Set 5

Explanation

We are given the first two terms of the sequence, $a_1 = -6$ and $a_2 = -8$, and the recursive equation:

Our goal is to find the value of $a_3 + 2a_4$. We will find the values of $a_3$ and $2a_4$ step by step.

Finding the Third Term

We use the recursive equation by substituting $n = 1$:

Substitute the known values of $a_1$ and $a_2$:

Move $-18$ to the left side:

Finding Two Times the Fourth Term

Next, we use the recursive equation again with $n = 2$ to introduce the term $a_4$:

Substitute the values $a_3 = 5$ and $a_2 = -8$:

We can directly solve for $2a_4$ (since the question asks for $a_3 + 2a_4$):

Final Result

Now we sum the obtained values:

Thus, the value of $a_3 + 2a_4$ is 34.

Explanation

We are given a determinant equation of $2 \times 2$ matrices. Recall that the formula for the determinant of a matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is $ad - bc$.

Calculating the Left Hand Side Determinant

First, we calculate the determinant of the matrix on the left side:

Calculating the Right Hand Side Determinant

Next, we calculate the determinant of the matrix on the right side which contains the variable $x$:

Determining the Variable Value

Now we equate both determinant results according to the original equation:

Divide both sides by 10 to find the value of $x$:

Thus, the value of $x$ that satisfies the equation is -3.

Explanation

We are given two conditions for the integer $n$:

1. $n \equiv 2 \pmod{4}$ (remainder is $2$ when divided by $4$)
2. $n \equiv 2 \pmod{6}$ (remainder is $2$ when divided by $6$)

We will check each statement to see if the value of $n$ satisfies both conditions.

Checking Statement 1

Value $n = 18$.

• Divided by $4$: $18 \div 4 = 4$ remainder $2$. (Satisfied)
• Divided by $6$: $18 \div 6 = 3$ remainder $0$. (Not Satisfied)

Thus, statement $(1)$ is incorrect.

Checking Statement 2

Value $n = 24$.

• Divided by $4$: $24 \div 4 = 6$ remainder $0$. (Not Satisfied)

Thus, statement $(2)$ is incorrect.

Checking Statement 3

Value $n = 36$.

• Divided by $4$: $36 \div 4 = 9$ remainder $0$. (Not Satisfied)

Thus, statement $(3)$ is incorrect.

Checking Statement 4

Value $n = 38$.

• Divided by $4$: $38 \div 4 = 9$ remainder $2$. (Satisfied)
• Divided by $6$: $38 \div 6 = 6$ remainder $2$. (Satisfied)

Thus, statement $(4)$ is correct.

Conclusion

Of the four options, only statement $(4)$ satisfies both conditions.

Explanation

We will simplify the forms of $P$ and $Q$ first to compare their values.

Simplifying P Value

Given $P = \frac{1 - x^6}{1 - x^3}$. Recall the difference of squares $a^2 - b^2 = (a - b)(a + b)$. We can view $1 - x^6$ as $1^2 - (x^3)^2$.

Simplifying Q Value

Given $Q = (1 + x)^3$. We expand it using binomial expansion:

Comparing P and Q

We find the difference between $Q$ and $P$:

Given the condition $0 < x < 1$. Since $x$ is positive ($x > 0$), then:

• $3x > 0$
• $3x^2 > 0$

Thus, their sum must be positive:

Or it can be written as $P < Q$.

Conclusion

Thus, the correct relationship is $P < Q$.

Explanation

We want to find the composite function $(g \circ f)(x)$.

Understanding Function Composition

The composite function $(g \circ f)(x)$ is defined as:

This means we will substitute the entire function $f(x)$ into the variable $x$ in the function $g(x)$.

Function Substitution

Given:

• $f(x) = x^2 + 4x - 5$
• $g(x) = 2x - 1$

We substitute $f(x)$ into $g(x)$:

Since $g(x) = 2x - 1$, we replace every $x$ in $g(x)$ with $(x^2 + 4x - 5)$:

Simplifying the Result

Now we distribute the multiplication and simplify:

Conclusion

Thus, the composite function is $2x^2 + 8x - 11$.

Explanation

We will find the value of the added numbers, determine the new median, and then compare the values of $P$ and $Q$.

