Recent questions in 2005

If the pair of lines $ax^2 + 2(a+b) xy + by^2 = 0 $ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector, then :

asked Nov 5, 2018 by pady_1

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The plane $x + 2y - z = 4$ cuts the sphere $x^2 + y^2 + z^2 - x - 2 = 0 $ in a circle of radius :

asked Nov 5, 2018 by pady_1

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The value of $\begin{align} \int_{-\pi}^{\pi} \frac{\cos^2 x}{1+a^x} dx, a >0 \end{align}$, is :

asked Nov 5, 2018 by pady_1

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If the equation $a_n\; x^n + a_{n-1}\; x^{n-1} + ....+ a_1x = 0$, $a_1 \neq 0, n \geq 2$, has a positive root $x=\alpha$, then the equation $n a_n x^{n-1} + (n-1) a_{n-1}\; x^{n-2} + . . . + a_1 = 0$ has a positive root, which is :

asked Nov 5, 2018 by pady_1

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A real valued function $f(x)$ satisfies the functional equation $f(x-y) = f(x) f(y) - f(a-x) f(a+y)$ where $a$ is a given constant and $f(0) = 1, f(2a - x)$ is equal to :

asked Nov 5, 2018 by pady_1

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If $a_1, a_2, a_3, . . . .a_n,...$ are in GP, then the determinant $\Delta = \begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}$ is equal to :

asked Nov 5, 2018 by pady_1

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If both the roots of the quadratic equation $x^2-2kx + k^2+k - 5 = 0$ are less than 5, then $k$ lies in the interval :

asked Nov 5, 2018 by pady_1

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A particle is projected from a point $O$ with velocity $u$ at an angle of $60^{\circ}$ with the horizontal. When it is moving in a direction at right angle to its direction at O, then its velocity is given by :

asked Nov 5, 2018 by pady_1

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Let $x_1, x_2 , . . . . x_n$ be $n$ observation such that $ \Sigma x_i^2 = 400 $ and $ \Sigma x_i = 80$. Then a possible value of $n$ among the following is :

asked Nov 5, 2018 by pady_1

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The sum of the series $1 + \frac{1}{4.2!} + \frac{1}{16.4!} + \frac{1}{64.6!}. . . . \infty$ is :

asked Nov 5, 2018 by pady_1

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$A$ and $B$ are two like parallel forces. A couple of moment $H$ lies in the plane of $A$ and $B$ and is contained with them. The resultant of $A$ and $B$ after combining is displaced through a distance :

asked Nov 5, 2018 by pady_1

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If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} $ are non-coplanar vectors and $\lambda$ is a real number, then $[\lambda(\overrightarrow{a} + \overrightarrow{b}) \;\lambda^2 \overrightarrow{b} \;\lambda \overrightarrow{c} ] = [\overrightarrow{a} \overrightarrow{b} + \overrightarrow{c} \overrightarrow{b} ] $ for :

asked Nov 5, 2018 by pady_1

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Let $a, b$ and $c$ be distinct non-negative numbers. If the vectors $a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}$ and $c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane, then $c$ is :

asked Nov 5, 2018 by pady_1

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Let $\overrightarrow{a} = \hat{i} - \hat{k}, \; \overrightarrow{b} = x \hat{i} + \hat{j} + (1-x) \hat{k}$ and $\overrightarrow{c} = y \hat{i} +x \hat{j} + (1+x -y) \hat{k}$. Then $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ depends on :

asked Nov 5, 2018 by pady_1

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The resultant $R$ of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is :

asked Nov 5, 2018 by pady_1

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A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of $2\; cm/s^2 $ and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. The the lizard will catch the insect after :

asked Nov 5, 2018 by pady_1

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Two points $A$ and $B$ move from rest along a straight line with constant acceleration $f$ and $f'$ respectively. If $A$ takes $m$ sec more than $B$ and describes $'n'$ unit more than $B$ in acquiring the same speed, then :

asked Nov 5, 2018 by pady_1

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A random variable $X$ has Poisson distribution with mean 2. Then $P(X > 1.5)$ equals :

asked Nov 5, 2018 by pady_1

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Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the apply for the same house, is :

asked Nov 5, 2018 by pady_1

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Let $A$ and $B$ be two events such that $P (\overline{A \cup B}) = \frac{1}{6}, ( A \cap B) = \frac{1}{4}$ and $P (\overline{A}) = \frac{1}{4}$, where $\overline{A}$ stands for complement of event $A$. Theus events $A$ and $B$ are :

asked Nov 5, 2018 by pady_1

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The angle between the lines $2x = 3y = -z$ and $6x = -y = -4z$ is :

asked Nov 5, 2018 by pady_1

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If the angle $\theta$ between the line $\frac{x+1}{1} = \frac{y-1}{2} = \frac{z-2}{2}$ and the plane $ 2x - y + \sqrt{\lambda} z+4 = 0$ is such that $\sin \theta = \frac{1}{3}$. The value of $\lambda$ is :

asked Nov 5, 2018 by pady_1

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The locus of a point $P(\alpha, \beta)$ moving under the condition that the line $y = \alpha x+ \beta$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is :

asked Nov 5, 2018 by pady_1

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An ellipse has OB as semi minor axis, $F$ and $F'$ its foci and th angle $FBF'$ is a right angle. Then the eccentricity of the ellipse is :

asked Nov 5, 2018 by pady_1

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If a circcle passes through the point $(a, b)$ and cuts the circle $x^2 + y^2 = p^2$ orthogonally, then the equation of the locus of its centre is :

asked Nov 5, 2018 by pady_1

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A circle touches the $x-axis$ and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is :

