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If the equations $$x^2 + bx - 1 = 0$$ and $$x^2 + x + b = 0$$ have a common root different from $$-1$$, then $$|b|$$ is equal to:
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The point represented by $$2 + i$$ in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $$2\sqrt{2}$$ units in the south-west wards direction. Then its new position in the Argand plane is at the point represented by:
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If the four letter words (need not be meaningful) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is R and the fourth letter is E, then the total number of all such words is:
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Let $$x$$, $$y$$, $$z$$ be positive real numbers such that $$x + y + z = 12$$ and $$x^3y^4z^5 = (0.1)(600)^3$$. Then $$x^3 + y^3 + z^3$$ is equal to
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The value of $$\sum_{r=1}^{15} r^2 \left(\frac{^{15}C_r}{{}^{15}C_{r-1}}\right)$$ is equal to:
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For $$x \in R$$, $$x \neq -1$$, if $$(1+x)^{2016} + x(1+x)^{2015} + x^2(1+x)^{2014} + \ldots + x^{2016} = \sum_{i=0}^{2016} a_i x_i$$, then $$a_{17}$$ is equal to
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If $$m$$ and $$M$$ are the minimum and the maximum values of $$4 + \frac{1}{2}\sin^2 2x - 2\cos^4 x$$, $$x \in R$$, then $$M - m$$ is equal to:
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The number of $$x \in [0, 2\pi]$$ for which $$\left|\sqrt{2\sin^4 x + 18\cos^2 x} - \sqrt{2\cos^4 x + 18\sin^2 x}\right| = 1$$ is:
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If a variable line drawn through the intersection of the lines $$\frac{x}{3} + \frac{y}{4} = 1$$ and $$\frac{x}{4} + \frac{y}{3} = 1$$, meets the coordinate axes at A and B, $$(A \neq B)$$, then the locus of the midpoint of AB is:
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The point $$(2, 1)$$ is translated parallel to the line $$L : x - y = 4$$ by $$2\sqrt{3}$$ units. If the new point $$Q$$ lies in the third quadrant, then the equation of the line passing through $$Q$$ and perpendicular to $$L$$ is
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A circle passes through $$(-2, 4)$$ and touches the y-axis at $$(0, 2)$$. Which one of the following equations can represent a diameter of this circle?
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If the tangent at a point on the ellipse $$\frac{x^2}{27} + \frac{y^2}{3} = 1$$ meets the coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is
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Let $$a$$ and $$b$$ respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation $$9e^2 - 18e + 5 = 0$$. If $$S(5, 0)$$ is a focus and $$5x = 9$$ is the corresponding directrix of this hyperbola, then $$a^2 - b^2$$ is equal to
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If $$f(x)$$ is a differentiable function in the interval $$(0, \infty)$$ such that $$f(1) = 1$$ and $$\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1$$, for each $$x \gt 0$$, then $$f\left(\frac{3}{2}\right)$$ is equal to
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If $$\lim_{x \to \infty} \left(1 + \frac{a}{x} - \frac{4}{x^2}\right)^{2x} = e^3$$, then $$a$$ is equal to
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Consider the following two statements:
$$P$$: If 7 is an odd number, then 7 is divisible by 2.
$$Q$$: If 7 is a prime number, then 7 is an odd number.
If $$V_1$$ is the truth value of the contrapositive of $$P$$ and $$V_2$$ is the truth value of contrapositive of $$Q$$, then the ordered pair $$(V_1, V_2)$$ equals
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If the mean deviation of the numbers $$1, 1+d, \ldots, 1+100d$$ from their mean is 255, then a value of $$d$$ is:
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If $$P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$$, $$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ and $$Q = PAP^T$$, then $$P^T Q^{2015} P$$ is:
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The number of distinct real roots of the equation, $$\begin{vmatrix} \cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\ \sin x & \sin x & \cos x \end{vmatrix} = 0$$ in the interval $$\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$$ is:
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For $$x \in R$$, $$x \neq 0$$, $$x \neq 1$$, let $$f_0(x) = \frac{1}{1-x}$$ and $$f_{n+1}(x) = f_0(f_n(x))$$, $$n = 0, 1, 2, \ldots$$. Then the value of $$f_{100}(3) + f_1\left(\frac{2}{3}\right) + f_2\left(\frac{3}{2}\right)$$ is equal to:
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If the function $$f(x) = \begin{cases} -x, & x < 1 \\ a + \cos^{-1}(x+b), & 1 \leq x \leq 2 \end{cases}$$ is differentiable at $$x = 1$$, then $$\frac{a}{b}$$ is equal to
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The minimum distance of a point on the curve $$y = x^2 - 4$$ from the origin is
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If the tangent at a point P, with parameter $$t$$, on the curve $$x = 4t^2 + 3$$, $$y = 8t^3 - 1$$, $$t \in R$$, meets the curve again at a point Q, then the coordinates of Q are:
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If $$\int \frac{dx}{\cos^3 x \sqrt{2\sin 2x}} = (\tan x)^A + C(\tan x)^B + k$$, where k is a constant of integration, then $$A + B + C$$ equals
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If $$2\int_0^1 \tan^{-1} x\,dx = \int_0^1 \cot^{-1}(1 - x + x^2)\,dx$$, then $$\int_0^1 \tan^{-1}(1 - x + x^2)\,dx$$ is equal to
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The area (in sq. units) of the region described by $$A = \{(x, y) | y \geq x^2 - 5x + 4, x + y \geq 1, y \leq 0\}$$ is
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In a triangle $$ABC$$, right angle at vertex $$A$$, if the position vectors of $$A$$, $$B$$ and $$C$$ are respectively $$3\hat{i} + \hat{j} - \hat{k}$$, $$-\hat{i} + 3\hat{j} + p\hat{k}$$ and $$5\hat{i} + q\hat{j} - 4\hat{k}$$, then the point $$(p, q)$$ lies on a line:
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The shortest distance between the lines $$\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$$ and $$\frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4}$$, lies in the interval:
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The distance of the point $$(1, -2, 4)$$ from the plane passing through the point $$(1, 2, 2)$$ and perpendicular to the planes $$x - y + 2z = 3$$ and $$2x - 2y + z + 12 = 0$$, is:
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If A and B are any two events such that $$P(A) = \frac{2}{5}$$ and $$P(A \cap B) = \frac{3}{20}$$, then the conditional probability, $$P(A|(A' \cup B'))$$, where $$A'$$ denotes the complement of A, is equal to:
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