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Equation of the auxiliary circle of the ellipse

$\frac{x^2}{12}+\frac{y^2}{18}=1$ is

If $f\left(x\right)={\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)-{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\forall |x|<\frac{1}{\sqrt{3}},$ then the value of $f^{\prime }\left(0\right)$ is

Find the maximum area of an isosceles triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with its vertex at one end of the major axis.

The range of the function $f\left(x\right)=|x-1|+|x-8|$ is

The mean and standard deviation of random variable $X$ are $10$ and $5$ respectively. Then $E{\left(\frac{X-15}{5}\right)}^2=$______

Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left(\right(y+1)e^{\frac{x^2}{2}}-x),0<x<2,$ such that $y\left(2\right)=0.$ Then the value of $\frac{dy}{dx}$ at $x=1$ is

The sum of the series $1+2\times 3+3\times 5+4\times 7+\dots \text{upto}{11}^{\text{th}}\text{term}$ is :

Let $p,q,r$ are lengths of an acute angled triangle $△PQR$ opposite to sides $QR,PR,PQ$ respectively. The perpendiculars are drawn from the angles $P$, $Q$ and $R$ on opposite sides and produced to meet the circumscribing circle. If these produced parts be ${\theta }_1,{\theta }_2,{\theta }_3$ respectively, then the value of $\frac{\left(p/{\theta }_1\right)+\left(q/{\theta }_2\right)+\left(r/{\theta }_3\right)}{\tan P+\tan Q+\tan R}$ is

(correct answer + 1, wrong answer - 0.25)

Prove that cos4x = cos2x

The number of observations in a group is 40. If the average of first 10 is 4.5 and that of the remaining 30 is 3.5, then the average of the whole group is

Integrate the following functions.
$\int \frac{2x}{1+x^2}dx.$

If the straight lines joining the origin and the point of the intersection of the curve $x^2+12xy-y^2+4x-2y+3=0$ and $x+ky-1=0$ are equally inclined to $x$-axis then the value of $k$ is

The average of $5$ distinct positive integers is $33$. If the average of the three largest numbers within this set is $39$ then the difference of the maximum and minimum possible values of the median of the $5$ numbers is

If $\left[x\right]$ be the greatest integer less than or equal to $x,$ then $\sum _{n=8}^{100}\left[\frac{{(-1)}^{n}n}{2}\right]$ is equal to

The least positive integer n which will reduce ${\frac{i-1}{i+1}}^n$to a real number , is

The approximate value of $(25{)}^{\frac{1}{3}}$ is

If $A=\{1,2,3,4,5,6,7,8\},B=\{1,3,5,6,7,8,9\}$ then $n\left(\right(A\Delta B)\times (B\Delta A\left)\right)$ is equal to

If $(2x-1{)}^{20}-(ax+b{)}^{20}=(x^2+px+q{)}^{10}$ holds true $\forall x\in R$ where $a,b,p$ and $q$ are real numbers, then the value of $p$ is

Show that the expansion of ${(x^2+\frac{1}{x})}^{12}$ does not contian any term involving $x^{-1}.$

Which of the following value(s) of $\alpha$ satisfy the equation $\left|\begin{array}{ccc}(1+\alpha {)}^2& (1+2\alpha {)}^2& (1+3\alpha {)}^2\\ (2+\alpha {)}^2& (2+2\alpha {)}^2& (2+3\alpha {)}^2\\ (3+\alpha {)}^2& (3+2\alpha {)}^2& (3+3\alpha {)}^2\end{array}\right|=-648\alpha$

The value of integral $\underset{2}{\overset{8}{\int }}\frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}}dx$ is

If the tangent to the curve $y=x^3$ at the point $P(t,t^3)$ meets the curve again at $Q$, then the ordinate of the point which divides $PQ$ internally in the ratio $1:2$ is :

We have to choose $11$ players for cricket team from $8$ batsmen, $6$ bowlers, $4$ all rounders and $2$ wicket keepers. Number of selections, when two particular batsmen do not want to play when a particular bowler will play, is

If $f\left(x\right)=\frac{1}{x-1}$ and $g\left(x\right)=\frac{x-1}{x+1}$, then the domain of $(f\circ g)\left(x\right)$ is

If the centre of the sphere $x^2+y^2+z^2-2x-4y-6z=0$ is (a,b,c) , find the value of a+b+c

The area (in sq.units) of the region $\left\{\right(x,y)\in R^2:x^2\le y\le 3-2x\}$, is:

${\int }_0^{\pi /4}\frac{si{n}^{9}x}{co{s}^{11}x}dx=$

Two lines whose direction ratios are a1, b1, c1 and a2, b2, c2 are parallel, if

Along a railway line, there are 20 stations. The number of different tickets required in order so that it may be possible to travel from every station to every station is what ?