Let $\mathrm{m}, \mathrm{n}, \mathrm{p}, \mathrm{q}$ be four positive integers. If $\int_0^{2 \pi} \sin ^m x \cos ^n x d x=4 \int_0^{\pi / 2} \sin ^m x \cos ^n x d x$, $\int_0^{2 \pi} \sin ^p x \cos ^n x d x=0, \int_0^\pi \sin ^p x \cos ^q x d x=0, \mathrm{a}=\mathrm{m}+\mathrm{n}+\mathrm{p}$ and $\mathrm{b}=\mathrm{m}+\mathrm{n}+\mathrm{q}$, then

• A

  $a$ is even number and $b$ is odd number
• B

  $a$ is odd number and $b$ is even number
• C

  Both $a$ and $b$ are even numbers
• Correct answer
  D

  Both $a$ and $b$ are odd numbers

Hence, the correct answer is option: 'D'.