Set 4

Explanation

Given the function $f(x) = 2x - 1$. We want to find the value of $x_1 + x_2$ that satisfies the equation $6f(x^2) - f^2(x) = 6x^2 - x$.

First, we find the forms of $f(x^2)$ and $f^2(x)$ or $(f(x))^2$.

Next, we substitute these results into the given equation.

We move all terms to the left side to form a quadratic equation $ax^2 + bx + c = 0$.

From the quadratic equation $2x^2 + 5x - 7 = 0$, we get $a = 2$, $b = 5$, and $c = -7$.

The sum of the roots of the quadratic equation ($x_1 + x_2$) can be calculated using the formula $-\frac{b}{a}$.

So, the value of $x_1 + x_2$ is $-\frac{5}{2}$.

Explanation

First, let's find the sum of the initial data.

Let $x$ be the datum that is replaced. Since it is replaced by the number $34$, the new sum of data becomes:

The number of data points ($n$) remains $7$. The formula for the new average ($\bar{x}_{new}$) is:

It is given that the new average is equal to twice the new median ($\bar{x}_{new} = 2 \times \text{Median}_{new}$). We need to check the possible values for the new median.

The initial data is ordered: $1, 2, 3, 4, 5, 6, 7$. The initial median is $4$. The replacement number $34$ will definitely be in the last position because it is the largest.

Case 1: If the replaced number is 1, 2, 3, or 4

For example, if we replace $1$, the data becomes $2, 3, 4, 5, 6, 7, 34$. The median becomes $5$. The same applies if we replace $2, 3$, or $4$. The data will shift so that the median becomes $5$.

Let's test this in the equation:

The result $x = -8$ is invalid because the data must be from the initial set ($1$ to $7$).

Case 2: If the replaced number is 5, 6, or 7

For example, if we replace $7$, the data becomes $1, 2, 3, 4, 5, 6, 34$. The median remains $4$. The same applies if we replace $5$ or $6$. The median remains $4$.

Let's test this in the equation:

The result $x = 6$ is valid because $6$ is in the initial data.

So, the replaced number is $6$. Thus, $P = 6$ and $Q = 6$.

In conclusion, $P = Q$.

Explanation

Let's analyze the project conditions before and after workers get sick.

Initial Plan:

The work is planned to be completed by $12$ people in $16$ days. Total workload = $12 \times 16 = 192$ work units.

After 4 Days:

The work has been done for $4$ days by $12$ people. Completed work = $12 \times 4 = 48$ units. Remaining workload = $192 - 48 = 144$ units.

Also, the remaining normal time is $16 - 4 = 12$ days if done by $12$ people.

Changed Condition:

There are $4$ sick people, so the remaining workers are $12 - 4 = 8$ people. We use inverse proportion to find the additional time ($w$) needed by $8$ people to complete the remaining work.

So, $8$ workers need $18$ more days to complete the remaining project.

The total project completion time is the time already elapsed plus the remaining time needed.

So, the project can be completed in $22$ days.

Explanation

Given the functions $f(x) = x^2 - x$ and $g(x) = 2x - 1$. We are asked to find the composite function $(fog)(x)$.

We substitute $g(x)$ into the function $f(x)$.

Next, we expand the quadratic form and simplify the equation.

So, the value of $(fog)(x)$ is $4x^2 - 6x + 2$.

Explanation

Let's analyze each statement one by one based on the given graph.

1. Checking the Y-Axis Intercept (Statement 1)

From the image, the intersection point with the $y$-axis appears to be at $(0, -1)$ and another point at the top.

Let's find the equation first. The vertex of the parabola is at $(4, 1)$.

The general form of a horizontal parabola equation with vertex $(x_p, y_p)$ is $x = a(y - y_p)^2 + x_p$.

Substitute $x_p = 4$ and $y_p = 1$:

The curve passes through the point $(0, -1)$. We substitute this point to find the value of $a$.

So, the equation of the curve is $x = -(y - 1)^2 + 4$.

Now we find the intersection points with the $y$-axis (when $x = 0$):

Solution 1: $y - 1 = 2 \Rightarrow y = 3$. The intersection point is $(0, 3)$.

Solution 2: $y - 1 = -2 \Rightarrow y = -1$. The intersection point is $(0, -1)$.

Statement (1) states $(0, 2)$, which is FALSE.

2. Checking the Curve Equation (Statement 2)

From the calculation above, we have obtained the equation of the curve is:

In function notation $f(y)$, this is written as $f(y) = -(y - 1)^2 + 4$.

Statement (2) is TRUE.

3. Checking the X-Axis Intercept (Statement 3)

The intersection point with the $x$-axis occurs when $y = 0$.

