Q55 / 90 · KVPY 2013 (SB/SX) · Mathematics
For $\mathrm{x}, \mathrm{t} \in \mathrm{R}$ let
$$
p_{t}(x)=(\operatorname{sint}) x^{2}-(2 \operatorname{cost}) x+\operatorname{sint}
$$
be a family of quadratic polynomials in $x$ with variable coefficients. Let $A(t)=\int_{0}^{1} p_{t}(x) d x$. Which of the following statements are true?
(I) $\mathrm{A}(\mathrm{t}) < 0$ for all $\mathrm{t}$.
(II) A(t) has infinitely many critical points.
(III) $\mathrm{A}(\mathrm{t})=0$ for infinitely many $\mathrm{t}$.
(IV) $A^{\prime}(t) < 0$ for all $t$.
- A(I) and (II) only
- Correct answerB(II) and (III) only
- C(III) and (IV) only
- D(IV) and (I) only
Solution
$$
\begin{array}{l}
A(t)=\int_{0}^{1}\left\{(\sin t) x^{2}-(2 \cos t) x+\sin t\right\} d x \\
A(t)=\frac{\sin t}{3}-\cos t+\sin t=\frac{4}{3} \sin t-\cos t \\
A^{\prime}(t)=\frac{4 \cos t}{3}+\sin t
\end{array}
$$
St. I and $\mathrm{N}$ are False
Hence, the correct answer is option: 'B'.