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If $$\alpha$$ and $$\beta$$ are the roots of the equation, $$7x^2 - 3x - 2 = 0$$, then the value of $$\frac{\alpha}{1-\alpha^2} + \frac{\beta}{1-\beta^2}$$ is equal to:
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The value of $$\left(\frac{-1+i\sqrt{3}}{1-i}\right)^{30}$$ is:
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There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:
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If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is:
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If the sum of the first 20 terms of the series $$\log_{(7^{1/2})} x + \log_{(7^{1/3})} x + \log_{(7^{1/4})} x + \ldots$$ is 460, then $$x$$ is equal to:
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If $$L = \sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$$ and $$M = \cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$$
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If the length of the chord of the circle, $$x^2 + y^2 = r^2$$ $$(r > 0)$$ along the line, $$y - 2x = 3$$ is $$r$$, then $$r^2$$ is equal to:
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If the line $$y = mx + c$$ is a common tangent to the hyperbola $$\frac{x^2}{100} - \frac{y^2}{64} = 1$$ and the circle $$x^2 + y^2 = 36$$, then which one of the following is true?
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$$\lim_{x \to 0} \frac{x\left(e^{(\sqrt{1+x^2+x^4}-1)/x} - 1\right)}{\sqrt{1+x^2+x^4} - 1}$$
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The statement $$(p \to (q \to p)) \to (p \to (p \vee q))$$ is:
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If the mean and the standard deviation of the data $$3, 5, 7, a, b$$ are $$5$$ and $$2$$ respectively, then $$a$$ and $$b$$ are the roots of the equation:
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If the system of linear equations
$$x + y + 3z = 0$$
$$x + 3y + k^2z = 0$$
$$3x + y + 3z = 0$$
has a non-zero solution $$(x, y, z)$$ for some $$k \in \mathbb{R}$$, then $$x + \left(\frac{y}{z}\right)$$ is equal to:
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If $$a + x = b + y = c + z + 1$$, where $$a, b, c, x, y, z$$ are non-zero distinct real numbers, then $$\begin{vmatrix} x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c \end{vmatrix}$$ is equal to:
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The derivative of $$\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$$ with respect to $$\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$$ at $$x = \frac{1}{2}$$ is:
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If $$x = 1$$ is a critical point of the function $$f(x) = (3x^2 + ax - 2 - a)e^x$$, then:
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Which of the following points lies on the tangent to the curve $$x^4 e^y + 2\sqrt{y+1} = 3$$ at the point $$(1, 0)$$?
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If $$\int \frac{\cos\theta}{5 + 7\sin\theta - 2\cos^2\theta}\,d\theta = A\log_e|B(\theta)| + C$$, where $$C$$ is a constant of integration, then $$\frac{B(\theta)}{A}$$ can be:
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The area (in sq. units) of the region $$A = \{(x,y) : (x-1)[x] \leq y \leq 2\sqrt{x},\; 0 \leq x \leq 2\}$$, where $$[t]$$ denotes the greatest integer function, is:
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Let $$y = y(x)$$ be the solution of the differential equation $$\cos x\frac{dy}{dx} + 2y\sin x = \sin 2x$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. If $$y(\pi/3) = 0$$, then $$y(\pi/4)$$ is equal to:
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If for some $$\alpha \in \mathbb{R}$$, the lines $$L_1 : \frac{x+1}{2} = \frac{y-2}{-1} = \frac{z-1}{1}$$ and $$L_2 : \frac{x+2}{\alpha} = \frac{y+1}{5-\alpha} = \frac{z+1}{1}$$ are coplanar, then the line $$L_2$$ passes through the point:
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The coefficient of $$x^4$$ in the expansion of $$(1 + x + x^2 + x^3)^6$$ in powers of $$x$$, is.......
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Let $$A = \{a, b, c\}$$ and $$B = \{1, 2, 3, 4\}$$. Then the number of elements in the set $$C = \{f : A \to B \mid 2 \in f(A)$$ and $$f$$ is not one-one$$\}$$ is...
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If the lines $$x + y = a$$ and $$x - y = b$$ touch the curve $$y = x^2 - 3x + 2$$ at the points where the curve intersects the $$x$$-axis, then $$\frac{a}{b}$$ is equal to...
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Let the vectors $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ be such that $$|\vec{a}| = 2$$, $$|\vec{b}| = 4$$ and $$|\vec{c}| = 4$$. If the projection of $$\vec{b}$$ on $$\vec{a}$$ is equal to the projection of $$\vec{c}$$ on $$\vec{a}$$ and $$\vec{b}$$ is perpendicular to $$\vec{c}$$, then the value of $$|\vec{a} + \vec{b} - \vec{c}|$$ is...
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In a bombing attack, there is 50% chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is.....
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