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If $z_1=(2-i)$ and $z_2=(1+i),$ find $\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|$.

The locus of a point which moves so that its distance from x-axis is double of its distance from y-axis is

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



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Let}z,\omega ,\alpha \text{be complex numbers such that}|z|=|\omega |=4\text{and}\alpha =\frac{z-\overline{\omega }}{16+z\overline{\omega }}\text{. Then Re}\left(\alpha \right)\text{is equal to}& \text{(P)}& 0\\ \text{(B)}& \text{If}x=p+iq\text{is a complex number such that}{x}^2=3+4i\text{and}{x}^3=2+11i\text{where}i=\sqrt{-1},& \text{(Q)}& 3\\ & \text{then}(p+q)\text{is equal to}& & \\ \text{(C)}& \text{Number of complex numbers}z\text{satisfying the equation}\overline{z}=i{z}^2\text{, where}i=\sqrt{-1}\text{is equal to}& \text{(R)}& 4\\ \text{(D)}& \text{If}z\in C\text{satisfies}|z+2-i|=5\text{, then the maximum value of}\frac{|3z+9-7i|}{4}\text{is equal to}& \text{(S)}& 5\end{array}$



Which of the following combinations is CORRECT?

Let $f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|.$ The value of $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}$ is equal to

The average of n numbers $x_1,x_2,x_3,..........,x_n$ is M. If $x_n$ is replaced by x’, then new average is

If $2|x+1{|}^2-3|x+1|+1=0$, then $x=$

Let $a_1$, $a_2$, ..., $a_n$ be fixed real numbers and define a function f(x) = $(x-a_1)(x-a_2)...(x-a_n)$. What is $\underset{x\to a_1}{\text{lim}}$ f(x)?
For some a $\ne a_1,a_2...a_n$, compute $\underset{x\to a}{\text{lim}}$ f(x).

If the centroid of $△ABC$ whose vertices are $A(-3,2)$ and $B(-2,1),$ lies on the line $3x+4y+2=0$, then the locus of vertex $C$ is

Find the equation of the hyperbola satisfying the given condition,

Foci $(\pm 5,0)$, the transverse axis is of length 8.

Find the minimum and maximum value of the function $y=x^3-3x^2+6$. Find the values of x at which it occurs.

$f\left(x\right)=2x^2+bx+c$ and $f\left(0\right)=3$ and $f\left(2\right)=1$, then $f\left(1\right)$ is equal to

What is the coefficient of ${}^{\prime }a{b}^{\prime }$ in the expansion of $(a+b+c{)}^2$ ___?

A man running on a race course notes that sum of its distances from two flags posts from him is always 10m and the distance between the flag posts is 8m. He notes that he can read the messages of value system 'HONESTY'and 'RESPECT FOR OTHER'on the poles which ever side he moves. Find the equation of the path traced by the man. Do you think these value systems are necessary in life?

The value(s) of k for which equations $3x^2+4kx+2=0and2x^2+3x-2=0$ will have a common root can be:

A rectangular hyperbola whose centre is C, is cut by any circle of radius r in four points P, Q, R and S. Then, $C{P}^2+C{Q}^2+C{R}^2+C{S}^2$ is equal to

If the sum of the first $40$ terms of the series, $3+4+8+9+13+14+18+19+\dots$ is $\left(102\right)m$, then $m$ is equal to :

Find the degree measures corresponding to the following radian measures $(\text{use}\pi =\frac{22}{7})$:

(i) $\frac{11}{16}$ (ii) -4 (iii) $\frac{5\pi }{3}$ (iv) $\frac{7\pi }{6}$

The general solution of the equation $si{n}^{50}x-co{s}^{50}x=1$ is

If $x^2+7x+1=0and(a^2-4a)x+bx-3=0$ have both the roots in common, then find the value of a

Consider a polynominal $p\left(x\right)=x^6+2x^2+1$. If $x_1,x_2,\dots ,x_6$ are the roots of $p\left(x\right)=0$ and $q\left(x\right)=x^3-1$, then the value of $\prod _{i=1}^{6}q\left(x_i\right)$ is

(where $\prod$ stands for product of terms)

How many of the following trigonometric ratios are correct?

