Mathematics practice questions

Answer: D) \(14.1\angle 45^\circ\)

Convert to rectangular form and add: 10 + j10. Converting back gives about 14.1∠45°.

• AIncorrect. The magnitude is right, but the angle is arctan(10/10) = 45°. Equal real and imaginary parts cannot give 90°.
• BIncorrect. This adds the magnitudes 10 + 10 as if the phasors pointed the same way. Phasors 90° apart add as vectors, so the magnitude is \(\sqrt{10^2+10^2}\) ≈ 14.1.
• CIncorrect. This averages the angles and keeps a single magnitude. The resultant 10 + j10 has magnitude ≈ 14.1, not 10.
• DCorrect. In rectangular form the sum is 10 + j10. Its magnitude is \(\sqrt{200}\) ≈ 14.1 and its angle is arctan(10/10) = 45°.

Answer: C) \(3 - j\)

• AIncorrect. Expand all four terms.
• BIncorrect. \(-j^2 = +1\).
• CCorrect. \(2 - 2j + j - j^2 = 3 - j\).
• DIncorrect. Recheck the imaginary terms.

Answer: A) 1

• ACorrect. \([x^2]_0^1 = 1\).
• BIncorrect. That's the integrand at x = 1.
• CIncorrect. Recheck the antiderivative.
• DIncorrect. Recheck the antiderivative.

Answer: C) \((1 + t)e^{t}\)

• AIncorrect. The product rule adds a term.
• BIncorrect. Use the product rule.
• CCorrect. By the product rule.
• DIncorrect. That isn't the derivative.

Answer: B) 3.32

• AIncorrect. That isn't a logarithm.
• BCorrect. \(\ln 10/\ln 2\).
• CIncorrect. That's base 10.
• DIncorrect. That isn't a logarithm.

Answer: A) −2 and −3

Replacing the derivatives with powers of s gives s² + 5s + 6 = 0. This factors as (s + 2)(s + 3) = 0, so the roots are −2 and −3.

• ACorrect. The characteristic equation is s² + 5s + 6 = (s + 2)(s + 3) = 0, so s = −2 and s = −3. The roots are real and distinct, so the response is overdamped.
• BIncorrect. These roots add up to −5 but multiply to −6. They solve s² + 5s − 6 = 0, which means the sign of the constant term was flipped. The roots here must multiply to +6.
• CIncorrect. Their product is 6, but their sum is −7. They solve s² + 7s + 6 = 0, not s² + 5s + 6 = 0.
• DIncorrect. These are just the coefficients with minus signs added. The roots come from factoring the characteristic polynomial, and these values do not satisfy s² + 5s + 6 = 0.

Answer: C) 12.57

• AIncorrect. That's \(8\pi\).
• BIncorrect. That's \(2\pi\).
• CCorrect. \(\pi r^2 = 4\pi\).
• DIncorrect. That's for r = 1.

Answer: B) 1

• AIncorrect. The integrand decays fast enough.
• BCorrect. \([-e^{-t}]_0^\infty = 0 - (-1)\).
• CIncorrect. The area is positive.
• DIncorrect. That isn't the integral.

Answer: A) \(2\cos t - t\sin t\)

Apply the product rule twice: f' = sin t + t cos t, and then f'' = cos t + cos t − t sin t = 2cos t − t sin t.

• ACorrect. f'(t) = sin t + t cos t. Differentiating again: cos t + (cos t − t sin t) = 2cos t − t sin t.
• BIncorrect. This differentiates only the t cos t term of f'(t) and drops the cos t that comes from differentiating sin t. The two cos t terms add to 2cos t.
• CIncorrect. This is only the first derivative, from one use of the product rule. It has to be differentiated once more.
• DIncorrect. This treats t as a constant and differentiates only sin t twice. It leaves out the product-rule terms that involve the derivative of t.

Answer: C) \(4.24\angle -45^\circ\)

• AIncorrect. Magnitudes don't add.
• BIncorrect. Recheck the magnitude.
• CCorrect. Magnitude \(\sqrt{18}\); angle \(\tan^{-1}(-3/3)\).
• DIncorrect. The real part is positive (fourth quadrant).