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The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:

If $x\in (0,\frac{\pi }{6})$, then the number of solutions of the equation $\left\{2x\right\}+\left\{\tan x\right\}=0$ is



(Here, $\left\{x\right\}$ denotes the fractional part of $x$.)

If $f\left(x\right)=(2x-3\pi {)}^5+\frac{4}{3}x+\cos x$ and $g$ is the inverse function of $f$, then $g^{\prime }\left(2\pi \right)$ is equal to

The sum of $\sum _{r=0}^n(-1{)}^r\frac{{}^n{C}_r}{{}^{r+2}{C}_r}$ is

Is the function f
defined by

continuous at x
= 0? At x
= 1? At x
= 2?

Circle drawn through the point $(2,0)$ to cut intercept of length $‘5^{\prime }$ units on the x-axis. If its centre lie in the first quadrant then the equation of family of such circles is

If $A$ is the area (in $\text{sq.units}$) enclosed between the curve $y=3\sqrt{2x}$ and $x=2\sqrt{3y},$ then which of the following is/are correct $?$

(where $[.]$ denotes $G.I.F.$)

Find the
general solution of the equation

The equation of normal to the hyperbola $3x^2-y^2=1$ having slope $\frac{1}{3}$ is

If $\left[\begin{array}{ccc}2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& x& 3\\ 2& 4& 5\\ 3& 2& x\end{array}\right]\left[\begin{array}{c}x\\ 2\\ 0\end{array}\right]=0$, then $x=$

The locus of a point whose chord of contact with respect to parabola
$y^2=8x$ passes through focus is

Multiply $(2x^2-6x+10)$ by $(x+y^2)$ .

Which average would be suitable in the following cases?

(i) Average size of readymade garments.

(ii) Average intelligence of students in a class.

(iii) Average production in a factory per shift.

(iv) Average wage in an industrial concern.

(v) When the sum of absolute deviations from average is least.

(vi) When quantities of the variable are in ratios.

(vii) In case of open-ended frequency distribution.

Let $T_1,T_2,T_3,\dots$ be terms of an A.P. If $S_1=T_1+T_2+T_3+\cdots +T_n$ and $S_2=T_2+T_4+T_6+\cdots +T_{n-1}$, where $n$ is odd, then the value of $\frac{S_1}{S_2}$ is

The number of common tangents to circles $x^2+y^2-4x+2y-4=0$ and $x^2+y^2+2x-6y+6=0$ is

Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is:

The value of $\tan (2{\tan }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right))$ is equal to:

Number of positive integral solutions of $abc=30$ is

If matrix $A$ is given by $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the determinant of $A^{2005}-6A^{2004}$ is

If $p\left(x\right)$ is a polynomial of degree greater than $2$ such that $p\left(x\right)$ leaves remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively. If $p\left(x\right)$ is divided by $x^2-a^2$ then remainder is

Find the Cartesian equation of the line that passes through the point (-3,-4,-2) and (3,4,2).

if $\sum _{r=0}^n\left\{a_r\right(x-y+2{)}^r-b_r(y-x-1{)}^r\}=0,\forall$ n $\in$ N, then $b_n$ is equal to

The distance moved by the particle in time t is given by $x=t^3-12t^2+6t+8$. At the instant when its acceleration is zero, then the velocity is

In a triangle $ABC$ medians $AD$ and $CE$ are drawn. If $AD=5,\angle DAC=\frac{\pi }{8},\angle ACE=\frac{\pi }{4}$ then area of $\Delta ABC$ is: