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Let $$p(x)$$ be a quadratic polynomial such that $$p(0) = 1$$. If $$p(x)$$ leaves remainder 4 when divided by $$x - 1$$ and it leaves remainder 6 when divided by $$x + 1$$ then:
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Let $$z \in C$$, the set of complex numbers. Then the equation, $$2|z + 3i| - |z - i| = 0$$ represents:
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If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is:
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If the arithmetic mean of two numbers $$a$$ and $$b$$, $$a > b > 0$$, is five times their geometric mean, then $$\frac{a+b}{a-b}$$ is equal to:
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If the sum of the first $$n$$ terms of the series $$\sqrt{3} + \sqrt{75} + \sqrt{243} + \sqrt{507} + \ldots$$ is $$435\sqrt{3}$$, then $$n$$ equals:
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If $$(27)^{999}$$ is divided by 7, then the remainder is:
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The locus of the point of intersection of the straight lines, $$tx - 2y - 3t = 0$$ and $$x - 2ty + 3 = 0$$ ($$t \in R$$), is:
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If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $$\cos^{-1}\left(\frac{1}{7}\right)$$ and $$\sec^{-1}(7)$$ at the center respectively, then the distance between these chords is:
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If the common tangents to the parabola, $$x^2 = 4y$$ and the circle, $$x^2 + y^2 = 4$$ intersect at the point $$P$$, then the distance of $$P$$ from the origin (units), is:
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If a point $$P(0, -2)$$ and $$Q$$ is any point on the circle, $$x^2 + y^2 - 5x - y + 5 = 0$$, then the maximum value of $$(PQ)^2$$ is:
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Consider an ellipse, whose center is at the origin and its major axis is along the $$x$$-axis. If its eccentricity is $$\frac{3}{5}$$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:
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$$\displaystyle\lim_{x \to 3} \frac{\sqrt{3x-3}}{\sqrt{2x-4} - \sqrt{2}}$$ is equal to:
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The proposition $$(\sim p) \vee (p \wedge \sim q)$$ is equivalent to:
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The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is:
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Let $$A$$ be any $$3 \times 3$$ invertible matrix. Then which one of the following is not always true?
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The number of real values of $$\lambda$$ for which the system of linear equations, $$2x + 4y - \lambda z = 0$$, $$4x + \lambda y + 2z = 0$$ and $$\lambda x + 2y + 2z = 0$$, has infinitely many solutions, is:
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If $$S = \left\{x \in [0, 2\pi] : \begin{vmatrix} 0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{vmatrix} = 0 \right\}$$, then $$\displaystyle\sum_{x \in S} \tan\left(\frac{\pi}{3} + x\right)$$ is equal to:
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The value of $$\tan^{-1}\left[\frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}}\right]$$, $$|x| \lt \frac{1}{2}$$, $$x \neq 0$$, is equal to:
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Let $$f(x) = 2^{10}x + 1$$ and $$g(x) = 3^{10}x - 1$$. If $$(fog)(x) = x$$, then $$x$$ is equal to:
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If $$y = \left[x + \sqrt{x^2-1}\right]^{15} + \left[x - \sqrt{x^2-1}\right]^{15}$$, then $$(x^2-1)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$$ is equal to:
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The tangent at the point $$(2, -2)$$ to the curve, $$x^2y^2 - 2x = 4(1-y)$$ does not pass through the point:
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The integral $$\displaystyle\int \sqrt{1 + 2\cot x(\csc x + \cot x)}\,dx$$, $$\left(0 \lt x \lt \frac{\pi}{2}\right)$$ is equal to:
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The integral $$\displaystyle\int_{\pi/12}^{\pi/4} \frac{8\cos 2x}{(\tan x + \cot x)^3}\,dx$$ equals:
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The area (in sq. units) of the smaller portion enclosed between the curves, $$x^2 + y^2 = 4$$ and $$y^2 = 3x$$, is:
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The curve satisfying the differential equation, $$ydx - (x + 3y^2)dy = 0$$ and passing through the point $$(1, 1)$$ also passes through the point:
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The area (in sq. units) of the parallelogram whose diagonals are along the vectors $$8\hat{i} - 6\hat{j}$$ and $$3\hat{i} + 4\hat{j} - 12\hat{k}$$, is:
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The coordinates of the foot of the perpendicular from the point $$(1, -2, 1)$$ on the plane containing the lines $$\frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8}$$ and $$\frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}$$, is:
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The line of intersection of the planes $$\vec{r} \cdot (3\hat{i} - \hat{j} + \hat{k}) = 1$$ and $$\vec{r} \cdot (\hat{i} + 4\hat{j} - 2\hat{k}) = 2$$, is:
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An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:
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Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are $$\frac{3}{4}$$, $$\frac{1}{2}$$ and $$\frac{5}{8}$$ respectively, then the probability that the target is hit by P or Q but not by R is:
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