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If $Z$ is a non zero complex number such that $Re\left(Z\right)=Im\left(Z\right),$ then

$\int \frac{1}{\sqrt{si{n}^{3}xsin(x+\alpha )}}dx.$

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. Then, R is

Circles $C_1$ and $C_2$ , of radii $r$ and $R$ respectively, touch each other as shown in the figure. The line $A$, which is parallel to the line joining the centres of $C_1$ and $C_2$ , is tangent to $C_1$ at $P$ and intersects $C_2$ at $A,B$. If $R^2=2r^2$, then $\angle AOB$ equals

(correct answer + 2, wrong answer - 0.50)

Let $f:R\to R$ be an invertible and a differentiable function defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2+ax-6,& x\le 2\\ \alpha x^2+\beta ,& x>2\end{array}$ where $a,\beta \in Z,\alpha \in R$ and $a\ge -5.$ If $f^{-1}$ denotes the inverse function of $f,$ then

If two matrices $A=\left[\begin{array}{cc}0& 2\\ 1& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}x& y\\ 0& 0\end{array}\right]$ are such that $AB=O.$ Then $(x+1,y-2)$ is

In a triangle $ABC,\frac{r_1-r}{a}+\frac{r_2-r}{b}+\frac{r_3-r}{c}=$

How many terms of G.P.
3, 3², 3³, … are needed to give the sum
120?

The solution of $ta{n}^{-1}x+2co{t}^{-1}x=\frac{2\pi }{3}$ is

The minimum and maximum values of $f\left(x\right)=x^2+4x+17$ are

A= $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$which of the following is true?

The equation of the plane containing the straight line $\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z}{5}$ and perpendicular to the plane $x-y+z+2=0$ is

If $f\left(x\right)=\{\begin{array}{cc}{x}^2+3x+a,& x\le 1\\ bx+2,& x>1\end{array}$ is a differentiable function, then which of the following is correct about $a$ and $b$?

For the given functions in $x\in [0,2],$ Rolle's theorem can be applicable on

The derivative of $\frac{sinx}{x}$ will be equal to .

Evaluate $\left|\begin{array}{cc}cos15& sin15\\ sin75& cos75\end{array}\right|$

How many positive integers are there with distinct digits??

The value of $\int \frac{e^x+3}{2x{e}^x+3e^x+6x+9}dx$ is

(where $C$ is constant of integration)

$6^2+7^2+8^2+9^2+{10}^2=$ ___ .

Find
the inverse of each of the matrices, if it exists.

Find two positive numbers x and
y such that their sum is 35 and the product x²y⁵
is a maximum

Three persons $A,B$ and $C$ speak at a function along with $5$ other persons. If persons speak at random, then the probability that $A$ speaks before $B$ and $B$ speaks before $C$ is

Let a function $f\left(x\right)=x\ln x$ and $f\left(x\right)=0$ has atleast one real solution, then which of the following(s) is(are) correct

$If{z}_1=2-i,z_2=-2+i,\text{find}\left(i\right)Re\left(\frac{z_1{z}_2}{z_1}\right)\left(ii\right)Im\left(\frac{1}{z_1{z}_2}\right)$

If the line joining the points $(k,1,2),(3,4,6)$ is parallel to the line joining the points $(-4,3,-6),(5,12,l)$ then $(k,l)$ is:

Let $f$ is a continuous function for all $x\in R$ and $f\left(x\right)=f(x+2T),T>0.$ If $I=\underset{0}{\overset{2T}{\int }}f\left(x\right)dx,$ then $\underset{4}{\overset{4+40T}{\int }}f\left(\frac{x}{4}\right)dx$ is equal to

If $f\left(x\right)=|x{|}^{|\sin x|},$ then $f^{\prime }(-\frac{\pi }{4})$ equals

Coordinates of the centre of a circle whose radius is $2$ units and touches the pair of lines $x^2-y^2-2x+1=0$ are

Which of the following will have their graph plotted in first and second quadrant only?

The solution of the differential equation $\left[\right(\cos x)dx–dy](1+x^2)+e^x\left[\right(1+x^2){\tan }^{–1}x+1]dx=0$ is

(where $C$ is arbitrary constant)

Projection of the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ on the plane $3x-6y+3z=108$ is

A ball is dropped from a height of $900$ centimetres. Each time it rebounds, it rises to two-third of the height it has fallen through. The total distance travelled by the ball before it comes to rest in metres, is

If $\left(\overrightarrow{A}\&\overrightarrow{B}\right)$ are two vectors then find $(\overrightarrow{A}+2\overrightarrow{B}).(2\overrightarrow{A}-3\overrightarrow{B})$.If $\theta$ is the angle between them.