Recent questions and answers in 2015

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $|\overrightarrow{a} +\overrightarrow{b} |= \sqrt{3}$. If $\overrightarrow{c} = \overrightarrow{a} + 2\overrightarrow{b} + 3(\overrightarrow{a} \times \overrightarrow{b})^2$, then $2|\overrightarrow{c}|$ is equal to:

answered Apr 7, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 7, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 7, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 7, 2020 by priyanka.clay6

1 answer

If $f(x)=2 \tan^{-1}x+\sin^{-1} (\frac{2x}{1+x^2}),\; x>1$, then $f(5)$ is equal to:

answered Apr 7, 2020 by priyanka.clay6

1 answer

The contrapositive of the statement "If it is raining, then I will not come", is :

answered Apr 7, 2020 by priyanka.clay6

1 answer

If the shortest distance between the lines $\frac{x-1}{\alpha} = \frac{y+1}{-1} = \frac{z}{1},\; (\alpha \neq -1)$ and $x+y+z+1=0=2x-y+z+3$ is $\frac{1}{\sqrt{3}}$, then a value of $\alpha$ is :

answered Apr 3, 2020 by priyanka.clay6

1 answer

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 ?

answered Apr 3, 2020 by priyanka.clay6

1 answer

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:

answered Apr 3, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 2, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 2, 2020 by priyanka.clay6

1 answer

The points $(0, \frac{8}{3})\; (1,3)$ and $(82, 30)$

answered Apr 2, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 2, 2020 by priyanka.clay6

1 answer

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 :

answered Apr 2, 2020 by priyanka.clay6

1 answer

The area (in square units) of the region bounded by the curves $y + 2x^2 =0$ and $y+3x^2 =1$, is equal to :

answered Apr 2, 2020 by priyanka.clay6

1 answer

For $x>0$, let $f(x) = \begin{align} \int_1^x \frac{\log t}{1+t} dt \end{align}$. Then $f(x) + f(\frac{1}{x})$ is equal to :

answered Apr 2, 2020 by priyanka.clay6

1 answer

The integral $\begin{align} \int \frac{dx}{(x+1)^{3/4} (x-2)^{5/4}}\end{align}$ is equal to:

answered Apr 1, 2020 by priyanka.clay6

1 answer

Let the tangents drawn to the circle, $x^2+y^2 = 16$ from the point $P(0,h)$ meet the $x$-axis at point A and B. If the area of $\Delta APB$ is minimum, then $h$ is equal to :

answered Apr 1, 2020 by priyanka.clay6

1 answer

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 $2b+c$ equals :

answered Apr 1, 2020 by priyanka.clay6

1 answer

The distance, from the origin of the normal to the curve, $x = 2 \cos t + 2 t \sin t,\; y = 2\sin t - 2t \cos t$ at $ t = \frac{\pi}{4}$, is:

answered Apr 1, 2020 by priyanka.clay6

1 answer

$\displaystyle{\lim_{x \to 0} \frac{e^{x^2}-\cos x}{\sin^2x}}$ is equal to :

answered Apr 1, 2020 by priyanka.clay6

1 answer