$$ and $$\vec{c}$$, then
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Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is
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An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value k when k consecutive heads are obtained for $$k = 3, 4, 5$$, otherwise X takes the value $$-1$$. Then the expected value of X, is
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If the sum of the coefficients of all even powers of $$x$$ in the product $$(1 + x + x^2 + \ldots + x^{2n})(1 - x + x^2 - x^3 + \ldots + x^{2n})$$ is 61, then $$n$$ is equal to
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Let $$A(1, 0)$$, $$B(6, 2)$$ and $$C\left(\frac{3}{2}, 6\right)$$ be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point $$\left(-\frac{7}{6}, -\frac{1}{3}\right)$$, is
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$$\lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{-x/2} - 3^{1-x}}$$ is equal to
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If the variance of the first $$n$$ natural numbers is 10 and the variance of the first $$m$$ even natural numbers is 16, then the value of $$m + n$$ is equal to
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Let S be the set of points where the function $$f(x) = |2 - |x - 3||$$, $$x \in R$$, is not differentiable. Then $$\sum_{x \in S} f(f(x))$$ is equal to
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