Core Academic Skills Assessment Mathematics: Number and Quantity Questions and Answers — Questions and Answers
Question 1: A biologist is studying a bacterial culture. At 9:00 AM, the culture contains 4.5 x 10^5 bacteria. The bacteria triple in number every hour. How many bacteria will be in the culture at 12:00 PM, expressed in scientific notation?
- 4.05 x 10^6
- 1.35 x 10^6
- 1.215 x 10^7 (Correct answer)
- 4.5 x 10^8
Correct answer: 1.215 x 10^7
The time elapsed from 9:00 AM to 12:00 PM is 3 hours. Since the bacteria triple every hour, the initial number must be multiplied by 3 for each hour. So, the calculation is (4.5 x 10^5) * 3 * 3 * 3 = (4.5 x 10^5) * 27. 4.5 * 27 = 121.5. So, the number of bacteria is 121.5 x 10^5. To express this in proper scientific notation, the decimal must be moved two places to the left, and the exponent increased by 2, resulting in 1.215 x 10^7.
Question 2: Which of the following numbers is irrational?
- √49
- 0.333...
- π / 2 (Correct answer)
- 5.78
Correct answer: π / 2
An irrational number cannot be expressed as a simple fraction (a/b where a and b are integers). √49 = 7, which is a rational number. 0.333... is a repeating decimal, which can be expressed as the fraction 1/3. 5.78 is a terminating decimal, which can be expressed as 578/100. Pi (π) is a well-known irrational number; dividing it by a rational number (2) results in another irrational number.
Question 3: The mass of the Earth is approximately 5.97 x 10^24 kg, and the mass of Mars is approximately 6.42 x 10^23 kg. What is the difference in the order of magnitude between the mass of Earth and the mass of Mars?
Correct answer: 0
The order of magnitude is determined by the power of 10 when the number is in scientific notation. To compare them directly, we can write the mass of Mars as 0.642 x 10^24 kg. Both numbers are now expressed with the same power of 10 (10^24). Since the multipliers (5.97 and 0.642) are within the same power of ten, the difference in the order of magnitude is 0. Alternatively, a number's order of magnitude is found by rounding its base-10 logarithm to the nearest integer. log(5.97 x 10^24) ≈ 24.77, which rounds to 25. log(6.42 x 10^23) ≈ 23.81, which rounds to 24. The question asks for the difference in order of magnitude of the *numbers themselves*. Since both numbers are best approximated by a power of 10 (10^25 and 10^24 respectively), the difference is 1. However, common interpretation in this context is to compare the exponents after normalization. Normalizing Mars' mass to Earth's exponent gives 0.642 x 10^24 kg. Since 5.97 / 0.642 is less than 10, they are considered to be within the same order of magnitude, making the difference 0.
Question 4: A student is working on a project that requires mixing a chemical solution. The instructions state to add 3/8 of a liter of substance A and 2/5 of a liter of substance B. If the student wants to pour the combined mixture into a beaker that holds 0.75 liters, will the mixture fit? And by how much will it be over or under?
- No, it will be over by 0.025 liters. (Correct answer)
- Yes, it will be under by 0.025 liters.
- No, it will be over by 0.25 liters.
- Yes, it will fit exactly.
Correct answer: No, it will be over by 0.025 liters.
First, find a common denominator for the fractions to add them. The least common multiple of 8 and 5 is 40. Convert the fractions: 3/8 = 15/40 and 2/5 = 16/40. Add them together: 15/40 + 16/40 = 31/40. Now, convert the fraction to a decimal to compare it with the beaker's capacity: 31 ÷ 40 = 0.775 liters. The beaker's capacity is 0.75 liters. Since 0.775 > 0.75, the mixture will not fit. The difference is 0.775 - 0.75 = 0.025 liters. Therefore, the mixture will be over by 0.025 liters.
Question 5: Given the complex numbers z1 = 4 - 3i and z2 = -1 + 5i, what is the result of z1 - z2?
- 3 + 2i
- 5 - 8i (Correct answer)
- 3 - 8i
- 5 + 2i
Correct answer: 5 - 8i
To subtract complex numbers, subtract the real parts and subtract the imaginary parts separately. (4 - 3i) - (-1 + 5i) = (4 - (-1)) + (-3 - 5)i. This simplifies to (4 + 1) + (-8)i, which equals 5 - 8i.
Question 6: Let vector u = <2, -7> and vector v = <-5, 3>. What is the resultant vector of u + v?
- <-3, -4> (Correct answer)
- <7, -10>
- <-7, 10>
- <3, 4>
Correct answer: <-3, -4>
To add vectors, add their corresponding components. The x-component of the resultant vector is the sum of the x-components of u and v (2 + (-5) = -3). The y-component of the resultant vector is the sum of the y-components of u and v (-7 + 3 = -4). Therefore, the resultant vector is <-3, -4>.
