Question

Consider three functions, $f\left(x\right)=x^3+x^2+x+1,g\left(x\right)=\frac{2x}{x^2+1}andh\left(x\right)=si{n}^{-1}x-co{s}^{-1}x+ta{n}^{-1}x-co{t}^{-1}x$ and let p(x) be a differentiable function on R defined as $p\left(x\right)=\{\begin{array}{c}a{\int }_0^x\sqrt{p\left(t\right)}dt+b;x>0\\ x^2+4x+1;x\le 0\end{array}where,a,bϵ(0,\infty )$ and tangent drawn to the graph of p(x) at x = 1 is y = mx + c
$\begin{array}{cccccc}& Column1& & Column2& & Column3\\ \left(I\right)& Ifrangeoff\left(g\right(x\left)\right)is[l,m],& \left(i\right)& a=& \left(P\right)& 1\\ & then(l+m)=& & & & \\ \left(II\right)& Thenumberofintegersinthe& \left(ii\right)& b=& \left(Q\right)& 3\\ & rangeofg\left(f\right(x\left)\right)isequalto& & & & \\ \left(III\right)& Themaximumvalueof& \left(iii\right)& |c|=& \left(R\right)& 4\\ & g\left(h\right(x\left)\right)isequalto& & & & \\ \left(IV\right)& Iftheminimumvalueof& \left(iv\right)& (m-7)=& \left(S\right)& 5\\ & h\left(g\right(f\left(x\right)\left)\right)is\frac{k\pi }{2},then|k|isequalto& & & & \end{array}$
Which of the following option is the only correct combination?

• A. (IV), (iv), (S)
• B. (III), (ii), (S)
• C. (IV), (ii), (S)
• D. (III), (iii), (Q)

Answer 1

The correct option is A

(IV), (iv), (S)

Range of f(g(x)) = [f(–1), f(1)]
= [0, 4]
Range of g(f(x)) = [–1, 1]
$g\left(h\right(x\left)\right){|}_{max}=1,wherex=1$
$h\left(g\right(f\left(x\right)\left)\right){|}_{min}=-\frac{5\pi }{2}$, when g(f(x)) =-1
$p\left(x\right)=\{\begin{array}{ccc}4x^2+4x+1& ;& x>0\\ x^2+4x+1& ;& x\le 0\end{array}$
$\therefore$ a = 4, b = 1, c = –3, m = 12