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If $n$ arithmetic means are inserted between $20$ and $80$ such that the ratio of the first mean to the last mean is $1:3,$ then the value of $n$ is

A tangent and a normal are drawn at the point $P(2,-4)$ on the parabola $y^2=8x$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If $Q(a,b)$ is a point such that $AQBP$ is a square, then $2a+b$ is equal to

Differentiate the following functions with respect to x

$cos\left(x^3\right)si{n}^2\left(x^5\right)$

Let $\alpha ,\beta$ are the angle of inclination of the tangents to the axis of the parabola $y^2=4ax$ drawn from the point $P.$



Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\cot \alpha \cot \beta =k,\text{then locus of}P\text{is}& \text{(P)}& kx=a\\ \text{(B)}& \text{If}\tan \alpha +\tan \beta =k,\text{then locus of}P\text{is}& \text{(Q)}& y=k(x-a)\\ \text{(C)}& \text{If}\tan (\alpha +\beta )=k,\text{then locus of}P\text{is}& \text{(R)}& kx=y\\ \text{(D)}& \text{If}\tan \alpha \tan \beta =k,\text{then locus of}P\text{is}& \text{(S)}& xy=k\\ & & \text{(T)}& x=ka\end{array}$



Which of the following is the only CORRECT combination?

A balloon, which always remains spherical, has a variable diameter $\frac{3}{2}(2x+1)$ . Find the rate of change of its volume with respect to x.

Find the sum of the GP $(1+x{)}^{21}+(1+x{)}^{22}+......(1+x{)}^{30}$

If $4^x-2^{2+x}+5+||b-1|-3|=|\sin y|$, where $x,y,b\in R$ has a real solution, then the maximum possible value of $b$ is

Let $A=\left[\begin{array}{ccc}2& b& 1\\ b& b^2+1& b\\ 1& b& 2\end{array}\right]$ where $b>0$. Then the minimum value of $\frac{det\left(A\right)}{b}$ is :

Differentiate the following functions with respect to x :

$\frac{(x^3+1)(x-2)}{x^2}$

Find the value of ${lim}_{x\to 3}\frac{x^2-8x+15}{x^2-9x+18}$

A point on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at a distance equal to the mean of the lengths of the semi major axis and semi minor axis from the centre is

Let $S$ and $S^{\prime }$ be foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S^{\prime }BS$ is a right angled triangle with right angle at $B$ and area of $△S^{\prime }BS=8\text{sq. units}$, then the length of a latus rectum of the ellipse (in units) is :

If $|z_2+i{z}_1|=|z_1|+|z_2|$ and $|z_1|=3$ and $|z_2|=4$, then area of $△ABC$, if affixes $A,B$ and $C$ are $z_1,z_2$ and $\left(\frac{z_2-i{z}_1}{1-i}\right)$ respectively, is

Let $y=\frac{x^2}{(x+1{)}^2(x+2)}$. Then $\frac{d^{2}y}{d{x}^2}$ is

Which company does D work in?

If $sin\alpha +sin\beta =aandcos\alpha +cos\beta =b$, prove that

(i) $sin(\alpha +\beta )=\frac{2ab}{a^2+b^2}$

(ii) $cos(\alpha -\beta )=\frac{a^2+b^2-2}{2}$

If R be relation ‘<' from A = {1, 2, 3, 4} to B = {1, 3, 5} ie, (a, b) $ϵ$ R iff a < b, then $Ro{R}^{-1}$ is

(i) Which term of the A.P. 3, 8, 13, .... is 248 ?

(ii) Which term of the A.P. 84, 80, 76, .... is 0 ?

(iii) Which term of the A.P. 4, 9, 14, ..... is 254 ?

If ${\log }_3(3x-8)=2-x,$ then the value of $x$ is

Reflection of the line $\frac{x-1}{-1}=\frac{y-2}{3}=\frac{z-4}{1}$ in the plane x +y +z =7 is :

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Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively, in the ratio 2:1
externally.

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$

The true set of values of $a$ for which the inequality $\underset{a}{\overset{0}{\int }}(3^{-2x}-{2\cdot 3}^{-x})dx\ge 0$ is true is

Consider a circle with its centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola. Then the point of intersection of the circle and parabola can be

Let $(1+x+2x^2{)}^{20}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{40}{x}^{40}.$ Then, $a_1+a_3+a_5+\cdots +a_{37}$ is equal to:

Equation of the parabola obtained by taking reflection of $y=4x^2-4x+3$ about the line $y=x,$ will be

If A and B are square matrices of the same order and A is non-singular, then for a positive integer $n,(A^{-1}BA{)}^{n}isequalto$

Let A be a non singular, symmetric matrix of order three such that $A=adj(A+A^T)$, then

Let ${\overrightarrow{p}}_1=6x\hat{i}+2m\hat{j}-\hat{k}$ and ${\overrightarrow{p}}_2=-m^{2}x\hat{i}+3x\hat{j}+2\hat{k}$, if the angle between them is obtuse, then $m$ belongs to

If $x^4$ occurs in the $r^{th}$ term in the expansion of $(x^4+\frac{1}{x^3}{)}^{16}$, then r =