5. Regarding the number of intersection points between the graph of the function $y = \sin x$ and the graph of $y = \frac{x}{10\pi}$ on the coordinate plane, which of the following options is correct?
(1) The number of intersection points is infinite
(2) The number of intersection points is odd and greater than 20
(3) The number of intersection points is odd and less than 20
(4) The number of intersection points is even and greater than or equal to 20
(5) The number of intersection points is even and less than 20

II. Multiple-Choice Questions (30 points)

Instructions: For questions 6 to 11, each of the five options is independent, and at least one option is correct. Select the correct options and mark them on the "Answer Sheet". No deduction for wrong answers. Five points are awarded for all five options correct, 2.5 points for only one wrong option, and no points for two or more wrong options.

13. How many real roots does the equation $x ^ { 4 } - 4 x ^ { 3 } + 4 x ^ { 2 } - 10 = 0$ have?
A 0
B 1
C 2
D 3
E 4

How many real roots does the equation $3 x ^ { 5 } - 10 x ^ { 3 } - 120 x + 30 = 0$ have?
A 1 B 2 C 3 D 4 E 5

Consider these simultaneous equations, where $c$ is a constant:
$$\begin{aligned} & y = 3 \sin x + 2 \\ & y = x + c \end{aligned}$$
Which of the following statements is/are true?
1 For some value of $c$ : there is exactly one solution with $0 \leq x \leq \pi$ and there is at least one solution with $- \pi < x < 0$.
2 For some value of $c$ : there is exactly one solution with $0 \leq x \leq \pi$ and there are no solutions with $- \pi < x < 0$.
3 For some value of $c$ : there is exactly one solution with $0 \leq x \leq \pi$ and there are no solutions with $x > \pi$.

It is given that $\mathrm { f } ( x ) = x ^ { 3 } + 3 q x ^ { 2 } + 2$, where $q$ is a real constant.
The equation $\mathrm { f } ( x ) = 0$ has 3 distinct real roots.
Which of the following statements is/are necessarily true?
I The equation $\mathrm { f } ( x ) + 1 = 0$ has 3 distinct real roots.
II The equation $\mathrm { f } ( x + 1 ) = 0$ has 3 distinct real roots.
III The equation $\mathrm { f } ( - x ) - 1 = 0$ has 3 distinct real roots.

What is the complete range of values of $k$ for which the curves with equations
$$y = x^3 - 12x$$
and
$$y = k - (x-2)^2$$
intersect at three distinct points, of which exactly two have positive $x$-coordinates?

How many real solutions are there to the equation
$$3\cos x = \sqrt{x}$$
where $x$ is in radians?
A $0$
B $1$
C $2$
D $3$
E $4$
F $5$
G infinitely many

$\quad p$ and $q$ are real numbers, and the equation
$$x | x | = p x + q$$
has exactly $k$ distinct real solutions for $x$. Which one of the following is the complete list of possible values for $k$ ?
A $0,1,2$
B $0,1,2,3$
C $0,1,2,3,4$
D 0, 2, 4
E 1, 2, 3 F 1,2,3,4

It is given that
$$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } ( x - 1 ) ^ { 2 } ( x - 2 ) \\ & \mathrm { g } ( x ) = - p ( x - q ) ^ { 2 } ( x - r ) ^ { 2 } \end{aligned}$$
where $p , q$ and $r$ are positive and $q < r$
Find the set of values of $q$ and $r$ that guarantees the greatest number of distinct real solutions of the equation $\mathrm { f } ( x ) = \mathrm { g } ( x )$ for all $p$.

$n$ is the number of points of intersection of the graphs
$$y = \left| x ^ { 2 } - a ^ { 2 } \right| \text { and } y = a ^ { 2 } | x - 1 |$$
where $a$ is a real number.
What is the smallest value of $n$ that is not possible?

1

Let $a$, $b$ be real numbers. The parabola $C: y = x^2 + ax + b$ in the coordinate plane has exactly 2 intersection points with the parabola $y = -x^2$, where the $x$-coordinate of one intersection point satisfies $-1 < x < 0$, and the $x$-coordinate of the other intersection point satisfies $0 < x < 1$.

1. [(1)] Illustrate in the coordinate plane the region of all possible points $(a,\, b)$.
2. [(2)] Illustrate in the coordinate plane the region through which the parabola $C$ can pass.

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