Explore topic-wise InterviewSolutions in Current Affairs.

A vector $\overrightarrow{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}(\alpha ,\beta \in R)$ lies in the plane of the vectors, $\overrightarrow{b}=\hat{i}+\hat{j}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\overrightarrow{a}$ bisects the angle between $\overrightarrow{b}$ and $\overrightarrow{c},$ then

The point which divides the join of $(1,2)$ and $(3,4)$ externally in the ratio 1 : 1

The equation of the line passing through the point (1, 1, –1) and perpendicular to the plane x – 2y – 3z = 7 is

For all permissible values of $A,2A$, following holds true.
$\left(i\right)\cot A+\tan A=\frac{1}{\sin A\cos A}=2\text{cosec}2A\left(ii\right)\cot A-\tan A=\frac{{\cos }^{2}A-{\sin }^{2}A}{\sin A\cos A}=2\cot 2A\left(iii\right)2\cot A=2(\text{cosec}2A+\cot 2A)\Rightarrow \text{cosec}2A+\cot 2A=\cot A$

Also to evaluate a series of form $f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)$ when $f\left(x\right)$ can be expressed as $g\left(x\right)-g\left(2x\right)$, we can use the following technique,
$f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)=\left(g\right(x)-g(2x\left)\right)+\left(g\right(2x)-g(4x\left)\right)+\cdots \left(g\right(2^{n}x)-g(2^{n+1}x\left)\right)=g\left(x\right)-g\left(2^{n+1}x\right)$

Based on the above information, solve the following questions for all permissible values of $x$.

The value of $\text{cosec}2x+\text{cosec}4x+\text{cosec}8x+\text{cosec}16x+\text{cosec}32x$

If $z^3+(3+2i)z+(-1+ia)=0$ has one real root, then the value of the $a$ lies in the interval $aϵR$

The projection of the line joining the points $(3,4,5)$ and $(4,6,3)$ on the line joining the points $(-1,2,4)$ and $(1,0,5)$ is

Find the angle of intersection of the curves $y=4-x^{2}andy=x^2$

The equation $x^2-3xy+\lambda y^2+3x-5y+2=0$ When $\lambda$ is a real number, represents a pair of straight lines.If $\theta$ is the angle between the lines, then $cose{c}^2\theta =$

If $\alpha ,\beta$ are the two roots of the quadratic polynomial $f\left(x\right)=x^2+\frac{b}{a}x+\frac{c}{a};a\ne 0.$ And if exactly one of the roots lies between $k_1,k_2.$ Then select the correct statements where $\alpha ,\beta \ne k_1,k_2$ .

Check whether the following function is strictly decreasing on $(0,\frac{\pi }{2})$ or not.

$cos3x$

The d.r's of two parallel lines are $4,-3,-1$ and $\lambda +\mu ,1+\mu ,2$ then $(\lambda ,\mu )$ is:

Find the values of m for which the equation $x^4-(1-2m)x^2+(m^2-1)=0$ has three real distinct root

The value of $\sin \frac{17\pi }{36}\cos \frac{11\pi }{36}\cos \frac{13\pi }{36}\sin \frac{11\pi }{36}\sin \frac{13\pi }{36}\cos \frac{17\pi }{36}$ is

The parametric form of the circle $x^2+y^2-4(x+y)=8$ is

The range of f(x) = $\frac{x^2+x+1}{x^2+x-1}$

If $\frac{(a^2+1{)}^2}{2a-i}=x+iy$, then $x^2+y^2$ is equal to

The interval on which the function f(x) = 2x³ + 9x² + 12x - 1 is decreasing is

(a) [-1, ∞) (b) [-2, -1] (c) (∞, -2] (d) [-1, 1]

Let $P$ be a plane $lx+my+nz=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. If pane $P$ divides the line segments $AB$ joining pionts $A(-3,-6,1)$ and $B(2,4,-3)$ in ratio $k:1$ then the value of $k$ is equal to :

Total number of solution of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$ in $x\in [-\pi ,\pi ]$ is :

The number of solutions of $5{\cos }^2\theta -3{\sin }^2\theta +6\sin \theta \cos \theta =7$ in the interval $[0,2\pi ]$ is

Differentiate the following functions with respect to x

sec ($tan\sqrt{x}$)

38. If f(x) and g(x) be two given function with all real numbers as their domain, then h(x) = {f(x)+f(-x)}{g(x)-g(-x)} is

1. Even

2. Odd

3. Even as well as odd

4. None

The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. find the probability that out of 5 such bulbs

(i) none

(ii) not more than one,

will fuse after 150 days of use.

Which of the following defines Range of a given data set.

Solution of the system of equations $x+18=2y,y=2x-9$ is:

An underpass bridge is in the shape of a semi-ellipse. It is $400$ metres long and has a maximum depth of $10$ metres at the middle point. The depth of the bridge at a point which is at a distance of $80$ metres from one end is

A normal at $P\left(\theta \right)$ to the hyperbola $\frac{x^2}{4}-\frac{y^2}{1}=1$ has equal intercepts on the positive $x,y-$axes. If this normal touches the circle $x^2+y^2=r^2,$ then the value of $12r^2$ is

The value of $\underset{n\to \infty }{lim}(\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2-1^2}}+\frac{1}{\sqrt{n^2-2^2}}+\cdots +\frac{1}{\sqrt{n^2-(n-1{)}^2}})$ is