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The mean of first three terms is 14 and mean of next two terms is 18. The mean of all the five terms is

If the letters in the word $CALCULUS$ are arranged in all possible manners (any arrangement is considered a word, for example $CCLLAUUS$ would be considered a word), then the probability that the $2U^{\prime }s$ are together, is

The number of six letter words that can be formed using the letters of the word "ASSIST" in which S's alternate with other letters is

1. 4

The equation of a line passing through $(-2,3)$ and parallel to the tangent at origin for circle $x^2+y^2+x-y=0$ is:

If z is a non-zero complex number, then $\left|\frac{|\overline{z}{|}^2}{z\overline{z}}\right|$ is equal to

The anti-derivative of$(\sqrt{x}+\frac{1}{\sqrt{x}})$ equals to (where $C$ is the constant of integration)

Find the rational values of $\lambda \&\mu$ if both the quadratic equations $4x^2-x-1=0\&3x^2+(\lambda +\mu )x+(\lambda -\mu )=0$ have a common root.

Write the equation of the hyperbola of eccentricity $\sqrt{2}$.if it is know that the distance between its foci is 6.

The point(s) of contact of the tangent(s) drawn from the origin to the curve $y=x^2+3x+4$ is/are

If the line, $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane, lx+my-z = 9, then $l^2+m^2$ is equal to

Range of the function $f\left(x\right)=\frac{1}{3|x|+2}$ is

The equation of the line joining origin to the points of intersection of the curve $x^2+y^2=a^2$ and $x^2+y^2-ax-ay=0$ is

Three planes, viz the XY Plane, XZ Plane and the YZ Plane divide the space into eight parts. Each part is called an OCTANT. What is the relation between these three planes

Prove
that the coefficient of xⁿ
in the expansion of (1 + x)²ⁿ
is twice the coefficient of xⁿ
in the expansion of (1 + x)^2n–1
.

If $\int \frac{({\cos }^{3}x+{\cos }^{5}x)}{({\sin }^{2}x+{\sin }^{4}x)}dx=A\sin x+B\text{cosec}x+C{\tan }^{-1}\left(\sin x\right)+K,$ then the value of $A+B-C$ is equal to

(Here, $A,B,C$ are fixed constants and $K$ is arbitrary constant of integration)

If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\arg \left(z\right)-\arg \left(\omega \right)=\frac{3\pi }{2},$ then $\arg \left(\frac{1-2\overline{z}\omega }{1+3\overline{z}\omega }\right)$ is

(Here $\arg \left(z\right)$ denotes the principal argument of complex number $z$ )

A die is rolled and the outcomes are observed. Event A is an even number turns up and event B is a prime number turns up. Match the events on left with the outcomes on right.

$\begin{array}{cc}1.\text{Event A or B}& P.\{4,6\}\\ 2.\text{Event A and B}& Q.\left\{2\right\}\\ 3.\text{Event A but not B}& R.\{2,3,4,5,6,\}\\ 4.\text{Complementary event of B}& S.\{1,4,6\}\end{array}$

If 1 = , 2 = and 3 = , then 5 = ‘?’.



यदि 1 = , 2 = तथा 3 = , तब 5 = ‘?’.

If $z$ is a complex number satisfying $|z-3-2i|\le 1$, then the maximum value of $|iz+2|$ is

If rolle's theorem is applicable on$f\left(x\right)=x^{\alpha }(x-1{)}^2$ in $[0,1],$ then the value of $\alpha$ can be

The function $f\left(x\right)=x^3+a{x}^2+bx+c,a^2\le 3b$ has

Find the area bounded by the curve x²
= 4y and the line x = 4y – 2

The radius of the circle $x^2+y^2+4x+6y+13=0$is

If $A=\left[\begin{array}{cc}3& x\\ y& 0\end{array}\right]$ and $A=A^T$, then which of the following is always correct?

If $y={\cot }^{-1}\left(\frac{\sqrt{1+x^2}+1}{x}\right),x>0,$ then $\frac{dy}{dx}$ is equal to

If ${\int }_0^a\frac{1}{1+4x^2}dx=\frac{\pi }{8},then$ a = ............