Finding the Added Numbers

Let the two added natural numbers be $x$. The initial data consists of $7$ numbers: $2, 3, 5, 7, 8, 9, 11$. Sum of initial data:

After adding $2$ numbers of value $x$, the count of data becomes $7 + 2 = 9$. The new average is given as $9$.

So, the added number is $18$.

Determining the New Median

The new data after adding $18$ and $18$ is:

The data is already sorted. Since the number of data points $n = 9$ (odd), the median is the $\frac{9+1}{2} = 5$-th data point.

Calculating P Value

The value of $P$ is defined as the difference between the new median and the new mean. The new mean is known to be $9$.

Comparing P and Q

Given $Q = -12$.

Since $-1 > -12$, then:

Conclusion

Thus, the correct relationship is $P > Q$.

Explanation

We are given the function $f(x) = 2x + 3$ and the equation $3f^2(x) = f(x) + 7$. We want to find the sum of the roots of the equation, which is $x_1 + x_2$.

Substituting the Function

The first step is to substitute $f(x)$ into the given equation.

Simplifying the Equation

Next, we expand the square and simplify the equation into the general quadratic form $ax^2 + bx + c = 0$.

Move all terms to the left side:

Using the Sum of Roots Formula

From the quadratic equation $12x^2 + 34x + 17 = 0$, we obtain the coefficients:

• $a = 12$
• $b = 34$
• $c = 17$

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

Simplify the fraction by dividing the numerator and denominator by $2$:

Conclusion

Thus, the value of $x_1 + x_2$ is $-\frac{17}{6}$.

Explanation

We can solve this problem using the concept of inverse proportion or total workload (man-days).

Calculating Workload

The total planned workload is:

The work is divided into two phases:

1. First phase: $24$ workers work for $8$ days.
2. Second phase: Half of the workers stop, so $24 / 2 = 12$ workers remain working for $P$ days.

The equation is:

Finding the Value of P

We solve the equation above to find $P$ (the duration for the remaining project).

So, the time needed to complete the remaining work with $12$ workers is $54$ days.

Calculating Additional Time

Based on the provided solution, the additional time is calculated as the difference between the duration of the remaining work ($P$) and the total initial target ($35$ days).

Thus, $19$ additional days are needed.

Explanation

We verify each statement based on the functions provided in the graph: Exponential function: $y = 2^x - 2$ Linear function: $y = -2x + 4$

Checking Point A

Point $A$ is the y-intercept of the exponential curve ($x=0$).

So, $A(0, -1)$. Statement $(1)$ is TRUE.

Checking Point B

Point $B$ is the x-intercept of the exponential curve ($y=0$).

So, $B(1, 0)$. Statement $(2)$ is TRUE.

Checking Point C

Point $C$ is the x-intercept of the linear line ($y=0$).

So, $C(2, 0)$. Statement $(3)$ is TRUE.

Checking Point D

Point $D$ is the intersection point or a point on the line. The statement claims $D(2, 2)$. If we test coordinate $(2, 2)$ into the line equation $y = -2x + 4$:

The result is $y = 0$, not $2$. Thus, point $(2, 2)$ does not lie on the line $y = -2x + 4$. Statement $(4)$ is FALSE.

Conclusion: Statements $(1)$, $(2)$, and $(3)$ are correct.

Explanation

We analyze the lines of symmetry (reflectional symmetry) and rotational symmetry order for each given plane figure:

1. Regular Hexagon

   • Lines of symmetry: $6$ (lines through vertices and midpoints of sides).
   • Rotational symmetry order: $6$ (angles of rotation $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$).
   • Conclusion: $6 = 6$. Figure $(1)$ satisfies the condition.

2. Isosceles Right Triangle

   • Lines of symmetry: $1$ (altitude from the right angle to the hypotenuse).
   • Rotational symmetry order: $1$ (only the initial position $360^\circ$).
   • Conclusion: $1 = 1$. Figure $(2)$ satisfies the condition.

3. Right Trapezoid

   • Lines of symmetry: $0$ (no axis of symmetry).
   • Rotational symmetry order: $1$ (only the initial position $360^\circ$).
   • Conclusion: $0 \neq 1$. Figure $(3)$ does not satisfy the condition.

4. Non-square Rectangle

   • Lines of symmetry: $2$ (vertical and horizontal lines through the center).
   • Rotational symmetry order: $2$ (angles of rotation $180^\circ$ and $360^\circ$).
   • Conclusion: $2 = 2$. Figure $(4)$ satisfies the condition.

