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Find number of value of $xϵ[0,\pi ]$ satisfying the relation cos 3x + sin 2x - sin 4x = 0

If a, b, c are distinct positive numbers, then the expression (b + c - a) (c + a - b) (a + b - c) - abc

Let A = {a, b} and B = {a, b, c}. Is $A\subset B$ ? What is $A\cup B$ ?

A rectangle with sides 2m -1 and 2n -1 is divided into squares of unit length by drawing parallel lines as shows in the diagram, then the number of rectangles possible with odd side lengths is

Plot the graph of $f(x-1)$ if the graph of $f\left(x\right)$ looks likes

If ${}^n{C}_8={}^n{C}_6,$ find ${}^n{C}_3.$

If the distance between the earth and the sun were reduces to half its present value, Then the number of days in one year would have been

If P = (1, 0), Q = (-1, 0) and R = (2, 0) are three given points, then the locus of a point S satisfying the relation

$S{Q}^2+S{R}^2=2S{P}^2$ is

To find:
$1^2+2^2+3^2...................+n^2$

The sum of (n-1) terms of 1+(1+3)+(1+3+5)+.............is

A die is thrown. Describe the following events.
(i) A: a number less than 7
(ii) B: a number greater than 7
(iii) C: a multiple of 3
(iv) D: a number less than 4
(v) E: an even number greater than 4
(vi) F: a number not less than 3
Also find $A\cup B,A\cap B,B\cup C,E\cap F,D\cap E,A-C,D-E,E\cap F^{\prime },F^{\prime }$

The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1. Find the common ratio and the terms.

$\text{Let f (a) =g (a)= k and their nth derivatives}{\text{f}}^n\left(a\right),{\text{g}}^n\left(a\right)\text{exist and are not equal for some n. Further if}\underset{x\to a}{lim}\frac{f\left(a\right)g\left(x\right)-f\left(a\right)-g\left(a\right)f\left(x\right)+g\left(a\right)}{g\left(x\right)-f\left(x\right)}=4,\text{then the value of k is:}$

Prove that, sin 3 A $si{n}^3$ A + cos 3 A $co{s}^3$ A = $co{s}^{3}2$A

The number of solutions of equation, sin5x cos 3x = sin 6x cos 2x, in the

interval [0, $\pi$] are

A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching X-axis, the equation of the reflected ray is

If $\frac{(1-3x{)}^{\frac{1}{2}}+(1-x{)}^{\frac{5}{3}}}{\sqrt{4-x}}$ is approximately equal to

a + bx for small values of x, then (a, b) =

Let $A$ be the set of first ten natural numbers and let $R$ be a relation on $A\times A$ defined by $R=\left\{\right(x,y):x+2y=10;x,y\in A\}.$ Then range of $R^{-1}$ is

If $S_1,S_2,S_3\dots \dots ,S_n,\dots$ are the sums of infinite geometric series whose first terms are $1,2,3,\dots \dots n,\dots \dots$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \dots \frac{1}{n+1}\dots \dots$ respectively, then the value of $\sum _{r=1}^{2n-1}{S}_r^2$ is

If $\sum _{i=1}^{10}(x_i-5)=20$ and $\sum _{i=1}^{10}(x_i-5{)}^2=660$ $y=S.D.$ of $10$ items $x_1,x_2,x_3.....,x_{10}$, then $\left[y\right]$ is equal to (where $[.]$ represents the greatest integer function)

What is the dimensions of $\frac{A}{B}$ in the relation $F=A\sqrt{x}+B{t}^2$, where $F$ is the force, $x$ is the distance and $t$ is the time?

