Set 4

If $f(x) = 2x - 1$ and $6f(x^2) - f^2(x) = 6x^2 - x$ are satisfied by $x_1$ and $x_2$, then the value of $x_1 + x_2 =$ ....

Given the following data: $1, 2, 3, 4, 5, 6, 7$. If one datum is replaced by the number $34$, then the average becomes twice the median.

A total of $12$ workers can complete a project in $16$ days. If after $4$ days of work, there are $4$ workers who are sick and cannot continue working. Under these conditions, the project can be completed in ... days.

If $f(x) = x^2 - x$ and $g(x) = 2x - 1$, then $fog(x) =$ ....

The correct statement regarding the curve is:

1. one of the intersection points with the $y$-axis is $(0, 2)$
2. $f(y) = -(y - 1)^2 + 4$
3. the intersection point of the curve with the $x$-axis is $(1, 0)$
4. the abscissa of the vertex is $x = 4$

Given $y = x^2 - x + 2$.

The equation of the tangent line at the point with abscissa $1$ is ....

Given 4 numbers as follows:

1. $94$
2. $80$
3. $38$
4. $17$

If these numbers are divided by $7$, the remainder is $3$, and if divided by $2$, the remainder is $1$.

Among the four numbers, which ones are incorrect?

Given 4 plane figures as follows:

1. Scalene triangle.
2. Parallelogram (not a rhombus).
3. Ellipse.
4. Regular hexagon.

How many plane figures have at least 2 rotational symmetries and at least 3 reflectional symmetries?

Given a sequence $b_1, b_2, b_3, \dots$.

If $b_{n+2} = b_{n+1} - 3b_n$, and $b_4 = 5$ and $b_5 = -1$, then the value of $b_2$ is ....

Given matrix $A = \begin{pmatrix} 5 & 3 - x \\ -1 & 1 \end{pmatrix}$.

If $\det(A) = 8$, then the value of $x$ is ....

Given $f(x) = 2x - 4$ and $(f(x))^2 - 6f(x) = -8$ is satisfied by $x_1$ or $x_2$.

What is the value of $x_1 \cdot x_2$?

At "Harapan Bangsa" High School, which has $800$ students, there are $400$ students who like Mathematics and $500$ students who like Physics.

If $x$ is the maximum number of people who like both and $y$ is the minimum number of people who like both, then the value of $x - y$ is ... people.

A password consists of $2$ vowels and $2$ digits with alternating order and starts with a digit.

The password formation follows these rules:

1. The first digit is an odd number.
2. The first letter is not the letter A.
3. The second digit is chosen from $2, 4, 6, 8$.
4. The second letter is the letter O.

How many passwords can be formed if letters and digits cannot be repeated?

There are $8$ pieces of data in the form of sorted positive integers, namely $1, 1, 2, x, 10, y, 17$ and $18$.

If the mean and median of the data are $9$.

Which is the correct relationship between quantities $P$ and $Q$ based on the given information?

Given $1 < x < 2$.

Which is the correct relationship between quantities $P$ and $Q$ based on the given information?

Given an operation on the set of integers defined by the rule $\beta(w \oplus x \ominus y \otimes z) = z(y + w) - \frac{z}{x}$.

The value of $\beta(1 \oplus 2 \ominus 3 \otimes -4)$ is ....

Triangle $ABC$ is an equilateral triangle. If lines $AP$, $BQ$, and $CR$ intersect at a single point, which is point $O$.

Determine the measure of angle $\angle BOC$?

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

1. $BC = 10 \text{ cm}$.
2. Lines $AP$, $BQ$, and $CR$ are angle bisectors of triangle $ABC$.

Most residents of Aman Village work as farmers and collect their harvests collectively in the village rice barn every year. During the harvest season, the residents of Aman Village are able to collect between $0.04$ tons and $0.09$ tons of rice every day. At the beginning of the 2020 harvest season, the rice barn contained $55$ tons of remaining rice from last year's harvest.

Which is the correct relationship between quantities $P$ and $Q$ based on the given information?

The ratio of alcohol volume to water in a solution is $4 : 3$. Then the solution is diluted with $5$ liters of water, so the ratio becomes $4 : 5$.

How much water is currently contained in the solution?

If $P(a \odot b \# c) = a^2 + 4\sqrt{b} - 2c$, then the value of $\frac{P(3 \odot 9 \# 6)}{3} =$ ...