Explore topic-wise InterviewSolutions in Current Affairs.

If the equation ax^2+bx+c=0 does not have 2 distinct real roots and a+c>b,

then prove that f(x)>=0,for all x belongs to real number.

The radical axis of the circles $x^2+y^2+4x-6y-12=0$ and $x^2+y^2+2x-2y-1=0$ divides the line segment joining the centres of the circles in the ratio

The complete set of values of 'k' for which $\frac{x^2-x}{1-kx}$ attains all real values is

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of 1 kg food is given below.

$\begin{array}{cccc}Food& VitaminA& VitaminB& VitaminC\\ X& 1& 2& 3\\ Y& 2& 2& 1\end{array}$

1 kg of food X costs of Rs. 16 and 1 kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet ?

Let $f\left(x\right)=\{\begin{array}{cc}a{\cot }^{-1}\left(\frac{b+x}{4}\right),& \frac{-2}{3}<x<0\\ 2,& x=0\\ \frac{\ln (1-cx)}{x},& 0<x<\frac{2}{3}\end{array}.$

If the function $f$ is differentiable at $x=0,$ then the value of $b^2-2a+c^6$ is

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n,$ then
the value of $\underset{0\le r<s\le n}{\sum \sum }(r+s)C_r{C}_s$ is

$={lim}_{n\to \infty }[\frac{1}{n}+\frac{1}{\sqrt{n^2+n}}+\frac{1}{\sqrt{n^2+2n}}+\cdots +\frac{1}{\sqrt{n^2+(n-1)n}}]$ is equal to [RPET 2000]

Write the first five terms of the following sequence and obtain the corresponding series:

Find the approximate value of f(5.001), where $f\left(x\right)=x^3-7x^2+15$.

The equations of the tangents to the ellipse $x^2+16y^2=16,$ each one of which makes an angle of ${60}^{\circ }$ with the $x-$axis, is

The greatest and least values of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ are

Let $f,g$ and $h$be real valued functions defined on the interval [0,1] by$f\left(x\right)=e^{x^2}+e^{-x^2},g\left(x\right)=x{e}^{x^2}+e^{-x^2}$ and $h\left(x\right)=x^2{e}^{x^2}+e^{-x^2}$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0,1], then

The vectors $\overrightarrow{a}=\hat{i}+\hat{j}+m\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}+(m+1)\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+m\hat{k}$ are coplanar, if $m$ is equal to

Let A $(\frac{1}{2},0),B(\frac{3}{2},0),C(\frac{5}{2},0)$ be the given points and P be a point satisfying max (PA+PB,PB+PC)< 2.

P lies inside

Show that the points $(-2,3,5),(1,2,3)$ and $(7,0,-1)$ are collinear.

The probability density functions of two independent random variables X and Y are given by,

Where a, b are positive real constants and u(•) represents the unit step function. The probability density functibn of the random variable Z = X + Y will be

$f_x\left(x\right)=a{e}^{-ax}u\left(x\right)and{f}_y\left(y\right)=b{e}^{-by}u\left(y\right)$

Let $f\left(x\right)=\{\begin{array}{c}(x-1{)}^{\frac{1}{2-x}},x>1,x\ne 2\\ \\ k,x=2\end{array}$
The value of $k$ for which $f$ is continuous at $x=2$ is :

If $\sum _{i=1}^{20}{\left(\frac{{}^{20}{C}_{i-1}}{{}^{20}{C}_i+{}^{20}{C}_{i-1}}\right)}^3=\frac{k}{21},$ then $k$ equals :

What is the equation of normal to the ellipse

$\frac{x^2}{25}+\frac{y^2}{16}=2\text{at}(5,4).$

Consider $f\left(x\right)=1+\underset{0}{\overset{2x}{\int }}\frac{e^{\frac{t}{2}}\cdot f\left(\frac{t}{2}\right)\left(2t\right)}{\sqrt{16-t^4}}dt-\underset{x}{\overset{0}{\int }}f\left(t\right)\cdot e^t{\sin }^{-1}\left(t^2\right)dt$ and $h\left(x\right)=\sin \left(e^{-x}\ln \right(f\left(x\right)\left)\right)$.

Then the range of $y=h\left(x\right)+4x+5$ is

A die is thrown. Describe the following events:

(i) A: a number less than 7 (ii) B: a number greater than 7 (iii) C: a multiple of 3

(iv) D: a number less than 4 (v) E: an even number greater than 4 (vi) F: a number not less than 3

Also find

There are four men and six women on the city councils. If one council member is selected for a committee at random, how likely is that it is a women?

$A=\{x:x\in R,x^2=16\text{and}2x=6\}$ can be represented in the roster form as _________ .

If $co{t}^{-1}\left(0.2\right)=c,then{\tan }^{-1}(-5)=$

The acute angle between the common tangents of two circles $x^2+y^2=25$ and $(x-10{)}^2+y^2=100$ is

1. 10

If $\frac{dy}{dx}=\frac{2^{x}y+2^y\cdot 2^x}{2^x+2^{x+y}{\log }_e{}^2},$ $y\left(0\right)=0,$ then for $y=1$, the value of $x$ lies in the interval

Sum of the series $\sum _{r=1}^n\frac{1}{(ar+b)(ar+a+b)}$, (where $a\ne 0$) is

Let $f\left(x\right)=e^{2x+1}$ and $g\left(x\right)=\ln x.$ Then $(f\circ g)\left(x\right)$ is



[2 marks]

Let $A+2B=\left|\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right|$ and $2A-B=\left|\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right|.$ It $Tr\left(A\right)$ denotes the sum of all diagonal elements of the matrix $A,$ then $T_r\left(A\right)–T_r\left(B\right)$ has value equal to:

If $(n-2)x^2+8x+(n+4)<0$, $\forall x\in R$, find the range of $n.$

The random
variable X has probability distribution P(X) of the following form,
where k is some number:

(a) Determine the value of k.

(b) Find
P(X < 2), P(X ≥ 2), P(X ≥ 2).

If $f:[0,\pi /2)\to R$ is defined as
$f\left(\theta \right)=\left|\begin{array}{ccc}1& \tan \theta & 1\\ -\tan \theta & 1& \tan \theta \\ -1& -\tan \theta & 1\end{array}\right|$. Then the range of $f$ is

The relation on the set $A=\{x:1<|x|\le 4,x\in Z\}$ is defined by $R=\left\{\right(x,y):y=|x|\}.$ Then the number of elements in the power set of $R$ is