Explore topic-wise InterviewSolutions in Current Affairs.

Show that lines:

$\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}+\hat{k})\overrightarrow{r}=4\hat{j}+2\hat{k}+\mu (\widehat{2i}-\hat{j}+\widehat{3k})$

are coplanar. Also, find the equation of the plane containing these lines.

The equation of common tangent(s) to the hyperbola $9x^2-16y^2=144$ and circle $x^2+y^2=9$ is/are

The equation of the straight line passing through the point of intersection of the lines $x-y=1$ and $2x-3y+1=0$ and parallel to the line $3x+4y=14$ is

If the abscissa and ordinates of two points $P$ and $Q$ are the roots of the equations $x^2+2ax-b^2=0$ and $x^2+2px-q^2=0$ respectively, then the equation of the circle with $PQ$ as diameter is

Reduce the lines 3 x - 4 y + 4 = 0 and 2 x + 4 y - 5 = 0 to the normal form and hence find which line is nearer to the origin.

Write the maximum value of sin (cos x).

How many elements are there in the reflexive relation defined on the set A = {1}?

In a class of 50 students, 30 students play cricket and 30 students play football and everyone plays at least one of these sports and no one plays any other sport. A relation is defined on the set of students such that aRb if ‘a’ and ‘b’ play a same sport. How many equivalence classes will be formed by this relation

If $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ determine the vertices of a triangle, show that $\frac{1}{2}[\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ are collinear. Also, find the vector normal to the plane of the triangle.

Maximize $Z=5x+3y$, subject to constraints $3x+5y\le 15,5x+2y\le 10,x\ge 0$ and $y\ge 0$.

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin:

2x +3y +4z -12=0

3y +4z -6 =0

x +y +z =1

Find the integrals of the functions.
$\int \frac{cos2x+2si{n}^{2}x}{co{s}^{2}x}dx.$

Integrate the rational functions.
$\int \frac{1}{x^2-9}dx$

Find the vector and Cartesian equations of the plane,

that passes through the point (1,4,6) and the normal to the plane is $\hat{i}-2\hat{j}+\hat{k}.$

If $\overrightarrow{a}=\hat{i}+\hat{j}+2\hat{k}\text{and}\overrightarrow{b}=2\hat{i}+\hat{j}-2\hat{k},$ then find the unit vector in the direction of $2\overrightarrow{a}-\overrightarrow{b}$

Let $E$ be an standard ellipse (center at the origin and major axis as $x-$axis) whose length of major axis is $2a$ and length of minor axis is $2b$. Let $C$ be a circle with centre at origin and cutting the ellipse at an angle $\alpha$, then the value of radius of $C$ for which the angle $\alpha$ is maximum is

There are 7 chocolates be distributed among 10 children subject to condition that child can take any number of chocolates. Find the required number of ways to do this if chocolates are identical.

If ${\log }_{0.2}(x-1)>{\log }_{0.04}(x+5)$ then

A line having slope 1 is drawn from a point A (-3,0) cuts a curve y=x​​​​​x^{​​​​​2}+x+1 at P and Q . Find l(AP) and l(AQ).

If $C_r$ stands for ${}^n{C}_r$ then the sum of the series
$\frac{2\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)}{n!}[C_0^2-2C_1^2+3C_2^2-\cdots +(-1{)}^n(n+1)C_n^2]$, where n is an even positive integer is equal to

Solve the solution: $1+4+7+10..........................+x=287$

The smallest positive $x$ satisfying the equation ${\cos }^{3}3x+{\cos }^{3}5x=8{\cos }^{3}4x\cdot {\cos }^{3}x$ is

Find domain and range of f(x) = - |x|

The equation x- y = 4 and $x^2+4xy+y^2=0$ represent the sides of

α, β are the roots of the equation $x^2$ + 14x + 10 = 0. Find the value of (${\alpha }^2$ + ${\beta }^2$).

The value of coefficient of variation of the first $7$ natural numbers is

In a $\Delta ABC,A=\frac{2\pi }{3},b-c=3\sqrt{3}$ cm and Area$\left(\Delta ABC\right)$$=\frac{9\sqrt{3}}{2}c{m}^2$ then a is

If $x_1,x_2$ are the roots of $a{x}^2$ + bx + c = 0 and $x_1+d,x_2+d$ are the roots of $p{x}^2$ + qx + r = 0, d $\ne$ 0 then

If $\tan {25}^{\circ }$ and $\tan {20}^{\circ }$ are the roots of $x^2+2px+q=0$, then the value of $2p-q$ is equal to

Show that the function $f:R\to R$ defined by $f\left(x\right)=\frac{x}{x^2+1},\forall xϵR$ is neither one-one nor onto. Also, if $g:R\to R$ is defined as $g\left(x\right)=2x-1$, find $fog\left(x\right)$.

The value of $\frac{C_1}{2}+\frac{C_3}{4}+\frac{C_5}{6}+\cdots$ equals to

The rank of the matrix, $\begin{array}{c}A=\left[\begin{array}{cccc}2& 3& 1& 4\\ 0& 1& 2& -1\\ 0& -2& -4& 2\end{array}\right]is\end{array}$

Prove that:

$\frac{sin2\theta }{1-cos2\theta }=cot\theta$

Tangent to a curve intercepts the y-axis at a point $P$. A line perpendicular to this tangent through $P$ passes through another point (1,0) the differential equation of the curve is

if i = $\sqrt{-1}$ , then $1+i^2+i^3+i^6+i^8$ is equal to

Let A be a square matrix of order n and B be its adjoint, then for a scalar K, $|AB+K{I}_n|$ is ___

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ where $x_1<0$ and $x_2>0$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1$ suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.
The orthocentre of $\Delta F_{1}MN$ is

From the following information, show the salaries item in the Income and Expenditure Account for the year ending 31st March, 2015 and in the Balance Sheet as at 31st March, 2014 and 31st March, 2015 :

An Extract of Receipts and Payments Account for the year ending 31st March, 2015
$\begin{array}{cccc}\text{Receipts}& \text{Rs}& \text{Payments}& \text{Rs}\\ & & \text{Salaries}& 2,00,000\end{array}$

Additional Information

Rs

(i) Salaries Outstanding on 31st March, 2014 18,000

(ii) Salaries Outstanding on 31st March, 2015 24,000

(iii) Salaries paid in advance on 31st March 2014 10,000

(iv) Salaries paid in advance on 31st March, 2015 15,000

The solution of the equation $\frac{dy}{dx}-e^{x-y}+x^2{e}^{-y}$ is
[MP PET 2004]

Let $f:R\to R$ be the function defined by $f\left(x\right)=2x-3,\forall x\in R$. Write $f^{-1}$.

Find the coordinates of the foot of perpendicular drawn from the point A(-1, 8, 4) to the line joining the points B(0,-1,3) and C(2, -3, -1). Hence find the image of point A in line BC.

If A =$\left[\begin{array}{cc}\propto & 2\\ 2& \propto \end{array}\right]$ and $|A^3|$ = 125 then $\propto$ =