Recent questions in 2003

A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration $f$ and in the second part with constant retardation $r$. The value of $t$ is given by:

asked Dec 11, 2018 by pady_1

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Two stones are projected from the top of a cliff $h$ metres high, with the same speed $u$ so as to hit the ground at the same spot. If one of the stones is projected horizontally and the other is projected at an angle $\theta$ to the horizontal, then $\tan \theta$ equals :

asked Dec 11, 2018 by pady_1

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Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\overrightarrow{u}$ and the other from rest with uniform acceleration $\overrightarrow{f}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time :

asked Dec 11, 2018 by pady_1

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A couple is of momen $\overrightarrow{G}$ and the force forming the couple is $\overrightarrow{P}$. If $\overrightarrow{P}$ is turned through a right angle, the moment of the couple thus formed is $\overrightarrow{H}$. If instead, the forcce $\overrightarrow{P}$ is turned through an angle $\alpha$, then the moment of couple becomes :

asked Dec 11, 2018 by pady_1

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Let $R_1$ and $R_2$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $R_1, \;R, \; R_2$ are in :

asked Dec 11, 2018 by pady_1

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The resultant of forces $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$. If $\overrightarrow{Q}$ is doubled, then $\overrightarrow{R}$ is doubled. If the direction of $\overrightarrow{Q}$ is reversed, then $\overrightarrow{R} $ is again doubled, then $p^2 : Q^2 : R^2$ is :

asked Dec 11, 2018 by pady_1

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The mean and variance of a random variable $X$ having a binomial distribution are 4 and 2 respectively, then $P(X=1)$ is :

asked Dec 11, 2018 by pady_1

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Events A, B, C are mutually exclusive events such that $P(A) = \frac{3x +1}{3}, \; P(B) = \frac{1-x}{4} $ and $P(C) = \frac{1-2x}{2}$. The set of possible values of $x$ are in the interval :

asked Dec 11, 2018 by pady_1

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Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probabiliy that Mr. A selected the winning horse, is :

asked Dec 11, 2018 by pady_1

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In an experiment with 15 observations on $x$, the following results were available <br> $\displaystyle\sum x^2 = 2830, \; \displaystyle\sum x = 170$ <br> One observation that was 20, was found to be wrong and was replaced by the correct value 30. Then the corrected variance is :

asked Dec 11, 2018 by pady_1

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The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set :

asked Dec 11, 2018 by pady_1

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A particle acted on by constant forces $4 \hat{i} + \hat{j} - 3 \hat{k}$ and $3 \hat{i} + \hat{j} - \hat{k}$ is displaced from the point $\hat{i} + 2 \hat{j} + 3 \hat{k}$ to the point $5\hat{i} + 4\hat{j} + \hat{k}$. The total work done by the forces is :

asked Dec 11, 2018 by pady_1

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The vectors $\overrightarrow{AB} = 3\hat{i} + 4 \hat{k}$, and $\overrightarrow{AC} = 5 \hat{i} -2\hat{j} + 4 \hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is :

asked Dec 11, 2018 by pady_1

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Consider points A, B, C and D with position vectors $7 \hat{i} - 4 \hat{j} + 7 \hat{k}, \; \hat{i} - 6 \hat{j} + 10 \hat{k}, \; -\hat{i} - 3 \hat{j} + 4 \hat{k}$ and $5 \hat{i} - \hat{j} + 5 \hat{k}$ respectively. Then ABCD is a :

asked Dec 11, 2018 by pady_1

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If $\overrightarrow{u}, \overrightarrow{v} $ and $\overrightarrow{w}$ are three non-coplanar vectors, then $(\overrightarrow{u} + \overrightarrow{v} - \overrightarrow{w}) . [(\overrightarrow{u} - \overrightarrow{v}) \times (\overrightarrow{v} - \overrightarrow{w} ) ] $ equals :

