The plane is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider the points $\mathrm{A}(-1;1)$, $\mathrm{B}(0;1)$, $\mathrm{C}(4;3)$, $\mathrm{D}(7;0)$, $\mathrm{E}(4;-3)$, $\mathrm{F}(0;-1)$ and $\mathrm{G}(-1;-1)$.
The part of the curve located above the x-axis is decomposed as follows:

• the portion located between points A and B is the graph of the constant function $h$ defined on the interval $[-1;0]$ by $h(x) = 1$;
• the portion located between points B and C is the graph of a function $f$ defined on the interval $[0;4]$ by $f(x) = a + b\sin\left(c + \frac{\pi}{4}x\right)$, where $a$, $b$ and $c$ are fixed non-zero real numbers and where the real number $c$ belongs to the interval $\left[0; \frac{\pi}{2}\right]$;
• the portion located between points C and D is a quarter circle with diameter [CE].

The part of the curve located below the x-axis is obtained by symmetry with respect to the x-axis.

1. a. We call $f'$ the derivative function of function $f$. For every real number $x$ in the interval $[0;4]$, determine $f'(x)$. b. We require that the tangent lines at points B and C to the graph of function $f$ be parallel to the x-axis. Determine the value of the real number $c$.
2. Determine the real numbers $a$ and $b$.

Question 180
A figura mostra o gráfico da função seno no intervalo $[0, 2\pi]$.
[Figure]
O número de soluções da equação $\operatorname{sen}(x) = 0{,}5$ no intervalo $[0, 2\pi]$ é
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Find all $x \in [ - \pi , \pi ]$ such that $\cos 3 x + \cos x = 0$.

The number of $\theta$ with $0 \leq \theta < 2 \pi$ such that $4 \sin ( 3 \theta + 2 ) = 1$ is
(A) 2
(B) 3
(C) 6
(D) none of the above

Find the number of real solutions to the equation $x = 99 \sin ( \pi x )$.

Recall that $\sin^{-1}$ is the inverse function of $\sin$, as defined in the standard fashion. (Sometimes $\sin^{-1}$ is called $\arcsin$.) Let $f(x) = \sin^{-1}(\sin(\pi x))$. Write the values of the following. (Some answers may involve the irrational number $\pi$. Write such answers in terms of $\pi$.)
(i) $f(2.7)$
(ii) $f'(2.7)$
(iii) $\int_0^{2.5} f(x)\, dx$
(iv) the smallest positive $x$ at which $f'(x)$ does not exist.

For $x$ satisfying the equation $3 \cos 2 x + 17 \cos x = 0$, find the value of $\tan ^ { 2 } x$. [3 points]

For a natural number $n$, let $a _ { n }$ be the $n$-th smallest $x$-coordinate among the intersection points of the line $y = n$ and the graph of the function $y = \tan x$ in the first quadrant.
What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$? [4 points]
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { 3 } { 4 } \pi$
(4) $\pi$
(5) $\frac { 5 } { 4 } \pi$

When $0 < x < 2 \pi$, the sum of all real roots of the equation $\cos ^ { 2 } x - \sin x = 1$ is $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [3 points]

For $0 < x < 2 \pi$, what is the sum of all values of $x$ that simultaneously satisfy the equation $4 \cos ^ { 2 } x - 1 = 0$ and the inequality $\sin x \cos x < 0$? [3 points]
(1) $\frac { 10 } { 3 } \pi$
(2) $3 \pi$
(3) $\frac { 8 } { 3 } \pi$
(4) $\frac { 7 } { 3 } \pi$
(5) $2 \pi$

