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If $$\frac{1}{\sqrt{\alpha}}$$, $$\frac{1}{\sqrt{\beta}}$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, $$(a \neq 0, a, b \in R)$$, then the equation $$x(x + b^3) + (a^3 - 3abx) = 0$$ has roots:
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If equations $$ax^2 + bx + c = 0$$, $$(a, b, c \in R, a \neq 0)$$ and $$2x^2 + 3x + 4 = 0$$ have a common root, then $$a : b : c$$ equals:
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Let $$w(Im\ w \neq 0)$$ be a complex number. Then, the set of all complex numbers $$z$$ satisfying the equation $$w - \bar{w}z = k(1 - z)$$, for some real number $$k$$, is:
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The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers 3, 4, 5 and 6, without repetition is:
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Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its 4$$^{th}$$ term is:
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If the sum $$\frac{3}{1^2} + \frac{5}{1^2+2^2} + \frac{7}{1^2+2^2+3^2} + \ldots$$ + up to 20 terms is equal to $$\frac{k}{21}$$, then $$k$$ is equal to:
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The number of terms in the expansion of $$(1+x)^{101}(1-x+x^2)^{100}$$ in powers of $$x$$ is:
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If $$\operatorname{cosec} \theta = \frac{p+q}{p-q}$$ ($$p \neq q, p \neq 0$$), then $$\left|\cot\left(\frac{\pi}{4} + \frac{\theta}{2}\right)\right|$$ is equals to:
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The number of values of $$\alpha$$ in $$[0, 2\pi]$$ for which $$2\sin^3\alpha - 7\sin^2\alpha + 7\sin\alpha = 2$$, is:
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Given three points $$P$$, $$Q$$, $$R$$ with $$P(5, 3)$$ and $$R$$ lies on the $$x$$-axis. If the equation of $$RQ$$ is $$x - 2y = 2$$ and $$PQ$$ is parallel to the $$x$$-axis, then the centroid of $$\Delta PQR$$ lies on the line:
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Let $$a$$ and $$b$$ be any two numbers satisfying $$\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4}$$. Then, the foot of perpendicular from the origin on the variable line $$\frac{x}{a} + \frac{y}{b} = 1$$ lies on:
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If the point $$(1, 4)$$ lies inside the circle $$x^2 + y^2 - 6x + 10y + p = 0$$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $$p$$ is the interval:
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If $$OB$$ is the semi-minor axis of an ellipse, $$F_1$$ and $$F_2$$ are its foci and the angle between $$F_1B$$ and $$F_2B$$ is a right angle, then the square of the eccentricity of the ellipse is:
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If $$f(x)$$ is continuous and $$f\left(\frac{9}{2}\right) = \frac{2}{9}$$, then $$\lim_{x \to 0} f\left(\frac{1-\cos 3x}{x^2}\right)$$ equals to:
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The contrapositive of the statement "I go to school if it does not rain" is:
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In a set of $$2n$$ distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then, the mean of the new set of observations:
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Let $$P$$ be the relation defined on the set of all real numbers such that $$P = \{(a, b) : \sec^2 a - \tan^2 b = 1\}$$. Then, $$P$$ is:
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If $$B$$ is a $$3 \times 3$$ matrix such that $$B^2 = 0$$, then $$\det[(I + B)^{50} - 50B]$$ is equal to:
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If $$a$$, $$b$$, $$c$$ are non-zero real numbers and if the system of equations
$$(a-1)x = y + z$$
$$(b-1)y = x + z$$
$$(c-1)z = x + y$$
has a non-trivial solution, then $$ab + bc + ca$$ equals:
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If $$y = e^{nx}$$, then $$\frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2}$$ is equal to:
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If $$f(x) = \left(\frac{3}{5}\right)^x + \left(\frac{4}{5}\right)^x - 1$$, $$x \in R$$, then the equation $$f(x) = 0$$ has:
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If the Rolle's theorem holds for the function $$f(x) = 2x^3 + ax^2 + bx$$ in the interval $$[-1, 1]$$ for the point $$c = \frac{1}{2}$$, then the value of $$2a + b$$ is:
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$$\int \frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x} dx$$ is equal to:
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The integral $$\int_0^{\frac{1}{2}} \frac{\ln(1+2x)}{1+4x^2} dx$$ equals:
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Let $$A = \{(x, y) : y^2 \leq 4x, y - 2x \geq -4\}$$. The area of the region $$A$$ in square units is:
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If the differential equation representing the family of all circles touching $$x$$-axis at the origin is $$(x^2 - y^2)\frac{dy}{dx} = g(x)y$$, then $$g(x)$$ equals:
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If $$|\vec{a}| = 2$$, $$|\vec{b}| = 3$$ and $$|2\vec{a} - \vec{b}| = 5$$, then $$|2\vec{a} + \vec{b}|$$ equals:
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Equation of the plane which passes through the point of intersection of lines $$\frac{x-1}{3} = \frac{y-2}{1} = \frac{z-3}{2}$$ and $$\frac{x-3}{1} = \frac{y-1}{2} = \frac{z-2}{3}$$ and has the largest distance from the origin is:
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A line in the 3-dimensional space makes an angle $$\theta$$ ($$0 < \theta \leq \frac{\pi}{2}$$) with both the X and Y-axes. Then, the set of all values of $$\theta$$ is in the interval:
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If $$A$$ and $$B$$ are two events such that $$P(A \cup B) = P(A \cap B)$$, then the incorrect statement amongst the following statements is:
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