asked Dec 11, 2018 by pady_1

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$\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} $ are three vectors, such that $\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{0}, \; |\overrightarrow{a}| = 1, \; |\overrightarrow{b} |= 2, \; |\overrightarrow{c} | = 3$, then $\overrightarrow{a}. \overrightarrow{b} + \overrightarrow{b}.\overrightarrow{c} + \overrightarrow{c}.\overrightarrow{a}$ is equal to :

asked Dec 11, 2018 by pady_1

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Two systems of rectangular axes have the same origin. If a plane cuts them at distances $a, \; b,\; c$ and $a',\;b',\;c'$ from the origin, then :

asked Dec 11, 2018 by pady_1

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The shortest distance from the plane $12x +4y +3z = 327$ to the sphere $x^2 + y^2 + z^2 + 4x - 2y - 6z = 155$ is :

asked Dec 11, 2018 by pady_1

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The two lines $x = ay + b, \; z = cy + d$ and $x = a' y + b', \; z = c' y + d'$ will be perpendicular, if and only if :

asked Dec 11, 2018 by pady_1

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The lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar if :

asked Dec 11, 2018 by pady_1

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The radius of the circle in which the sphere $x^2 + y^2 + z^2 + 2x - 2y - 4z - 19 = 0$ is cut by the plane $x + 2y + 2z + 7 = 0$ is :

asked Dec 11, 2018 by pady_1

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A tetrahedron has vertices at $O(0, 0, 0),\; A(1,2,1),\; B(2,1,3)$ and $C(-1, 1, 2)$. Then the angle between the faces $OAB$ and $ABC$ will be :

asked Dec 11, 2018 by pady_1

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The foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}$ coincide. Then the value of $b^2$ is :

asked Dec 11, 2018 by pady_1

1 answer

The normal at the point $(bt_1^2, 2bt_1)$ on a parabola meets the parabola again in the point $(bt_2^2 , 2bt_2)$ then :

asked Dec 11, 2018 by pady_1

1 answer

The lines $2x - 3y = 5$ and $3x - 4y =7$ are diameters of a circle having area as $154\; sq. unit$. Then the equation of the circle is :

asked Dec 11, 2018 by pady_1

1 answer

If the two circles $(x-1)^2 + (y-3)^2 = r^2$ and $x^2 + y^2 - 8x + 2y + 8 =0$ intersect in two distinct points, then :

asked Dec 11, 2018 by pady_1

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A square of side $a$ lies above the $x$-axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha ( 0 < \alpha < \frac{\pi}{4})$ with the positive direction of $x$-axis. The equation of its diagonal not passing through the origin is :

asked Dec 11, 2018 by pady_1

1 answer

If the pair of straight line $x^2 -2pxy - y^2 = 0$ and $x^2 - 2qxy - y^2 = 0$ be such that each pair bisects the angle between the other pair, then :

asked Dec 11, 2018 by pady_1

0 answers

Locus of centroid of the triangle whose vertices are $( a \cos t, \; a \sin t), \; (b \sin t, -b \cos t)$ and $(1, 0)$ where $t$ is a parameter, is :

asked Dec 11, 2018 by pady_1

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If the equation of the locus of a point equidistant from the points $(a_1, b_1)$ and $(a_2, b_2)$ is $(a_1 - a_2) x + (b_1 - b_2) y + c = 0$, then the value of $'C'$ is :

asked Dec 11, 2018 by pady_1

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The solution of the differential equation $(1+y^2) + (x - e^{\tan^{-1} y} )\frac{dy}{dx} = 0 $ is :

asked Dec 11, 2018 by pady_1

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The degree and order of the differential equation of the family of all parabolas whose axis is $x$-axis, are respectively :

asked Dec 11, 2018 by pady_1

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Let $f(x)$ be a function satisfying $f'(x) = f(x)$ with $f(0) = 1$ and $g(x)$ be a function that satisfies $f(x) + g(x) = x^2$. Then the value of the integral $\begin{align} \int_0^1 f(x) \; g(x) \;dx \end{align} $ is :

