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Evaluate tan $\frac{13\pi }{12}$

Find the modulus and argument of the following complex numbers and hence express each of then in polar form:
$-\sqrt{3}-i$

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$36x^2+4y^2=144$

Find the general solution for the following equation:

sin x + sin 3x + sin 5x = 0

If $2ta{n}^{-1}x=si{n}^{-1}\frac{2x}{1+x^2}$, then:

The statement $p\Rightarrow \sim q$ is equivalent to

If α, β and γ are the roots of the equation $x^3$ - 3 $x^2$ + 5x - 9 = 0 then the value of the expression

( α + β - γ) ( β + γ -α) (γ + α - β).

Let $\alpha ,\beta$ be real and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Re\left(z\right)=1$, then it is necessary that

Identify the graph of $-|-x+3|$

Team of four students is to be selected from a total of 12 students for a quiz competition. In how many ways it can be selected if one particular student should be included and three particular students don't want to be in the team.

$\text{Let f}\left(x\right)=3x^{10}-7x^8+5x^8-21x^3+3x^2-7\text{THEN}{lim}_{h\to 0}\frac{f(1-h)-f\left(1\right)}{h^3+3h}$

For $\frac{-\pi }{2}$ < $\theta$ < $\frac{\pi }{2}$ , $\frac{sin\theta +sin2\theta }{1+cos\theta +cos2\theta }$ lies in the interval.

Find the differentiation of $y$ w.r.t $x$ ,if $y=\frac{\sin x}{x}$

The range of the function $f\left(x\right)=\frac{x^2+x+2}{x^2+x+1},x\in R$ is

The Boolean expression $\sim (p\vee q)\vee (\sim p\wedge q)$ is equivalent to

Which of the following contain(s) the highest number of atoms?

The value of $\cot \left(\sum _{n=1}^{19}{\cot }^{-1}(1+\sum _{p=1}^{n}2p)\right)$ is :

A rectangle ABCD, A = (0,0), B = (4, 0), C = (4, 2), D = (0, 2) undergoes the following transformations successively.
(i) $f_1(x,y)\to (y,x)$
(ii) $f_2(x,y)\to (x+3y,y)$
(iii) $f_3(x,y)\to (\frac{x-y}{2},\frac{x+y}{2})$
The final figure will be

If $f(x+y,x-y)=xy$, then $\frac{f(x,y)+f(y,x)}{2}$ is

At ${90}^{\circ }$C pure water has [${H_{3}O}^{+}$] = ${10}^{-6}$mol ${litre}^{-1}$. The value of $K_W$ at ${90}^{\circ }$C is

If each of the observations $x_1,x_2,x_3,\cdots ,x_n$ is increased by an amount a, where a is a negative or positive number, then show that the variance remains unchanged.

If a, b, c, d are in G.P., prove that $(a^n+b^n),(b^n+c^n),(c^n+d^n)$ are in G.P.

A knight is placed somewhere on the chess board. If you have 2 consecutive chances then taking the initial position of knight as the origin which of these position can be the maximum displaced position of the knight?

$Rangeoff\left(x\right)=\frac{tan\left(\pi [x^2-x]\right)}{1+sin\left(cosx\right)}iswhere\left[x\right]denotesthegreatestintegerfunction)$

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$, the correct statements about the equation a${(x-2)}^2$ + b(x - 2) + c = 0 is

An object is thrown at an angle of ${45}^{\circ }$ with the horizontal direction. The horizontal range of the particle is equal to _____

The points (a, b), (c, d) and $(\frac{kc+la}{k+l},\frac{kd+lb}{k+l})$ are

The total number of ways of dividing 15 different things into groups of 8,4 and 3 respectively is

Express $\frac{(cos\theta +isin\theta {)}^4}{(sin\theta +icos\theta {)}^5}$ in a+ib form where i=$\sqrt{-1}$

The maximum value of $cos{\alpha }_1.cos{\alpha }_2......cos{\alpha }_n,$
under the restrictions $0\le {\alpha }_1{\alpha }_2,.....,{\alpha }_n\le \frac{\pi }{2}andcot{\alpha }_1.cot{\alpha }_2......cot{\alpha }_n=1$ is

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

$\begin{array}{cccccc}{x}_i& 15& 21& 27& 30& 35\\ f_i& 3& 5& 6& 7& 8\end{array}$

Prove that $sinx+sin3x+sin5x+sin7x=4cosxcos2xsin4x.$

If cos x +cosy = $\frac{1}{3}$, sin x + sin y = $\frac{1}{4}$ then sin (x + y) =

If equation $a{x}^2+bx+c$ = 0 has only imaginary roots and c < 0, then a is ______.

If $a_1,a_2,a_3,a_4$ be the coefficeints of four consecutive terms in the expansion of $(1+x{)}^n$ then prove that
$\frac{a_1}{a_1+a_2}+\frac{a_3}{(a_3+a_4)}=\frac{2a_2}{(a_2+a_3)}.$

The number of ways in which any four letters can be selected from the word "EXAMINATION” if two letters are alike and other two are distinct.

Find the value of ${}^n{C}_1+2^n{C}_2+3^n{C}_3.........n^n{C}_n$

The pair of straight lines joining the origin to the points of intersection of the line y = $2\sqrt{2}x+c$ and the circle $x^2+y^2=2$ are at right angles, if

If ${}^{56}{P}_{r+6}.{}^{54}{P}_{r+3}$ = 30800:1, then r =

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

$\frac{x^2}{49}+\frac{y^2}{36}=1$

Given standard equation of ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, with eccentricity e

Match the following
$\begin{array}{cc}\text{a)Focus}& \text{i) (ae,0)}\\ \text{b)Directrix}& \text{ii) (a,0)}\\ \text{c)Eccentricity}& \text{iii) x=}\frac{\text{a}}{\text{e}}\\ \text{d)Vertices}& \text{iv) (-ae,0)}\\ & \text{v) x=}\frac{\text{-a}}{\text{e}}\\ & \text{vi)}\sqrt{\text{1-}\frac{{\text{b}}^{\text{2}}}{{\text{a}}^{\text{2}}}}\end{array}$

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

$3(7+i7)+i(7+i7)$

”Each of Rajat's students either scored distinction in chemistry or physics” is represented by which of the following expressions

X : set of Rajat's students

P : x scored distinction in chemistry

Q : x scored distinction in physics

The function f(x) is continuous in the interval (a, b) then which among the following is true?

If I < x < I +1, Find [-x], where I is an integr

Evaluate $\underset{x\to 2}{lim}\left\{\frac{(x^2-4)}{\sqrt{3x-2}-\sqrt{x+2}}\right\}$

If $z=i^i$,where i = $\sqrt{-1}$, then Re(z) is

If $a_1,a_2,a_3$,.................,$a_n$ are in H.P., then $a_1{a}_2+a_2{a}_3+$...........+$a_{n-1}{a}_n$ will be equal to

Find the value of 4 tan 4A + 8 cot 8A