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If $y=\frac{a{x}^2}{(x-a)(x-b)(x-c)}+\frac{bx}{(x-b)(x-c)}+\frac{c}{x-c}+1,$ then $\frac{y^{\prime }}{y}$ is equal to

$(\text{Here},y^{\prime }=\frac{dy}{dx})$

If $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$
then area of the triangle whose vertices are $z_1,z_2,z_3$ is

Maximum number of total nodes is present in:

Seven athletes are participating in a race. If there are three prizes $Gold,Silver$ and $Bronze$, then the number of ways in which the prizes can be distributed to the athletes is

If sin A = n sin B, then $\frac{n-1}{n+1}$ tan $\frac{A+B}{2}$ =

Equation of plane which is parallel to XY-plane is

The vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ are in the direction of north-east and north-west respectively and $|\overrightarrow{b}|=|\overrightarrow{c}|=4$. The magnitude and direction of the vector $\overrightarrow{d}=\overrightarrow{c}-\overrightarrow{b},$ is

For a, b, c to be in GP, what is the value of $\frac{a-b}{b-c}$

$\text{Range of the function}$ $f\left(x\right)=x^2+\frac{1}{x^2+1},is$

The locus of the points representing complex number $z=x+iy$ for which $|z+5{|}^2-|z-5{|}^2=10$ is

4 HMs are inserted between 2 and 5, the 4th term in the H.P so formed is

${\int }_0^{10}se{c}^2(3x+6)dx$

If p: Arjun is the fastest q: Azad is the captain. Then which of the following denotes the compound statement: "Arjun is the fastest OR Azad is not the captain”

If $\frac{m}{n}=\frac{tanA}{tanB}$, find the value of $\frac{m+n}{m-n}$ is

Reflection of the complex number $\frac{2-i}{3+i}$ in the straight line $z(1+i)$ = $\overline{z}(i-1)$ is

The sum of coefficients of last 15 terms of the exapnsion $(1+x{)}^{29}$, when expanded in ascending powers of x is

If $k>0,|z|=k$ and $w=\frac{z-k}{z+k}$, then $Re\left(w\right)$ equals

Find the equation of the line passing through (-3, 5) and perpendicular to the line through the points (2, 5) and (-3, 6).

The inegral $\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$ is equal to :

(where $C$ is a constant of integration)

If $z=i\log (2-\sqrt{3})$, then $\cos z=$

Let $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0.$ Let $S_n={\alpha }^n+{\beta }^n$ for $n\ge 1$ and $\Delta =\left|\begin{array}{ccc}3& 1+S_1& 1+S_2\\ 1+S_1& 1+S_2& 1+S_3\\ 1+S_2& 1+S_3& 1+S_4\end{array}\right|$.If $a,b,c$ are rational and one of the roots of the equation is $1+\sqrt{2}$, then the value of $\Delta$ is

The focus of the parabols $x^2=-16y$ is

Prove the following:
cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1

The sum of an infinite geometric series with positive terms is $3$ and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is :

Polar form of $z=\frac{(1+7i)}{(2-i{)}^2}$ is

The positive integer $n$ for which $2\times 2^2+3\times 2^3+4\times 2^4+\dots +n\times 2^n=2^{n+10}$ is

For any two sets $M\&N,\text{if}M\subseteq N,$ then $M△N=$

Which of the following doesn't lie in any octant?

If $y=x^2+5x^{\frac{3}{2}}+\frac{2}{x},then\frac{dy}{dx}=$

The intercepts on $x$-axis made by tangents to curve, $y=\underset{0}{\overset{x}{\int }}|t|dt,x\in R$, which are parallel to the line $y=2x$, are equal to:

What is the product of roots of the cubic equation $x^3$-9 $x^2$+16x-30=0 __

If sec$\theta$ = $\frac{5}{4}$, then tan $\frac{\theta }{2}$ =

The standard deviation of the first n natural numbers is

If $z$ is a complex number satisfying $|z|=1$, then the range of $\arg \left(\frac{1}{1-z}\right)$ is

Find the value of ($cos{20}^{\circ }-sin{20}^{\circ }$) (1+4$sin{20}^{\circ }cos{20}^{\circ }$)

Bulk modulus was first defined by

Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$. The $(AB{)}^2$ is equal to :

The co-ordinates of the point in which the line joining the points (3, 5, -7) and (-2, 1, 8) is intersected by the plane yz are given by

[MP PET 1993]

Identify the correct statement(s).

The ends of latus rectum of parabola $x^2+8y=0$ are

If $f^{\prime }\left(x\right)={\tan }^{-1}(\sec x+\tan x),-\frac{\pi }{2}<x<\frac{\pi }{2},$ and $f\left(0\right)=0$, then $f\left(1\right)$ is equal to :

For $1\le r\le n$ , the value of \( {}^nC_r + ^{{}^{n-1}C_r}+ ^{{}^{n-2}C_r}.....+...+....+ ^{{}^rC_r} \space is : \)

For any two complex numbers $z_1$ and $z_2$ prove that

$Re\left(z_1{z}_2\right)=Re\left(z_1\right)Re\left(z_2\right)-Im\left(z_1\right)Im\left(z_2\right)$

If p and q are simple statements, $p\iff \sim q$ is true when

$$\begin{array}{cc}coloumn1& coloumn2\\ a& p)\frac{1}{x}\\ b& q)\frac{1}{x^2}\\ c& r)\frac{1}{x^3}\\ d& s)\frac{1}{x^4}\end{array}$$