asked Dec 11, 2018 by pady_1

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The area of the region bounded by the curves $y = |x-1| $ and $y=3 - |x|$ is :

asked Dec 11, 2018 by pady_1

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Let $\begin{align}\frac{d}{dx} F(x) =(\frac{e^{\sin x}}{x}), \; x > 0 \end{align}$ <br> If $\begin{align} \int_1^4 \frac{3}{x} e^{\sin x^3} dx = F(k) - F(1) \end{align}$, <br> then one of the possible values of $k$, is

asked Dec 11, 2018 by pady_1

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$\displaystyle\lim_{n \to \infty} \frac{1 + 2^4 + 3^4 + ...+ n^4}{n^5} - \displaystyle\lim_{n \to \infty} \frac{1+2^3 + 3^3 + ....+ n^3}{n^5}$ is :

asked Dec 11, 2018 by pady_1

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The value of the integral $\begin{align}I = \int_0^1 x(1-x)^n dx \end{align}$ is :

asked Dec 11, 2018 by pady_1

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The value of $\displaystyle\lim_{x \to 0} \frac{ \int_0^{x^2} \sec^2t \; dt}{x \sin x}$ is :

asked Dec 11, 2018 by pady_1

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If $f(a + b - x) = f(x) $, then $\int_a^b x f(x) dx$ is equal to :

asked Dec 11, 2018 by pady_1

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If $f(y) = e^y,\; g(y) = y; \; y>0$ and $F(t) = \int^t_0 f(t-y) \;g(y) \;dy$, then :

asked Dec 11, 2018 by pady_1

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If the function $f(x) = 2x^3 - 9 ax^2 + 12 a^2x + 1$, where $a>0$, attains its maximum and minimum at $p$ and $q$ respectively such that $p^2 = q$, then $a$ equals :

asked Dec 11, 2018 by pady_1

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If $ f(x) = \begin{cases} xe^{-\begin{bmatrix}\frac{1}{|x|} - \frac{1}{x}\end{bmatrix}}, & \quad \text{$x$ $\neq$ 0} \text{ then $f(x)$ is :}\\ \; \; \; 0 & \quad \text{$x=0$} \end{cases}$

asked Dec 11, 2018 by pady_1

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The function $f(x) = \log (x +\sqrt{x^2+1})$, is :

asked Dec 11, 2018 by pady_1

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If $f^n(a),g^n(a)$ exist and are not equal for some $n$. Further is $f(a) = g(a) =k$ and $\displaystyle\lim_{x \to a} \frac{f(a)g(x) - f(a) - g(a) f(x) + g(a) }{g(x) - f(x)} = 4$, then the value of $k$ is equal to :

asked Dec 11, 2018 by pady_1

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If $\displaystyle\lim_{x \to 0} \frac{\log(3+x) - \log(3-x)}{x} = k$, the value of $k$ is :

asked Dec 11, 2018 by pady_1

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$\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{[1- \tan(\frac{x}{2})][1 - \sin x]}{[1+\tan(\frac{x}{2})][\pi - 2x]^3}$ is :

asked Dec 11, 2018 by pady_1

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Domain of definition of the function $f(x) = \frac{3}{4-x^2} + \log_{10} (x^3 - x)$, is :

asked Dec 11, 2018 by pady_1

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If $f(x) = x^n$, then the value of $f(1) - \frac{f'(1)}{1!} + \frac{f''(1)}{2!} - \frac{f'''(1)}{3!} + ....+\frac{(-1)^nf^n(1)}{n!} $ is :

asked Dec 11, 2018 by pady_1

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If $f : R \to R$ satisfies $f(x+y) = f(x) + f(y)$, for all $x, \; y \in R$ and $f(1) = 7$, then $\displaystyle\sum_{r=1}^{n} f(r)$ is :

asked Dec 11, 2018 by pady_1

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The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equals to :

asked Dec 11, 2018 by pady_1

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