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The largest value of $$r$$, for which the region represented by the set $$\{\omega \in C | |\omega - 4 - i| \leq r\}$$ is contained in the region represented by the set $$\{z \in C | |z - 1| \leq |z + i|\}$$, is equal to:
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If $$2 + 3i$$ is one of the roots of the equation $$2x^3 - 9x^2 + kx - 13 = 0$$, $$k \in R$$, then the real root of this equation (where $$i^2 = -1$$):
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The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman is
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The value of $$\sum_{r=16}^{30}(r+2)(r-3)$$ is equal to:
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Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is:
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If the coefficient of the three successive terms in the binomial expansion of $$(1 + x)^n$$ are in the ratio 1 : 7 : 42, then the first of these terms in the expansion is
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In a $$\Delta ABC$$, $$\frac{a}{b} = 2 + \sqrt{3}$$, and $$\angle C = 60^\circ$$. Then the ordered pair $$(\angle A, \angle B)$$ is equal to:
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Let $$L$$ be the line passing through the point $$P(1, 2)$$ such that its intercepted segment between the co-ordinate axes is bisected at $$P$$. If $$L_1$$ is the line perpendicular to $$L$$ and passing through the point $$(-2, 1)$$, then the point of intersection of $$L$$ and $$L_1$$ is
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The points $$\left(0, \frac{8}{3}\right)$$, $$(1, 3)$$ and $$(82, 30)$$
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If $$y + 3x = 0$$ is the equation of a chord of the circle $$x^2 + y^2 - 30x = 0$$, then the equation of the circle with this chord as diameter is:
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Let the tangents drawn to the circle, $$x^2 + y^2 = 16$$ from the point $$P(0, h)$$ meet the x-axis at points $$A$$ and $$B$$. If the area of $$\Delta APB$$ is minimum, then positive value of $$h$$ is:
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If the tangent to the conic, $$y - 6 = x^2$$ at $$(2, 10)$$ touches the circle, $$x^2 + y^2 + 8x - 2y = k$$ (for some fixed $$k$$) at a point $$(\alpha, \beta)$$; then $$(\alpha, \beta)$$ is
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An ellipse passes through the foci of the hyperbola, $$9x^2 - 4y^2 = 36$$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $$\frac{1}{2}$$, then which of the following points does not lie on the ellipse?
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$$\lim_{x \to 0} \frac{e^{x^2} - \cos x}{\sin^2 x}$$ is equal to
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The contrapositive of the statement "If it is raining, then I will not come", is
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A factory is operating in two shifts, day and night, with 70 and 30 workers, respectively. If per day mean wage of the day shift workers is Rs. 54 and per day mean wage of all the workers is Rs. 60, then per day mean wage of the night shift workers (in Rs.) is:
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In a certain town, 25% of the families own a phone and 15% own a car; 65% families own neither a phone nor a car and 2000 families own both a car and a phone. Consider the following three statements:
(i) 5% families own both a car and a phone.
(ii) 35% families own either a car or a phone.
(iii) 40000 families live in the town.
Then,
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If $$A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$, then which one of the following statements is not correct?
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The least value of the product $$xyz$$ (such that $$x$$, $$y$$ and $$z$$ are positive real numbers) for which the determinant $$\begin{vmatrix} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{vmatrix}$$ is non-negative is
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If $$f(x) = 2\tan^{-1} x + \sin^{-1}\left(\frac{2x}{1+x^2}\right)$$, $$x > 1$$, then $$f(5)$$ is equal to
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If Rolle's theorem holds for the function $$f(x) = 2x^3 + bx^2 + cx$$, $$x \in [-1, 1]$$ at the point $$x = \frac{1}{2}$$, then $$2b + c$$ is equal to
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The distance from the origin, of the normal to the curve, $$x = 2\cos t + 2t\sin t$$, $$y = 2\sin t - 2t\cos t$$ at $$t = \frac{\pi}{4}$$, is:
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The integral $$\int \frac{dx}{(x+1)^{3/4}(x-2)^{5/4}}$$, is equal to
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For $$x > 0$$, let $$f(x) = \int_1^x \frac{\log t}{1-t} dt$$. Then $$f(x) + f\left(\frac{1}{x}\right)$$ is equal to
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The area (in square units) of the region bounded by the curves $$y + 2x^2 = 0$$ and $$y + 3x^2 = 1$$, is equal to
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If $$y(x)$$ is the solution of the differential equation $$(x + 2)\frac{dy}{dx} = x^2 + 4x - 9$$, $$x \neq -2$$ and $$y(0) = 0$$, then $$y(-4)$$ is equal to
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Let $$\vec{a}$$ and $$\vec{b}$$ be two unit vectors such that $$|\vec{a} + \vec{b}| = \sqrt{3}$$. If $$\vec{c} = \vec{a} + 2\vec{b} + (\vec{a} \times \vec{b})$$, then $$2|\vec{c}|$$ is equal to:
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If the points $$(1, 1, \lambda)$$ and $$(-3, 0, 1)$$, are equidistant from the plane, $$3x + 4y - 12z + 13 = 0$$, then $$\lambda$$ satisfies the equation:
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If the shortest distance between the line $$\frac{x-1}{\alpha} = \frac{y+1}{-1} = \frac{z}{1}$$, $$(\alpha \neq -1)$$, and $$x + y + z + 1 = 0 = 2x - y + z + 3$$ is $$\frac{1}{\sqrt{3}}$$, then value of $$\alpha$$ is:
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Let X be a set containing 10 elements and P(X) be its power set. If A and B are picked up at random from P(X), with replacement, then the probability that A and B have equal number of elements is:
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