Exercise 1

Let $\alpha \in R$ be such that the function $f\left(x\right)=\left\{\begin{array}{ll}\frac{{\cos }^{-1}\left(1-{\left\{x\right\}}^2\right){\sin }^{-1}\left(1-\left\{x\right\}\right)}{\left\{x\right\}-{\left\{x\right\}}^3},& x\ne 0\\ \alpha ,& x=0\end{array}\right.$ is continuous at $x=0,$ where $\left\{x\right\}=x-\left[x\right],\left[x\right]$ is the greatest integer less than or equal to $x$. Then :

$$ Given $f\left(x\right)=\left\{\begin{array}{ccc}{\log }_a{\left(a\left|\left[x\right]+[-x]\right|\right)}^x\left(\frac{a^{{\displaystyle \frac{2}{\left(\frac{\left[x\right]+\left[-x\right]}{\left|x\right|}\right)}-5}}}{3+a^{\frac{1}{\left|x\right|}}}\right)& \text{for}& x\ne 0\\ 0& \text{for}& x=0\end{array}\right.$; $a>1$ where [.] represents the integral part function, then

If a differentiable function $f$ satisfies $f\left(\frac{x+y}{3}\right)=\frac{4-2\left(f\left(x\right)+f\left(y\right)\right)}{3} \forall x, y\in R$, then $f\left(x\right)$ is equal to

Let $x_1<\alpha <\beta <\gamma <x_4, x_1<x_2<x_3$. If $f\left(x\right)$ is a cubic polynomial with real coefficients such that $\left(f\right(\alpha ){)}^2+(f\left(\beta \right){)}^2+\left(f\right(\gamma ){)}^2=0, f\left(x_1\right)f\left(x_2\right)<0, f\left(x_2\right)f\left(x_3\right)<0$ and $f\left(x_1\right)f\left(x_3\right)>0$, then which of the following are CORRECT?

Suppose that $f$ is a differentiable function with the property that $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+xy$ and $\underset{h\to 0}{\lim }\frac{1}{h}f\left(h\right)=3$ (where $\left[.\right]$ represents greatest integer function), then

If $f\left(x\right)=x^2-2\left|x\right|$ and $g\left(x\right)=\left\{\begin{array}{c}\min .\left\{f\right(t):-2\le t\le x,-2\le x\le 0\\ \max .\left\{f\right(t):0\le t\le x,0\le x\le 3\end{array}\right\}$. Then

Let $f:R\to R$ be a function such that $f\left(0\right)=1$ and for any $x,y\in R, f\left(xy+1\right)=f\left(x\right)f\left(y\right)-f\left(y\right)-x+2$. Then, $f$ is
 

$A\left(0,1\right),B\left(\frac{\pi }{2},1\right)$ are two points on the graph given by $y=2 \sin x+\cos 2x$. If there exists a point $P$ on the curve between $A \& B$ such that tangent at $P$ is parallel to $AB$. Having co-ordinates $\left(\frac{\pi }{a}, \frac{3}{b}\right),$ find $\left(a+b\right)$.