Recent questions tagged 2004

If $t_1$ and $t_2$ are the times of flight of two particles having the same velocity $u$ and range $R$ on the horizontal, then $t^2_1 + t^2_2$ is equal to

asked Nov 5, 2018 by pady_1

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A velocity $\frac{1}{4}\;m/s$ is resolved into two components along OA and OB making angle $30^{\circ}$ and $45^{\circ}$ respectively with the given velocity. Then the component along OB is

asked Nov 5, 2018 by pady_1

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A particle moves towards east from a point A to a point B at the rate of $4\; km/h$ and then towards north from B to C at the rate of $%\; km/h$. If $AB = 12\; km$ and $BC = 5\; km$, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively

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Three forces $\overrightarrow{P}, \overrightarrow{Q}$ and $\overrightarrow{R}$ acting along $IA, \; IB$ and $IC$, where $I$ is the incentre of a $\Delta ABC$, are in equilibrium. Then $\overrightarrow{P} : \overrightarrow{Q} : \overrightarrow{R}$ is

asked Nov 5, 2018 by pady_1

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In a right angle $\Delta ABC, \; \angle{A}=90^{\circ}$ and sides a, b, c respectively, 5 cm, 4 cm and 3 cm. If a force $\overrightarrow{F}$ has moments 0, 9 and 16 in N cm units respectively about vertices A, B and C, then magnitude of $\overrightarrow{F}$ is

asked Nov 5, 2018 by pady_1

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With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then the forces are

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The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is

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A random variable $X$ has the probability distribution :

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The probability that A speaks truth is $\frac{4}{5}$, while this probability for B is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact is

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In a series of 2n observations, half of them equal a and remaining half equal -a. If the standard deviation of the observations is 2, then |a| equals

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Consider the following statements : (a) Mode can be computed from histogram (b) Median is not independent of change of scale (c) Variance is independent of change of origin and scale. Which of these is/are correct ?

asked Nov 5, 2018 by pady_1

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Let $\overline{a},\; \overline{b}$ and $\overline{c}$ be non-zero vectors such that $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3} |\overline{b}| |\overline{c}| \overline{a}$. If $\theta$ is the acute angle between the vectors $\overline{b}$ and $\overline{c}$, then $\sin \theta$ equals

asked Nov 5, 2018 by pady_1

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Let $\overline{u}, \; \overline{v},\; \overline{w}$ be such that $|\overline{u}| = 1, \; |\overline{v}|=2, \; |\overline{w}|=3$ If the projection $\overline{v}$ along $\overline{u}$ is equal to that of $\overline{w}$ along $\overline{u}$ and $\overline{v}\; , \overline{w}$ are perpendicular to each other then $|\overline{u} - \overline{v} + \overline{w}|$ equals

asked Nov 5, 2018 by pady_1

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If $\overline{a}, \overline{b}, \overline{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\overline{a} + 2 \overline{b} + 3 \overline{c}, \; \lambda \overline{b} + 4 \overline {c}$ and $(2 \lambda -1) \overline{c}$ are non-coplanar for

asked Nov 5, 2018 by pady_1

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A particle is acted upon by constant forces $4 \hat{i} +\hat{j}-3 \hat{k}$ and $3 \hat{i} + \hat{j} - \hat{k}$ which displace it from a point $\hat{i} + 2 \hat{j} + 3 \hat{k}$ to the point $5 \hat{i} + 4 \hat{j} + \hat{k}$. The work done in standard units by the forces is given by

asked Nov 5, 2018 by pady_1

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Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\overrightarrow{a} + 2 \overrightarrow{b} $ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b} + 3 \overrightarrow{c}$ is collinear with $\overrightarrow{a}$ ($\lambda$ being some non-zero scalar) then $\overrightarrow{a} + 2\overrightarrow{b} + 6 \overrightarrow{c}$ equals

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The intersection of the spheres $x^2+y^2+z^2+7x-2y-z=13$ and $x^2+y^2+z^2 -3x+3y+4z=8$ is the same as the intersection of one of the sphere and the plane

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If the straight lines $x=1+s,\;y = -3 -\lambda s, \; z = 1 + \lambda s$ and $x= \frac{t}{2},\; y=1+t,\;z=2-t$ with parameters s and t respectively, are co-planar then $\lambda$ equals

asked Nov 5, 2018 by pady_1

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A line with direction cosines proportional to 2, 1, 2 meets each of the lines $x=y+a=z$ and $x+a=2y=2z$. The co-ordinates of each of the point of intersection are given by

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Distance between two parallel planes $2x+y+2z=8$ and $4x+2y+4z+5=0$ is

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A line makes the same angle $\theta$, with each of the $x$ and $z$-axis. If the angle $\beta$, which it makes with $y-axis$, is such that $\sin^2 \beta = 3 \sin^2 \theta$, then $\cos^2 \theta$ equals

asked Nov 5, 2018 by pady_1

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The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x=4$, then the equation of the ellipse is

asked Nov 5, 2018 by pady_1

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If $a \neq 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay$, then

asked Nov 5, 2018 by pady_1

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The intercept on the line $y=x$ by the circle $x^2 + y^2 -2x=0$ is AB. Equation of the circle on AB as a diameter is

