Multiple Choice

1. In a group of 30 students, 18 like to play soccer and 15 like to play basketball. If 7 students like to play both soccer and basketball, how many students like to play \emph{neither} soccer nor basketball?

   2

   4

   3

   5

2. Let $P = \{x \mid x \text{ is a natural number and } x < 10\}$ and $Q = \{x \mid x \text{ is an even number and } 0 < x \le 12\}$. Which of the following sets represents $P \cap Q$?

   $\{2, 4, 6, 8\}$

   $\{2, 4, 6\}$

   $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\}$

   $\{10, 12\}$

3. Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Let $A = \{x \mid x \text{ is an odd number}\}$ and $B = \{x \mid x \text{ is a prime number}\}$. What is the set $A' \cap B$?

   $\{2, 3, 4, 5, 6, 7, 8, 10\}$

   $\{2\}$

   $\{1, 9\}$

   $\{3, 5, 7\}$

4. A survey of 100 students showed that 40 liked Math, 35 liked Science, and 30 liked English. 15 liked Math and Science, 12 liked Science and English, 10 liked Math and English, and 5 liked all three subjects. How many students liked \emph{only Math}?

   25

   15

   20

   30

5. Which of the following statements is \textbf{false}? Let $A = \{x, y\}$ and $B = \{y, x\}$.

   $\emptyset \subseteq A$

   $A \subset B$

   $A \subseteq A$

   $A = B$

6. Let $A = \{x \mid x \in \mathbb{R}, 2 \le x < 7\}$ and $B = \{x \mid x \in \mathbb{R}, 5 < x \le 10\}$. What is $A \cap B$?

   $\{x \mid x \in \mathbb{R}, 2 \le x < 5\}$

   $\{x \mid x \in \mathbb{R}, 2 \le x \le 10\}$

   $\{x \mid x \in \mathbb{R}, 7 \le x \le 10\}$

   $\{x \mid x \in \mathbb{R}, 5 < x < 7\}$

7. Let $U = \{1, 2, 3, ..., 10\}$ be the universal set. Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{4, 5, 6, 7\}$. What is $(A \setminus B)'$?

   $\{4, 5, 6, 7\}$

   $\{6, 7, 8, 9, 10\}$

   $\{4, 5, 6, 7, 8, 9, 10\}$

   $\{1, 2, 3\}$

8. Which of the following is equivalent to $(X \cup Y)'$?

   $X' \cap Y'$

   $X \cup Y'$

   $X \cap Y$

   $X' \cup Y'$

9. Let $A = \{red, blue\}$ and $B = \{small, medium, large\}$. What is the cardinality of the Cartesian product $A \times B$? ($|A \times B|$)

   5

   3

   6

   2

10. A water tank is being filled. A straight line graph shows the volume of water in liters in the tank over time in minutes. The graph starts with $0$ liters at $0$ minutes. For every $1$ minute that passes, $5$ liters of water are added. How many liters of water are in the tank after $4$ minutes?

    $20$ liters

    $5$ liters

    $10$ liters

    $1$ liter

11. A linear graph displays the cost of renting a bicycle for different durations. The cost is in euros, and the duration is in hours. The graph shows that even if you rent for $0$ hours, there is a fixed cost of $5$ euros. After $2$ hours, the total cost is $15$ euros. What is the fixed cost (y-intercept) in euros, before any rental time?

    $5$ euros

    $2$ euros

    $0$ euros

    $15$ euros

12. A linear graph shows the amount of juice remaining in a pitcher. The x-axis is time in minutes, and the y-axis is volume in milliliters. At $0$ minutes, the pitcher has $1000$ milliliters of juice. Every minute, $50$ milliliters are poured out. How many milliliters of juice are left after $3$ minutes?

    $800$ milliliters

    $850$ milliliters

    $950$ milliliters

    $1000$ milliliters

13. Two robots, A and B, move in a straight line. Robot A's graph shows it starts at $0$ meters and travels $10$ meters in $2$ seconds. Robot B's graph also starts at $0$ meters and travels $12$ meters in $3$ seconds. Which robot is moving faster?

    Cannot be determined

    Robot B

    Robot A

    They are moving at the same speed

14. A linear graph shows the weight of a baby over its first few months. At $1$ month old, the baby weighed $4$ kilograms. At $3$ months old, the baby weighed $6$ kilograms. If the baby's weight gain was steady, what was the baby's weight in kilograms at $2$ months old?

    $6$ kilograms

    $5$ kilograms

    $4$ kilograms

    $5.5$ kilograms

15. A straight line graph shows the cost of apples. $1$ kilogram of apples costs $2$ euros. $3$ kilograms of apples cost $6$ euros. Which of these options describes a point that does NOT fit on this linear graph?

