Set 1

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Given the operation $\triangle(a, b) = ka + \frac{b}{m}$ where $k$ and $m$ are positive integers. If $\triangle(2, 4) = 8$ and $\triangle(3, 5) = 11$ , then the value of $\triangle(5, 9)$ is ....

Given $A$ , $B$ , and $D$ are matrices with

A = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} a & 2 \\ b & 1 \end{pmatrix}, \quad D = 3A + B

If $AB = \begin{pmatrix} 1 & 3 \\ 17 & 11 \end{pmatrix}$ , then matrix $D$ is ....

$\begin{pmatrix} 4 & 1 \\ 7 & 9 \end{pmatrix}$

$\begin{pmatrix} 8 & -1 \\ 15 & 10 \end{pmatrix}$

$\begin{pmatrix} 9 & 4 \\ 14 & 10 \end{pmatrix}$

$\begin{pmatrix} 8 & 5 \\ 13 & 6 \end{pmatrix}$

$\begin{pmatrix} 11 & 5 \\ 10 & 6 \end{pmatrix}$

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

Which statements are true based on the information above?

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

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

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

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

$(1), (2), \text{ and } (3)$

$(1) \text{ and } (3)$

$(2) \text{ and } (4)$

$(4)$

$(1), (2), (3), \text{ and } (4)$

If the size of each small square on the map is $2 \text{ cm} \times 2 \text{ cm}$ , the map scale is $1 : 200,000$ , and an airplane takes $12 \text{ minutes}$ to fly from City B to City A with a map distance of $12 \text{ cm}$ , then the average speed of the airplane is ....

$24 \text{ km/h}$

$48 \text{ km/h}$

$72 \text{ km/h}$

$96 \text{ km/h}$

$120 \text{ km/h}$

Rhombus $PQRS$ has a perimeter of $20$ units. Diagonal $PR$ is perpendicular to diagonal $QS$ . The gradient of $QR$ is ....

Rhombus PQRS

Diagram of rhombus

PQRS

on the coordinate plane.

Rhombus $PQRS$ has a perimeter of $20$ units. Diagonal $PR$ is perpendicular to diagonal $QS$ . The equation of the line that passes through $Q$ and is perpendicular to $PS$ is ....

Rhombus PQRS

Diagram of rhombus

PQRS

on the coordinate plane.

Given the trigonometric ratios of two acute angles $A$ and $B$ with $\sin A = \frac{5}{13}$ and $\cos B = \frac{8}{17}$ .

How many of the following four statements are true based on the information above?

\text{(1)} \quad \tan A + \tan B = \frac{19}{20}

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

How many of the following four statements are true based on the information above?

$(1)$ $a$ is an even number

$(2)$ The unit digit of $a$ is always $0$

$(3)$ $a + b$ is divisible by $5$

$(4)$ $3a + 2b$ is an odd number

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

Based on the information above, the true statements are

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

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

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

$\text{I, II, and III}$

$\text{I and II}$

$\text{II and III}$

$\text{I}$

$\text{III}$

Function $f$ is expressed as $f(x) = \frac{1}{\sqrt{-x^2 + 4x + 21}}$ .

Based on the given information, which relationship between quantities $P$ and $Q$ is correct?

$P$	$Q$
$7$	$x$ such that $f(x)$ is defined as a real number

Quantity $P$ is greater than $Q$

Quantity $P$ is less than $Q$

Quantity $P$ is equal to $Q$

The information is insufficient to determine the relationship

Cannot be determined

The following table presents the mathematics scores of three student groups.

Group 1	Group 2	Group 3
$8$	$6$	$3$
$2$	$9$	$b$
$5$	$7$	$10$
$8$		$8$
$7$

The sum of the mode of Group 1's scores and the median of Group 2's scores is equal to $2$ times the average of Group 3's scores.

Based on the given information, which of the following relationships between quantities $P$ and $Q$ is correct?

$P$	$Q$
$b$	$8$

Quantity $P$ is greater than $Q$

Quantity $P$ is less than $Q$

Quantity $P$ is equal to $Q$

The information is insufficient to determine the relationship

Cannot be determined

Given a regular pyramid $T.ABCD$ with $AB = 4 \text{ cm}$ and point $O$ is the intersection point of lines $AC$ and $BD$ . What is the volume of the pyramid $T.ABCD$ ?

