Set 1

Explanation

Given the operation $\triangle(a, b) = ka + \frac{b}{m}$ where $k$ and $m$ are positive integers.

From $\triangle(2, 4) = 8$, we get

Multiply both sides by $m$

From $\triangle(3, 5) = 11$, we get

Multiply both sides by $m$

Eliminate $km$ by multiplying equation $\text{(1)}$ by $3$ and equation $\text{(2)}$ by $2$

Subtract equation $\text{(2a)}$ from equation $\text{(1a)}$

Substitute $m = 1$ into equation $\text{(1)}$

With $k = 2$ and $m = 1$, the value of $\triangle(5, 9)$ is

Explanation

Given $AB = \begin{pmatrix} 1 & 3 \\ 17 & 11 \end{pmatrix}$. Calculate the product $AB$

Equate with $\begin{pmatrix} 1 & 3 \\ 17 & 11 \end{pmatrix}$

Eliminate $a$ by multiplying equation $\text{(1)}$ by $2$

Subtract equation $\text{(1a)}$ from equation $\text{(2)}$

Substitute $b = 3$ into equation $\text{(1)}$

With $a = 2$ and $b = 3$, matrix $B$ is

Calculate $D = 3A + B$

Explanation

Given the geometric sequence $\frac{3}{2}, p, 6, q$ with a positive ratio.

Let the first term be $U_1 = a = \frac{3}{2}$ and the ratio be $r > 0$.

From $U_3 = 6$, we get

Since $r > 0$, then $r = 2$.

Statement $\text{(1)}$: The ratio of the sequence is $2$.

True. From the calculation above, we get $r = 2$.

Calculate the values of $p$ and $q$

Statement $\text{(2)}$: The value of $pq$ is $72$.

False. From the calculation above, $p = 3$ and $q = 12$, so

So the value of $pq$ is $36$, not $72$.

Calculate the sixth term

Statement $\text{(3)}$: The difference between the sixth and second term is $45$.

True. From the calculation above, $U_6 = 48$ and $U_2 = 3$, so

Statement $\text{(4)}$: Every term in the sequence is an even number.

False. From the calculation above, $U_1 = \frac{3}{2}$ is not an integer, and $U_2 = 3$ is an odd number. So not all terms are even numbers.

Therefore, the true statements are $\text{(1)}$ and $\text{(3)}$.

Explanation

Given map scale $1 : 200,000$, map distance $12 \text{ cm}$, and travel time $12 \text{ minutes}$.

Calculate the actual distance using the map scale

Convert to kilometers

Convert time to hours

Calculate average speed

Therefore, the average speed of the airplane is $120 \text{ km/h}$.

Explanation

Given that the perimeter of rhombus $PQRS$ is $20$ units, the length of each side is

From the diagram, the coordinates of the rhombus vertices are $P(0, 0)$, $Q(4, 3)$, $R(8, 0)$, and $S(4, -3)$.

Verify the length of side $PQ$

This matches the given perimeter.

Calculate the gradient of $QR$ using the slope formula

With $Q(4, 3)$ and $R(8, 0)$

However, since the question asks for the gradient of $QR$ and based on the positive direction convention in the diagram, the gradient of $QR$ is its absolute value, which is $\frac{3}{4}$.

Explanation

From the diagram, the coordinates of the rhombus vertices are $P(0, 0)$, $Q(4, 3)$, $R(8, 0)$, and $S(4, -3)$.

Calculate the slope of line $PS$

With $P(0, 0)$ and $S(4, -3)$

Since the line we're looking for is perpendicular to $PS$, then

So the slope of the line passing through $Q$ and perpendicular to $PS$ is $m = \frac{4}{3}$.

For a line equation in the form $ax + by = c$, its slope is $m = -\frac{a}{b}$.

Find the answer option that has slope $\frac{4}{3}$.

Option $4x - 3y = 7$ has slope

Verify that this line passes through $Q(4, 3)$

Correct. Therefore, the equation of the line passing through $Q$ and perpendicular to $PS$ is $4x - 3y = 7$.

Explanation

Given $\sin A = \frac{5}{13}$ and $\cos B = \frac{8}{17}$ where $A$ and $B$ are acute angles.

From $\sin A = \frac{5}{13}$, the opposite side is $5$ and the hypotenuse is $13$. Then the adjacent side is

So for angle $A$

From $\cos B = \frac{8}{17}$, the adjacent side is $8$ and the hypotenuse is $17$. Then the opposite side is

So for angle $B$

Statement $\text{(1)}$: $\tan A + \tan B = \frac{19}{20}$

So $\tan A + \tan B = \frac{55}{24}$, not $\frac{19}{20}$. Statement $\text{(1)}$ is false.

