KVPY SX SB Exam 2013 Question Paper

For $\mathit{x},\mathit{t}\in \mathit{R}$ let ${\mathit{p}}_{\mathit{t}}\left(\mathit{x}\right)=\left(\sin \mathit{t}\right){\mathit{x}}^2-\left(2\cos \mathit{t}\right)\mathit{x}+\sin \mathit{t}$
be a family of quadratic polynomials in x with variable coefficients. Let $\mathit{A}\left(\mathit{t}\right)=\underset{0}{\overset{1}{\int }}{\mathit{p}}_{\mathit{t}}\left(\mathit{x}\right)\mathit{d}\mathit{x}.$ Which of the following statements are true?
(I) A (t) < 0 for all t.
(II) A (t) has infinitely many critical points.
(III) A (t) = 0 for infinitely many t.
(IV) A'(t) < 0 for all t.