# The values of $\dpi{100} p\; and\; q$ for which the function $\dpi{100} f(x)=\left\{\begin{matrix} \frac{\sin (p+1)x+\sin x}{x} &,x< 0 \\ q&,x=0 \\ \frac{\frac{q}{\sqrt{x+x^{2}-\sqrt{x}}}}{x^{3/2}}&,x> 0 \end{matrix}\right.$is continuous for all $\dpi{100} x$ in R,are Option 1) $p=-\frac{3}{2},q=\frac{1}{2}\;$ Option 2) $\; \; p=\frac{1}{2},q=\frac{3}{2}\;$ Option 3) $\; p=\frac{1}{2},q=-\frac{3}{2}\;$ Option 4) $\; \; p=\frac{5}{2},q=\frac{1}{2}$

As we learnt in

Continuity -

If the function is continuous, Its graph does not break but for discontinuous functions there is a break in the graph.

- wherein

$f(x)\:=\left\{\begin{matrix} \frac{sin(p+1)x+sinx}{x} & x< 0\\ q & =0\\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^\frac{3}{2}} & x> 0 \end{matrix}\right.$

$f(x)\:=\left\{\begin{matrix} \frac{2sin\frac{(p+2)x}{2}.cos\frac{px}{2}}{x} & x< 0\\ q & x=0\\ \frac{(x+x)^{2}-x}{x.x^\frac{1}{2}}.\frac{1}{\sqrt{x+x^{2}}+\sqrt{x}} & x> 0 \end{matrix}\right.$

$f(x)=\left\{\begin{matrix} \frac{2sin\frac{(p+2)x}{2}}{\frac{(p+2)x}{2}\times2}(p+2).cos\frac{px}{2}& x< 0\\ q & x=0\\ \frac{x^{2}}{x.x^\frac{1}{2}.\sqrt{x}(1+\sqrt{1+x})} & x> 0 \end{matrix}\right.$

Since f(x) is continuous so that

=$>$ P+2 = q = $\frac{1}{2}$

$\therefore$ P = $-\frac{3}{2}$

q = $\frac{1}{2}$

Option 1)

$p=-\frac{3}{2},q=\frac{1}{2}\;$

This solution is correct.

Option 2)

$\; \; p=\frac{1}{2},q=\frac{3}{2}\;$

This solution is incorrect

Option 3)

$\; p=\frac{1}{2},q=-\frac{3}{2}\;$

This solution is incorrect

Option 4)

$\; \; p=\frac{5}{2},q=\frac{1}{2}$

This solution is incorrect

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