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Let $A$ and $B$ be any two points on the lines represented by $4x^2-9y^2=0.$ If the area of triangle $OAB$ is $5$ $\left(O\text{is origin}\right)$ then which of the following is the possible equation of the locus of midpoint of $AB?$

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$). Then if ${\alpha }_1<\alpha <{\beta }_1<\beta$, then

If in $\Delta ABC,\tan \frac{A}{2}=\frac{5}{6}$ and $\tan \frac{C}{2}=\frac{2}{5},$ then $a,b,c$ are in:

The equation(s) of angular bisector between the intersecting lines $L_1:\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and $L_2:\frac{x}{-3}=\frac{y}{-2}=\frac{z}{-1}$ is/are

sin
2x
+ 2sin 4x
+ sin 6x
= 4cos²
x
sin 4x

Find a quadratic equation with $\frac{1}{8}$ as the sum and 2 as product of its rootss.

The area (in $\text{sq.units}$) enclosed between the curve $|x|+|y|<3$ and $x^2-6x+y^2<0$ is

Using vectors, find the area of the $△$ABC with vertices A(1, 2, 3), B (2, -1, 4) and C(4, 5, -1).

Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f(2-x)$ and $f^{\prime \prime }\left(x\right)<0,$ $\forall x\in (0,2).$ Then $g\left(x\right)$ is strictly decreasing in

Find X, if $Y=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]and2X+Y=\left[\begin{array}{cc}1& 0\\ -3& 2\end{array}\right]$

The general solution of the differential equation $\frac{dy}{dx}+\sin \frac{x+y}{2}=\sin \frac{x-y}{2}$ is (where $c$ is a constant of integration)

The number of real roots of the equation $e^{4x}+{2e}^{3x}-e^x-6=0$ is

The equation of the bisectors of the angles between the lines represented by $x^2+2xycot\theta +y^2=0$, is

If ${\left(\frac{a}{b}\right)}^{\frac{1}{3}}+{\left(\frac{b}{a}\right)}^{\frac{1}{3}}=4$, then the acute angle $\left(\theta \right)$ of intersection of the parabolas $y^2=4ax$ and $x^2=4by$ at a point other than the origin is

$\text{Find z,~if}|z|=4andarg\left(z\right)=\frac{5\pi }{6}$

If $y=x^{(\ln x{)}^{\ln \left(\ln x\right)}},$ then $\frac{dy}{dx}=$

The number of solutions of ${\sum }_{r=1}^{5}cosrx=5$ in the interval $[0,2\pi ]$ is

A reversible adiabatic path on a $P-V$ diagram for an ideal gas passes through state $A$, where $P=0.7\times {10}^5{\text{Nm}}^{–2}$ and $V=0.0049{\text{m}}^3$. The ratio of specific heats of the gas is $1.4$. The slope of $P-V$ diagram at $A$ is

If $\frac{a}{\sqrt{bc}}-2=\sqrt{\frac{b}{c}}+\sqrt{\frac{c}{b}}$, where $a,b,c>0$ then the family of line $\sqrt{a}x+\sqrt{b}y+\sqrt{c}=0$ passes through a fixed point given by

If vector $\overrightarrow{a}=2\hat{i}-3\hat{j}+6\hat{k}$ and vector $\overrightarrow{b}=-2\hat{i}+2\hat{j}-\hat{k}$, then ratio of projection of $\overrightarrow{a}$ on vector $\overrightarrow{b}$ to projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is equal to

If f(x) = (x-1), what is f(0)?

The interval in which $\theta$ belongs, such that the inequality $2{\sin }^2(\theta -\frac{\pi }{3})-\sin (\theta -\frac{\pi }{3})-1\le 0$ is satisfied and $\theta \in [-\pi ,\pi ]$ is

What is the condition for a function y = f(x) to be a strictly increasing function.

Examine
the continuity of the function.

If $2p+q=48$ (where $p,q$ are prime), then

The probability that the birthday of six different persons will fall in exactly two calendar months is

The solution set of the inequality $\left({\cot }^{-1}x\right)\left({\tan }^{-1}x\right)+(2-\frac{\pi }{2}){\cot }^{-1}x-3{\tan }^{-1}x-3(2-\frac{\pi }{2})>0$ is

If the general solution of some differential equation is $y=a_1(a_2+a_3)\cos (x+a_4)-a_5{e}^{x+a_6}$, then order of the differential equation is

If $2f\left(sinx\right)+f\left(cosx\right)=x\forall xϵ$ R then range of f(x) is

The sum of the series $\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}\cdots$ up to $\infty$ is equal to

The minimum number of elements that must be added to the relation $R=\left\{\right(1,2\left)\right(2,3\left)\right\}$ on the set of natural numbers so that it is an equivalence is:

If both the roots of $a{x}^2+bx+c=0$ are negative and $b<0$, then which of the following statements is always true?

The sum of $10$ terms of the series $\frac{3}{1^2\times 2^2}+\frac{5}{2^2\times 3^2}+\frac{7}{3^2\times 4^2}+\cdots$ is