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If x lies in third quadrant and 5 sin x + 3 = 0, find the value of

$\frac{2tanx-5sinx+cotx}{2sin\frac{x}{2}cos\frac{x}{2}}$

Solve the inequalities:

$-12<4-\frac{3x}{-5}\le 2$

Find the value of ${\log }_2.{\log }_3.........{\log }_{100}.100.99.98..............2.1$

If α and β are two real roots of the equation $x^3$ - p $x^2$ + qx + r = 0 satisfying the relation αβ + 1 = 0, then find the value of $r^2$ + pr + q.

A complex number z is said to be unimodular, if $|z|=1$. If and $z_1$ and $z_2$ are complex numbers such that $\frac{z_1-2z_2}{2-\left(z_1\overline{z_2}\right)}$ is unimodular and $z_2$ is not unimodular.
Then, the point $z_1$ lies on a

The value of ${lim}_{x\to \infty }\frac{x^5}{5^x}$ is

Without using the Pythogoras theorem, show that the points (4, 4), (3, 5) and (-1, -1) are the vertices of a right angled triangle.

In a single throw of three dice, find the probability of getting (i) a total of 5 (ii) a total of at most 5.

What is the equation of the chord centered at (1, 2) in the circle $x^2+y^2-4x-6y-10=0$

The remainder when ${1690}^{2608}+{2608}^{1609}$ is divided by 7, is

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 two particular students have to be included in the team.

The value of tan67 ${\frac{1}{2}}^0$ + cot67 ${\frac{1}{2}}^0$ is

If sin(A+B+C)=1, tan(A-B)=$\frac{1}{\sqrt{3}}$ and Sec(A+C)=2, then

In triangle ABC, $1-tan\frac{B}{2}.tan\frac{C}{2}$ is equal to

If |z| = 4 and arg z = -$\frac{\pi }{4}$, then z is:

If $n^{th}$ term of the series 1 + 5 + 12 + 22 + 35 .........can be written as $T_n$ = a$n^2$ + bn + c Find the sum of the 16 terms of the series.

The function f : R → R defined by y = f(x) = 5 for each x ϵ R is

If $8{\sin }^2\theta +10\sin \theta \cos \theta -3{\cos }^2\theta =0$ then $\theta =$

$\mathrm{1.2.3}+\mathrm{2.3.4}+\cdots +n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}$

Number of integral solutions of the equation
${\log }_{0.5x}\left(x^2\right)-14{\log }_{16x}\left(x^3\right)+40{\log }_{4x}\sqrt{x}=0$ is

Evaluate $\underset{x\to 0}{lim}\frac{(1-cosx)}{x^2}$

$\underset{n\to \infty }{lim}\frac{1.n^2+2.(n-1{)}^2+3.(n-2{)}^2+.....n{.1}^2}{1^3+2^3+....n^3}$

How many of the following pairs of function are identcal

(a) f (x) = $\frac{x^2}{x},g\left(x\right)=x$

(b) f(x) = $\frac{x^2-1}{x-1}$ g(x) = x + 1

(c) f(x) = $se{c}^{2}x-ta{n}^{2}x$ , g(x) = 1

(d) $f\left(x\right)=\frac{x}{x^2},g\left(x\right)=\frac{1}{x}$

If Domain of f(x) is A and domain of g(x) is B, then the domain of f(x).g(x) is

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+...+10(11{)}^9=k(10{)}^9,$ then k is equals to:
(IIT JEE Main 2014)

Find the remainder when $(1+8{)}^n+7$ is divided by 8

The minimum value of $2cos\theta +\frac{1}{sin\theta }+\sqrt{2}tan\theta$ in the interval $(0,\frac{\pi }{2})$ is

Consider two quadratic expressions f(x)=$a{x}^2+bx+c$ and g(x)=$a{x}^2+px+q$ (a,b,c,p,q R, b≠p)such that their discriminents are equal. If f(x)=g(x) has a root x=a then.

If the points $(0,7,10),(-1,6,6)and(-4,9,6)$ are the vertices of a triangle, then the triangle is _____

If $\frac{cosx}{a}=\frac{sinx}{b}$ then $|acos2x+bsin2x|=$

Find the equation of the curve formed by the set of all points whose distances from the points (3, 4, -5) and (-2, 1, 4) are equal.

Eighteen guests have to be seated half on each side of a long table. Four particular guests desire to sit on one particular side and three on the other side.Determine the number of ways in which the sitting arrangements can be made.

Or

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw?

If x,y,z are in H.P and $a^x$ = $b^y$ = $c^z$ , then a,b,c are in

If the axes are shifted to the point (1, -2) without rotation. What do the equation $2x^2+y^2-4x+4y=0$ becomes?

If $S$ is the set of all real $x$ such that $\frac{2}{x^2-x+1}-\frac{1}{x+1}-\frac{2x-1}{x^3+1}\ge 0$, then $S$ contains

If $\alpha$ is the value of $xϵ[0,2\pi ]$ which is a solution of the equation

2 $co{s}^2$ $\frac{x^2+x}{6}$ = $2^x$ + $2^{-x}$, then find the value of $\frac{2\alpha }{3\pi }$ .

Consider a weak monoacidic base $BOH$ having a concentration C with dissociation $\alpha <<1.$ The pH of this weak base in terms of base dissociation constant $K_b$ and concentration $C$ is :

In the sum fo first n terms of an A.P. is $c{n}^2$ then the sum of squares of these n terms is

$\text{If}\tan \frac{\theta }{2}=\sqrt{\frac{a-b}{a+b}}\tan \frac{\varphi }{2}\text{prove that}\cos \theta =\frac{a\cos \varphi +b}{a+b\cos \varphi }$

If $\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}(x\ne 0)$, then show that a, b, c and d are in G.P.

The number of ways to put 6 letters in 6 addressed envelopes so that all are in wrong envelopes?

Find the general solution of $sinx+\sqrt{2}=-sinx$

If A, B and C be three non empty sets given in such a way that $A\times B=A\times C$, then prove that B = C.

A random variable $X$ has the following probability distribution:
$$\begin{array}{cccccc}X& 1& 2& 3& 4& 5\\ P\left(X\right)& K^2& 2K& K& 2K& 5K^2\end{array}$$
Then $P(X>2)$ is equal to:

An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. If the probability of getting 2 red balls is P(A), find $121\times P\left(A\right)$.

An experiment is called random experiment if

With usual notations, $C_1+2^2{C}_2+3^2{C}_3+.......=$