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From the prices of shares X and Y below, find out which is more stable in value :

$\begin{array}{ccccccccccc}X& 35& 54& 52& 53& 56& 58& 52& 50& 51& 49\\ Y& 108& 107& 105& 105& 106& 107& 104& 103& 104& 101\end{array}$

The product of infinite terms in $x^{\frac{1}{2}}.x^{\frac{1}{4}}.x^{\frac{1}{8}}...........\infty$ is

$\underset{n\to \infty }{\overset{}{lim}}$ $\sum _{x=1}^{20}$ $cos{}^{2n}(x-10)$ is equal to

In a town of 10,000 families it was found that 40% family buy newspaper A, 20% buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers, then number of families which buy A only is __.

Match the column

$\begin{array}{cc}\text{Equation}& \text{Name of the curve}\\ \text{1)}{x}^2-2x-y-3=0& \text{P) Circle}\\ \text{2)}{x}^2+3xy+2y^2-x-4y-6=0& \text{Q) Parabola}\\ \text{3)}{x}^2+y^2-20=0& \text{R) Ellipse}\\ \text{4)}7x^2+7y^2+2xy+10x-10y+7=0& \text{S) Hyperbola}\\ \text{5)}6x^2-xy-y^2-23x+4y+15=0& \text{T) Pair of straight lines}\end{array}$

The position vectors of three particles are (i + 2j + 3k), (5i - 2j + 2k) and (3i - 6j + 4k) and their masses are respectively 2 kg, 1 kg and 3 kg. The position vector of a fourth mass of 4 kg, so that the centre of mass of the system may be at the origin, is

If $x+y=\pi +z$, then $si{n}^{2}x+si{n}^{2}y-si{n}^{2}z=$

The fourth term in the expansion of $(1-2x{)}^{\frac{3}{2}}$ will be

If the $m^{th}$ term of a H.P. be n and $n^{th}$ be m, then the $r^{th}$ term will be

$\overrightarrow{A}=2\hat{i}+4\hat{j}+6\hat{k}$

$\overrightarrow{B}=-4\hat{i}+3\hat{k}Find\overrightarrow{A}.\overrightarrow{B}$

Evaluate $\underset{x\to 2}{lim}\frac{x^{10}-1024}{x^5-32}$

The end points of latus rectum of the parabola $x^2=4ay$ are

The coefficient of $x^6$ in $\left\{\right(1+x{)}^6+(1+x{)}^7+.....+(1+x{)}^{1}5\}$is

Given n observations x₁, x₂ ......x_n and their central tendency a, the mean deviation about a, M.D(a) =

Find the number of middle terms in the expansion of $(a+b{)}^{20}$.

$1+\frac{2}{3}.\frac{1}{2}+\frac{2.5}{3.6}{\left(\frac{1}{2}\right)}^2.+\frac{\mathrm{2.5.8}}{\mathrm{3.6.9}}{\left(\frac{1}{2}\right)}^3$ + .... =

In a survey of 300 android mobile users, who make video calls, 75 people said they use only Viber, 45 use only Skype and 90 people use both. The number of persons who use neither Viber nor skype is __.

If the graph of a $x^2$ + bx + c is given as

Then the graph of a ${(x-h)}^2$ + b ${(x-h)}^2$ + c = 0

Where h > 0, will be

The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection, is

$\frac{d}{dx}(x{.2}^x-1)=$

If $(A\cup B)=(A\cap B)$ then prove that A=B

Prove that:

${\text{cot}}^2\frac{\pi }{6}+\text{cosec}\frac{5\pi }{6}+3{\text{tan}}^2\frac{\pi }{6}=6$

Let $T_n$ be the $n^{\text{th}}$ term and $S_n$ be the sum of $n$ terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots n\text{terms}$. Then which of the following is/are true?

Let z and w be two complex numbers such that

$|z|\le 1$, $|w|\le 1$ and |z+iw|= |z- i$\overline{w}$|= 2. Then z is equal to

Prove the following:

= $\frac{\text{cos}(\pi +x)\text{cos}(-x)}{\text{sin}(\pi -x)cos(\frac{\pi }{2}+2)}=co{t}^{2}x$

Draw the curve $y=\left[sinx\right]$

A value of $\theta$ for which $z=\frac{2+3isin\theta }{1-2isin\theta }$ is purely imaginary, is

If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

If $z_1\ne 0,z_2\ne 0,|z_1|$ = $|z_2|$ and arg $z_1+$ arg $z_2$ = $\pi$, then:

Through the vertex O of the parabola $y^2=4ax$ two chords OP & OQ are drawn and the circles on OP &OQ as diameter intersect in R. If ${\theta }_1$,${\theta }_2$ & $\varphi$ are the inclinations of the tangents at P & Q on the parabola and the line through O, R respectively, then the value of $cot{\theta }_1+cot{\theta }_2$ is

Find the ${10}^{th}$ term and sum of the 10 terms of the series $1,2,2^2,2^3,2^4....$

find the range of rational expression $y=\frac{x^2+34x-71}{x^2+2x-7}$ if x is real

Suppose A1,A2,.....................A30 are thirty sets each having 5 elements and B1,B2,................Bn are n sets each with 3 elements, let $\bigcup _{i=1}^{30}$$A_i$=$\bigcup _{j=1}^n$$B_j$ = S and each element of S belongs to exactly 10 of $A_i$'S and exactly 9 of the $B_j$'S. Then n is equal to

$\frac{d}{dx}\left\{co{s}^{-}1(\frac{4x^3}{27}-x)\right\}=$

If ${}^n{C}_r$ = ${}^n{C}_{r-1}$ and ${}^n{P}_r$ = ${}^n{P}_{r+1}$, then the value of n is

The equation of the directrices of the ellipse $16x^2+25y^2=400$ are

The statement P(n) = $4^n-1\ge 3^n$ is true for which of the following options?

Let $f:R\to R$ be such that for all $x\in R,(2^{1+x}+2^{1-x}),f\left(x\right)$ and $(3^x+3^{-x})$ are in A.P., then the minimum value of $f\left(x\right)$ is:

Find the equation of a line whose perpendicular distance from the origin is $\sqrt{8}$ units and the angle between the positive direction of the x-axis and the perpendicular is ${135}^{\circ }$.

Fill in the blanks:

(i) The x-axis and y-axis taken together determine a plane known as ___

(ii) The coordinates of points in the XY-plane are of the form ___

(iii) Coordinate planes divide the space into ___ octants

How many elements has $P\left(A\right),$ if $A=\varphi ?$

if $(1+x+\frac{1}{x}{)}^4={\sum }_{r_1+r_2+r_3}=4\frac{4!}{r_1!\times r_2!\times r_3!}\times (1{)}^{r_1}\left(x{)}^{r_2}\right(\frac{1}{x}{)}^{r_3}$ how many values can $r_1$ take so that the terms in the expansion will be independent of x.

$\underset{x\to 0}{lim}\frac{tan2x-x}{3x-sinx}$

e and $e^1$ are the eccentricities of the hyperbolas $16x^2-9y^2=144$ and $9x^2-16y^2$= - 144 then e - $e^1$ =