Thus, the figures that have the same number of lines of symmetry as the order of rotational symmetry are figures $(1)$, $(2)$, and $(4)$. There are $3$ figures.

Explanation

Given:

• Total students $= 700$.
• Students choosing sports $= 350$.
• Students choosing arts $= 450$.

We want to find the value of $P$ (maximum intersection) and $Q$ (minimum intersection).

1. Finding $P$ (Maximum Intersection) The maximum number of students participating in both occurs if one set is a subset of the other (or as many members of the smaller set as possible are included in the larger set).

   $$P = \min(350, 450) = 350 \text{ students}$$

2. Finding $Q$ (Minimum Intersection) The minimum number of students participating in both occurs if the distribution of students is as spread out as possible so that the union of the two sets is maximized (equal to the total universe, which is $700$). Using the Inclusion-Exclusion Principle:

   $$\text{Intersection} = (\text{Sports} + \text{Arts}) - \text{Total}$$

   $$Q = (350 + 450) - 700$$

   $$Q = 800 - 700 = 100 \text{ students}$$

3. Finding the Difference Between $P$ and $Q$

   $$P - Q = 350 - 100 = 250 \text{ students}$$

Thus, the difference between $P$ and $Q$ is $250$ students.

Explanation

We analyze each statement based on the graph:

1. Statement $(1)$: The graph intersects the $y$-axis at $4$ and $-2$. So the intersection points are $(0, 4)$ and $(0, -2)$. Statement $(1)$ is CORRECT.

2. Statement $(2)$: Since the graph intersects the $y$-axis at $y = 4$ and $y = -2$, the equation is of the form:

   $$x = a(y - 4)(y + 2)$$

   The graph passes through the point $(8, 0)$ (x-intercept). Substitute into the equation:

   $$8 = a(0 - 4)(0 + 2)$$

   $$8 = a(-4)(2)$$

   $$8 = -8a \implies a = -1$$

   So the function equation is:

   $$x = -(y - 4)(y + 2)$$

   $$x = -(y^2 - 2y - 8)$$

   $$x = -y^2 + 2y + 8$$

   Converting to vertex form:

   $$x = -(y^2 - 2y + 1 - 1 - 8)$$

   $$x = -((y - 1)^2 - 9)$$

   $$x = -(y - 1)^2 + 9$$

   Or in function notation $f(y)$:

   $$f(y) = -(y - 1)^2 + 9$$

   Statement $(2)$ is CORRECT.

3. Statement $(3)$: The axis of symmetry of a horizontal parabola is $y_p$ (ordinate of the vertex). From the form $x = a(y - y_p)^2 + x_p$, we get $y_p = 1$. Or from the midpoint of the roots:

   $$y = \frac{4 + (-2)}{2} = \frac{2}{2} = 1$$

   So the axis of symmetry is $y = 1$. Statement $(3)$ is CORRECT.

4. Statement $(4)$: The intersection with the $x$-axis occurs when $y = 0$. From the graph, it clearly intersects at $x = 8$. By calculation: $x = -(0 - 1)^2 + 9 = -1 + 9 = 8$. So the intersection point is $(8, 0)$. Statement $(4)$ is CORRECT.

Conclusion: All statements $(1)$, $(2)$, $(3)$, and $(4)$ are correct.

Explanation

First, we find the actual width of the field. Given the area of the field $L = 384\text{ m}^2$ and length $p = 32\text{ m}$. Then the actual width ($l$) is:

Next, we convert the actual width to cm to match the drawing unit.

Finally, we calculate the width of the field in the drawing using the scale ratio $3 : 600$.

So, the width of the field in the drawing is $6\text{ cm}$.

Explanation

Given the quadratic function $y = -2x^2 - 5x + 7$. We need to find the equation of the tangent line at the point with abscissa $x_1 = 2$.

First, we find the ordinate ($y_1$) of the tangent point by substituting $x_1 = 2$ into the curve equation:

So, the tangent point is $(2, -11)$.

Second, we find the gradient of the tangent line ($m$) using the first derivative ($y'$):

Substitute $x = 2$ into the first derivative to get the gradient:

Third, we determine the tangent line equation using the formula $y - y_1 = m(x - x_1)$: