A scientist, in his studies to model a person's blood pressure, uses a function of the type $P(t) = A + B\cos(kt)$ where $A$, $B$, and $K$ are positive real constants and $t$ represents the time variable, measured in seconds. Consider that a heartbeat represents the time interval between two successive maximum pressures.
When analyzing a specific case, the scientist obtained the data:

Minimum pressure	78
Maximum pressure	120
Number of heartbeats per minute	90

The function $P(t)$ obtained by this scientist when analyzing the specific case was
(A) $P(t) = 99 + 21\cos(3\pi t)$
(B) $P(t) = 78 + 42\cos(3\pi t)$
(C) $P(t) = 99 + 21\cos(2\pi t)$
(D) $P(t) = 99 + 21\cos(t)$
(E) $P(t) = 78 + 42\cos(t)$

On the coordinate plane, let $\theta _ { 1 }$ be the acute angle that the line $y = m x ( 0 < m < \sqrt { 3 } )$ makes with the $x$-axis, and let $\theta _ { 2 }$ be the acute angle that the line $y = m x$ makes with the line $y = \sqrt { 3 } x$. What is the value of $m$ that maximizes $3 \sin \theta _ { 1 } + 4 \sin \theta _ { 2 }$? [4 points]
(1) $\frac { \sqrt { 3 } } { 6 }$
(2) $\frac { \sqrt { 3 } } { 7 }$
(3) $\frac { \sqrt { 3 } } { 8 }$
(4) $\frac { \sqrt { 3 } } { 9 }$
(5) $\frac { \sqrt { 3 } } { 10 }$

When the maximum value of the function $f ( x ) = 2 \cos ^ { 2 } x + k \sin 2 x - 1$ is $\sqrt { 10 }$, what is the value of the positive constant $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the maximum value of the function $f ( x ) = \sin x + \sqrt { 7 } \cos x - \sqrt { 2 }$? [2 points]
(1) $\sqrt { 2 }$
(2) $\sqrt { 3 }$
(3) 2
(4) $\sqrt { 5 }$
(5) $\sqrt { 6 }$

As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 14 } { 9 }$

4. The maximum value of the function $y = 2\sin x - \cos x$ is $\_\_\_\_$

3. As shown in the figure, the water depth change curve of a certain port from 6 to 18 o'clock approximately satisfies the function $y = 3 \sin \left( \frac { \pi } { 6 } x + \varphi \right) + k$. Based on this function, the maximum water depth (in meters) during this period is
A. 5
B. 6
C. 8
D. 10 [Figure]

Given the function $f ( x ) = \sin x - 2 \sqrt { 3 } \sin ^ { 2 } \frac { x } { 2 }$\n(I) Find the minimum positive period of $f ( x )$;\n(II) Find the minimum value of $f ( x )$ on the interval $\left[ 0 , \frac { 2 \pi } { 3 } \right]$.

The maximum value of the function $f(x) = 2\cos x + \sin x$ is \_\_\_\_

Let the function $f(x) = 5\cos x - \cos 5x$.
(1) Find the maximum value of $f(x)$ on $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a$ is a real number, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) If there exists $\varphi$ such that for all $x$, $5\cos x - \cos(5x + \varphi) \leq b$, find the minimum value of $b$.

Find the range of $y = \cos\theta\left(\sin\theta + \sqrt{\sin^2\theta + 3}\right)$.

The maximum value of the expression $\frac { 1 } { \sin ^ { 2 } \theta + 3 \sin \theta \cos \theta + 5 \cos ^ { 2 } \theta }$ is

If $\int \frac { 1 } { \mathrm { a } ^ { 2 } \sin ^ { 2 } x + \mathrm { b } ^ { 2 } \cos ^ { 2 } x } \mathrm {~d} x = \frac { 1 } { 12 } \tan ^ { - 1 } ( 3 \tan x ) +$ constant, then the maximum value of $\mathrm { a } \sin x + \mathrm { b } \cos x$, is:
(1) $\sqrt { 40 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 39 }$
(4) $\sqrt { 42 }$

4. Which of the following numbers is closest to $\sqrt { 2 }$?
(1) $\sqrt { 3 } \cos 44 ^ { \circ } + \sin 44 ^ { \circ }$
(2) $\sqrt { 3 } \cos 54 ^ { \circ } + \sin 54 ^ { \circ }$
(3) $\sqrt { 3 } \cos 64 ^ { \circ } + \sin 64 ^ { \circ }$
(4) $\sqrt { 3 } \cos 74 ^ { \circ } + \sin 74 ^ { \circ }$
(5) $\sqrt { 3 } \cos 84 ^ { \circ } + \sin 84 ^ { \circ }$