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) $P(t) = 78 + 42\cos(t)$

A container has a shape such that, when filled with water at a constant flow rate, the distance $D$ from the water surface to the table top, in centimeter, increases in relation to time $T$, in minute, according to a function of the type $$D = k + \operatorname{tg}[p(T + m)],$$ where the parameters $k$, $p$, and $m$ are real numbers, for $T$ varying from 0 to 4 minutes, as illustrated in the figure, in which the vertical asymptotes of the tangent function used in the definition of $D$ are presented.
The algebraic expression that represents the relationship between $D$ and $T$ is
(A) $D = 2.5 + \operatorname{tg}\left[30\left(T - \dfrac{5 - 2\pi}{2}\right)\right]$
(B) $D = 4 + \operatorname{tg}\left[30\left(T + \dfrac{5}{2}\right)\right]$
(C) $D = 4 + \operatorname{tg}\left[2.5\left(T + \dfrac{5 + 2\pi}{2}\right)\right]$
(D) $D = 30 + \operatorname{tg}\left[\dfrac{1}{2}(T - 5)\right]$
(E) $D = 30 + \operatorname{tg}\left[\dfrac{1}{2}\left(T - \dfrac{5}{2}\right)\right]$

There is a line $l$ passing through the origin O with slope $\tan \theta$. Let $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ be the feet of the perpendiculars from two points $\mathrm { A } ( 0,2 ) , \mathrm { B } ( 2 \sqrt { 3 } , 0 )$ to line $l$, respectively. What is the value of $\theta$ that maximizes the sum of the distances from the origin O to point $\mathrm { A } ^ { \prime }$ and to point $\mathrm { B } ^ { \prime }$, $\overline { \mathrm { OA } ^ { \prime } } + \overline { \mathrm { OB } ^ { \prime } }$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [3 points]
(1) $\frac { \pi } { 12 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 3 }$
(5) $\frac { 5 } { 12 } \pi$

Find the maximum value $a$ of the function $f ( x ) = 2 \cos \left( x - \frac { \pi } { 3 } \right) + 2 \sqrt { 3 } \sin x$. Find the value of $a ^ { 2 }$. [3 points]

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) $\sqrt { 6 }$

The function $$f ( x ) = a - \sqrt { 3 } \tan 2 x$$ has a maximum value of 7 and a minimum value of 3 on the closed interval $\left[ - \frac { \pi } { 6 } , b \right]$. What is the value of $a \times b$? (Here, $a$ and $b$ are constants.) [4 points]
(1) $\frac { \pi } { 2 }$
(2) $\frac { 5 \pi } { 12 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 4 }$
(5) $\frac { \pi } { 6 }$

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

Given the function $f(x) = \cos(2x + \varphi)$ $(0 \leq \varphi < \pi)$, $f(0) = \frac{1}{2}$.
(1) Find $\varphi$;
(2) Let $g(x) = f(x) + f\left(x - \frac{\pi}{6}\right)$. Find the range and monotonic intervals of $g(x)$.

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

In the triangle $ABC$, the angle $\angle BAC$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $ABC$ is
(A) obtuse angled
(B) right angled
(C) acute angled but not equilateral
(D) equilateral

Given real numbers $x$ and $y$ that satisfy
$$\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1 , \quad x \geqq 0 , \quad y \geqq 0$$
we are to find the maximum value of
$$P = x ^ { 2 } + x y + y ^ { 2 } .$$
Let $x$ and $y$ satisfy the conditions. When we set $x = \sqrt { 2 } \cos \theta \left( 0 \leqq \theta \leqq \frac { \pi } { 2 } \right)$, we have
$$y = \mathbf { A } \sin \theta .$$
Thus $P$ can be represented as
$$\begin{aligned} P & = \sqrt { \mathbf { B } } \sin 2 \theta - \cos 2 \theta + \mathbf { C } \\ & = \sqrt { \mathbf { D } } \sin ( 2 \theta - \alpha ) + \mathbf { E } \end{aligned}$$
where
$$\sin \alpha = \frac { \sqrt { \mathbf { F } } } { \mathbf { G } } , \quad \cos \alpha = \frac { \sqrt { \mathbf { H } } } { \mathbf { I } } \quad \left( 0 < \alpha < \frac { \pi } { 2 } \right) .$$
Hence the maximum value of $P$ is $\sqrt { \square \mathbf { J } } + \mathbf { K }$. Let us denote the $\theta$ at which the value of $P$ is maximized by $\theta _ { 0 }$. Then we have
$$2 \theta _ { 0 } = \alpha + \frac { \pi } { \square } ,$$
and hence
$$\sin 2 \theta _ { 0 } = \frac { \sqrt { \mathbf { M } } } { \mathbf { N } } , \quad \cos 2 \theta _ { 0 } = - \frac { \sqrt { \mathbf { O } } } { \mathbf { N } } .$$