Show the inequality $t\cos(t) \leqslant \sin(t)$, for every $t$ in $[0, \pi/2]$.

Let $\theta \in [-\pi, \pi]$.
a. Show that $\cos(\theta) \geq 1 - \frac{\theta^2}{2}$.
b. Show that $\left|\frac{e^{i\theta} - (1-p)}{p}\right| \leq \exp\left(\frac{1-p}{2p^2} \cdot \theta^2\right)$.
Hint. One may calculate $\left|\frac{e^{i\theta} - (1-p)}{p}\right|^2$.

Show that $\sin^5 x + \cos^3 x \geq \sin^3 x + \cos^2 x$ implies the expression equals $1$, and find when equality holds.

Between the following two statements: Statement I : Let $\vec { a } = \hat { i } + 2 \hat { j } - 3 \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \hat { k }$. Then the vector $\vec { r }$ satisfying $\vec { a } \times \vec { r } = \vec { a } \times \vec { b }$ and $\vec { a } \cdot \vec { r } = 0$ is of magnitude $\sqrt { 10 }$. Statement II : In a triangle $A B C , \cos 2 A + \cos 2 B + \cos 2 C \geq - \frac { 3 } { 2 }$.
(1) Statement I is incorrect but Statement II is correct.
(2) Both Statement I and Statement II are correct.
(3) Statement I is correct but Statement II is incorrect.
(4) Both Statement I and Statement II are incorrect.