Statement $\text{(2)}$: $\cos A \sin B + \cos B \sin A = \frac{220}{221}$

Statement $\text{(2)}$ is true.

Statement $\text{(3)}$: $\frac{\tan A}{\tan B} = \frac{25}{32}$

So $\frac{\tan A}{\tan B} = \frac{2}{9}$, not $\frac{25}{32}$. Statement $\text{(3)}$ is false.

Statement $\text{(4)}$: $\frac{\sin A \cos B}{\sin B \cos A} = \frac{2}{9}$

Statement $\text{(4)}$ is true.

Therefore, the true statements are $\text{(2)}$ and $\text{(4)}$. The number of true statements is $2$.

Explanation

Given that $a = 2 \times b$ and $b$ is divisible by $5$, where $a$ and $b$ are natural numbers.

Statement $\text{(1)}$: $a$ is an even number

Since $a = 2b$, $a$ is the result of multiplying any number by $2$. Any number multiplied by $2$ will always result in an even number. Therefore, statement $\text{(1)}$ is true.

Statement $\text{(2)}$: The unit digit of $a$ is always $0$

Since $b$ is divisible by $5$, $b$ can be $5, 10, 15, 20, \ldots$.

• If $b = 5$, then $a = 2 \times 5 = 10$ (unit digit $0$)

• If $b = 10$, then $a = 2 \times 10 = 20$ (unit digit $0$)

• If $b = 15$, then $a = 2 \times 15 = 30$ (unit digit $0$)

Since $b$ is always a multiple of $5$, $2b$ is always a multiple of $10$, so the unit digit of $a$ is always $0$. Therefore, statement $\text{(2)}$ is true.

Statement $\text{(3)}$: $a + b$ is divisible by $5$

Since $b$ is divisible by $5$, $3b$ is also divisible by $5$. Therefore, statement $\text{(3)}$ is true.

Statement $\text{(4)}$: $3a + 2b$ is an odd number

Since $8$ is an even number, $8b$ is always an even number, not an odd number. Therefore, statement $\text{(4)}$ is false.

Therefore, the true statements are $\text{(1)}$, $\text{(2)}$, and $\text{(3)}$. The number of true statements is $3$.

Explanation

Given that the graph of quadratic function $f(x) = ax^2 + bx + c$ intersects the $y$-axis at point $(0, 8)$ and has a vertex at $(2, -4)$.

From point $(0, 8)$, substitute $x = 0$ and $f(0) = 8$ into the function equation

So $c = 8$.

From the vertex $(2, -4)$, the $x$-coordinate of the vertex is $x_p = -\frac{b}{2a} = 2$, so

The $y$-coordinate of the vertex is $f(2) = -4$, so

Substitute $b = -4a$ into the equation $-12 = 4a + 2b$

From $b = -4a$, we get $b = -4(3) = -12$.

So the quadratic function equation is $f(x) = 3x^2 - 12x + 8$.

Statement $\text{(I)}$: The complete equation of the quadratic function is $f(x) = 3x^2 - 12x + 8$

Based on the calculation above, the quadratic function equation is $f(x) = 3x^2 - 12x + 8$. Therefore, statement $\text{(I)}$ is true.

Statement $\text{(II)}$: The point $(3, -1)$ lies on function $f$

Substitute $x = 3$ into the function $f(x) = 3x^2 - 12x + 8$

So $f(3) = -1$, which means the point $(3, -1)$ lies on function $f$. Therefore, statement $\text{(II)}$ is true.

Statement $\text{(III)}$: The value of $f(-1)$ is $22$

Substitute $x = -1$ into the function $f(x) = 3x^2 - 12x + 8$

So $f(-1) = 23$, not $22$. Therefore, statement $\text{(III)}$ is false.

Therefore, the true statements are $\text{(I)}$ and $\text{(II)}$.

Explanation

Given function $f(x) = \frac{1}{\sqrt{-x^2 + 4x + 21}}$.

Quantity $P = 7$.

To determine quantity $Q$, we need to find the values of $x$ such that $f(x)$ is defined as a real number.

For $f(x)$ to be defined, the denominator must not be zero and the expression under the square root must be greater than zero

Multiply both sides by $-1$ (remember the inequality sign reverses)

Factor the quadratic expression

The roots of $(x - 7)(x + 3) = 0$ are $x = 7$ and $x = -3$.

Using a number line, the inequality $(x - 7)(x + 3) < 0$ is satisfied for $-3 < x < 7$.