The integer $$k$$, for which the inequality $$x^2 - 2(3k - 1)x + 8k^2 - 7 > 0$$ is valid for every $$x$$ in $$R$$ is:
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The integer $$k$$, for which the inequality $$x^2 - 2(3k - 1)x + 8k^2 - 7 > 0$$ is valid for every $$x$$ in $$R$$ is:
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Let the lines $$(2 - i)z = (2 + i)\bar{z}$$ and $$(2 + i)z + (i - 2)\bar{z} - 4i = 0$$, (here $$i^2 = -1$$) be normal to a circle $$C$$. If the line $$iz + \bar{z} + 1 + i = 0$$ is tangent to this circle $$C$$, then its radius is:
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The total number of positive integral solutions $$(x, y, z)$$ such that $$xyz = 24$$ is:
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If $$0 < \theta, \phi < \frac{\pi}{2}$$, $$x = \sum_{n=0}^{\infty} \cos^{2n}\theta$$, $$y = \sum_{n=0}^{\infty} \sin^{2n}\phi$$ and $$z = \sum_{n=0}^{\infty} \cos^{2n}\theta \cdot \sin^{2n}\phi$$ then:
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All possible values of $$\theta \in [0, 2\pi]$$ for which $$\sin 2\theta + \tan 2\theta > 0$$ lie in:
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The image of the point (3, 5) in the line $$x - y + 1 = 0$$, lies on:
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A tangent is drawn to the parabola $$y^2 = 6x$$ which is perpendicular to the line $$2x + y = 1$$. Which of the following points does NOT lie on it?
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If the curves, $$\frac{x^2}{a} + \frac{y^2}{b} = 1$$ and $$\frac{x^2}{c} + \frac{y^2}{d} = 1$$ intersect each other at an angle of 90°, then which of the following relations is TRUE?
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$$\lim_{n \to \infty} \left(1 + \frac{1 + \frac{1}{2} + \ldots + \frac{1}{n}}{n^2}\right)^n$$ is equal to
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The statement $$A \to (B \to A)$$ is equivalent to:
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A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30° (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45°. Then the time taken (in seconds) by the boat from B to reach the base of the tower is:
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Let $$f, g : N \to N$$ such that $$f(n + 1) = f(n) + f(1)$$ $$\forall n \in N$$ and $$g$$ be any arbitrary function. Which of the following statements is NOT true?
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If Rolle's theorem holds for the function $$f(x) = x^3 - ax^2 + bx - 4$$, $$x \in [1, 2]$$ with $$f'\left(\frac{4}{3}\right) = 0$$, then ordered pair $$(a, b)$$ is equal to:
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The value of the integral $$\int \frac{\sin\theta \cdot \sin 2\theta (\sin^6\theta + \sin^4\theta + \sin^2\theta)\sqrt{2\sin^4\theta + 3\sin^2\theta + 6}}{1 - \cos 2\theta} d\theta$$ is (where $$c$$ is a constant of integration)
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The value of $$\int_{-1}^{1} x^2 e^{[x^3]} dx$$, where $$[t]$$ denotes the greatest integer $$\leq t$$, is:
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If a curve passes through the origin and the slope of the tangent to it at any point $$(x, y)$$ is $$\frac{x^2 - 4x + y + 8}{x - 2}$$, then this curve also passes through the point:
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Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations $$l + m - n = 0$$ and $$l^2 + m^2 - n^2 = 0$$. Then the value of $$\sin^4\alpha + \cos^4\alpha$$ is:
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The equation of the line through the point (0, 1, 2) and perpendicular to the line $$\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{-2}$$ is:
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The coefficients $$a, b$$ and $$c$$ of the quadratic equation, $$ax^2 + bx + c = 0$$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
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When a missile is fired from a ship, the probability that it is intercepted is $$\frac{1}{3}$$ and the probability that the missile hits the target, given that it is not intercepted, is $$\frac{3}{4}$$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:
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The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is ______
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Let $$A_1, A_2, A_3, \ldots$$ be squares such that for each $$n \geq 1$$, the length of the side of $$A_n$$ equals the length of diagonal of $$A_{n+1}$$. If the length of $$A_1$$ is 12 cm, then the smallest value of $$n$$ for which area of $$A_n$$ is less than one, is ______
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The locus of the point of intersection of the lines $$\left(\sqrt{3}\right)kx + ky - 4\sqrt{3} = 0$$ and $$\sqrt{3}x - y - 4\left(\sqrt{3}\right)k = 0$$ is a conic, whose eccentricity is ______
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If $$A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$$ and $$(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$, then $$13(a^2 + b^2)$$ is equal to ______.
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Let $$A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$$, where $$x, y$$ and $$z$$ are real numbers such that $$x + y + z > 0$$ and $$xyz = 2$$. If $$A^2 = I_3$$, then the value of $$x^3 + y^3 + z^3$$ is ______
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If the system of equations
$$kx + y + 2z = 1$$
$$3x - y - 2z = 2$$
$$-2x - 2y - 4z = 3$$
has infinitely many solutions, then $$k$$ is equal to ______.
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The number of points, at which the function $$f(x) = |2x + 1| - 3|x + 2| + |x^2 + x - 2|$$, $$x \in R$$ is not differentiable, is ______
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Let $$f(x)$$ be a polynomial of degree 6 in $$x$$, in which the coefficient of $$x^6$$ is unity and it has extrema at $$x = -1$$ and $$x = 1$$. If $$\lim_{x \to 0} \frac{f(x)}{x^3} = 1$$, then $$5 \cdot f(2)$$ is equal to ______
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The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $$A$$. Then $$A^4$$ is equal to ______
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Let $$\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$$, $$\vec{b} = \hat{i} - \hat{j}$$ and $$\vec{c} = \hat{i} - \hat{j} - \hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$\vec{r} \cdot \vec{a}$$ is equal to ______
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