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Factorize $x^4+x^2+1$

${lim}_{x\to 0}(\cos x+\sin x{)}^{\frac{1}{x}}$

Let $S=S_1\cap S_2\cap S_3,$ where

$S_1=\{z\in C:|z|<4\},$ $S_2=\{z\in C:\text{Im}\left(\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right)>0\}$ and $S_3=\{z\in C:\text{Re}z>0\}$



Area of $S=$

Range of function $y={\log }_2\left(\sin x\right)$ is :

Find the equation of the tangents draws form the point (5,3) to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1.$

If a, b, c belongs to Q then roots of the equation (b+c-2a)x²+(c+a-2b)x+(a+b-2c)=0 are

a)rational b) non real c) irrational d) equal

The eccentricity of a hyperbola whose transverse axis is along $x-$ axis and passes through the point $(6,4)$ is $\sqrt{\frac{5}{3}}$. The equation of tangent to this hyperbola at $(6,4)$ is

The point on the hyperbola $x^2-9y^2=9$, where the line 5x + 12y = 9 touches it, is .

Prove that:

$\frac{sin(90-\theta )cos(90-\theta )}{cot(90-\theta )}$ + $si{n}^2\theta$ = 1.

A transverse periodic wave on a string with a linear mass density of $0.200$kg/m is described by the following equation $y=0.05\sin (420t-21.0x)$ where $x$ and $y$ are in metres and $t$ is in seconds.

A point on the line $\overrightarrow{r}=(1-t)(2\hat{i}+3\hat{j}+4\hat{k})+t(3\hat{i}-2\hat{j}+2\hat{k})$

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1,-1)$ then the radius of $C$ is :

The smallest positive value of $x$ which satisfies the equation ${\log }_{\cos x}\sin x+{\log }_{\sin x}\cos x=2$ is

The smallest positive root of the equation $\tan x-x=0$ lies in

$X\left(s\right)=\frac{2+2s{e}^{-2s}+4e^{-4s}}{(s^2+4s+3)},Re\left(s\right)>-1$ is of the form

$x\left(t\right)=(e^{-t}+e^{-3t}u(t)+[-e^{-t-a}-3e^{-3(t-a)}\left]u\right(t-a)+2[e^{-t-2a}-e^{-3(t-2a)}\left]u\right(t-2a)$

value of a is ______

1. 2

sin³ (2x
+ 1)

If $\sqrt{(x+y)}+\sqrt{(y-x)}=a$ then $\frac{d^{2}y}{d{x}^2}$ equals

Expand
the expression

$\text{The limiting value of}{\left(cosx\right)}^{1/sinx}\text{as~x}\to 0is$

If $y_1\left(x\right)$ is a solution of the differential equation $\frac{dy}{dx}+f\left(x\right)y=0$, then a solution of differential equation $\frac{dy}{dx}+f\left(x\right)y=r\left(x\right)$ is

Equation of line passing through the point (2, 3, 1) and parallel to the line of intersection of the plane x – 2y – z + 5 = 0 and x + y + 3z = 6 is

If the sum of n terms of an A.P is $2n^2+3n$, then write its nth term.

Evaluate $\int x^{x^3+2x}((3x^2+2)\ln x+x^2+2)dx$

(where $C$ is constant of integration)

The abscissae of the two points A and B are the roots of the equation $x^2+2ax-b^2=0$ and their ordinates are the roots of the equation $x^2+2px-q^2=0$. Find the equation of the circle with AB as diameter. Also, find its radius.

If origin is shifted to the point $(2,3)$ without rotation of axes then the coordinates of the point $P$ which divides the join of $A(4,8)$ and $B(7,14)$ in the ratio $1:2$ or $2:1$ with respect to the new system of coordinates can be

$\begin{array}{cccc}& List-I& List-II& \\ P.& I{f}_{\alpha }=\frac{\pi }{7}then\frac{1}{cos\alpha }+\frac{2cos\alpha }{cos2\alpha }=& 1.& 2\\ Q.& undersetx\to \infty Lt\left[\right(x-1\left)\right(x-2\left)\right(x+3\left)\right(x+5\left)\right(x+10){]}^{\frac{1}{5}}-x=& 2.& 3\\ R.& {\int }_{-2}^2\frac{3x^{2}dx}{1+e^x}=& 3.& 4\\ S.& Letf\left(x\right)=x^3+x^2+2x-1.Theminimumintegralvalueofxifxsatisfiesf\left(f\right(x\left)\right)>f(2x+1)is& 4.& 8\end{array}$

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$

1. 0.133

Consider a binary operation $\ast$ on N defind as $a\ast b=a^3+b^3$. Choose the correct answer.
$\left(a\right)\ast$ both associative and commutative
$\left(b\right)\ast$ commutative but not associative
$\left(c\right)\ast$ is associative but not commutative
$\left(d\right)\ast$ is neither commutative nor associative

Let $f\left(x\right)$ be an even function such that $f\left(x\right)+f(x-3)=x(x-3)+1.$ Then $\underset{0}{\overset{3}{\int }}\frac{f\left(x\right)dx}{(x^2-3x+1)}$ is

Using properties of determinants prove the following questions.

$\left|\begin{array}{ccc}x& x^2& 1+p{x}^3\\ y& y^2& 1+p{y}^3\\ z& z^2& 1+p{z}^3\end{array}\right|=(1+pxyz)(x-y)(y-z)(z-x)$, where p is any scalar.