Continuity in an Interval

Q. If $f\left(x\right)=sinx+cosx,g\left(x\right)=x^2-1$, then g{f(x)} is invertible in the domain

1. $[-\frac{\pi }{4},\frac{\pi }{4}]$
2. $[-\frac{\pi }{2},\frac{\pi }{2}]$
3. $[0,\pi ]$

4. $[0,\frac{\pi }{2}]$

Q. $Letf\left(x\right)=\{\begin{array}{cc}(1+|cosx|{)}^{\frac{ab}{|cosx|}},& n\pi <x<(2n+1)\frac{\pi }{2}\\ e^a.e^b,& x=(2n+1)\frac{\pi }{2}\\ e^{\frac{cot2x}{cot8x}},& (2n+1)\frac{\pi }{2}<x<(n+1)\pi \end{array}Iff\left(x\right)iscontinuousin\left(\right(n\pi ),(n+1)\pi ,then)$

1. a = 1, b = 2
2. a = 2, b = 2
3. a = 2, b = 3
4. a = 3, b = 4

Q.

The function, $f\left(x\right)=min\{x,x^2\}\text{/}\mho xϵR$ is continuous?

1. True

2. False

Q. If f is continuous on its domain D, then | f | is also continuous on D.

1. True
2. False

Q. Let $a,b,c\in R$ such that $a+b+c=\pi .$
If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (a{x}^2+bx+c)}{x^2-1},& \text{if}x<1\\ -1,& \text{if}x=1\\ a\text{sgn}(x+1)\cos (2x-2)+b{x}^2,& \text{if}1<x\le 2\end{array}$
is continuous at $x=1,$ then the value of $\frac{a^2+b^2}{5}$ is
( Here, $\text{sgn}\left(k\right)$ denotes signum function of $k$ )

Q. The function f(x) defined by $f\left(x\right)=(\begin{array}{c}\frac{{\sin }^2\mathrm{ax}}{x^2},x\ne 0\\ 1,x=0\end{array}$ is continuous at x = 0. The value(s) of a is/are

1. \N
2. 1
3. -1
4. 2