So, the intersection point with the $x$-axis is $(3, 0)$.

Statement (3) states $(1, 0)$, which is FALSE.

4. Checking the Vertex (Statement 4)

From the graph, it is clearly visible that the vertex is $(4, 1)$.

The abscissa ($x$ value) of the vertex is $4$.

Statement (4) is TRUE.

Conclusion:

The correct statements are (2) and (4).

Thus, the answer is C.

Explanation

To determine the equation of the tangent line, we need to find the gradient ($m$) and the tangent point first.

Determining the Gradient

The gradient is the first derivative of the curve function ($y'$).

Substitute the abscissa $x = 1$ into $y'$ to get the gradient.

Determining the Tangent Point

We find the ordinate value ($y$) when the abscissa $x = 1$.

So, the tangent point is $(1, 2)$.

Determining the Equation of the Tangent Line

Use the formula for the equation of a tangent line $y - y_1 = m(x - x_1)$ with $m = 1$ and point $(1, 2)$.

So, the equation of the tangent line is $y = x + 1$.

Explanation

We will check each number based on the two given conditions:

• Condition 1: Divided by $7$, the remainder is $3$.
• Condition 2: Divided by $2$, the remainder is $1$ (meaning it is an odd number).

Checking Number 1 (94)

• Divided by 7: $94 \div 7 = 13$ remainder $3$ (Satisfies condition 1).
• Divided by 2: $94$ is an even number, so $94 \div 2$ remainder $0$ (Does not satisfy condition 2).

Conclusion: Number 94 is incorrect.

Checking Number 2 (80)

• Divided by 7: $80 \div 7 = 11$ remainder $3$ (Satisfies condition 1).
• Divided by 2: $80$ is an even number, so $80 \div 2$ remainder $0$ (Does not satisfy condition 2).

Conclusion: Number 80 is incorrect.

Checking Number 3 (38)

• Divided by 7: $38 \div 7 = 5$ remainder $3$ (Satisfies condition 1).
• Divided by 2: $38$ is an even number, so $38 \div 2$ remainder $0$ (Does not satisfy condition 2).

Conclusion: Number 38 is incorrect.

Checking Number 4 (17)

• Divided by 7: $17 \div 7 = 2$ remainder $3$ (Satisfies condition 1).
• Divided by 2: $17$ is an odd number, so $17 \div 2$ remainder $1$ (Satisfies condition 2).

Conclusion: Number 17 is correct.

Final Conclusion

The question asks which numbers are incorrect. Based on the check above, the numbers that do not satisfy both conditions are numbers (1), (2), and (3).

Explanation

Let's analyze each plane figure based on the number of its reflectional and rotational symmetries.

The required conditions are:

1. Rotational symmetry $\ge 2$
2. Reflectional symmetry $\ge 3$

Scalene Triangle

• Reflectional symmetry: 0 (has no axis of symmetry).
• Rotational symmetry: 1 (only returns to original position after 360-degree rotation).

Conclusion: Does not satisfy both conditions.

Parallelogram Not a Rhombus

• Reflectional symmetry: 0 (has no axis of symmetry).
• Rotational symmetry: 2 (rotational symmetry of order 2).

Conclusion: Satisfies the rotational symmetry condition, but does not satisfy the reflectional symmetry condition.

Ellipse

• Reflectional symmetry: 2 (major and minor axes).
• Rotational symmetry: 2 (rotational symmetry of order 2).

Conclusion: Satisfies the rotational symmetry condition, but does not satisfy the reflectional symmetry condition (only has 2, requires at least 3).

Regular Hexagon

• Reflectional symmetry: 6 (has 6 axes of symmetry).
• Rotational symmetry: 6 (rotational symmetry of order 6).

Conclusion: Satisfies both conditions (rotational symmetry $6 \ge 2$ and reflectional symmetry $6 \ge 3$).

Final Conclusion

Of the four plane figures, only 1 plane figure satisfies the criteria, which is the Regular Hexagon.

Explanation

Given the recursive formula of the sequence is $b_{n+2} = b_{n+1} - 3b_n$. We also know the values $b_4 = 5$ and $b_5 = -1$. We are asked to find the value of $b_2$. We will find it by working backwards from $b_5$ to $b_3$, and then to $b_2$.

Finding the Value of b3

Use the formula with $n = 3$:

Substitute the values $b_5 = -1$ and $b_4 = 5$:

Finding the Value of b2

Use the formula with $n = 2$:

Substitute the values $b_4 = 5$ and $b_3 = 2$:

So, the value of $b_2$ is $-1$.

