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If $\int x^5{e}^{-4x^3}dx=\frac{1}{48}{e}^{-4x^3}f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to :

The values of $x\in (-\frac{\pi }{2},\frac{\pi }{2})$ for which both the conditions ${\tan }^{2}x-\tan x<0$ and $2|\sin x|<1$ satisfy simultaneously are

Let $A=\{x\in R:x^2-|x|-2=0\}$ and $B=\{\alpha +\beta ,\alpha \beta \}$ where $\alpha ,\beta$ are real roots of the quadratic equation $x^2+|x|-2=0.$ If $(a,b)\in A\times B,$ then the quadratic equation whose roots are $a,b$ is

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2-x+3,& x<1\\ \lambda ,& x\ge 1\end{array}.$

If $f\left(x\right)$ has local minimum at $x=1,$ then set of real values of $\lambda$ is

The differential equation for the equation y = Acos αx + Bsin αx is:

The maximum radius of a sphere that can be fitted in the octahedral hole of cubical closed packing of the sphere of radius $r$

The value of $\underset{0}{\overset{100}{\int }}(x-[x\left]\right)dx$,

where $[\cdot ]$ represents the greatest integer function, is

If $a=3^{1/223}+1$ and $f\left(n\right)=\sum _{r=1}^n(-1{)}^{r-1}{}^n{C}_{r-1}\cdot a^{n-r}$ for all $n\ge 3$, then the value of $f\left(2007\right)+f\left(2008\right)$ is

The general solution of the differential equation $(x\cos \left(\frac{y}{x}\right)+y\sin \left(\frac{y}{x}\right))ydx=(y\sin \left(\frac{y}{x}\right)-x\cos \left(\frac{y}{x}\right))xdy$ is

(where $c$ is constant of integration)

If the transformed equation of $xy=c^2$ when the axis are rotated through an angle of $\frac{\pi }{4}$ (in the anti clockwise direction), is $p{X}^2+q{Y}^2=r{c}^2$, then $p-q+r$ is equal to

Find the equation of the parabola that satisfies the following conditions: Focus (6, 0); directrix x = –6

The total number of tangents to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ that are perpendicular to the line $5x+2y-3=0$ is

Let $f\left(x\right)=max\{2\sin x,1-\cos x\},\forall x\in (0,\pi ).$

The parametric coordinates of the point $(8,3\sqrt{3})$ on the hyperbola $9x^2-16y^2=144$ is

If $|z_1|=1,|z_2|=2,|z_3|=3$ and $|9z_1{z}_2+4z_1{z}_3+z_2{z}_3|=12,$ then the value of $|z_1+z_2+z_3|$ is

A circle has its centre at the vertex of the parabola $x^2=4y$ and the circle cuts the parabola at the ends of its latus rectum. The equation of the circle is

Find $\frac{dy}{dx}$in the following questions:

$y=si{n}^{-1}\left(\frac{2x}{1+x^2}\right)$

$\underset{x\to \infty }{lim}\frac{sinax}{bx}=$

If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$, then $A^n=$

$Theinverseoff\left(x\right)={(5-{(x-8)}^5)}^{\frac{1}{3}}is$

The equation of the circle with centre on the x-axis, radius 4 and passing through the origin, is

The normal to the rectangular hyperbola $xy=c^2$ at the point '$t_1$' meets the curve again at the point '$t_2$'. Then the value of $t_1^3{t}_2$is

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{j}-\hat{k}.$ If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}$ and $\overrightarrow{a}\cdot \overrightarrow{c}=3,$ then $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})$ is equal to

Let $g\left(x\right)=\left[x\right]$, where $[.]$ represents greatest integer function. Then the function $f\left(x\right)=\left(g\right(x){)}^2-g(x)$ is discontinuous at :

What is the value of ∆ if (cos6x + 6cos4x + 15cos2x + 10)/(cos5x + 5cos3x + 10cosx)= ∆cosx ?

If the coefficient of $(2r+4{)}^{th}$ and $(r-2{)}^{th}$ terms in the expansion of ($1+x{)}^{18}$ are equal, then r =

Find the value of the expression
$3{\sec }^{-1}\left(2\right)-6{\sin }^{-1}\left(\frac{1}{2}\right).$

The angle of intersection of the curves $y^2=32x$ and $x^2=108y$ at a point of intersection (other than origin) is

Given the matrix $$\left[\begin{array}{cc}-4& 2\\ 4& 3\end{array}\right]$$, the eigen vector is