Mathematics practice questions

Answer: C) \(5/s\)

• AIncorrect. That's a ramp of slope 5.
• BIncorrect. A step transforms to a constant over s.
• CCorrect. L{u(t)} = 1/s, scaled by 5.
• DIncorrect. That's \(e^{-5t}\).

Answer: B) 7.39

• AIncorrect. That's for \(\log_{10} x = 2\).
• BCorrect. \(x = e^2\).
• CIncorrect. That's ln 2.
• DIncorrect. Exponentiate with base e.

Answer: A) \(2 \pm \sqrt{3}\)

• ACorrect. \(x = [4 \pm \sqrt{16 - 4}]/2\).
• BIncorrect. Divide the whole numerator by 2.
• CIncorrect. Their product would be 3, not 1.
• DIncorrect. The discriminant is 16 − 4 = 12.

Answer: B) 10 years

• AIncorrect. Divide ln 2 by the rate.
• BCorrect. ln 2/0.0693.
• CIncorrect. Recheck: 0.693/0.0693.
• DIncorrect. Divide by 0.0693.

Answer: D) 4

• AIncorrect. That isn't the calculation.
• BIncorrect. \(\partial f/\partial y = x^2 = 1\).
• CIncorrect. f(1, 2) = 2.
• DCorrect. \(\partial f/\partial x = 2xy = 2(1)(2)\).

Answer: C) 1

• AIncorrect. Evaluate at x = 0, where \(e^0 = 1\).
• BIncorrect. Apply the product rule.
• CCorrect. Product rule: \(e^x + xe^x\), which is 1 at x = 0.
• DIncorrect. The second term \(xe^x\) is 0 at x = 0.

Answer: C) 1

The antiderivative of cos x is sin x, so the integral is sin(π/2) − sin(0) = 1.

• AIncorrect. π/4 is the integral of cos²x over 0 to π/2, not of cos x. The antiderivative of cos x is sin x, which gives 1.
• BIncorrect. This is the integrand's value at the upper limit, cos(π/2) = 0, not the area under the curve.
• CCorrect. [sin x] from 0 to π/2 = sin(π/2) − sin(0) = 1 − 0 = 1.
• DIncorrect. This is the width of the interval. It would equal the integral only if the integrand were a constant 1.

Answer: A) ln y versus x

• ACorrect. ln y = ln a + bx.
• BIncorrect. That's a curve.
• CIncorrect. That linearizes a logarithmic relation.
• DIncorrect. That linearizes a power law.

Answer: C) x = 3, y = 2

• AIncorrect. It gives x − y = 3.
• BIncorrect. That fails x − y = 1.
• CCorrect. Adding the equations gives 2x = 6.
• DIncorrect. That fails x − y = 1.

Answer: A) 5∠126.87°

The magnitude is √(a² + b²) = 5. Because the real part is negative and the imaginary part is positive, the angle is in the second quadrant: 180° − 53.13° = 126.87°.

• ACorrect. r = √((−3)² + 4²) = 5. The point (−3, 4) lies in the second quadrant, so θ = 180° − arctan(4/3) = 180° − 53.13° = 126.87°.
• BIncorrect. This is the calculator result arctan(4/−3) = −53.13°, rewritten as a positive angle (360° − 53.13°). That angle points into the fourth quadrant and represents 3 − 4i. The point (−3, 4) is in the second quadrant.
• CIncorrect. This ignores the negative real part. 5∠53.13° is 3 + 4i, not −3 + 4i.
• DIncorrect. The angle is right, but the magnitude adds the absolute values of the parts (3 + 4). The magnitude is the square root of the sum of their squares, which is 5.