Set 6

Explanation

We are asked to determine the intersection point of two lines:

1. $y = 2x + 3$
2. $y = ax + b$

To determine a unique intersection point $(x, y)$, we must be able to solve this system of equations. Let's equate the two equations:

From the equation above, the value of $x$ can be determined if and only if the coefficient of $x$ is not zero, i.e., $2 - a \neq 0$ or $a \neq 2$. If $a \neq 2$, then:

After finding $x$, we can find $y$ by substituting it into one of the equations. So, to determine the intersection point specifically, we need to know:

1. That $a \neq 2$ (so the lines intersect at one point).
2. The value of $b$ (to calculate the coordinate values).

Analysis of Statement 1

Statement $(1)$ provides the information $a \neq 2$. This guarantees that the two lines have different gradients ($m_1 = 2$ and $m_2 = a \neq 2$), so they definitely intersect at one point. However, we do not know the value of $b$. Without the value of $b$, we cannot determine the coordinates of the intersection point (values of $x$ and $y$ still depend on $b$). Therefore, statement $(1)$ alone is not sufficient.

Analysis of Statement 2

Statement $(2)$ provides the information $b = 3$. The second equation becomes $y = ax + 3$. We do not know the value of $a$.

• If $a = 2$, then the line becomes $y = 2x + 3$, which coincides with the first line (infinitely many intersection points).
• If $a \neq 2$, the lines intersect at one point.

Since $a$ is unknown, we cannot ensure there is a single intersection point, nor can we determine its coordinates. Therefore, statement $(2)$ alone is not sufficient.

Analysis of Both Statements Together

If we combine both statements:

1. $a \neq 2$
2. $b = 3$

We substitute into the equation to find $x$:

(Division by $2 - a$ is valid because $a \neq 2$). Then we find $y$:

We get a unique intersection point $(0, 3)$. Since we can determine the intersection point, both statements together are sufficient.

Therefore, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation

Given:

• Initial weight (wet wood) = $90$ grams.
• Final weight (dry wood) = $60$ grams.

The water content in the wood is the difference between the initial weight and the final weight:

The percentage of water content is calculated relative to the initial weight (wet wood):

Therefore, the water content in the wet wood is $33\frac{1}{3}\%$.

Explanation

Let $B$ be a boy and $G$ be a girl.

The sample space for a family with $3$ children (without conditions) has $2^3 = 8$ possibilities:

It is given that "at least one child is a girl". This means the event "all children are boys" ($BBB$) is impossible.

The new sample space ($S$) consists of all possibilities except $BBB$:

The number of elements in the sample space is $n(S) = 7$.

The expected event ($A$) is having $2$ girls and $1$ boy.

From the sample space $S$, the outcomes that satisfy this condition are:

The number of expected outcomes is $n(A) = 3$.

The probability of the event is:

Therefore, the probability that the family has $2$ girls and $1$ boy is $\frac{3}{7}$.

Explanation

Sort the data from smallest to largest:

Number of data points $n = 7$.

Statement 1 Analysis

The median is the $\frac{n+1}{2} = \frac{8}{2} = 4$-th value. The $4$-th value is $4$.

Statement $1$ is CORRECT.

Statement 2 Analysis

The range is the difference between the maximum and minimum values.

Statement $2$ is INCORRECT (should be $5$, not $6$).

Statement 3 Analysis

Difference between range and median:

Statement $3$ is CORRECT.

Statement 4 Analysis

The mean ($\bar{x}$) is the sum of all data divided by the number of data points.

The value $3\frac{5}{7}$ is not equal to $4.5$. Statement $4$ is INCORRECT.

Conclusion

The correct statements are $(1)$ and $(3)$.

Explanation

Given the equation $x^2 + y^2 = 169$ where $x$ and $y$ are integers. The value $169$ is the square of $13$ ($13^2 = 169$). The integer pairs $(x, y)$ satisfying this circle equation can be found using Pythagorean triples or properties of perfect squares.

The possible pairs are:

1. If one variable is $0$, then the other is $\pm 13$. Pairs: $(13, 0), (-13, 0), (0, 13), (0, -13)$. Possible values for $x + y$:

   $$\pm 13 + 0 = \pm 13$$

2. If both variables are non-zero, we look for combinations of squares summing to $169$. The Pythagorean triple involving $13$ as the hypotenuse is $(5, 12, 13)$ because $5^2 + 12^2 = 25 + 144 = 169$. Possible pairs $(x, y)$ (including negative values): $(\pm 5, \pm 12)$ and $(\pm 12, \pm 5)$. Possible values for $x + y$ from these combinations:

   $$\begin{aligned} 5 + 12 &= 17 \\ 5 + (-12) &= -7 \\ -5 + 12 &= 7 \\ -5 + (-12) &= -17 \end{aligned}$$

   Thus, $x + y$ can be $\pm 17$ or $\pm 7$.

The set of all possible values for $x + y$ is $\{\pm 13, \pm 17, \pm 7\}$.

Statement 1 Analysis

The value $\pm 7$ is in the solution set. Statement $1$ is CORRECT.

Statement 2 Analysis

The value $\pm 10$ is NOT in the solution set. Statement $2$ is INCORRECT.

Statement 3 Analysis

The value $\pm 13$ is in the solution set. Statement $3$ is CORRECT.

Statement 4 Analysis

The value $\pm 14$ is NOT in the solution set. Statement $4$ is INCORRECT.

Conclusion

The correct statements are $(1)$ and $(3)$.

Explanation

We are given the function $f(x) = 2x - 3$ and the composite function $(g \circ f)(x) = 4x^2 - 12x + 10$. We are asked to find the value of $g(1)$.