    $0$ kg costs $0$ euros

    $5$ kg costs $9$ euros

    $4$ kg costs $8$ euros

    $2$ kg costs $4$ euros

16. A linear graph tracks the amount of fuel in a car's tank. The x-axis is distance traveled in kilometers, and the y-axis is fuel remaining in liters. The car starts with $50$ liters of fuel. For every $100$ kilometers driven, $5$ liters of fuel are used. How many liters of fuel remain after driving $300$ kilometers?

    $35$ liters

    $50$ liters

    $40$ liters

    $45$ liters

17. The cost of sending a parcel (in euros) for certain weights (in kilograms) is given by the following pattern: $1$ kg costs $3$ euros, $2$ kg costs $5$ euros, and $3$ kg costs $7$ euros. What would be the cost, in euros, to send a parcel weighing $4$ kilograms?

    $10$ euros

    $9$ euros

    $8$ euros

    $11$ euros

18. A linear graph shows the temperature of water being heated, then kept at a steady temperature. After $5$ minutes of heating, the water reached $100$ degrees Celsius. For the next $10$ minutes (from $5$ minutes to $15$ minutes), the graph shows a horizontal line at $100$ degrees Celsius. What was the temperature of the water in degrees Celsius at $12$ minutes?

    $0$ degrees Celsius

    $5$ degrees Celsius

    $100$ degrees Celsius

    $10$ degrees Celsius

19. A robot starts at point $(1, 2)$. It then moves to point $(3, 4)$, and then to point $(5, 6)$. If the robot continues to move following the exact same straight line pattern, what will be the coordinates of its next stop?

    $(6, 7)$

    $(7, 6)$

    $(8, 8)$

    $(7, 8)$

20. On a map, a hidden treasure is located exactly in the middle of two landmarks. Landmark A is at $8$ blocks to the east, and Landmark B is at $16$ blocks to the east. How many blocks to the east is the treasure located?

    $10$ blocks

    $12$ blocks

    $13$ blocks

    $11$ blocks

21. Two roads are parallel, meaning they never meet. Road 1 starts at $(1, 1)$ and goes to $(5, 3)$. Road 2 starts at $(1, 4)$ and goes in the exact same direction and pattern as Road 1. Which point could Road 2 pass through?

    $(5, 6)$

    $(5, 5)$

    $(5, 7)$

    $(6, 5)$

22. A bird flies from its nest at $(2, 2)$ to a feeder at $(5, 6)$. If the bird must fly along the grid lines (only horizontal or vertical paths), which of these describes a path it could take to reach the feeder?

    The total distance flown along grid lines would be $7$ units.

    All of the above are correct descriptions of a path along grid lines.

    Fly $4$ units up, then $3$ units right.

    Fly $3$ units right, then $4$ units up.

23. A remote-controlled car starts at coordinates $(2, 3)$. It moves $1$ unit right and $2$ units up, then repeats this movement $2$ more times. What are the final coordinates of the car after these $3$ movements?

    $(3, 6)$

    $(5, 9)$

    $(4, 7)$

    $(4, 7)$

24. Line P is drawn on a graph from $(1, 2)$ to $(5, 4)$. Line Q is parallel to Line P and starts at $(2, 5)$. If Line Q is the same length as Line P, what are the coordinates of the other end of Line Q?

    $(5, 6)$

    $(6, 7)$

    $(7, 7)$

    $(5, 6)$

25. A planet is $5 \times 10^3$ miles away. Another star is $3 \times 10^4$ miles away. Which one is further? (Remember $10^3 = 1,000$ and $10^4 = 10,000$)

    They are the same distance.

    You cannot compare them.

    The star is further.

    The planet is further.

26. A farmer has $8 \times 10^2$ apples. He sells $3 \times 10^2$ apples. How many apples does he have left? (Remember $10^2 = 100$)

    $5 \times 10^2$ apples

    $5 \times 10^1$ apples

    $11 \times 10^2$ apples

    $5 \times 10^4$ apples

27. Simplify the expression $\frac{(3x^{-2}y^3)^2}{ (9x^4y^{-1})^{-1} }$ completely, assuming $x, y \neq 0$.

    $81x^8y^7$

    $\frac{y^5}{81}$

    $9y^5$

    $81y^5$

28. Evaluate the expression $(-2)^3 - 4^{-2} + (-1)^0 \times 2^1$.

    $-10$

    $-\frac{99}{16}$

    $-\frac{95}{16}$

    $-\frac{97}{16}$

29. Calculate the value of $\frac{(4.5 \times 10^7) \times (3.0 \times 10^{-3})}{ (9.0 \times 10^2) }$. Express your answer in scientific notation.

