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If $|x|\ge 6,$ then $x$ can be represented on the number line by

Let $Z_1$ and $Z_2$ be complex numbers such that $z_1^2-4z_2$ = 16 + 20i. Suppose that $\alpha ,\beta$ rae roots of $t^2+z_{1}t+z_2+m$ = 0 for some complex number 'm' satisfying $|\alpha -\beta |$ = $2\sqrt{7}$. Then greatest value of |m| is

Let $f\left(x\right)=\frac{\sqrt{x^2+kx+1}}{x^2-k}$ is continuous for every $x\in R$ then set of values of $k$ is

Which of the following biconditional statements are true?

Let $\alpha +1$ and $\frac{1}{\alpha }+1$ be the roots of $x^2-2(p+1)x+5p-p^2=0$ for non-zero $\alpha .$ If $f:R\to [0,\infty )$ defined by $f\left(x\right)=x^2-2(p+1)x+5p-p^2$ is surjective function, then $p$ can be

If $\int \frac{1-x^7}{x(1+x^7)}dx=A\ln |x|+B\ln |1+x^7|+C$, then which of the following is/are true (where $A,B$ are fixed constants and $C$ is constant of integration)

A tangent to the ellipse $x^2+4y^2=4$ meets the ellipse $x^2+2y^2=6$ at $P$ and $Q$. Then the angle between pair of tangents at $P$ and $Q$ of the ellipse $x^2+2y^2=6$ is

The value(s) of $a$ for which the roots of $2x^2+(a^2-1)x+a^2+3a+4=0$ are reciprocal to each other is/are

A die is thrown, a man C gets a prize of $Rs.5$ if the die turns up $1$ and gets a prize of $Rs.3$ if the die turns up $2$, else he gets nothing, A man A whose probability of speaking the truth is $\frac{1}{2}$ tells C that the die has turned up $1$ and another man B whose probability of speaking the truth is $\frac{2}{3}$ tells C that the die has turned up $2$. Find the expectation value of C.

The direction ratio of the normal to the plane passing through the points $(1,2,-3),$ $(-1,-2,1)$ and parallel to $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z}{4}$ is

The equation of the circle having centre at the origin and passing through the point $(3,4),$ is

A solid disk of radius $R$ is suspended from a spring of linear spring constant $k$ and torsional constant $C$, as shown in figure. In terms of $k$ and $C$, what value of $R$ will give the same period for the vertical and torsional oscillations of this system?

The letters of the word 'CLIFTON' arc placed at random in a row. What is the chance that two vowels come together?

The equation of plane passes through the point $(-1,0,1)$ and containing the line $\frac{x+1}{2}=\frac{y-3}{-1}=\frac{z+1}{1},$ is

If the difference of the roots of the equation $(k-2)x^2-(k-4)x-2=0,k\ne 2$ is $3$, then the sum of all the values of $k$ is

Which of the following function is an into function if all are defined on f: R $\to$ R

A function $f\left(x\right)=1-x^2+x^3$ is defined in the closed interval $[-1,1]$. The value of $x$, in the open interval $(-1,1)$ for which the mean value theorem is satifsied, is

If ${\tan }^{-1}x=\frac{\pi }{10}\mathrm{for}\mathrm{some}x\in R,$

then the value of $co{t}^{-1}x\mathrm{is}$

Aron scored $2$ goal in an ice - hockey match. Steve scored $2$ times as many goals as Aron . How many goals did Steve score in the match ?

Let $A(1,2),B(\text{cosec}\alpha ,-2)$ and $C(2,\sec \beta )$ are $3$ points such that $(OA{)}^2=OB\cdot OC$,($O$ is the origin) then the value of $2{\sin }^2\alpha -{\tan }^2\beta$ is

$\int \frac{dt}{\sqrt{3t-2t^2}}=$

If the number of integral terms in the expansion of ${(3^{1/2}+5^{1/8})}^n$ is exactly $33,$ then the least value of $n$ is:

$\underset{x\to 0}{lim}\frac{(1+x{)}^{\frac{1}{3}}-(1-x{)}^{\frac{1}{3}}}{x}=$

जानवरों की बोलियाँ तो तुमने सुनी ही होंगी। कोयल की बोली को जैसे कूकना कहते हैं और मक्खी की बोली को भिनभिनाना, वैसे ही अन्य जानवरों की बोलियों के भी नाम हैं।

नीचे दिए गए खाने में एक तरफ़ जानवरों के नाम हैं, दूसरी तरफ़ बोलियों के। ढूँढ़ निकालो कौन-सी बोली किसकी है?

Let the function, $f:[-7,0]\to R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }\left(x\right)\le 2,$ for all $x\in (-7,0),$ then for all such functions $f,f(-1)+f\left(0\right)$ lies in the interval:

Consider the differential equation, $y^{2}dx+(x-\frac{1}{y})dy=0.$ If value of $y$ is $1$ when $x=1,$ then the value of $x$ for which $y=2,$ is:

Value of $\underset{x\to 0}{lim}\frac{1-\cos x}{x\sin 3x}$ is

$\int {\tan }^{2}x.se{c}^{4}xdx=$

If $sin\alpha ,sin\beta ,cos\alpha$ are in GP, then roots of equation $x^{2}sin\beta +2xcos\beta +sin\beta =0$ are

Let $a,b,c$ and $d$ be non-zero numbers such that $x=c$ and $x=d$ are the roots of the equation $x^2+ax+b=0$ and $x=a$ and $x=b$ are the roots of the equation $x^2+cx+d=0.$ Then the value of $a+b+c+d$ is

Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that n (A) = 200, n(B) =300 and $n(A\cap B)$= 100. Then , $n(A^{\prime }\cap B^{\prime })=$

Integrate the following functions w.r.t. x.

$\int \frac{1}{x^{1/2}+x^{1/3}}dx.$