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Let $f\left(x\right)=\sin x,g\left(x\right)={\log }_e|x|$. If the ranges of the composite functions $fog$ and $gof$ are $R_1$ and $R_2$ respectively, then

In ΔABC, right angled at B. If, find the value of

(i) sin A cos C + cos A sin C

(ii) cos A cos C − sin A sin C

Let $f:R\to R$ be a continuously differentiable function such that $f\left(2\right)=6$ and $f^{\prime }\left(2\right)=\frac{1}{48}.$

If $\underset{6}{\overset{f\left(x\right)}{\int }}4t^{3}dt=(x-2)g\left(x\right),$ then $\underset{x\to 2}{lim}g\left(x\right)$ is equal to

$\int \frac{x}{x^4-1}dx=$

The equation of circle touching the line $2x+3y+1=0$ at $(1,-1)$ and cutting orthogonally the circle having line segment joining $(0,3)$ and $(-2,-1)$ as diameter is

Suppose $a^2=5a-8$ and $b^2=5b-8$, then equation whose roots are $\frac{a}{b}$ and $\frac{b}{a}$ is

If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R-Q) ?

Which of these is the shape of graph of $y=x^2-2x+5$.

The number of all possible values of $\theta ,where0<\theta <\pi$, for which the system of equations $(y+z)cos3\theta =\left(xyz\right)sin3\theta$
$xsin3\theta =\frac{2cos3\theta }{y}+\frac{2sin3\theta }{z}\left(xyz\right)sin3\theta =(y+2z)cos3\theta +ysin3\theta$
have a solution $(x_0,y_0,z_0)with{y}_0{z}_0\ne 0$ is___

Let set $R=\{P:B\subseteq P\subseteq A\}.$ If $A=\{1,2,3,4,5\}$ and $B=\{1,2\},$ then the number of elements in set $R$ is

cos68^o cos8^o + sin68^o sin8^o =?

If matrix $A$ is a square matrix, then the possible number of elements in $A$ is

Graph of f(x) is given. Draw the graph of f${}^{-1}$(x)

18 guests have to be seated , half on each side of a long table.4 particular guests desire to sit on 1 particular side and 3 others on the other side .Deter Dete the number of ways in which the sitting arrangement can be done.

Area under the circle x² + y² = 16 is

The resultant of $P$ and $Q$ is $R$. If $Q$ is doubled, $R$ is also doubled and if $Q$ is reversed, $R$ is again doubled. Then, $P^2:Q^2:R^2$ given by

Find the set of values of $m$ for which exactly one root of the equation $x^2+mx+(m^2+6m)=0$ lie in $(-2,0)$

The equation of the circle which passes through the focus of the parabola $x^2=4y$ and touches it at (6, 9) is

The value of $\underset{z\to 4}{lim}\frac{\sqrt{z}-2}{z-4}$ is

If for a triangle $ABC$,

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$

then ${\sin }^{3}A+{\sin }^{3}B+{\sin }^{3}C=$

Which of the following is/are quadratic equation?

$bcosB+ccosC=acos(B-C)$

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$, then the length of its latus rectum is

If $1+2+2^2+2^3+\dots +2^{1999}$ is divided by $5$, then the remainder is

The solutions of the equation $z^2+|z|=0$ are

Consider a right angled triangle $ABC$


If the sides of the triangle are $a=6,b=10,c=8$ units, then the distance between incentre and circumcentre will be

Let $f\left(x\right)=|x-2|$ and $g\left(x\right)=f\left(f\right(x\left)\right),x\in [0,4].$ Then ${\int }_0^3\left(g\right(x)-f(x\left)\right)dx$ equal to