asked Nov 5, 2018 by pady_1

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If the circles $x^2 + y^2 + 2ax + cy + a = 0 $ and $x^2 + y^2-3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $Q$, then the line $5x+by-a = 0$ passes through $P$ and $Q$ for :

asked Nov 5, 2018 by pady_1

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If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of a triangle is :

asked Nov 5, 2018 by pady_1

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If non-zero numbers $a, b, c$ are in HP, then the straight line $\frac{x}{a} +\frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is :

asked Nov 5, 2018 by pady_1

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For any vector $\overrightarrow{a}$, the value of $(\overrightarrow{a} \times \hat{i})^2+ (\overrightarrow{a} \times \hat{j})^2 + ( \overrightarrow{a} \times \hat{k})^2$ is equal to :

asked Nov 5, 2018 by pady_1

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The distance between the line $\overrightarrow{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda (\hat{i} - \hat{j} + 4 \hat{k}) $ and the plane $\overrightarrow{r} . (\hat{i} + 5 \hat{j} + \hat{k}) = 5$ is :

asked Nov 5, 2018 by pady_1

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If the plane $2ax - 3ay + 4az + 6 = 0$ passes through the mid point of the line joining the centres of the spheres $x^2 + y^2 +z^2 + 6x - 8y - 2z = 13$ and $ x^2 + y^2 + z^2 - 10x +4y - 2z = 8 $, than $a$ equals :

asked Nov 5, 2018 by pady_1

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The parabolas $y^2 = 4x$ and $x^2 = 4y$ divide the square region bounded by the lines $x = 4, y = 4$ and the co-ordinate axes. If $S_1, S_2, S_3$ are respectively the areas of these parts numbered from top to bottom, then $S_1 : S_2 : S_3$ is :

asked Nov 5, 2018 by pady_1

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The area enclosed between the curve $y= \log_e (x+e)$ and the co-ordinate axes is :

asked Nov 5, 2018 by pady_1

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If $\begin {align}I_1 = \int_0^1 2^{x^2} \; \; dx , I_2 = \int_0^1 2 ^{x^3} \;\; dx, I_3 = \int_1^2 2^{x^2} \; \; dx \; and \; I_4 = \int_1^2 2^{x^3} \;\; dx, \end{align}$ then :

asked Nov 5, 2018 by pady_1

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Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x-axis$ and the ordinates $x=\frac{\pi}{4}$ and $x=\beta > \pi/4$ is $( \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta)$. Then $f(\frac{\pi}{2})$ is :

asked Nov 5, 2018 by pady_1

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Let $f:R \to R$ be a differentiable function having $f(2) = 6, f'(2) =(\frac{1}{48}) $. Then $\displaystyle\lim_{x \to 2} \int_6^{f(x)} \frac{4t^3}{x-2} dt$ equals :

asked Nov 5, 2018 by pady_1

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$\begin{align}\int \left\{\frac{(\log x -1)}{1 + (\log x)^2}\right\}^2 dx \end{align}$ is equal to :

asked Nov 5, 2018 by pady_1

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A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\; cm^2/min$. When the thickness of ice is 15 cm, then the rate at which the thickness of ice decreases, is :

asked Nov 5, 2018 by pady_1

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The line parallel to the $x-axis$ and passing through the intersection of the lines $ax+2by+3b =0$ and $bx-2ay - 3a = 0$, where $(a,b) \neq (0,0)$ is :

asked Nov 5, 2018 by pady_1

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If $x \frac{dy}{dx} = y (\log y - \log x + 1)$, then the solution of the equation is :

asked Nov 5, 2018 by pady_1

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Let $\alpha$ and $\beta$ be the distinct roots of $ax^2 + bx + c = 0 $, then $\displaystyle{\lim_{x \to \alpha} \frac{1- \cos (ax^2 + bx + c)}{(x-\alpha)^2}}$ is equal to :

asked Nov 5, 2018 by pady_1

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A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched ?

asked Nov 5, 2018 by pady_1

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The normal to the curve $x = a (\cos \theta + \theta \sin \theta), y = a (\sin \theta - \theta \cos \theta)$ at any point $'\theta'$ is such that :

asked Nov 5, 2018 by pady_1

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If in a $\Delta ABC$, the altitudes from the vertices $A, B, C$ on opposite sides are in HP, then $\sin A, \sin B, \sin C$ are in :

asked Nov 5, 2018 by pady_1

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If $\cos^{-1} x- \cos^{-1} \frac{y}{2} = \alpha$, then $4x^2 - 4xy \cos \alpha + y^2$ is equal to :

asked Nov 5, 2018 by pady_1

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In a triangle ABC, let $\angle{C} = \pi / 2$, if $r$ is the inradius and $R$ is the circumradius of the triangle ABC, then $2(r+R)$ equals :

asked Nov 5, 2018 by pady_1

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If $x = \displaystyle\sum_{n=0}^{\alpha} a^n,y=\displaystyle\sum_{n=0}^{\alpha} b^n,z= \displaystyle\sum_{n=0}^{\alpha} c^n$ where $a, b, c$ are in AP and $|a| < 1, |b| < 1, |c|<1$, then $x, y, z$ are in :

asked Nov 5, 2018 by pady_1

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If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3/2} - (1+\frac{1}{2}x)^3}{(1-x)^{1/2}}$ may be approximated as :

asked Nov 5, 2018 by pady_1

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If $f$ is a real-valued differentiable function satisfying $|f(x) - f(y)| \leq (x-y)^2, x, y \in R$ and $f(0) = 0$, then $f(1)$ equals :

asked Nov 5, 2018 by pady_1

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