Explore topic-wise InterviewSolutions in .

Evaluate the following limit:
$\underset{x\to \pi }{\text{lim}}(x-\frac{22}{7})$

If in a triangle a = $\sqrt{3}+1,b=\sqrt{3}-1,C={60}^{\circ }$ then the value of A is ?

Given that $tanA,tanB$ are the roots of the equation $x^2-px+q=0$, then the value of $si{n}^2(A+B)$ is

$\{x\in R:cos2x+2co{s}^{2}x=2\}$ is equal to

The probability that at least one of the events $E_1$ and $E_2$ occurs is 0.6 If the probability of the simultaneous occurrence of $E_1$ and $E_2$ is 0.2, find $P\left({\overline{E}}_1\right)+P\left({\overline{E}}_2\right).$

Let f(x) = $(1+x{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$ $x^2$ + ${}^n{C}_2$ $x^3$ ........${}^n{C}_n$ $x^n$.

If f(1) = $S_1$, f(w) = $S_2$ and $f\left(w^2\right)$ = $S_3$, find the value of ${}^n{C}_0$ + ${}^n{C}_3$ + ${}^n{C}_6$ + .......... in terms of $S_1$, $S_2$ and

$S_3$.[W is the complex cube root of unity]

Express $(1+i{)}^i$ in the A+iB form, Where i = $\sqrt{-1}$

If $\frac{1}{1^2}$ + $\frac{1}{2^2}$ + $\frac{1}{3^2}$ +$\frac{1}{4^2}$ + ........ = $\frac{{\pi }^2}{6}$ then $\frac{1}{1^2}$ + $\frac{1}{3^2}$ +$\frac{1}{5^2}$ + ....... =

Using the principle of mathematical induction prove that $(1^2+2^2+\cdots +n^2)>\frac{n^3}{3}$ for all values of n $ϵ$ N.

Or

Evaluate $\sqrt{16-30i}$

Match the following
Given $sinA=\frac{2}{3}andsinB=\frac{1}{4}$
$\begin{array}{cc}\left(1\right)sin(A+B)& \left(p\right)\frac{2\sqrt{15}-\sqrt{5}}{2+5\sqrt{3}}\\ \left(2\right)Cos(A-B)& \left(q\right)\frac{55}{144}\\ \left(3\right)Tan(A-B)& \left(r\right)\frac{2\sqrt{15}+\sqrt{5}}{12}\\ \left(4\right)Sin(A+B)sin(A-B)& \left(s\right)\frac{5\sqrt{3}+2}{12}\end{array}$

A person standing at the junction (crossing) of two straight paths represented by the equations $2x-3y+4=0$ and $3x+4y-5=0$ wants to reach the path whose equation is $6x-7y+8=0$ in the least time. Find equation of the path that he should follow.

The area of triangle formed by the lines x = 0, y = 0 and $\frac{x}{a}+\frac{y}{b}=1$, a and b are positive , is

$\frac{1}{2}(\frac{3x}{5}+4)\ge \frac{1}{3}(x-6)$.

If the angles of a triangle are in the ratio $4:1:1,$ then the ratio of the longest side to the perimeter is

If one root of $5x^2$ + 13x + k = 0 is reciprocal of the other, then k is equal to :

Let $x_n,y_n,z_n,w_n$ denote $n^{th}$ terms of four different arithmetic progressions with positive terms. If $x_4+y_4+z_4+w_4=8$ and $x_{10}+y_{10}+z_{10}+w_{10}=20$ then the maximum value of $x_{20}\cdot y_{20}\cdot z_{20}\cdot w_{20}$ is

(A) If there is high variability in the distribution of income and wealth of the country then which value is compromised?

(B) Mean and standard deviation of two distributions of 100 and 150 items are 50, 5 and 40, 6 respectively. Find the standard deviation of all the 250 items taken together.

