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The value of \((1+i)^2 × (1- i)^2 is

If tan $\frac{\varphi }{2}$ = 2,find the value of $\frac{5}{4}sin\varphi$.

Let $a_1,a_2,a_3,...$ be the terms of an AP. If $\frac{a_1+a_2+a_3+...+a_p}{a_1+a_2+a_3+...+a_q}=\frac{p^2}{q^2},p\ne q,$ find the value of $\frac{a_6}{a_{21}}$.

Find the rational value of x if 6(${\log }_x$2-${\log }_4$ x)+7=0

Which of the following options are correct,

where i, j and k are unit vectors along the x, y and z axis?

Which one of the following relations on Z is equivalence relation?

If $a,b,c$ are in A.P. and $a^2,b^2,c^2$ are in G.P. such that $a<b<c$ and $a+b+c=\frac{3}{4}$, then the value of $a$ is :

P is a moving point, S is the focus and L is the directrix as shown in figure.

Which of the following represents the equation of a conic?

The value of ${16}^{{\log }_{4}3}-3^{{\log }_{27}512}$ is

Find the domain of $f\left(x\right)=\frac{1}{\sqrt{\frac{1}{\sqrt{x+|x|}}}}$

How many numbers between 2000 and 3000 can be formed from the digits 2, 3, 4, 5, 6, 7 when repetition of digits is not allowed ?

Find the coordinates of the foci, the vertices, the accentricity and the length of the latus rectum of the hyperbola,

$\frac{y^2}{9}-\frac{x^2}{27}=1$

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$

Which one of the following remain constant with change of $\alpha$ ?

The sum of coefficients in the expansion of $(1+x-3x^2{)}^{171}$ is

The value of (1 + cos $\frac{\pi }{8}$)(1 + cos $\frac{3\pi }{8}$)(1 + cos $\frac{5\pi }{8}$)(1 + cos $\frac{7\pi }{8}$) is equal to

If $a_1,a_2,a_3,a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x{)}^n$, then $\frac{a_1}{a_1+a_2}+\frac{a_3}{a_3+a_4}=$

The angle between the pair of straight lines $y^{2}si{n}^2\theta -xysi{n}^2\theta +x^2(co{s}^2\theta -1)=1,$ is

The Greatest co-efficient in the expansion of $(1+x{)}^{2n+2}$ is

Write the first five terms of the sequences whose $n^{th}$ term is :

$a_n=n\frac{n^2+5}{4}$

Evaluate: $\underset{x\to 0}{lim}\frac{cos2x-1}{cosx-1}$

Identify the false statement.

The roots $Z_1,Z_2,Z_3$ of the equation $x^3+3p{x}^2+3qx+r=0$ (p, q, r are complex numbers) correspond to
points A, B and C, then triangle ABC is equilateral, if

If $acos2\theta +bsin2\theta =c$ has $\alpha$ and $\beta$ as its roots, prove that $\alpha +tan\beta =\frac{2b}{a+c}$ and $\alpha -tan\beta =\frac{2}{a+c}\sqrt{b^2-c^2+a^2.}$

If $\alpha ,\beta$ are the roots of the equation $3x^2+5x+4=0$, then the quadratic equation whose roots are ${\alpha }^2,{\beta }^2$

The number of solutions of equation [sin x ] = [1 + sin x] + [ 1 - cos x] where [.] denotes the greatest integer function and is

Find the common interval among x < 0, x < -2 and x $\le$ $\frac{-5}{2}$

Suppose a,b,c are in A.P and $a^2$,$b^2$,$c^2$ are in GP. If a < b < c and a+b+c = $\frac{3}{2}$, then the value of a is

In Triangle ABC
(a+b-c)(b+c-a)(c+a-b) -abc is always

If the Cartesian product A $\times$ A has 16 elements among which few elements are found to be (-1, 0), (0, 1), (1, 2). Find set A.

If $2sec\left(2\alpha \right)=tan\beta +cot\beta$ ,then one of the values of $\alpha +\beta$is

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 $0$ point 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

If sin A = $\frac{4}{5}$ and cos B = - $\frac{12}{13}$, where A and b lie in first

and third quadrant respectively, then cos(A + B) =

If two roots of the equation $x^3-3x+2=0$ are same, then the roots will be

Three integers $a,b,c$ are in $G.P.$ If $a,b,c-64$ are in $A.P.$, and $a,b-8,c-64$ are in $G.P.$, then $(a+b+c)$ is equal to

Let $\left[k\right]$ denotes the greatest integer less than or equal to $k.$ Then the number of positive integral solutions of the equation $\left[\frac{x}{\left[{\pi }^2\right]}\right]=\left[\frac{x}{\left[11\frac{1}{2}\right]}\right]$ is

The number of real roots of the equation $e^{sinx}-e^{-sinx}-4=0$ are

The distances of the roots of the equation $|\sin {\theta }_1|z^3+|\sin {\theta }_2|z^2+|\sin {\theta }_3|z+|\sin {\theta }_4|=3$, from $z=0$, are

For every natural number n, n$(n^2-1)$ is divisible by

The value of $5^{{\log }_{\frac{1}{5}}\left(\frac{1}{2}\right)}+{\log }_{\sqrt{2}}\left(\frac{4}{\sqrt{3}+\sqrt{7}}\right)+{\log }_{\frac{1}{2}}\left(\frac{1}{10+2\sqrt{21}}\right)$ is

If $p\Rightarrow (q\vee r)$ is false, then the truth values of p,q,r are respectively

The sum of series 1.2.4 + 2.3.5 + 3.4.6 + ----------- 20 terms.

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