The definition of a composite function is $(g \circ f)(x) = g(f(x))$. Let $u = f(x) = 2x - 3$. We rearrange this equation to express $x$ in terms of $u$:

Substitute $x$ into the equation $(g \circ f)(x)$:

Thus, we have $g(x) = x^2 + 1$.

Then the value of $g(1)$ is:

So, the value of $g(1)$ is $2$.

Explanation

We are given the following determinant equation:

The first step is to calculate the determinant of the matrices on the left and right sides. Recall that the determinant of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $ad - bc$.

Determinant of the left side:

Determinant of the right side:

Now we equate both results:

Move $5$ to the left side to form a quadratic equation:

We factor this quadratic equation. We look for two numbers that multiply to $3 \times (-5) = -15$ and sum to $-2$. These numbers are $-5$ and $3$.

Thus, the solutions are:

So, the values of $x$ that satisfy the equation are $\frac{5}{3}$ and $-1$.

Explanation

Given the function operation definition $W(P@Q\#R) = 2P^2 - Q + 3\sqrt{R}$.

We are asked to find the value of $W(5@8\#4)$.

Based on the position of variables in the function definition, we can identify the following values:

• $P = 5$
• $Q = 8$
• $R = 4$

Substitute these values into the function equation:

Perform the calculation step by step:

1. Calculate the square: $5^2 = 25$
2. Calculate the square root: $\sqrt{4} = 2$

Then the equation becomes:

So, the value of $W(5@8\#4)$ is $48$.

Explanation

Let's analyze the number of lines of symmetry (folding symmetry) and rotational symmetries for each given plane figure:

1. Isosceles right triangle

   • Lines of symmetry: $1$ (altitude from the right angle to the hypotenuse).
   • Rotational symmetries: $1$ (rotation of $360^\circ$).
   • Conclusion: The numbers are equal.

2. Right trapezoid

   • Lines of symmetry: $0$.
   • Rotational symmetries: $1$ (rotation of $360^\circ$).
   • Conclusion: The numbers are not equal.

3. Rectangle (not a square)

   • Lines of symmetry: $2$ (vertical and horizontal axes).
   • Rotational symmetries: $2$ (rotation of $180^\circ$ and $360^\circ$).
   • Conclusion: The numbers are equal.

4. Rhombus (not a square)

   • Lines of symmetry: $2$ (both diagonals).
   • Rotational symmetries: $2$ (rotation of $180^\circ$ and $360^\circ$).
   • Conclusion: The numbers are equal.

The figures that have the same number of rotational symmetries and lines of symmetry are:

1. Isosceles right triangle.
2. Rectangle.
3. Rhombus.

So, there are $3$ figures that satisfy the condition.

Explanation

Given the recursive equation:

We can rearrange the equation to find $a_{n+2}$:

Step 1: Find the value of $a_3$ by substituting $n = 1$. Given $a_1 = 8$ and $a_2 = -5$.

Step 2: Find the value of $a_4$ by substituting $n = 2$.

Step 3: Calculate the sum $a_3 + a_4$.

So, the value of $a_3 + a_4$ is $34$.

Explanation

Step 1: Calculate the actual width of the object. Given the area of the rectangle is $540\text{ m}^2$ and the length is $30\text{ m}$.

Convert to cm:

Step 2: Calculate the width of the miniature using the given ratio. The ratio of miniature size to actual size is $5 : 2500$ or $\frac{5}{2500}$.

So, the width of the miniature object is $3.6\text{ cm}$.

Explanation

Consider the small triangle at the bottom. The sum of angles in a triangle is $180^\circ$. Given the top vertex angle is $100^\circ$. So, the sum of the two base angles is:

Next, consider the large triangle. Its base angles are $(x+y)$ and $(y+x)$. Since $x+y = 80^\circ$, both base angles of the large triangle are $80^\circ$. The sum of angles in the large triangle is $180^\circ$:

Thus, the value of $a$ is $20^\circ$.

Explanation

We will analyze quantities $P$ and $Q$ separately.

Analyzing Quantity P

The expression for $P$ can be simplified as follows:

Given $0 < x < 1$. Thus:

Analyzing Quantity Q

Given $Q = 2+x$. Since $0 < x < 1$, then:

Comparing P and Q

From the analysis above, we have:

• $1 < P < 2$
• $2 < Q < 3$

It is clear that the maximum value of $P$ is less than the minimum value of $Q$.

Thus, $P < Q$.

Explanation

Given the function $g(x) = 3x + 2$. We will substitute this function into the given equation.

First, determine the forms of $g^2(x)$ and $g(x^2)$:

Substitute these forms into the initial equation:

Group like terms:

This quadratic equation has roots $x_1$ and $x_2$. The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is $x_1 + x_2 = -\frac{b}{a}$.

Thus, the value of $x_1 + x_2$ is $-\frac{3}{5}$.

Explanation

We will find the value of the replaced number (quantity $P$) first.

Analyzing Quantity P

Let the three numbers be $a, b,$ and $c$. The initial average is given as $24$:

Suppose the number $c$ is replaced by the number $14$. The new average decreases by $3$ from the initial average, so the new average becomes $24 - 3 = 21$.

We substitute the value $a + b = 49$ into the initial equation to find the value of $c$:

Thus, the replaced number is $23$. So, $P = 23$.

Comparing P and Q

Given $Q = 42$.

It is clear that $23 < 42$.

Thus, $P < Q$.

Explanation

This problem involves inverse proportion. If the number of workers increases, the time required decreases, and vice versa.

Given:

• $15$ people can complete the work in $35$ days.
• The work is desired to be completed in $3$ weeks $= 3 \times 7 = 21$ days.

Let $P$ be the total number of workers needed to complete the work in $21$ days.

So, a total of $25$ workers are needed.

The question asks for the additional workers. The initial number of workers is $15$.

Thus, $10$ additional workers are needed.