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If $co{s}^{2}A+co{s}^{2}C=si{n}^{2}B,$ then $△$ ABC is
[MP PET 1991]

Find the general solution of the equation sin 2x+sin 4x +sin 6x =0.

If f(x) = x² and g(x) = x are two functions from R to R then
(fg)(2) is

If the sum of the $3^3$ + $7^3$ + ${11}^3$ + ${15}^3$ ...................20 terms is $s_{20}$. Find the value of $\frac{s_{20}}{100}$

Express the following in standard form: (2+3i)(2-3i)(1+i)²

A straight line L through the poit $(3,-2)$ is inclined at an angle ${60}^0$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, then the equation of L is

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

How many real numbers satisfy the condition 0 $\le$$\frac{2}{3}\left[x\right]\le$1

The least positive integer n for which ${\left(\frac{1+i}{1-i}\right)}^n=\frac{2}{\pi }(se{c}^{-1}\frac{1}{x}+si{n}^{-1}x),x\ne 0,-1\le x\le 1$ is:

If $S_1$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$..................${}^n{C}_n$ and $S_2$ = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$....................${}^n{C}_n$

Find ${}^n{C}_1$ + ${}^n{C}_3$ + ${}^n{C}_5$..............

The point (4, 1) undergoes the following transformation successively.
(i) reflection about the line y = x
(ii) translation through a distance 2 units along the positive direction of x - axis
(iii) rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction.
(iv) reflection aout x = 0
The final position of the given point is

The function f(x) defined as f(x) = $\sqrt{(x-4{)}^2}$

If $\frac{c+i}{c-i}$ = a+ib, where a,b,c are real, then $a^2+b^2$ =

The value of the expression ${}^{10}{C}_0{10}^9{-}^{10}{C}_1{9}^9{+}^{10}{C}_2{8}^9.......{-}^{10}{C}_9$ is

For each point (x,y) on an ellipse, the sum of the distances from (x,y) to the points (2,0) and (-2,0) is 8. Then the positive value of x so that (x,3) lies on the ellipse is ___

Let the angle made by the line OA with respect to positive x-axis be ${\theta }_1$ and the angle made by OB be ${\theta }_2$. Find the value of ${\theta }_1+{\theta }_2(0\le |{\theta }_1|,|{\theta }_2|\le {180}^{\circ })$. Angle measured in clockwise direction is negative and angle measured in anti-clockwise direction is positive.

Prove $1+2+3+\cdots +n<\frac{1}{8}(2n+1{)}^2.$

The value of $x$ if ${\log }_{10}{x}^2-7x-6=1-{\log }_{10}5$

If relation R is defined as "Is of the same color" on set of objects. Then the total number of equivalence classes on the set with respect to R is equal to.

A function f is defined by f(x) = 2x - 5. Write down the values of:

(i) f(0) (ii) f(7) (iii) f(-3)

If the third term in the expansion of ${(\frac{1}{x}+x^{log}{10}^x)}^5$ is 1000, then x =

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

$(1-i{)}^4$

Find the principal solutions of each of the following equations.

(i)sin $x=\frac{1}{2}$

(ii)cos $x=\frac{1}{\sqrt{2}}$

(iii)tan $x=\frac{1}{\sqrt{3}}$

Let $X=\{2,3,4,5\}$ and $Y=\{7,9,11,13,15,17\}$. Define a relation f from X to Y by:
$f=\left\{\right(x,y):xϵX,yϵY\text{and}y=2x+3\}$

(i) Write f in roster form.

(ii) Find dom(f) and range (f).

(iii) Show that f is a function from X to Y.

Identify the function f(x) from the description given below.

1. x - 1 $<f\left(x\right)\le x$

2. Its domain is R and Range is I(set of integers)

3. f(x) = 3 $\Rightarrow 3\le x<4$

4. f(x) = -3 $\Rightarrow -3\le x<-2$

Find the sum of first 10 terms of the G.P 3, 6, 12, 24 ..........

The complex number $z_2$(2 + 3i) is rotated through an angle of ${90}^{\circ }$ anti-clockwise about $z_1$(1 + i). Find the new complex number $z_3$ obtained.

If a,b,c be positive, then minimum value of $\frac{b+c}{a}$ + $\frac{c+a}{b}$ + $\frac{a+b}{c}$ 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 x + $\frac{1}{x}$ = $2cos\theta$, then $x^3$ + $\frac{1}{x^3}$ =

The middle term in the expansion of $(1+x{)}^{2n}$ is

Find the coefficient of $x^{n-2}$ in
$({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)\times ({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)$

Position of the point (1,1) with respect to the circle $x^2+y^2-x+y-1=0$ is

If x be real, then the minimum value of $x^2-8x+17$ is

If z is a complex number such that $z^2$ = (${\overline{z}}^2)$,then

Calculate coefficient of variation of the following series:

$\begin{array}{ccccccccccc}\text{S. No}& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ \text{Frequency}& 53& 58& 25& 30& 54& 42& 32& 48& 46& 52\end{array}$

Find the sum of n terms of the series 0.3 + 0.33 + 0.333 + . . . .

The coefficient of the middle term in the binomial expansion of $(1+ax{)}^4$ and of $(1-ax{)}^6$ is the same, if a equals:

The valuesof $x^2+4x-3\forall x\in R$ lie in the interval

The range of $f\left(x\right)={\log }_e(3x^2-4x+5)$ is

Find the derivative of $x^2$ using first principle.

The $4^{th},7^{th}$ and ${10}^{th}$ term of a G.P. are $a,b,c$ respectively, then

Find the general solution of the equation $sin7\theta =sin3\theta +sin\theta$

Verify the following:

(i) $(0,7,-10)(1,6,-6)$ and $(4,9,-6)$ are the vertices of an isosceles triangle.

(ii) $(0,7,10),(-1,6,6)$ and $(-4,9,6)$ are the vertices of a right angled triangle.

(iii) $(-1,2,1)(1,-2,5),(4,-7,8)$ and $(2,-3,4)$ are the vertices of a parallelogram.

If $A$ and $B$ are two events such that $P\left(A\right)=\frac{3}{4}$ and $P\left(B\right)=\frac{5}{8}$, then

Evaluate $\underset{x\to \frac{\pi }{4}}{lim}\frac{(sinx-cosx)}{(x-\frac{\pi }{4})}$