Multiple Choice

1. Simplify the expression using the properties of exponents: $(2x^3y^{-2})^3 \cdot (4x^{-1}y^5)^{-1}$.

   $4x^8y^{-1}$

   $8x^8y^{-11}$

   $2x^{10}y^{-11}$

2. Calculate the product of $2.5 \times 10^5$ and $3 \times 10^{-8}$. Express your answer in scientific notation.

   $7.5 \times 10^{-4}$

   $7.5 \times 10^{-3}$

   $75 \times 10^{-3}$

   $7.5 \times 10^{-40}$

3. Evaluate the expression using the order of operations (BIDMAS): $5 [12 \div (-3) + 7^2]$.

   $225$

   $205$

   $21$

   $-225$

4. Evaluate the following expression involving fractions: $\frac{1}{2} + \frac{3}{4} \times (\frac{5}{6} - \frac{1}{3})$.

   $\frac{3}{4}$

   $\frac{7}{8}$

   $\frac{1}{2}$

   $\frac{5}{6}$

5. Simplify the expression involving surds: $2\sqrt{50} - 3\sqrt{8} + 4\sqrt{2} - \sqrt{18} + 5\sqrt{200}$.

   $61\sqrt{2}$

   $50\sqrt{2}$

   $55\sqrt{2}$

6. Rationalize the denominator of the fraction: $\frac{4}{\sqrt{5} - 1}$.

   $\sqrt{5} + 1$

   $\frac{\sqrt{5} + 1}{2}$

   $4(\sqrt{5} + 1)$

   $2\sqrt{5}$

7. Solve for $x$ in the linear equation: $3(x - 2) - 2(4 - x) = 11$.

   $x = 5$

   $x = 7$

   $x = 3$

   $x = \frac{11}{5}$

8. Solve the system of linear equations for $y$: $2x + y = 10$ and $x - 3y = -9$.

   $y = 4$

   $y = 5$

   $y = 3$

   $y = 2$

9. A line passes through the points $(1, 5)$ and $(3, 11)$. What is the equation of the line?

   $y = 3x + 2$

   $y = 3x + 5$

   $y = 2x + 3$

   $y = 3x - 2$

10. A car traveled 180 km in 2 hours and 30 minutes. What was its average speed in km/h?

    $90$ km/h

    $75$ km/h

    $72$ km/h

    $60$ km/h

11. The ratio of apples to bananas in a basket is $5:3$. If there are 40 pieces of fruit in total, how many apples are there?

    $20$

    $15$

    $25$

    $30$

12. A shirt is originally priced at $\$50$. It is on sale for $20\%$ off. A customer then uses an additional coupon for $10\%$ off the sale price. What is the final price of the shirt?

    $\$36.00$

    $\$37.50$

    $\$40.00$

    $\$35.00$

13. Sarah has $\$150$. She spends $1/5$ of her money on books and then $\$30$ on lunch. How much money does Sarah have left?

    $\$80$

    $\$100$

    $\$75$

    $\$90$

14. In a survey of 50 people, 25 read Book A, 20 read Book B, and 15 read Book C. 8 read A and B, 7 read B and C, 6 read A and C. 3 people read all three books. How many people read NONE of these books?

    $5$

    $10$

    $8$

    $12$

15. A student must choose one flavor of ice cream (Vanilla, Chocolate, Strawberry - 3 options), one topping (Nuts, Sprinkles - 2 options), and one syrup (Caramel, Fudge, Honey - 3 options). How many different combinations of ice cream are possible?

    $9$

    $12$

    $8$

    $18$

16. A bag contains 4 red marbles, 5 blue marbles, and 3 green marbles. If one marble is drawn randomly, what is the probability that it is NOT blue?

    $\frac{5}{12}$

    $\frac{1}{2}$

    $\frac{7}{12}$

    $\frac{1}{3}$

17. If $3x - 5 < 10$, and $x$ is a positive integer, what is the largest possible value of $x$?

    $5$

    $4$

    $3$

    $6$

18. If the sum of two consecutive odd integers is 64, what is the smaller of the two integers?

    $33$

    $35$

    $30$

    $31$

19. Calculate the value of the expression: $-(-2)^3 + 4^2 - 5^0$.

    $15$

    $25$

    $23$

    $13$

20. The cost $C$ (in dollars) of renting a moving truck is given by the formula $C = 0.50k + 40$, where $k$ is the distance driven in kilometers. What does the number 40 represent in this context?

    The cost per kilometer driven.

    The fixed rental fee before any distance is driven.

    The total maximum cost of the rental.

    The maximum distance the truck can be driven.

21. A survey was conducted among 150 high school students regarding their participation in three sports: Soccer (S), Basketball (B), and Tennis (T). 70 students play S, 80 play B, and 65 play T. 35 play S and B, 30 play B and T, and 25 play S and T. If 5 students play none of the three sports, how many students play all three sports?

