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If the coordinates of a point be given by the equation $x=a(1-cos\theta ),y=asin\theta$, then the locus of the point will be

A particle is executing simple harmonic motion with a period of T seconds and amplitude a metre. The shortest time it takes to reach a point $\frac{a}{\sqrt{2}}$ m from its mean position in seconds is

[EAMCET (Med.) 2000]

If A + B + C = $\pi$, prove that $co{s}^{2}A+co{s}^{2}B-co{s}^{2}C=1-2sinAsinBcosC$

If $sinx+cosecx=2$, then $si{n}^{n}x+cose{c}^{n}x$ is equal to

The plane XOZ divides the join of (1, -1, 5) and (2, 3, 4) in the ratio $\lambda :1$ then $\lambda$ is [JET 1988]

Find the area of the triangle formed by joining the mid points of the sides of the triangle formed with coordinates A (-4, 3), B (2, 3) and C (4, 5).

If α,β are the roots of $a{x}^2+bx+c$=0 then ${(a\alpha +b)}^{-3}$ +${(a\beta +b)}^{-3}$

If the value of $\frac{(1+i{)}^{1-i}}{(1-i{)}^{1+i}}$ = isin p + cos p. Find the value of p.

cos 2x =

If x, y, z are real and distinct, then $x^2+4y^2+9z^2-6yz-3zx-2xy$ =

Given below is the percentage of marks secured by 5 students in Economics and Statistics.

$\begin{array}{cccccc}\text{Students}& \text{A}& \text{B}& \text{C}& \text{D}& \text{E}\\ \text{Marks in Economics}& 60& 48& 49& 50& 55\\ \text{Marks in Statistics}& 85& 60& 55& 65& 75\end{array}$
Calculate the coefficient of rank correlation.

The equations of the two sides of a triangle are $3x-2y+6=0$ and $4x+5y-20=0$ respectively. If the orthocentre of the triangle is (1, 1), find the equation of third side.

Show that the coordinates of the centroid of the triangle with vertices $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(x_3,y_3,z_3)$ is $(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3})$.

If n is a positive integer, find the coefficient of $x^{-1}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$

Differentiation of $\frac{-x^3}{3}$ w.r.t $x$ is

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number } and D = {x : x is a prime number}, find :

(i) $A\cap B$ (ii) $A\cap C$ (iii) $A\cap D$

(iv) $B\cap C$ (v) $B\cap D$ (vi) $C\cap D$

Express the following expression in the form of a + ib

$\frac{(3+\sqrt{5}i)(3-\sqrt{5}i)}{(3+\sqrt{2}i)-(\sqrt{3}-\sqrt{2}i)}$

Find the points on the x - axis, where distances from the line $\frac{x}{3}+\frac{y}{4}=1$ are 4 units.

Prove by using the principle of mathematical induction $\forall n\in N$

$2+5+8+11+...+(3n-1)=\frac{1}{2}n(3n+1)$

Or

Using principle of mathematical induction, prove that $4^n+15n-1$is divisible by 9 for all natural numbers n.

Number of ways of selection of 8 umbrella from 24 umbrellas of which 8 are green color, 8 are blue color and the rest unlike, is given by

The product of the sines of the angles of a ΔABC is ϕ and the product of their cosines is q. Then the tangent of the angles is the roots of the equation

Find the values of $\frac{1}{x}$ for x $\in$ (3,$\infty$).

Let $n\in N$ and n < $(5\sqrt{3}+8{)}^4$. Then the greatest value of n is

If Z= $\frac{(13-5i)}{(4-9i)}$ find $z^6$

The sum of all three-digit natural numbers which are divisible by $7$ is

Number of positive integers which have characteristic $2$ when base is $10$, is

Find the sum to n terms

$1\times 2+2\times 3+3\times 4+4\times 5+\dots \dots$

If $T_{n+1}=\frac{2T_n+1}{2},n\in N$ and $T_{10}=\frac{19}{2}$, then the ${101}^{th}$ term of the sequence is

If the sum of the first $15$ terms of the series
${\left(\frac{3}{4}\right)}^3+{\left(1\frac{1}{2}\right)}^3+{\left(2\frac{1}{4}\right)}^3+3^3+{\left(3\frac{3}{4}\right)}^3+....$

is equal to $225k$, then $k$ is equal to :

If $2^x+2^{f\left(x\right)}=2$ , then the domain of the function $f\left(x\right)$ is

Prove the following:
$\frac{\text{sin 5x + sin 3x}}{\text{cos 5x + cos 3x}}=\text{tan 4x}$

Record the following transaction in simple cash book for November 2010.
$\begin{array}{ccc}& & \text{Rs.}\\ 1& \text{Cash in hand}& 12,500\\ 4& \text{Cash paid to Hari}& 600\\ 7& \text{Purchased goods}& 800\\ 12& \text{Cash received from Amit}& 1,960\\ 16& \text{Sold goods for cash}& 800\\ 20& \text{Paid to Manish}& 590\\ 22& \text{Paid cartage}& 100\\ 28& \text{Paid salary}& 1,000\end{array}$

Find the sum of the series $1.2+{2.2}^2+{3.2}^2+......100{.2}^{100}$

$P\left(n\right):1+3+3^2+...+3^{n-1}=\frac{3^n-1}{2}$
The statement P(n)

Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 points for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2$, respectively, after two games.

$P(X>Y)$ is

Express each of the following as an algebraic sum of sines or cosines :

(i) 2 sin $3x$ cos $2x$

(ii) 2 cos $4x$ sin 2$x$

(iii) 2 cos $6x$ cos $4x$

(iv) 2 sin $3x$ sin $5x$

Find the value of $\sqrt{7\left(\sqrt{7\sqrt{7}}\right)}$

Let z and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and arg $z\omega =\pi$. Then arg z equals

If the chord joining the points $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$ of the parabola $y^2=4ax$ passes through the focus of the parabola, then

If $a+i{b}^{11}$ = p + iq, where i = $\sqrt{-1}$.Find the value of $(b+ia{)}^{11}$.

The solution set of $\frac{x^2-7x+10}{14x-x^2-45}\ge 0$ is