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$\text{If the sides a, b, c of a}\Delta ABC\text{are in H.P., prove that}si{n}^2\frac{A}{2},si{n}^2\frac{B}{2},si{n}^2\frac{C}{2}\text{are in H.P.}$

The symmetric difference of A and B is not equal to

Let a plane $P_1$ containing the line $L_1:\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-1}{-1}$ and another plane $P_2$ containing the line $L_1$ and passses through the point $(0,1,1).$ If d.r's of normal to the plane $P_1$ are $1,1,0,$ then the acute angle between the planes $P_1$ and $P_2$ is equal to

If $f\left(x\right)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ then $f^{\prime }\left(\frac{1}{2}\right)=$

Region represented by x ≥ 0, y ≥ 0 is

Sum of all real $x$ such that $\frac{4x^2+15x+17}{x^2+4x+12}=\frac{5x^2+16x+18}{2x^2+5x+13}$ is

A random variable X as the following probability distribution:

$\begin{array}{cccccc}X& 0& 1& 2& 3& 4\\ P\left(x\right)& 0.1& k& 2k& 2k& k\end{array}\text{Find}k.$

If the system of linear equations
$\begin{array}{c}x-4y+7z=7g\\ 3y-5z=h\\ -2x+5y-9z=k\end{array}$

is consistent, then :

${lim}_{x\to a}\frac{(x+2{)}^{\frac{3}{2}}-(a+2{)}^{\frac{3}{2}}}{x-a}$

${lim}_{x\to \pi }\frac{\sqrt{5+cosx}-2}{(\pi -x{)}^2}$

If the $A.M.$ between $p^{th}$ and $q^{th}$ terms of an $A.P.$ is equal to the $A.M.$ between $r^{th}$ and $s^{th}$ terms of the $A.P.$, then which of the following is true

Which of the following integral represents the summation ${\sum }^{}\left(4x^3\right)\delta x$ for the limit $\delta x\to 0$?

1. 10

If $R$ is a relation from $\{11,12,13\}$ to $\{8,10,12\}$ defined by $y=x-3.$ Then

The equation of the curve passing through (3, 9) which satisfies
​$\frac{dy}{dx}=\frac{x^3+1}{x^2}$is

Solve for $y$ if $\frac{d^{2}y}{d{t}^2}+2\frac{dy}{dt}+y=0$ with $y\left(0\right)=1$ and $y^{\prime }\left(0\right)=-2$

Prove that:

$\frac{cos2\theta }{1+sin2\theta }=tan(\frac{\pi }{4}-\theta )$

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

Which among the following is the correct graphical representation of $y=x^2+2x-3?$

Find the principal values of the following questions:

$se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)$

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$, then ${\cos }^{-1}x+{\cos }^{-1}y$ is equal to

1/2x+1/4x=x-6

If n positive integers are taken at random and multiplied together, the probability that the last digit of the product is 2, 4, 6 or 8, is

In a bank principal increases at the rate of 5% per year. An amount of Rs.1000 is deposited with this bank, how much will it be worth after 10yr $(e^{0.5}=1.648)$

Solve the
differential equation

Let $f\left(x\right)=\{\begin{array}{cc}{(1+\left|\sin x\right|)}^{\frac{a}{\left|\sin x\right|}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{\tan 2x}{\tan 3x}},& 0<x<\frac{\pi }{6}\end{array}$

If $f$ is continuous at $x=0$, then which of the following option(s) is correct

Out of $n$ students, a committee of $12$ students is formed. If number of such committees containing $2$ particular students $A,B$ is $3$ times the number of committees containing another $3$ particular students $D,E,F,$ then $n$ is

If $x+y+z=6,xy+xz+yz=11,xyz=6$,

then $\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy}$ equals

The value of $\int \frac{\sin x+\cos x}{2-\sin 2x}dx$ is

(where $C$ is constant of integration)