Set 7

Observe the curve below!

The correct statements regarding the curve are

1. The intersection point of the curve with the $x$-axis is $(4, 0)$.
2. The intersection points of the curve with the $y$-axis are $(0, -5)$ and $(0, 1)$.
3. The axis of symmetry is $y = -1$.
4. The axis of symmetry is $x = -3$.

Given $y = -x^2 - 2x + 8$, the equation of the tangent line at the point with abscissa $-2$ is...

Given the following $4$ numbers:

1. $19$
2. $29$
3. $39$
4. $49$

When these numbers are divided by $5$, the remainder is $4$, and when divided by $4$, the remainder is $3$. Which of the following is correct...

Given the following $4$ plane figures:

1. Right trapezoid.
2. Regular pentagon.
3. Equilateral triangle.
4. Kite (not a square).

How many of these plane figures have at most $2$ lines of symmetry and $2$ rotational symmetries?

Given that $a_n$ is the $n$-th term of a sequence. If $a_{n+1} = a_{n+2} + 2a_n$, $a_3 = -1$, and $a_5 = 3$, then the value of $a_2$ is ...

Given the matrix $A = \begin{pmatrix} 2 & x-2 \\ -1 & 1 \end{pmatrix}$. If $\det(A) = 8$, then the value of $x$ is ...

Given $f(x) = x + 1$ and $(f(x))^2 = 5f(x) - 4$ are satisfied by $x_1$ or $x_2$. What is the value of $x_1 + x_2$?

In a village with a population of $600$ people, there are $450$ people who raise goats and $300$ people who raise chickens. What is the sum of the maximum number of people who raise both animals and the minimum number of people who raise both animals?

A code consists of $2$ vowels and $2$ digits in an alternating sequence, starting with a letter. The formation of the code follows these rules:

1. The first letter is not $A$.
2. The first digit is an odd number.
3. The second letter is the letter $O$.
4. The second digit is chosen from $0, 1, 2, 4, \text{ and } 6$.

How many codes can be formed if no letters or digits can be repeated?

Given an ordered data set $4, 4, a, b, \text{ and } c$. The average of these five numbers is $8$. The largest data value is known to be $3$ times the median value. Which of the following relationships is correct between quantities $P$ and $Q$ based on the information provided?

Given $0 < x < 1$. Which of the following relationships is correct between quantities $P$ and $Q$ based on the information provided?

Given an operation on the set of integers defined by the rule $\beta(w \oplus x \ominus y \otimes z) = 2w + x \cdot y - (-z)$. The value of $\beta(2 \oplus 3 \ominus 4 \otimes (-2))$ is ...

Triangle $ABC$ is an isosceles triangle with $AC = BC$. Determine the measure of angle $\angle BPC$!

Decide whether statements $(1)$ and $(2)$ below are sufficient to answer the question!

1. $\angle CAB = 40^\circ$.
2. $PC$ is the angle bisector of triangle $ABC$.

Given $f(x) = 5 - x$ and $g(x) = 2^x - 1$. Determine which of the following statements are correct:

1. $g(x)$ is a linear line with a gradient of $2$.
2. $f(x)$ and $g(x)$ intersect at $x > 0$.
3. $f(x)$ is above $g(x)$ for all values of $x$.
4. The graphs of $f(x)$ and $g(x)$ intersect at $(2, 3)$.

Joko's water tank can hold $86$ liters of water. Unfortunately, the tank is cracked, causing a leak of between $0.3$ liters and $0.7$ liters per hour. At $07.00$ AM, Joko filled his tank completely. Between $07.00 - 11.00$, there was no refilling or water usage. Which of the following relationships between quantities $P$ and $Q$ is correct based on the given information?

It is known that the current ratio of Farhat's salary to Abas's salary is $2 : 3$. If Farhat and Abas both receive a salary increase of $40,000$ rupiah, the ratio of their salaries becomes $5 : 7$. What is Abas's current salary?

Given $4$ numbers as follows:

1. $18$
2. $38$
3. $58$
4. $78$

If these numbers are divided by $5$, the remainder is $3$, and if divided by $4$, the remainder is $2$. Among the four numbers, which one is correct?

Given $4$ plane figures as follows:

1. Isosceles trapezoid.
2. Rectangle (not a square).
3. Equilateral triangle.
4. Kite (not a square).

How many plane figures have a number of lines of symmetry that is not equal to their order of rotational symmetry?

Given $a_n$ is the n-th term of a sequence of numbers. If $a_n = a_{n+1} + 3a_{n+3}$, $a_1 = -1$, and $a_2 = 2$, then the value of $a_4 + a_1$ is

Given matrix $A = \begin{pmatrix} 2 & -2 \\ x-1 & 1 \end{pmatrix}$. If $\det(A) = 4$, then the value of $x$ is