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If f, $g:R\to R$ be defined by f(x) =2x+1 and $g\left(x\right)=x^2-2,\forall x\in R$, respectively. Then, find gof.

Consider the equation of curve $y=x^2-3x+3$ and $x\ne 3.$

Then equation of normal at point, where its ordinate and abscissa are equal, is



[2 marks]

Let $f$ be a biquadratic function of $x$ given by $f\left(x\right)=A{x}^4+B{x}^3+C{x}^2+Dx+E$ where $A,B,C,D,E\in R$ and $A\ne 0.$ If $\underset{x\to 0}{lim}{\left(\frac{f(-x)}{2x^3}\right)}^{1/x}=e^{-3},$ then

What value will you assign to the slope of the savings function S, when the slope of C-function is given to be $=0.6$?

Find all points of discontinuity of $f\left(x\right)wheref\left(x\right)$ is defined by $f\left(x\right)=\{\begin{array}{c}2x+3,ifx\le 2\\ 2x-3,ifx>2\end{array}$

The value of the integral $\underset{0}{\overset{1}{\int }}\frac{\sqrt{x}dx}{(1+x)(1+3x)(3+x)}$ is

If $2a+3b+6c=0$, then atleast one root of the equation $a{x}^2+bx+c=0$ lies in the interval

A line $L$ is passing through the point $P(0,1,-1)$ and perpendicular to both the lines $\frac{x-2}{2}=\frac{y-4}{1}=\frac{z+2}{4}$ and $\frac{x+2}{3}=\frac{y+4}{2}=\frac{z-2}{-2}.$ If the position vector of point $Q$ on line $L$ is $(a,b,c)$ such that $(PQ{)}^2=357,$ then possible value of $a+2b+3c$ is

For any angle $\theta \in (0,\pi ],\sin \theta =1,$ then $\sin 2\theta =$

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectivley. The variance of the combined data set is

Solve the following system of equations in R.

x-2 >0,3x<18

For which of the following values of $x$, $5$th term will be the numerically greatest term in the expansion of ${(1+\frac{x}{3})}^{10}$.

If the vector, $e_1=(1,0,2),e_2=(0,1,0)$ and $e_3=(-2,0,1)$ from an orthogonal basis of the three dimensional real space $R^3$, then the vector u = (4,3-3) $ϵR^3$ can be expressed as

The total number of terms which are dependent on the value of $x$, in the expansion ${(x^2-2+\frac{1}{x^2})}^n,n\in N$ is equal to

Show
that the statement “For any real numbers a
and b,
a²
= b²
implies that a
= b”
is not true by giving a counter-example.

If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are ${30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$ respectively, then the ratio AB:BC is

The graph of the function $y=f\left(x\right)$ is symmetrical about line $x=2,$ then

If $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$, then $A^2=$

Show that the function given byhas
maximum at x = e.

Find the general solution of the following .

2sin²x+3cosx=0

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

$\frac{x^2}{25}+\frac{y^2}{100}=1$

The length of the transverse axis of the hyperbola $9x^2-16y^2-18x-32y-151=0is$

If $(1+2x+x^2{)}^n=\sum _{r=0}^{2n}{a}_r{x}^r$, then $a_r=$

Let $f\left(x\right)=\{\begin{array}{cc}\left[x\right]+[-x],& x\ne 2\\ \lambda ,& x=2\end{array};$ where $[.]$ denotes the greatest integer function. If $f$ is continuous at $x=2,$ then the value of $\lambda$ is