Explanation

Given matrix $A = \begin{pmatrix} 5 & 3 - x \\ -1 & 1 \end{pmatrix}$ and its determinant is $8$.

The formula for the determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $\det(A) = ad - bc$.

Let's calculate the determinant of matrix $A$:

Given $\det(A) = 8$, then:

So, the value of $x$ is $0$.

Explanation

Given the equation $(f(x))^2 - 6f(x) = -8$ with $f(x) = 2x - 4$.

First, we rewrite the equation into the general form of a quadratic equation:

Then we substitute $f(x) = 2x - 4$ into the equation:

Expand the quadratic form and simplify the equation:

From the quadratic equation $ax^2 + bx + c = 0$, the product of the roots ($x_1 \cdot x_2$) is given by the formula $\frac{c}{a}$.

Given:

Then:

So, the value of $x_1 \cdot x_2$ is $12$.

Explanation

Given:

• Total students ($S$) = $800$ people.
• Students who like Mathematics ($M$) = $400$ people.
• Students who like Physics ($F$) = $500$ people.

We want to find the maximum value ($x$) and minimum value ($y$) of the intersection of these two sets (students who like both).

Finding Maximum Value x

The number of students who like both will be maximum if the smaller set is a subset of the larger set. Here, the number of students who like Mathematics ($400$) is less than those who like Physics ($500$). So, the maximum value of the intersection is the number of students in the smallest group.

Finding Minimum Value y

The number of students who like both will be minimum if the distribution is spread out as much as possible so that the total union is maximum (equal to the total number of students).

Use the set formula: $n(M \cup F) = n(M) + n(F) - n(M \cap F)$.

To get the minimum intersection, we assume all students ($800$) like at least one of the two subjects (no one dislikes both).

Calculating the Difference

Finally, we calculate the difference between the maximum and minimum values.

So, the value of $x - y$ is $300$ people.

Explanation

We will determine the number of possible passwords using the multiplication rule of combinatorics.

Password structure: Digit - Letter - Digit - Letter (starts with a digit and alternates).

Counting Choices for Position 1 (First Digit)

The first digit is an odd number ($1, 3, 5, 7, 9$).

Number of choices = $5$.

Counting Choices for Position 2 (First Letter)

Vowels: $A, E, I, O, U$ (total $5$ vowels).

The first letter is not the letter $A$, so the remaining choices are: $E, I, O, U$.

Number of choices = $4$.

Counting Choices for Position 3 (Second Digit)

The second digit is chosen from $2, 4, 6, 8$ (total $4$ choices).

Since digits cannot repeat, we must ensure the second digit is different from the first digit. However, the first digit is odd and the second digit is even, so they automatically do not repeat.

Number of choices = $4$.

Counting Choices for Position 4 (Second Letter)

The second letter is the letter $O$ (fixed).

Number of choices = $1$.

Calculating Total Passwords

Using the multiplication rule:

However, we need to check if letters can repeat. From the problem, letters cannot repeat. Let's check the case where the first letter is $O$ (same as the second letter).

If the first letter = $O$, then there is letter repetition. Valid choices for the first letter are $E, I, U$ (cannot be $A$ and cannot be $O$ because $O$ is already used in the second position).

So:

• First digit = $5$ choices.
• First letter = $3$ choices ($E, I, U$).
• Second digit = $4$ choices.
• Second letter = $1$ choice ($O$).

So, the number of passwords that can be formed is $60$.

Explanation

Given sorted data: $1, 1, 2, x, 10, y, 17, 18$ (total $8$ data points).

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

Finding the Value of x from Median

Since there are $8$ data points (even), the median is the average of the 4th and 5th data points.

Finding the Value of y from Mean

The total of all data divided by the number of data equals the mean.

Calculating the Average of the Last 4 Numbers

Complete data: $1, 1, 2, 8, 10, 15, 17, 18$.

The last 4 numbers are: $10, 15, 17, 18$.

Comparing P and Q

The value $P = 15$ and $Q = 15$.

Therefore, $P = Q$.

Explanation

We will simplify the forms of P and Q first.

Simplifying P

Recall that $a^2 - b^2 = (a - b)(a + b)$. Note that $16^x = (4^x)^2$.

Simplifying Q

Similarly, note that $16^{-x} = (4^{-x})^2$.

Analyzing the Value of P

Given $1 < x < 2$.

Analyzing the Value of Q

Given $1 < x < 2$. Multiply by $-1$ (inequality signs flip).

Comparing P and Q

From the results above, we get:

• $P > 5$
• $Q < \frac{5}{4} = 1.25$

It is clearly visible that the value of $P$ is always greater than $Q$ for the given range of $x$.