Regular Pyramid T.ABCD

Visualization of regular pyramid

T.ABCD

with square base

ABCD

and center of base at point

O

Decide whether the following statements $(1)$ and $(2)$ are sufficient to answer the question.

(1) \quad AC = 4\sqrt{2} \text{ cm}

Statement $(1)$ ALONE is sufficient, but statement $(2)$ ALONE is not sufficient

Statement $(2)$ ALONE is sufficient, but statement $(1)$ ALONE is not sufficient

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

EACH statement ALONE is sufficient

Statements $(1)$ and $(2)$ TOGETHER are NOT sufficient

Functions $f$ and $g$ are defined as $f(x) = \frac{x-a}{x+a}$ for $x \neq -a$ and $g(x) = bx^2 - x + 2$ . Is $\frac{g(-2)}{f(-2)} > 0$ ?

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

(1) \quad a < 5 \text{ and } b < 5

Statement $(1)$ ALONE is sufficient, but statement $(2)$ ALONE is not sufficient

Statement $(2)$ ALONE is sufficient, but statement $(1)$ ALONE is not sufficient

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

EACH statement ALONE is sufficient

Statements $(1)$ and $(2)$ TOGETHER are NOT sufficient

If $N$ satisfies $7 - N \times (-3) = 52$ , the value of $N$ is ....

Triangle $ABC$ is similar to triangle $ADE$ .

Triangle ABC and ADE

Visualization of two similar triangles with marked vertices.

The area of triangle $ADE$ is ....

The result of

\frac{x + 2y}{3} + \frac{2x - y}{4} - \frac{3x + 2y}{6}

is ....

$\frac{x + y}{12}$

$\frac{2x + y}{12}$

$\frac{3x + y}{12}$

$\frac{4x + y}{12}$

$\frac{5x + y}{12}$

Given the function $f(x) = (x+3)(x^2+6x+8)^4$ and $F(x) = \int f(x) \, dx$ . If $F(-3) = 2$ , then the function $F(x)$ is ....

$\frac{1}{10}(x^2+6x+8)^5 + \frac{1}{10}$

$\frac{1}{10}(x^2+6x+8)^5 + \frac{11}{10}$

$\frac{1}{10}(x^2+6x+8)^5 + \frac{21}{10}$

$\frac{1}{10}(x^2+6x+8)^5 + \frac{31}{10}$

$\frac{1}{10}(x^2+6x+8)^5 + \frac{41}{10}$

The number sequence $5, 8, 11, \ldots$ is an arithmetic sequence where $a$ is the first term, $b$ is the common difference, $U_n$ is the $n$ -th term, and $S_n$ is the sum of the first $n$ terms.

Which statements are true based on the information above?

(1) \quad \text{The value of } a \text{ is greater than the value of } b

$(1), (2), \text{ and } (3)$ ONLY are true

$(1) \text{ and } (3)$ ONLY are true

$(2) \text{ and } (4)$ ONLY are true

ALL statements are true

ALL statements are false

Given line $g_1$ passes through point $A(-1, 1)$ and $B(2, 3)$ . Also given the equation of line $g_2$ : $mx + ny - 7 = 0$ , where $m$ and $n$ are real numbers.

Which statements are true based on the information above?

$(1)$ Line $g_1$ and $g_2$ are parallel if $m = 4$ and $n = 6$ .

$(2)$ Point $B$ passes through line $g_2$ if $m = 2$ and $n = -1$ .

$(3)$ If $g_1$ and $g_2$ are parallel, then $\frac{m}{n} = -\frac{3}{2}$ .

$(4)$ If $m = 3$ and $n = 2$ , then $g_1$ and $g_2$ are perpendicular.

$(1), (2), \text{ and } (3)$ ONLY are true

$(1) \text{ and } (3)$ ONLY are true

$(2) \text{ and } (4)$ ONLY are true

$(4)$ ONLY is true

ALL statements are true

Given two real numbers with the following conditions:

Two times the first number minus three times the second number results in $14$ .

If the first number multiplied by $k$ plus two times the second number, then it results in $4$ , where $k$ is a real number.

If the sum of the first and second numbers results in $-3$ , the value of $k$ is ....

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