Let $a_1,a_2,\dots ,a_{30}$ be in A.P., $S=\sum _{i=1}^{30}{a}_i$ and $T=\sum _{i=1}^{15}{a}_{2i-1}$. If $a_5=27$ and $S-2T=75$, then the value of $a_{10}$ is

$\text{The value of}\underset{x\to 0}{lim}\frac{1}{x}si{n}^{-1}\left(\frac{2x}{1+x^2}\right)\text{is}$

If $\cos \theta =\frac{-5}{13}$ and $\frac{\pi }{2}<\theta <\pi$ then find the values of

(i) $\sin 3\theta +\sin 5\theta$

(ii) $\tan 3\theta$

Find the sum of the product of the corresponding terms of the sequences 2, 4 ,16 and 32 128, 32, 8 ,2 $\frac{1}{2}$.

5 harmonic means are inserted between 5 and 10. The common difference of corresponding A.P is.

if f(x) = $\frac{1}{x}$ and g(x) = $x^2$ find f(g(x))

Let ‘A’, ‘B’ and ‘C’ be three independent events with $P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{2}$ and $P\left(C\right)=\frac{1}{4}$. The probability of exactly 2 of these events occurring, is equal to:

$\text{In a triangle ABC,}\frac{cosA}{a}=\frac{cosB}{b}=\frac{cosC}{c}\text{If a=}\frac{1}{\sqrt{6}}\text{then the area of the triangle (in square unit) is}$

Let $f\left(x\right)=cosx(sinx+\sqrt{si{n}^{2}x+si{n}^2\theta }),\theta$ is a given const, then max of f(x)is

A complex number z is said to be unimodular, if $|z|=1$. If and $z_1$ and $z_2$ are complex numbers such that $\frac{z_1-2z_2}{2-\left(z_1\overline{z_2}\right)}$ is unimodular and $z_2$ is not unimodular.
Then, the point $z_1$ lies on a

For any two complex number $z_1$ and $z_2$ and any

real numbers a and b; $|(a{z}_1-b{z}_2){|}^2$ + $|(b{z}_1-a{z}_2){|}^2$ =

If a=$\sqrt[2]{22i}$ then which of the following is correct

The distance between two particles is decreasing at the rate of $6\frac{m}{sec}$. If these particles travel with same speeds and in the same direction, then the separation increase at the rate of $4\frac{m}{sec}$. The particles have speeds as

Let $a_1,a_2,a_3,.............$ be in harmonic progression with $a_1=5and{a}_{20}=25.$ The least positive integer n for which $a_n<0$ is

If the digits of the number 12345 are randomly rearranged, what is the probability that the new number is divisible by
(i) 3 (ii) 9 [4 MARKS]

For observations $x_1,x_2,x_3,..........,x_n,$ if ${\sum }_{i=1}^n(x_i+1{)}^2=9n$ and ${\sum }_{i=1}^n(x_i-1{)}^2=5n.$, then standard deviation of the data is

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+...+10(11{)}^9=k(10{)}^9$, then $k$ is equal to :

Find the value of ${\sum }_{r=1}^{10}r\frac{{}^n{C}_r}{{}^n{C}_{r-1}}$

The domain of the function ${\cos }^{-1}(3x-2)$ is

If the 6th, 7th and 8th terms in the expansion of $(x+a{)}^n$ are respectively 112, 7 and , $\frac{1}{4}$, find x, a and n.

Find the number of middle terms in the expansion of $(a+b{)}^{21}$

Find the asymptotes of hyperbola xy - 3y - 2x = 0

$\text{The value of}\underset{x\to 1}{lim}\frac{x^n+x^{n-1}+x^{n-2}+.......+x^2+x-n}{x-1}$

Two masses 90 kg and 160 kg are at a distance 5 m apart. Compute the magnitude of intensity of the gravitational field at a point distance 3m from the 90 kg and 4m from the 160 kg mass. G=6.67 $\times$ ${10}^{-11}$SI units..

$li{m}_{x\to 0}$ $\frac{\left(x^{3}cotx\right)}{1-cosx}$ =


[AI CBSE ; DSSE 1988]

If $A(a{t}^2,2at),B(\frac{a}{t^2},\frac{-2a}{t})$ and C(a, 0), then 2a is equal to

Angles made by the lines represented by the equation xy + y = 0 with y-axis are

If the ratio of the coefficient of third and fourth term in the expansion of $(x-\frac{1}{2x}{)}^n$ is $1:2$, then the value of $n$ will be

In how many ways a team of 10 players out of 22 players can be made if 6 particular players are always to be included and 4 particular players are always excluded.