(1)Sin (${15}^{\circ }$) = $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(2)Cos (${105}^{\circ }$) = $\frac{1-\sqrt{3}}{2\sqrt{2}}$

(3)Sin ${75}^{\circ }$ = $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(4)Tan ${15}^{\circ }$ = $2-\sqrt{3}$

Let $\overline{a}=2\overline{i}-3\overline{j}+\hat{k},\overline{b}=\hat{i}-\hat{j}+3\hat{k}and\overline{c}=\hat{i}+\hat{j}-\hat{k}$ Then the system of vectors $\overline{a},\overline{b}and\overline{c}$

Let $f:N\to R$ be a function satisfying the following conditions:
$f\left(1\right)=1$ and $f\left(1\right)+2f\left(2\right)+\dots +nf\left(n\right)=n(n+1)f\left(n\right)$ for $n\ge 2$.
If $f\left(999\right)=\frac{1}{K}$, then $K$ equals

A pair of unbiased dice are rolled together till a sum of either 5 or 7 is obtained. Probability that 5 comes before 7 is

Which of the following statements is/are correct for the above given figure?

1.For all the function A, B and C, domain is R and Range is $(0,\infty )$.

2.Function B is a graph for $F\left(x\right)=a^x,a>1,xϵR$.

3.Function A is a graph for $F\left(x\right)=a^{x}0<a<1,xϵR$

4.Function C is a graph for $F\left(x\right)=a^x,a<0,xϵR$

5.If the graph of B is $5^x$, then the graph of A is $y=8^x$ and the graph of C is

$y=2^x$ (you have to choose graph from $2^x,8^{x}and{5}^x)$

6.If the graph of B is $5^x$, then the graph of A is $y=2^x$ and the graph of C is

$y=8^x$ (you have to choose graph from $2^x,8^{x}and{5}^x)$

The middle term in the expansion of $(1+x{)}^{2n}$ is

What is the slope of the chord of contact of the point (6, -4) to the circle $x^2+y^2-2x-2y+1=0$

If $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2$ are such that the minimum value of $f\left(x\right)$ always exceeds maximum value of $g\left(x\right),$ then which of the following is/are correct?

Hot water cools from ${60}^{o}Cto{50}^{o}C$ in the first 10 minutes and to ${42}^{o}C$ in the next 10 minutes. The temperature of the surrounding is

Let $f:(4,6)\to (6,8)$ be defined by $f\left(x\right)=x+\left[\frac{x}{2}\right]$, where $[.]$ represents the greatest integer function. Then the range of $f$ is

Let set A = {a, b, c, d} and set B = {4, 8, 16, 32). If the total number of relations from A to B is $2^x$ find the value of $\frac{\left(x\right)}{2}$?

In which of the following cases is the function f(x) discontinuous at a?

If $\frac{\sqrt{1+cosx}+\sqrt{1-cosx}}{\sqrt{1+cosx}=\sqrt{1-cosx}}=cot(a+\frac{x}{2}),xϵ(\pi ,2\pi )$ then a___

If the function
$f\left(x\right)=\{\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|,}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

The number of real values of $x$, that satisfy the equation $x^{{\log }_{\sqrt{x}}2x}=4$ is

Find $\frac{d}{dx}$ of $y=tanx$.

In any $\Delta ABC,$ prove that

$\frac{(b-c)}{b+c}=\frac{tan\frac{1}{2}(B-C)}{tan\frac{1}{2}(B+C)}$

Find $\frac{dy}{dx}$ if $y=e^{x}sinx$

If z is any complex number then $\frac{\overline{z}}{|z{|}^2}$ =

Find the mean deviation about the mean for the data in Exercise 9 to 10

$\begin{array}{ccccccc}\text{Height in cm}& 95-105& 105-115& 115-125& 125-135& 135-145& 145-155\\ \text{Number of boys}& 9& 13& 26& 30& 12& 10\end{array}$

In a hyperbola the latusrectum equals to semitransverse axis,then its eccentricity is

Let $\alpha ,\beta$ be the roots of $x^2-x-1=0$ $(\alpha >\beta )$ and $m,n\in Z,k\in W$ such that $a_k=m{\alpha }^k+n{\beta }^k.$ If $a_4=35,$ then the value of $3m+2n$ is equal to

What can be inferred if the Range of one data set is higher than other derived from the same experiment?

If the roots of the equation

$(1-q+\frac{p^2}{2})x^2+p(1+q)x+q(q-1)+\frac{p^2}{2}=0$ are equal, then