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If the lines $2x +3y+1=0$ and $3x-y-4=0$ lie along diameters of a circle of circumference $10 \pi$, then the equation of the circle is

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A variable circle passes through the fixed point $A(p, q)$ and touches x-axis. The locus of the other end of the diameter through A is

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If a circle passes through the point (a, b) and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is

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If one of the lines given by $6x^2-xy+4cy^2=0$ is $3x+4y=0$, then $c$ equals

asked Nov 5, 2018 by pady_1

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If the sum of the slopes of the lines given by $x^2 - 2cxy -7y^2 =0$ is four times their product, then $c$ has the value

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The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

asked Nov 5, 2018 by pady_1

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Let $A(2, -3) $ and $B(-2, 1)$ be vertices of a triangle ABC. If the centroid of this triangle moves on the line $2x+3y=1$, then the locus of the vertex C is the line

asked Nov 5, 2018 by pady_1

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The solution of the differential equation $ y \;dx + (x +x^2y) dy=0$ is

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The differential equation for the family of curves $x^2 + y^2 - 2ay = 0$, where $a$ is an arbitary constant is

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The area of the region bounded by the curves $y = |x-2|, \; x=1, \;x=3$ and the $x-axis$ is

asked Nov 5, 2018 by pady_1

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If $ \begin{align} f(x) = \frac{e^x}{1+e^x}, \; I_1 = \int_{f(-a)}^{f(a)} xg \{ x (1-x) \} dx \end{align} $ and $\begin{align}I_2 = \int_{f(-a)}^{f(a)} g \{x(1-x)\} \end{align}$ then the value of $\frac{I_2}{I_1}$ is

asked Nov 5, 2018 by pady_1

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If $\begin{align} \int_0^{\pi} x f (\sin x) dx = A \int_0^{\pi/2} f(\sin x) dx \end{align}$, then A is

asked Nov 5, 2018 by pady_1

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The value of $\begin{align} I = \int_0^{\pi/2} \frac{(\sin x + \cos x )^2}{\sqrt{1+\sin 2x}} \;dx \end{align}$ is

asked Nov 5, 2018 by pady_1

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The value of $\begin{align} \int_{-2}^{3} |1 - x^2| dx \end{align}$ is

asked Nov 5, 2018 by pady_1

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$\begin{align} \frac{dx}{\cos x - \sin x} \end{align}$ is equal to

asked Nov 5, 2018 by pady_1

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If $\begin{align} \int \frac{\sin x}{\sin(x-\alpha)} dx = Ax + B \log \sin (x-\alpha) + C \end{align}$, then value of $(A, B)$ is

asked Nov 5, 2018 by pady_1

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$\begin{align} \displaystyle \lim_{n \to \infty} \displaystyle\sum_{r=1}^{n} \frac{1}{n} e^{^{\frac{r}{n}}} \end{align}$ is

asked Nov 5, 2018 by pady_1

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If $2a + 3b + 6c = 0$, then at least one root of the equation $ax^2 + bx +c =0$ lies in the interval

asked Nov 5, 2018 by pady_1

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The normal to the curve $x = a (1+ \cos \theta)\; y = a \; sin \theta$ at $'\theta'$ always passes through the fixed point

asked Nov 5, 2018 by pady_1

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A function $y = f(x)$ has a second order derivative $f"(x) = 6 (x=1)$. If its graph passes through the point $(2, 1)$ and at that point the tangent to the graph is $y = 3x -5$, then the function is

asked Nov 5, 2018 by pady_1

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A point on the parabola $y^2 =18 x$ at which the ordinate increases at twice the rate of the abscissa is

asked Nov 5, 2018 by pady_1

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If $ \begin{align} x = e^{{y+e}^{^{y+...to\; \alpha}}}, \; x >0 \end{align}$, then $\frac{dy}{dx}$ is

asked Nov 5, 2018 by pady_1

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Let $f(x) = \frac{1- \tan x}{4x - \pi}, \; x \neq \frac{\pi}{4} , \; x \in [0, \frac{\pi}{2}]$. If $f(x)$ is continuous in $[0, \frac{\pi}{2}]$, then $f(\frac{\pi}{4})$ is

asked Nov 5, 2018 by pady_1

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If $\displaystyle\sum_{x \to \alpha} \begin{pmatrix}1 + \frac{a}{x} + \frac{b}{x^2} \end{pmatrix}^{2x} = e^2 $ then the values of $a$ and $b$, are

asked Nov 5, 2018 by pady_1

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The domain of the function $f(x) = \frac{ \sin^{-1} (x-3) }{\sqrt{9-x^2}}$ is

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The graph of the function $y = f(x)$ is symmetrical about the line $x=2$, then

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