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Let PS, be the median of the triangle with vertices P(2,2), Q(6,-1) and R(7,3) . Find the equation of line passing through (-1,1) and parallel to PS.

Find four numbers forming a geometric progression in which the third term is greater than the first term by 9 and the second term is greater than by $4^{th}$ by a18.

Let $S\subset (0,\pi )$ denote the set of values of x satisfying the equation $8^{1+|cosx|+co{s}^{2}x+|co{s}^{3}x|+...to\infty }=4^3$ .Then, S=

Find

$\underset{x\to 1}{\text{lim}}$ f(x) where f(x)= $\{\begin{array}{cc}{x}^2-1,& x\le 1\\ -x^2-1& x>1\end{array}$

Let A(2, 1, -3) and B(5, -8, 3) be two given points. Find the coordinates of the points of trisection of the line segment AB.

Find the mean deviation about the mean for the given data.

$\begin{array}{cccccc}{x}_i& 5& 10& 15& 20& 25\\ f_i& 7& 4& 6& 3& 5\end{array}$

$y=cos{x}^3,then\frac{dy}{dx}$=?

if $sinx+\sqrt{3}cosx=\sqrt{2}$ then x is

The (x, y) coordinates of the corners of a square plate are (0, 0), (L, 0), (L, L) and (0, L). The edges of the plate are clamped and transverse standing waves are set up in it. If u(x, y) denotes the displacement of the plate at the point (x, y) at some instant of time, the possible expression(s) for u is(are) (a = positive constant)

$f\left(x\right)=\frac{1}{x+|x-1|},g\left(x\right)=\frac{1}{x+|x+1|}$

Let 0 < A,B < $\frac{\pi }{2}$ satisfying the equation $3si{n}^{2}A+2si{n}^{2}B=1$ and 3sin2A - 2sin2B = 0 then A + 2B is equal to

The coefficient of two consecutive terms in the

expansion of $(1+x{)}^n$ will be equal, if

If x is real, the expression $\frac{x+2}{2x^2+3x+6}$ takes all value in the interval

Find the acute angle (in degree) between the pair of straight lines $4x^2+14xy+6y^2=0$

For each n ∈ N, the correct statement is

In a triangle ABC, if a = 3, b = 4, c = 5, then the distance between its incentre and circumcentre is

Determine the domain and range of the relation R defined by:

$R=\left\{\right(x,x+5):xϵ\{0,1,2,3,4,5\left\}\right\}$

The term independent of x in the expansion of ${(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}})}^{10}$ is

If every pair of the equations $x^2+px+qr=0,x^2+qx+rp=0,x^2+rx+pq=0$ have a common root, then the sum of three common roots is

Match the statements of Column I with values of Column II
$\begin{array}{cc}ColumnI& ColumnII\\ \left(A\right)\int \frac{e^{2x}-2e^x}{e^{2x}+1}dx=Aln(e^{2x}+1)+Bta{n}^{-1}\left(e^x\right)+c& \left(p\right)A=-\frac{1}{2},B=-\frac{1}{4}\\ \left(B\right)\int \sqrt{x+\sqrt{x^2+2}}dx=A\{x+\sqrt{x^2+2}{\}}^{\frac{3}{2}}+\frac{B}{\sqrt{x+\sqrt{x^2+2}}}+c& \left(q\right)A=\frac{1}{2},B=-2\\ \left(C\right)\int \frac{cos8x-cos7x}{1+2cos5x}dx=Asin3x+Bsin2x+c& \left(r\right)A=\frac{1}{3},B=-2\\ \left(D\right)\int \frac{lnx}{x^3}dx=A\frac{lnx}{x^2}+\frac{B}{x^2}+c& \left(s\right)A=\frac{1}{3},B=-\frac{1}{2}\end{array}$

For natural number n, $2^n$(n-1) ! < $n^n$, if

Find the 20th term of the series $2\times 4+4\times 6+6\times 8+\dots \dots +n$ terms.

Minimum value of $5si{n}^2\theta +4co{s}^2\theta$ is

Find the Standard Deviation from the given data.

$\begin{array}{cccccc}\text{S.No}& 1& 2& 3& 4& 5\\ \text{X}& 10& 20& 30& 40& 50\end{array}$

If A lies in the third quadrant and 3 tan A - 4 = 0, then 5 sin 2A + 3 sin A + 4 cos A =

If the probability of the occurrence of a certain event E is $\frac{3}{11}$, find

(i) The odds in favour of its occurrence. and

(ii) The odds against its occurrence.

Express the complex numbers in the form of a + ib:

$(1-i)-(-1+6i)$

If the latus rectum of an hyperbola be 8 and eccentricity be $\frac{3}{\sqrt{5}}$ , then the equation of the hyperbola is

If ${}^n{P}_4$ = 24,${}^n{C}_5$, then the value of n is

Which of the following points lie on the x-z plane?

$\underset{x\to \infty }{lim}\frac{(2+x{)}^{40}(4+x{)}^5}{(2-x{)}^{45}}$

Which of the following gives a description about standard and mean deviation.

The equation of the locus of foot of perpendiculars drawn from the origin to the line passing through a fixed point (a,b), is

The combined equation of the bisectors of the angle between the lines represented by $(x^2+y^2)\sqrt{3}=4xy$ is

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from ${60}^{\circ }$ what is $A^{\prime }$?

What is the point of contact between the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and

in tangent $y=mx\pm \sqrt{a^2{m}^2-b^2}$.

The ratio of rates of diffusion of $S{O}_2,O_{2}andC{H}_4$ is __.

If $y=\frac{(x^2+2x)}{(3x-4)},then\frac{dy}{dx}=?$

Find the general solutions of $3ta{n}^{2}x-1=0$

A set contains (2n+1) elements. The number of sub-sets of the set which

contain at most n elements is

The locus of the mid-point of the line segment joining the focus ot a moving point on the parabola $y^2$ = 4ax is another parabola with directrix

The sum of two numbers is $\frac{13}{6}$.

An even number of AMs are being inserted between them. The sum of means inserted exceeds the number of means by one. Find the number of AMs inserted.

Or

Find the sum of infinite series. $\frac{1}{3}+\frac{1}{5^2}+\frac{1}{3^2}+\frac{1}{5^4}+\frac{1}{3^3}+\frac{1}{5^6}+.......$

If the equation of the locus of a point equidistant from the points $(a_1,b_1)and(a_2,b_2)is(a_1-a_2)x+(b_1-b_2)y+c=0$
then the value of c is

If sinA=sinB and cosA=cosB, then

[EAMCET 1994]

$Iff\left(x\right)=co{s}^{2}x+se{c}^{2}x,then$