Differentiable Function

Q.

The equation$2y^2-xy-x^2+6x-8=0$ represents

1. A pair of straight lines

2. A circle

3. An ellipse

4. A parabola

Q. Let $\left[t\right]$ denote the greatest integer less than or equal to $t.$ Let $f\left(x\right)=x-\left[x\right],g\left(x\right)=1-x+\left[x\right],$ and $h\left(x\right)=min\left\{f\right(x),g(x\left)\right\},x\in [-2,2].$ Then $h$ is

1. not continuous at exactly four points in $[-2,2]$
2. continuous in $[-2,2]$ but not differentiable at more than four points in $(-2,2)$
3. not continuous at exactly three points in $[-2,2]$
4. continuous in $[-2,2]$ but not differentiable at exactly three points in $(-2,2)$

Q. The repeated factor of the determinant $\left|\begin{array}{ccc}y+z& x& y\\ z+x& z& x\\ x+y& y& z\end{array}\right|$

1. $z-x$
2. $x-y$
3. $y-z$
4. none of these

Q. Check the continuity of the function.
$f\left(x\right)=\{\begin{array}{c}\frac{x^4+x^3+2x^2}{{\tan }^{-1}x}ifx\ne 0\\ 10ifx=0\end{array}$

Q. $f\left(x\right)=\{\begin{array}{cc}ax;& x<2\\ a{x}^2-bx+3;& x\ge 2\end{array}$ If $f$ is differentiable for all $x$ then

1. $a=\frac{3}{4},b=\frac{9}{4}$
2. $a=\frac{3}{2},b=\frac{9}{2}$
3. $a=1,b=2$
4. $a=\frac{3}{4},b=\frac{9}{2}$

Q. If $A=\left[\begin{array}{cc}4& x+2\\ 2x-3& x+1\end{array}\right]$ is symmetric, then x=

1. 3
2. 5
3. 2
4. 4

Q. Let $f\left(n\right)=\left|\begin{array}{ccc}n& n+1& n+2\\ {}^n{P}_n& {}^{n+1}{P}_{n+1}& {}^{n+2}{P}_{n+2}\\ {}^n{C}_n& {}^{n+1}{C}_{n+1}& {}^{n+2}{C}_{n+2}\end{array}\right|$ then $f\left(n\right)$ is divisible by:

1. $n+1)$
2. $n^2+n$
3. $(n+2)!$
4. $n!(n^2+n+1)$

Q. Given $\begin{array}{c}A\\ B\end{array}=A^2+B^2+2AB$, what is $A+B$, if $\begin{array}{c}A\\ B\end{array}=9$?

1. $3$
2. $-2$
3. $5$
4. $4$

Q. If ${\Delta }_1=\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|,{\Delta }_2=\left|\begin{array}{ccc}1& bc& a\\ 1& ca& b\\ 1& ab& c\end{array}\right|$ then

1. ${\Delta }_1+{\Delta }_2=0$
2. ${\Delta }_1={\Delta }_2$
3. ${\Delta }_1+2{\Delta }_2=0$
4. none of these

Q. Let $f\left(x\right)=\{\begin{array}{cc}4x^2+1,& x<\frac{1}{2}\\ 2(\ln x+1),& \frac{1}{2}\le x<1\\ x^2+1,& x\ge 1\end{array}$
then which of the following is/are true?

1. $f\left(x\right)$ has two points of extremum
2. $f\left(x\right)$ is continuous $\forall xϵR,$ except at one point
3. $f\left(x\right)$ is differentiable $\forall xϵR,$ except at one point
4. $f\left(x\right)$ has exactly one point of minima