Functions without Antiderivatives as Known Combination of Basic Functions

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Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)=\{\begin{array}{c}x;x\le 1\\ x^2+bx+c;x>1\end{array}\text{and}{f}^{\prime }\left(x\right)\text{exist finitely for}x\in R,& \text{(P)}& -1\\ & {\text{then}}^{\prime }b{c}^{\prime }\text{is equal to}& & \\ \text{(B)}& \text{The number of tangents to the}\text{curve}{y}^2-2x^3-4y+8=0\text{that}& \text{(Q)}& 1\\ & \text{pass through}(1,2)\text{is}& & \\ \text{(C)}& f\left(x\right)=2x^3-3a{x}^2+\frac{4}{3}{a}^{2}x+1(a>0)\text{attains its maxima and}& \text{(R)}& 2\\ & \text{minima at}x=k\text{and}x=k^2\text{respectively,}\text{then}a\text{is}& & \\ \text{(D)}& \text{If}f\left(x\right)=\underset{-1}{\overset{1}{\int }}\frac{\sin x}{1+t^2}dt\text{and}{f}^{\prime }\left(\frac{\pi }{3}\right)=\frac{\pi }{k},{\text{then}}^{\prime }{k}^{\prime }\text{is equal to}& \text{(S)}& 3\\ & & \text{(T)}& 4\\ & & \text{(U)}& 6\end{array}$

Which of the following option is INCORRECT ?

1. $\left(A\right)\to \left(P\right)$
2. $\left(B\right)\to \left(T\right)$
3. $\left(C\right)\to \left(U\right)$
4. $\left(D\right)\to \left(T\right)$

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