    $1.5 \times 10^6$

    $0.15 \times 10^3$

    $1.5 \times 10^2$

    $1.5 \times 10^3$

30. What is the sum of $(2.7 \times 10^5) + (3.4 \times 10^3)$? Express your answer in scientific notation.

    $6.1 \times 10^5$

    $2.734 \times 10^3$

    $2.734 \times 10^5$

    $6.1 \times 10^8$

31. Simplify the expression $\left( \frac{a^3 b^{-2}}{a^{-1} b^4} \right)^{-2} \times (a^2 b^3)^3$, assuming $a,b \neq 0$.

    $\frac{a^{14}}{b^{15}}$

    $a^2b^{21}$

    $\frac{b^{15}}{a^2}$

    $\frac{b^{21}}{a^2}$

32. If $x = 3^k$ and $y = 3^{k+2}$, what is the value of $\frac{y}{x^2}$ in terms of $k$?

    $3^{-k}$

    $3^{3k+2}$

    $3^{k-2}$

    $3^{2-k}$

33. A rectangular field has a length of $3.2 \times 10^3$ meters and a width of $1.5 \times 10^2$ meters. If a tractor can cultivate an area of $6.0 \times 10^4$ square meters per hour, how many hours will it take to cultivate the entire field? Express your answer in scientific notation.

    $4.8 \times 10^5$ hours

    $8.0 \times 10^0$ hours

    $1.2 \times 10^3$ hours

    $8.0 \times 10^1$ hours

34. Simplify the expression $\frac{ (x^2y^{-3})^4 }{ (x^{-1}y^2)^{-3} } \div \left( \frac{x^3}{y^{-1}} \right)^{-2}$ completely, assuming $x,y \neq 0$.

    $\frac{x^{11}}{y^{10}}$

    $x^{11}y^{-10}$

    $\frac{y^4}{x^{11}}$

    $\frac{x^{11}}{y^4}$

35. If $5^{2x-1} = 125^{x-3}$, what is the value of $x$?

    $8$

    $5$

    $2$

    $-8$

36. Which of the following expressions represents the largest value?

    $2.9 \times 10^4$

    $3.1 \times 10^3$

    $3 \times 10^4$

    $3.05 \times 10^4$

37. Factorise completely: $3xy - 24x + 5y - 40$.

    $(3x+5)(y+8)$

    $3x(y-8) + 5(y-8)$

    $(3x-5)(y-8)$

    $(3x+5)(y-8)$

38. Factorise the quadratic expression $6x^2 - 19x + 10$.

    $(2x-5)(3x-2)$

    $(6x-5)(x-2)$

    $(2x+5)(3x-2)$

    $(3x-5)(2x+2)$

39. Express $2x^2 + 12x - 7$ in the form $a(x+h)^2 + k$.

    $2(x+3)^2 - 25$

    $2(x+6)^2 - 25$

    $2(x+3)^2 - 1$

    $(2x+6)^2 - 25$

40. Solve the equation $3x^2 = 5x + 2$.

    $x=2, x=1/3$

    $x=-2, x=1/3$

    $x=2, x=-1/3$

    $x=-2, x=-1/3$

41. Solve $x^2 - 8x + 5 = 0$ by completing the square.

    $x = -4 \pm \sqrt{11}$

    $x = 8 \pm \sqrt{11}$

    $x = 4 \pm \sqrt{11}$

    $x = 4 \pm 11$

42. Solve $2x^2 - 7x - 3 = 0$ using the quadratic formula.

    $x = \frac{-7 \pm \sqrt{73}}{4}$

    $x = \frac{7 \pm \sqrt{73}}{4}$

    $x = \frac{7 \pm \sqrt{25}}{4}$

    $x = \frac{7 \pm \sqrt{73}}{2}$

43. A rectangular garden has a length that is $5 \text{ m}$ more than its width. If the area of the garden is $84 \text{ m}^2$, what is the width of the garden?

    $12 \text{ m}$

    $7 \text{ m}$

    $6 \text{ m}$

    $9 \text{ m}$

44. The height $h$ (in meters) of a projectile launched upwards from a platform is given by the equation $h = -5t^2 + 20t + 15$, where $t$ is the time in seconds after launch. At what time (to one decimal place) will the projectile hit the ground? (Assume $h=0$ at ground level).

    $5.2$ seconds

    $2.6$ seconds

    $3.0$ seconds

    $4.6$ seconds

45. Factorise completely: $2x^2y + 6x - xy - 3$.

    $(xy+3)(2x-1)$

    $(xy-3)(2x+1)$

    $(x+3)(2y-1)$

    $(2xy-3)(x+1)$

46. For what value of $k$ does the quadratic equation $x^2 - 6x + k = 0$ have exactly one real solution?

    $k=0$

    $k=6$

    $k=9$

    $k=3$