For a positive number $a$, there is a function $$f ( x ) = \tan \frac { \pi x } { a }$$ defined on the set $\left\{ x \left\lvert \, - \frac { a } { 2 } < x \leq a \right. , x \neq \frac { a } { 2 } \right\}$. As shown in the figure, there is a line passing through three points $\mathrm { O } , \mathrm { A } , \mathrm { B }$ on the graph of $y = f ( x )$. Let $\mathrm { C }$ be the point other than $\mathrm { A }$ where the line parallel to the $x$-axis passing through point $\mathrm { A }$ meets the graph of $y = f ( x )$. When triangle $\mathrm { ABC }$ is equilateral, what is the area of triangle $\mathrm { ABC }$? (Here, $\mathrm { O }$ is the origin.) [4 points]
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 17 \sqrt { 3 } } { 12 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 4 }$
(5) $\frac { 7 \sqrt { 3 } } { 6 }$

For the function $f(x) = \sin\frac{\pi}{4}x$, find the sum of all natural numbers $x$ satisfying the inequality $$f(2+x)f(2-x) < \frac{1}{4}$$ for $0 < x < 16$. [3 points]

When $\sin \theta + 3 \cos \theta = 0$ and $\cos ( \pi - \theta ) > 0$, what is the value of $\sin \theta$? [3 points]
(1) $\frac { 3 \sqrt { 10 } } { 10 }$
(2) $\frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $- \frac { \sqrt { 10 } } { 5 }$
(5) $- \frac { 3 \sqrt { 10 } } { 10 }$

The number of zeros of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 6 } \right)$ on $[ 0, \pi ]$ is $\_\_\_\_$.

Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$. It is known that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$. The following are four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$
The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)

The graph of the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 3 } \right) ( \omega > 0 )$ is shifted left by $\frac { \pi } { 2 }$ units. If the minimum value of the resulting curve is $-1$ and the distance between two consecutive minimum points is $\pi$, then the minimum value of $\omega$ is
A. $\frac { 1 } { 6 }$
B. $\frac { 1 } { 4 }$
C. $\frac { 1 } { 3 }$
D. $\frac { 1 } { 2 }$

Let the function $f(x) = \cos(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) have minimum positive period $T$. If $f(T) = \frac{\sqrt{3}}{2}$ and $x = \frac{\pi}{6}$ is a zero of $f(x)$, then the minimum value of $\omega$ is $\_\_\_\_$.

Given the function $f ( x ) = \sin ( \omega x + \varphi )$ is monotonically increasing on the interval $\left( \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right)$, and the lines $x = \frac { \pi } { 6 }$ and $x = \frac { 2 \pi } { 3 }$ are two axes of symmetry of the graph of $y = f(x)$, then $f \left( - \frac { 5 \pi } { 12 } \right) =$
A. $- \frac { \sqrt { 3 } } { 2 }$
B. $- \frac { 1 } { 2 }$
C. $\frac { 1 } { 2 }$
D. $\frac { \sqrt { 3 } } { 2 }$

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Prove that
$$\forall n \in \mathbb{N}, \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \backslash\{0\}, \quad S_{n}(x) = \frac{\sin((2n+1)\pi x)}{\sin(\pi x)}$$

The number of solutions of the equation $\sin ( \cos \theta ) = \theta$, $- 1 \leq \theta \leq 1$, is
(a) 0
(b) 1
(c) 2
(d) 3

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

The set of all solutions of the equation $\cos 2\theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n\pi + \frac{\pi}{2}$, where $n$ is any integer
(C) $\theta = 2n\pi$ or $\theta = 2n\pi - \frac{\pi}{2}$ or $\theta = n\pi - \frac{\pi}{4}$, where $n$ is any integer
(D) $\theta = 2n\pi$ or $\theta = n\pi + \frac{\pi}{4}$, where $n$ is any integer

The number of solutions of the equation $\sin^{-1} x = 2 \tan^{-1} x$ is
(A) 1
(B) 2
(C) 3
(D) 5

The number of roots of the equation $x^2 + \sin^2 x = 1$ in the closed interval $\left[ 0, \frac{\pi}{2} \right]$ is
(A) 0
(B) 1
(C) 2
(D) 3

Find the number of solutions of $\sec x + \tan x = 2\cos x$ in $[0, 2\pi]$.
(A) 0 \quad (B) 1 \quad (C) 2 \quad (D) 3