Question 1:
\[
\text{Evaluate } \lim_{x \to 0} \frac{\sin(x) – x}{x^3}.
\]
\[
\text{(a) } -\frac{1}{6}, \quad \text{(b) } 0, \quad \text{(c) } \frac{1}{6}, \quad \text{(d) } \frac{1}{2}
\]
Answer: A
Question 2:
\[
\text{Evaluate } \int_0^\infty \frac{x^2}{e^x – 1} dx.
\]
\[
\text{(a) } \frac{\pi^2}{6}, \quad \text{(b) } \frac{\pi^2}{3}, \quad \text{(c) } \frac{\pi^2}{2}, \quad \text{(d) } \frac{\pi^2}{4}
\]
Answer: B
Question 3:
\[
\text{Find the sum of the series } \sum_{n=1}^{\infty} (-1)^n \frac{\ln n}{n}.
\]
\[
\text{(a) } -\frac{\pi}{4}, \quad \text{(b) } \frac{\pi}{4}, \quad \text{(c) } 0, \quad \text{(d) } \frac{\pi}{2}
\]
Answer: A
Question 4:
\[
\text{Evaluate } \int_0^\pi x \ln(\sin x) dx.
\]
\[
\text{(a) } -\frac{\pi}{2} \ln 2, \quad \text{(b) } -\frac{\pi}{4} \ln 2, \quad \text{(c) } \frac{\pi}{2} \ln 2, \quad \text{(d) } \frac{\pi}{4} \ln 2
\]
Answer: A
Question 5:
\[
\text{Evaluate } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}.
\]
\[
\text{(a) } -\frac{\pi^2}{12}, \quad \text{(b) } \frac{\pi^2}{12}, \quad \text{(c) } -\frac{\pi^2}{6}, \quad \text{(d) } \frac{\pi^2}{6}
\]
Answer: A
Question 6:
\[
\text{Find } \lim_{x \to 0} \frac{\ln(1 + x) – \sin x}{x^3}.
\]
\[
\text{(a) } -\frac{1}{2}, \quad \text{(b) } \frac{1}{6}, \quad \text{(c) } -\frac{1}{6}, \quad \text{(d) } 0
\]
Answer: C
Question 7:
\[
\text{Evaluate } \int_0^1 \frac{\ln(1+x)}{x} dx.
\]
\[
\text{(a) } \frac{\pi^2}{12}, \quad \text{(b) } \frac{\pi^2}{6}, \quad \text{(c) } -\frac{\pi^2}{12}, \quad \text{(d) } -\frac{\pi^2}{6}
\]
Answer: A
Question 8:
\[
\text{Find } \lim_{x \to 0} \frac{e^x – e^{-x} – 2x}{x^3}.
\]
\[
\text{(a) } \frac{1}{3}, \quad \text{(b) } \frac{1}{6}, \quad \text{(c) } -\frac{1}{6}, \quad \text{(d) } \frac{1}{2}
\]
Answer: B
Question 9:
\[
\text{Evaluate } \sum_{n=1}^{\infty} \frac{1}{n^4}.
\]
\[
\text{(a) } \frac{\pi^4}{90}, \quad \text{(b) } \frac{\pi^4}{120}, \quad \text{(c) } \frac{\pi^4}{60}, \quad \text{(d) } \frac{\pi^4}{45}
\]
Answer: A
Question 10:
\[
\text{Evaluate } \int_0^\infty \frac{x^{s-1}}{e^x – 1} dx, \quad \text{where } s > 1.
\]
\[
\text{(a) } \Gamma(s) \zeta(s), \quad \text{(b) } \frac{\Gamma(s)}{2} \zeta(s), \quad \text{(c) } \Gamma(s) \zeta(s-1), \quad \text{(d) } \frac{\Gamma(s)}{2} \zeta(s-1)
\]
Answer: A
Question 11:
\[
\text{Find the residue of } f(z) = \frac{e^z}{(z-1)^2} \text{ at } z = 1.
\]
\[
\text{(a) } e, \quad \text{(b) } 0, \quad \text{(c) } -e, \quad \text{(d) } 2e
\]
Answer: D
Question 12:
\[
\text{Find } \lim_{n \to \infty} n \left( \int_0^1 x^n e^x dx \right).
\]
\[
\text{(a) } 1, \quad \text{(b) } e-1, \quad \text{(c) } e, \quad \text{(d) } 0
\]
Answer: B
Question 13:
\[
\text{Evaluate } \int_0^\infty e^{-x^2} dx.
\]
\[
\text{(a) } \frac{\sqrt{\pi}}{2}, \quad \text{(b) } \sqrt{\pi}, \quad \text{(c) } \frac{\pi}{2}, \quad \text{(d) } \frac{\pi}{4}
\]
Answer: A
Question 14:
\[
\text{Find } \lim_{x \to 0} \frac{x – \sin x}{x^3}.
\]
\[
\text{(a) } -\frac{1}{6}, \quad \text{(b) } \frac{1}{6}, \quad \text{(c) } 0, \quad \text{(d) } \frac{1}{2}
\]
Answer: A
Question 15:
\[
\text{Evaluate } \int_0^\infty \frac{x^{3/2}}{e^x + 1} dx.
\]
\[
\text{(a) } \Gamma(5/2) \zeta(5/2), \quad \text{(b) } \frac{\Gamma(5/2)}{2} \zeta(5/2), \quad \text{(c) } \Gamma(5/2) \zeta(3/2), \quad \text{(d) } \frac{\Gamma(5/2)}{2} \zeta(3/2)
\]
Answer: B
