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An auto mobile dealer provides motor cycles and scooters in 3 body patterns and 4 different colours each. The number of choices open to customer is

The sum $\sum _{k=1}^{20}k\frac{1}{2^k}$ is equal to :

The equation of the internal bisector of $\angle$BAC of $\Delta$ABC with vertices A(5, 2), B(2, 3) and C(6, 5), is

A circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to the circle?

If sinx + cosx = $\frac{1}{5}$ then the value of cos2x

From the data given below, find the no of items (N):$\sum xy=120,r=0.5$, standard deviation of Y = 8,$\sum x^2=90$, where x and y are deviations from arithmetic mean.

$\int e^{ta{n}^{-1}x}(1+x+x^2).d\left(co{t}^{-1}x\right)$ is equal to

What is the equation of a curve given by the parametric form $x=9+6sec\theta ;y=-2-4tan\theta$.

Let A = {1,2,3}, B = {1,3,5}. A relation R:A $\to$ B is

defined by R = {(1,3),(1,5),(2,1)}. Then $R^{-1}$ is defined by

Number of ways of selection of 8 letters from 24 letters of which 8 are a, 8 are b and the rest unlike, is given by

The number of four letter words that can be formed using the letters of the word $BARRACK$ is

If ${}^n{\text{C}}_4,{}^n{\text{C}}_5$ and ${}^n{\text{C}}_6$ are in A.P., then $n$ can be :

The conjugate of the complex number $\frac{(2+3i)}{4i}$ is _____.

The point on $X-$ axis at a distance of $10$ units from $(6,10)$ is

For x > 0, Let $A=\left[\begin{array}{ccc}x+\frac{1}{x}& 0& 0\\ 0& x& 0\\ 0& 0& 16\end{array}\right]B=\left[\begin{array}{ccc}\frac{5x}{x^2+1}& 0& 0\\ 0& \frac{3}{x}& 0\\ 0& 0& \frac{1}{4}\end{array}\right]$

$X=(AB{)}^{-1}+(AB{)}^{-2}+(AB{)}^{-3}+...\infty$

$Z=X^{-1}-2I$ (I is identity matrix of order 3)

$\begin{array}{cc}\text{(P) minimum value of [Tr(Ax)]}is& \left(1\right)24\\ \text{(when [.])}\to \text{represent integer function}& \\ \text{(Q) det}\left(X^{-1}\right)is& \left(2\right)12\\ \text{(R) If}Tr(z+z^2+---+z^{10})=2^a+b,& \\ (a,b\in N)\text{then a + b is}& \left(3\right)6\\ \text{(S) If value of}|adj\left(\sqrt{5}{X}^{-1}\right)|=k& \\ \text{then number of positive divisors}& \left(4\right)19\\ \text{of k which are odd is}& \end{array}$

If the sum of odd terms and the sum of even terms in the expansion of ${(x+a)}^n$ are p and q respectively then
$p^2+q^2$

In the expansion of ${(1-x-x^2+x^3)}^6,$ the coefficient of $x^7$ is

k = $\underset{x\to \infty }{lim}\left(\frac{\sum _{k=1}^{1000}(x+k{)}^m}{x^m+{10}^{1000}}\right)$ is (m > 101)

The distance between the points ($\cos$ q, $\sin$ q) and ($\sin$ q, – $\cos$ q) is ___.

The standard deviation of a variate x is $\sigma$. Then the standard deviation of the variate $\frac{ax+b}{c}$ where a, b, c are constants, is

Which of the following is/are true?

1) $(7\times 241+3{)}^{2608}=7k+3^{2608}$

2) $(7\times 372+4{)}^{1609}=7m+4^{1609}$

k and m are positive integers

The total number of matrices

$A=\left[\begin{array}{ccc}0& 2y& 1\\ 2x& y& -1\\ 2x& -y& 1\end{array}\right],(x,y\in R,x\ne y)$



for which $A^{T}A=3I_3$ is :

Let $S=\{x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2}\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x+\text{cosec}x+2(\tan x-\cot x)=0$ in the set $S$ is equal to:

The equations of the assymptotes of the hyperbola $3x^2+10\mathrm{xy}+8y^2+14x+22y+7=0$ are .

The set of real values for which the expression ${\log }_{0.1}\left({\log }_2\left(\frac{x^2+1}{|x-1|}\right)\right)$ is defined,

A circular wire of diameter 10cm is cut and placed along the circumference of a circle of diameter 1 metre. The angle subtended by the wire at the centre of the circle is equal to

Locus of complex number $z$ if $z,i$ and $iz$ are collinear is

The value of $\sum _{r=0}^{10}$ $co{s}^3$ $\frac{\pi r}{3}$ is equal to

If tangents to the curve $y=\frac{x^4}{4}+\frac{a{x}^3}{3}+\frac{a{x}^2}{2}+x+1,xϵR$

always lie below the curve, then range of $a$ is

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) $A\times (B\cap C)=(A\times B)\cap (A\times C)$

(ii) $A\times C$ is a subset of $B\times D.$

$\hat{i}$ and $\hat{j}$ are unit vector along x and y-axis respectively What is the component of a vector

A = 2 $\hat{i}$ + 3$\hat{j}$ along the direction $\hat{i}$ + $\hat{j}$?

In ( $\sqrt[3]{32}+\frac{1}{\sqrt[3]{33}}{)}^n$ if the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}$, then n =

The remainder when $3^{100}$ is divided by $100$ is

The coordinates of a particle moving in a plane are given by x(t) = a cos(pt) and y(t) = a sin (pt)

where a, and p are positive contants of appropriate dimensions. Then,

If $G$ be the geometric mean of $x$ and $y,$ where $x,y>0,$ then the value of $\frac{1}{G^2-x^2}+\frac{1}{G^2-y^2}$ is

Two finite sets have $m$ and $n$ number of elements. The total number of subsets of the first set is $56$ more than the total number of subsets of the second set. Then $m$ and $n$ are

Show that if $A\subset B$ then $C-B\subset C-A.$

$If\frac{si{n}^{4}A}{a}+\frac{co{s}^{4}A}{b}=\frac{1}{a+b},thenthevalueof\frac{si{n}^{8}A}{a^3}+\frac{co{s}^{8}A}{b^3}isequalto$