Number and Algebra practice questions.

USD 5000 is invested at 4% per annum, compounded quarterly. The value of the investment after 3 years is

• 5624.32
• 5634.13
• 5600.00
• 5665.29

Answer: 5634.13

FV = 5000(1 + 0.04/4)^(4×3) = 5000(1.01)¹² ≈ 5634.13. With compound interest, divide the rate and multiply the periods by the compounding frequency. 5624.32 is the annual-compounding trap.

An account pays a nominal interest rate of 5% per year while inflation is 2% per year. Using (1 + real) = (1 + nominal)/(1 + inflation), the real interest rate is closest to

• 2.94%
• 3.00%
• 7.10%
• 2.06%

Answer: 2.94%

1.05/1.02 = 1.02941…, so the real rate is about 2.94%. Simply subtracting (5% − 2% = 3%) is only an approximation.

Written in the form a × 10^k with 1 ≤ a < 10 and k an integer, the number 0.00042 is

• 4.2 × 10⁻³
• 42 × 10⁻⁵
• 4.2 × 10⁻⁴
• 0.42 × 10⁻³

Answer: 4.2 × 10⁻⁴

Move the decimal point 4 places right to get 4.2, so k = −4. Options like 42 × 10⁻⁵ have the right value but violate 1 ≤ a < 10, so they are not standard form.

USD 200 is deposited at the end of every month into an annuity paying 6% per annum, compounded monthly. After 10 years the account holds approximately

• 24000.00
• 32939.75
• 32775.87
• 30200.99

Answer: 32775.87

FV = 200 × [(1.005¹²⁰ − 1)/0.005] ≈ 32 775.87 (i = 0.06/12 = 0.005, n = 120). 24 000 is just the deposits with no interest; 32 939.75 assumes start-of-month payments.

Let A = [[2, 1], [0, 3]] and B = [[1, 4], [2, -1]] (rows shown). The product AB is

• [[4, 7], [6, -3]]
• [[2, 4], [0, -3]]
• [[2, 13], [4, -1]]
• [[4, 9], [2, -3]]

Answer: [[4, 7], [6, -3]]

Row × column: top-left 2·1 + 1·2 = 4, top-right 2·4 + 1·(−1) = 7, bottom-left 0·1 + 3·2 = 6, bottom-right 0·4 + 3·(−1) = −3. [[2,4],[0,−3]] is the element-wise trap and [[2,13],[4,−1]] is BA — order matters.

The inverse of the matrix A = [[3, 1], [5, 2]] (rows shown) is

• [[3, -1], [-5, 2]]
• [[-2, 1], [5, -3]]
• [[2, 1], [5, 3]]
• [[2, -1], [-5, 3]]

Answer: [[2, -1], [-5, 3]]

det A = 3×2 − 1×5 = 1, so A⁻¹ = (1/1)[[d, −b], [−c, a]] = [[2, −1], [−5, 3]]. Swap the main-diagonal entries, negate the others, divide by the determinant. Check: A·A⁻¹ = I.