    10

    15

    20

    25

22. Calculate the value of the following expression: $\frac{6^2 - 4 \times [-10 - 2(3 - 7)]}{-3 + \sqrt[3]{-27}}$.

    $-\frac{14}{3}$

    $-3$

    $-6$

    $3$

23. Evaluate the expression: $(\frac{5}{6} - \frac{1}{3})^2 \div (\frac{1}{2} - \frac{3}{4}) + (\frac{-2}{5})$.

    $-7/5$

    $4/5$

    $1/5$

    $-11/5$

24. Solve for $x$: $3[5x - 2(x + 4)] + 9 = 4(2x - 1) - 1$.

    $x = -12$

    $x = 14$

    $x = 10$

    $x = 12$

25. A vendor sells two types of coffee, Gourmet blend (G) at $\$12$ per kg and House blend (H) at $\$8$ per kg. If she sold a total of 50 kg of coffee and made $\$520$ in total sales, how many kilograms of the Gourmet blend (G) did she sell?

    $30$ kg

    $15$ kg

    $25$ kg

26. Simplify the surd expression containing five terms: $2\sqrt{50} - \sqrt{18} + \sqrt{200} - 5\sqrt{2} + \sqrt{8}$.

    $14\sqrt{2}$

    $10\sqrt{2}$

    $12\sqrt{2}$

27. Rationalize the denominator and simplify the expression: $\frac{14}{\sqrt{7} - \sqrt{5}}$.

    $7\sqrt{7} + 7\sqrt{5}$

    $\frac{7(\sqrt{7} + \sqrt{5})}{6}$

    $14\sqrt{7} + 14\sqrt{5}$

    $7(\sqrt{7} + \sqrt{5})$

28. Simplify the expression using the properties of exponents: $\frac{(3a^4b^{-2})^3}{(9a^5b^{-1})^2}$.

    $\frac{1}{3}a^2b^{-4}$

    $3a^2b^{-4}$

    $\frac{1}{3}a^2b^{-2}$

    $3a^2b^{-8}$

29. Calculate the result of $\frac{(8.1 \times 10^{11})}{(2.7 \times 10^{-4})}$ and express the answer in scientific notation.

    $3.0 \times 10^{7}$

    $3.0 \times 10^{15}$

    $3.0 \times 10^{14}$

30. A straight line passes through the point $(9, -1)$ and is parallel to the line given by the equation $2x + 3y = 6$. What is the equation of the new line?

    $y = -\frac{3}{2}x + \frac{25}{2}$

    $y = \frac{3}{2}x - 14$

    $y = -\frac{2}{3}x + 5$

31. A motorist drives 180 km. He drives the first 100 km at an average speed of 50 km/h. If his average speed for the entire 180 km journey must be 60 km/h, what must his average speed be for the remaining 80 km?

    $100$ km/h

    $90$ km/h

    $80$ km/h

    $72$ km/h

32. In a mixture, water, cement, and sand are combined in the ratio $1:3:5$. If 4 liters of water is added to the mixture, the new ratio of water to cement becomes $1:2$. What was the original volume of sand in the mixture?

    30 liters

    15 liters

    20 liters

    40 liters

33. An artifact appreciated in value by 10% in the first year, and then depreciated by 20% of its new value in the second year. If the final value after two years was $\$26,400$, what was the original value of the artifact?

    $\$32,000$

    $\$30,000$

    $\$28,800$

34. A locker combination uses 5 unique digits (0-9). The first digit must be prime (2, 3, 5, or 7), and the last digit must be even. How many possible unique combinations are there?

    10,080

    6,720

    6,384

    8,400

35. Given that $X$ is a positive integer greater than 1, and $Y$ is a negative integer less than $-5$. Which of the following expressions MUST yield a positive result?

    $X + Y$

    $X^2 Y$

    $XY^2$

    $X - Y^2$

36. A sequence starts with the number 3. The next number is determined by either squaring the current number or subtracting 4. If a sequence consists of exactly three steps, what is the largest number that can be reached?

    729

    25

    49

    625

37. If $p = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$, calculate the value of $p + \frac{1}{p}$.

    $8$

    $2\sqrt{15}$

    $4\sqrt{15}$

38. Evaluate: $\left[\frac{1}{2} + 3(4 - 5)^2\right] \times [(-2)^{-3} \div 4^{-1}]$.

    $-2$

    $-\frac{7}{4}$

    $14$

    $-5/16$

39. If $3(2y - 5x) = 12$, what is the value of $2y - 5x$?

    $6$

    $3$

    $4$

    $2$

40. In a group of 80 people, 45 like Coffee (C), 40 like Tea (T), and 30 like Juice (J). 20 like C and T, 15 like T and J, and 10 like C and J. If $x$ people like all three beverages and $x$ people like none of the beverages, what is the value of $x$?

    8

    10

    5