The length of the latus rectum and the length of the transverse axis of a hyperbola are $4\sqrt{3}\&2\sqrt{3}$ respectively, then the equation of the hyperbola can be

Prove that,

(i) $cos\alpha +cos\beta +cos\gamma +cos(\alpha +\beta +\gamma )=4cos\left(\frac{\alpha +\beta }{2}\right).cos\left(\frac{\beta +\gamma }{2}\right).cos\left(\frac{\gamma +\alpha }{2}\right).$

(ii) If $tanx=\frac{b}{a},$ then $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}=\frac{2cosx}{\sqrt{cos2x}}.$

If $2tan\frac{\alpha }{2}=tan\frac{\beta }{2},$ prove that $cos\alpha =\frac{3+5cos\beta }{5+3cos\beta }.$ A.so, determine the value of $cos\alpha$ when $\beta ={60}^{\circ }.$

Or

If a $sin\theta =bsin(\theta +\frac{2\pi }{3})=csin(\theta +\frac{4\pi }{3}),$ find the value of ab + bc + ca.

Find the real values of x satisfying ${\log }_{0.3}$ (10x + 3) < ${\log }_{0.3}$ (7x - 4).

If $\omega$ is a complex cube root of unity , find the equation , whose roots are 2 , 2$\omega$ , 2${\omega }^2$

$\overrightarrow{A}=3\hat{i}+2\hat{j}+7\hat{k}$

$\overrightarrow{B}=\hat{i}-6\hat{j}+5\hat{k}$

$\overrightarrow{C}=-2\hat{i}+\hat{j}-4\hat{k}$

Find $2\overrightarrow{A}-\overrightarrow{B}+3\overrightarrow{C}$

How many values of x in the interval [0, 2π] satisfies the equation $si{n}^6$ x = 1 + $co{s}^4$ 3x

The solution of the equation $[\sin x+\cos x{]}^{1+\sin 2x}=2,-\pi \le x\le \pi$ is

Let $p={lim}_{x\to 0^{+}}(1+ta{n}^2\sqrt{x}{)}^{\frac{1}{2x}}$ then log p is equal to:

If arg($z_1$) = 11 $\frac{\pi }{18}$ and arg ($z_2$) = 19 $\frac{\pi }{18}$, then the principal argument of $z_1$$z_2$ is

The distance of the point (15,8,13) from the z-axis is ___.

The solution set of $x^2-7x+12\ge 0$ is

If the different permutations of all the letters of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?

Let$P=(-1,0),Q=(0,0)$ and $R=(3,3\sqrt{3})$ be three points, Then the equation of the bisector of the angle PQR is -

If $x\in R$ satisfies $(lo{g}_{10}100x{)}^2+(lo{g}_{10}10x{)}^2+lo{g}_{10}x\le 14$ then $x$ contains the interval

If the circle $x^2+y^2=a^2$ intersect the hyperbola $xy=c^2$ in four points P$(x_1,y_2),Q(x_2,y_2),R(x_3,y_3),S(x_4,y_4),$ then which of the following does not hold

If a,b,c are distinct positive real numbers such that b(a+c) = 2ac then the roots a$x^2$ + 2bx + c = 0 are :

If $a_1,a_2,\cdot \cdot \cdot ,a_{15}$ are in A.P. and $a_1+a_8+a_{15}=15$, then $a_2+a_3+a_8+a_{13}+a_{14}$ equals

If ${}^n{P}_3+{}^n{C}_{n-2}$ = $14n$, then n =

The probability that a teacher will give an unannounced test during any class meeting is $\frac{1}{5}$.If a student is absent twice, the probability that he will miss atleast one test, is

If $(1-i{)}^n=2^n$ , then n is

Find the value of ${}^{47}{C}_4+\sum _{r=1}^5$ ${}^{52-r}{C}_3=?$

In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present?

If a, b, c and n are positive real numbrs other than unity , prove that

$lo{g}_{a}n.lo{g}_{b}n+lo{g}_{b}n.lo{g}_c.n+lo{g}_{c}n.lo{g}_{a}n$

$=\frac{lo{g}_{a}n.lo{g}_{b}n.lo{g}_{c}n}{lo{g}_{abc}n}$

Four particles $A$, $B$, $C$ and $D$ having masses $m$, $2m$, $3m$ and $4m$ respectively are placed in order at the corners of a square of side $a$. Locate the centre of mass

Find the value of $\theta$, if the equation $cos\theta x^2-2sin\theta x-cos\theta =0$ has real roots