Single Point Continuity

Q. If the function $f\left(x\right)=2p[2x+5]+q[3x-7]$ is continuous at $x=1$, then
$($where [.] denotes greatest integer function and $p,qϵR)$

1. $2p+q=0$
2. $p+2q=0$
3. $p+q=0$
4. $10p-7q=0$

Q. Let $a,b$ and $c$ be the side lengths of a triangle $ABC$ and assume that $a\le b$ and $a\le c.$ If $x=\frac{b+c-a}{2},$ then the minimum value of $\frac{ax}{rR},$ where $r$ and $R$ denote the inradius and circumradius, respectively of triangle $ABC,$ is

Q. If $f:R\to [-1,1]$ where $f\left(x\right)=\sin \left(\frac{\pi }{2}\right[x\left]\right)$ $\left(\right[.]$ denotes greatest integer function$)$, then $f\left(x\right)$ is:

1. many-one function
2. an onto function
3. an into function
4. a periodic function

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 $k$ )

1. $3$
2. $5$
3. $12$
4. $8$

Q. If $f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right],x\in R$, where $\left[x\right]$ denotes the greatest integer function, then :

1. $\underset{x\to 4^{+}}{lim}f\left(x\right)$ exists but $\underset{x\to 4^{-}}{lim}f\left(x\right)$ does not exist.
2. $\underset{x\to 4^{-}}{lim}f\left(x\right)$ exists but $\underset{x\to 4^{+}}{lim}f\left(x\right)$ does not exist.
3. Both $\underset{x\to 4^{-}}{lim}f\left(x\right)$ and $\underset{x\to 4^{+}}{lim}f\left(x\right)$ exist but are not equal.
4. $f$ is continuous at $x=4$.

Q.

If the function
$f\left(x\right)=\{\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|,}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

1. $a=lo{g}_{e}b,a=\frac{2}{3}$

2. $b=lo{g}_{e}a,a=\frac{2}{3}$

3. $a=lo{g}_{e}b,b=2$

4. none of these

Q. A continuous function can have some points where limit does not exist.

1. True
2. False