Mathematics practice questions

Answer: B) 50.3

• AIncorrect. That's the volume at r = 2.
• BCorrect. \(dV/dr = 4\pi r^2 = 16\pi\).
• CIncorrect. Recheck the derivative.
• DIncorrect. Recheck the derivative.

Answer: B) 10

Use d = √((x₂ − x₁)² + (y₂ − y₁)²). With Δx = 6 and Δy = −8, the distance is √100 = 10.

• AIncorrect. This is |Δx| + |Δy| = 6 + 8, the 'taxicab' distance. The straight-line distance is the square root of the sum of the squares.
• BCorrect. Δx = 4 − (−2) = 6 and Δy = −5 − 3 = −8, so d = √(6² + 8²) = √100 = 10.
• CIncorrect. This is d², the sum of the squared differences. The square root was never taken.
• DIncorrect. This adds the coordinates instead of subtracting them: (−2 + 4) = 2 and (3 + (−5)) = −2, which gives √8 ≈ 2.83.

Answer: A) 2

• ACorrect. \(\mathbf{A}\times\mathbf{B} = (0, 0, 2)\).
• BIncorrect. That's the dot product of these perpendicular vectors.
• CIncorrect. Magnitudes multiply for perpendicular vectors.
• DIncorrect. Recheck the cross product.

Answer: B) 18.85 m²

The area of an ellipse is A = πab, which reduces to πr² for a circle. Here A = π(3)(2) ≈ 18.85 m².

• AIncorrect. This is a·b = 3 × 2 with the factor π left out. The ellipse area is πab.
• BCorrect. A = πab = π(3)(2) = 6π ≈ 18.85 m².
• CIncorrect. This is 2πab, which doubles the true area. The area of an ellipse is πab.
• DIncorrect. This is π(a + b) = 5π, which adds the semi-axes instead of multiplying them. It is not an area formula and has units of length.

Answer: C) 4

• AIncorrect. Include the ½ factor.
• BIncorrect. That's the exact integral.
• CCorrect. \((2/2)(0 + 4)\); the exact value is 2.67.
• DIncorrect. Recheck the rule.

Answer: A) 15 m²

A trapezoid's area is the average of the parallel sides times the height: A = (4 + 6)/2 × 3 = 15 m².

• ACorrect. A = (a + b)h/2 = (4 + 6)(3)/2 = 15 m².
• BIncorrect. This is (a + b)h = 30 with the factor 1/2 left out, which counts the area twice.
• CIncorrect. This multiplies only the shorter side by the height (4 × 3), as if the shape were a rectangle 4 m wide.
• DIncorrect. This multiplies only the longer side by the height (6 × 3), as if the shape were a rectangle 6 m wide.

Answer: B) 2 and 3

• AIncorrect. Those are the trace and determinant.
• BCorrect. A diagonal matrix's eigenvalues are its diagonal entries.
• CIncorrect. Recheck the characteristic equation.
• DIncorrect. Recheck the characteristic equation.

Answer: C) \(1 - x^2/2\)

• AIncorrect. That's the start of \(e^x\).
• BIncorrect. That's the series for \(\sin x\).
• CCorrect. The series alternates with even powers.
• DIncorrect. The second term is \(x^2/2\).

Answer: B) 2

• AIncorrect. The geometric series converges since |r| < 1.
• BCorrect. \(a/(1 - r) = 1/(1 - 1/2)\).
• CIncorrect. Recheck the formula.
• DIncorrect. That's the first term.

Answer: B) 1

With u = 2x, ∫sin(2x) dx = −cos(2x)/2. Evaluating from 0 to π/2 gives 1/2 − (−1/2) = 1.

• AIncorrect. This comes from taking cos π = +1, so the two limits cancel. Since cos π = −1, the result is positive. sin(2x) ≥ 0 on [0, π/2], so the integral must be positive.
• BCorrect. The antiderivative is −cos(2x)/2. So [−cos(2x)/2] from 0 to π/2 = −cos(π)/2 + cos(0)/2 = 1/2 + 1/2 = 1.
• CIncorrect. This uses −cos(2x) as the antiderivative and drops the 1/2 factor from the chain rule: −cos π + cos 0 = 2.
• DIncorrect. This evaluates the antiderivative only at the upper limit (−cos π / 2 = 1/2) and skips subtracting its value at the lower limit, which is −1/2.