If $$\alpha$$ and $$\beta$$ are the roots of the equation $$2x(2x+1) = 1$$, then $$\beta$$ is equal to:
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If $$\alpha$$ and $$\beta$$ are the roots of the equation $$2x(2x+1) = 1$$, then $$\beta$$ is equal to:
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Let $$z = x + iy$$ be a non-zero complex number such that $$z^2 = i|z|^2$$, where $$i = \sqrt{-1}$$, then $$z$$ lies on the:
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The common difference of the A.P. $$b_1, b_2, \ldots, b_m$$ is 2 more than common difference of A.P. $$a_1, a_2, \ldots, a_n$$. If $$a_{40} = -159$$, $$a_{100} = -399$$ and $$b_{100} = a_{70}$$, then $$b_1$$ is equal to:
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If the constant term in the binomial expansion of $$\left(\sqrt{x} - \frac{k}{x^2}\right)^{10}$$ is 405, then $$|k|$$ equals:
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Let $$L$$ denote the line in the $$xy$$-plane with $$x$$ and $$y$$ intercepts as 3 and 1 respectively. Then the image of the point $$(-1, -4)$$ in the line is:
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The centre of the circle passing through the point $$(0, 1)$$ and touching the parabola $$y = x^2$$ at the point $$(2, 4)$$ is:
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If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $$e$$ of the ellipse satisfies:
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Consider the statement: "For an integer n, if $$n^3 - 1$$ is even, then $$n$$ is odd". The contrapositive statement of this statement is:
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The angle of elevation of the summit of a mountain from a point on the ground is $$45^\circ$$. After climbing up one km towards the summit at an inclination of $$30^\circ$$ from the ground, the angle of elevation of the summit is found to be $$60^\circ$$. Then the height (in km) of the summit from the ground is:
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Let $$\theta = \frac{\pi}{5}$$ and $$A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$$. If $$B = A + A^4$$, then $$\det(B)$$:
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For a suitably chosen real constant $$a$$, let a function, $$f : \mathbb{R} - \{-a\} \to \mathbb{R}$$ be defined by $$f(x) = \frac{a-x}{a+x}$$. Further suppose that for any real number $$x \neq -a$$, and $$f(x) \neq -a$$, $$(f \circ f)(x) = x$$. Then $$f\left(-\frac{1}{2}\right)$$ is equal to:
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = \max\{x, x^2\}$$. Let $$S$$ denote the set of all points in $$\mathbb{R}$$, where $$f$$ is not differentiable. Then:
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The set of all real values of $$\lambda$$ for which the function $$f(x) = (1 - \cos^2 x) \cdot (\lambda + \sin x)$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, has exactly one maxima and exactly one minima, is:
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For all twice differentiable functions $$f : \mathbb{R} \to \mathbb{R}$$, with $$f(0) = f(1) = f'(0) = 0$$,
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If the tangent to the curve $$y = f(x) = x\log_e x$$, $$(x > 0)$$ at a point $$(c, f(c))$$ is parallel to the line-segment joining the points $$(1, 0)$$ and $$(e, e)$$, then $$c$$ is equal to:
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The integral $$\int_1^2 e^x \cdot x^x(2 + \log_e x)\,dx$$ equals:
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The area (in sq. units) of the region enclosed by the curves $$y = x^2 - 1$$ and $$y = 1 - x^2$$ is equal to:
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If $$y = \left(\frac{2}{\pi}x - 1\right) \operatorname{cosec} x$$ is the solution of the differential equation, $$\frac{dy}{dx} + p(x)y = -\frac{2}{\pi} \operatorname{cosec} x$$, $$0 < x < \frac{\pi}{2}$$, then the function $$p(x)$$ is equal to:
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A plane P meets the coordinate axes at A, B and C respectively. The centroid of $$\triangle ABC$$ is given to be $$(1, 1, 2)$$. Then the equation of the line through this centroid and perpendicular to the plane P is:
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The probabilities of three events A, B and C are given $$P(A) = 0.6$$, $$P(B) = 0.4$$ and $$P(C) = 0.5$$. If $$P(A \cup B) = 0.8$$, $$P(A \cap C) = 0.3$$, $$P(A \cap B \cap C) = 0.2$$, $$P(B \cap C) = \beta$$ and $$P(A \cup B \cup C) = \alpha$$, where $$0.85 \leq \alpha \leq 0.95$$, then $$\beta$$ lies in the interval:
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The number of words (with or without meaning) that can be formed from all the letters of the word 'LETTER' in which vowels never come together is_____.
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Consider the data on x taking the values $$0, 2, 4, 8, \ldots, 2^n$$ with frequencies $$^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_n$$ respectively. If the mean of this data is $$\frac{728}{2^n}$$, then n is equal to_______.
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The sum of distinct values of $$\lambda$$ for which the system of equations:
$$(\lambda - 1)x + (3\lambda + 1)y + 2\lambda z = 0$$
$$(\lambda - 1)x + (4\lambda - 2)y + (\lambda + 3)z = 0$$
$$2x + (3\lambda + 1)y + 3(\lambda - 1)z = 0$$
Has non-zero solutions, is_______.
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Suppose that a function $$f : \mathbb{R} \to \mathbb{R}$$ satisfies $$f(x+y) = f(x)f(y)$$ for all $$x, y \in \mathbb{R}$$ and $$f(1) = 3$$. If $$\sum_{i=1}^{n} f(i) = 363$$, then $$n$$ is equal to_____.
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If $$\vec{x}$$ and $$\vec{y}$$ be two non-zero vectors such that $$|\vec{x} + \vec{y}| = |\vec{x}|$$ and $$2\vec{x} + \lambda\vec{y}$$ is perpendicular to $$\vec{y}$$, then the value of $$\lambda$$ is_____.
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