Practice Questions
-
If \(x + y = e^{x + y}\), then the value of \(\dfrac{dy}{dx}\) at the point \(\left(\dfrac{1}{2}, \dfrac{1}{2}\right)\) is:
- A) 1
- B) \(e - 1\)
- C) \(1 - e\)
- D) \(e\)
- E) \(-1\)
-
Evaluate the integral:
\[
\int_{1}^{3} \frac{\sqrt{(x-2)^{2}}}{x-2} \, dx
\]
- A) 0
- B) 1
- C) 2
- D) \(-1\)
- E) \(-2\)
-
The acceleration \(a(t)\) of a body in motion along a straight line is given by \(a(t) = 2t - 4\), where \(t\) is time. If \(x(t)\) is the distance of the body from the origin at time \(t\) and \(x(6) - x(2) = 10\), then the velocity \(v(t)\) of the body is given by:
- A) \(t^2 - 4t\)
- B) \(t^2 - 4t + \dfrac{7}{6}\)
- C) \(t\)
- D) \(t^2\)
- E) \(2t^2 - 4t\)
-
Evaluate:
\[ \sum_{n=1}^{\infty} \frac{1}{n!} \]
- A) 1
- B) \(e\)
- C) \(e - 1\)
- D) \(e + 1\)
- E) 0
-
Which of these series is/are convergent?
\[ A) \sum_{n=1}^{\infty} \frac{1}{n^{2}} \qquad B) \sum_{n=1}^{\infty} \frac{1}{2^{n}} \qquad C) \sum_{n=1}^{\infty} \frac{1}{n} \]
- A) A only
- B) B only
- C) A, B and C
- D) A and B only
- E) B and C only
-
Evaluate:
\[ \sum_{n=1}^{\infty} \ln\left(\frac{n(n+2)}{(n+1)^{2}}\right) \]
- A) \(\ln\left(\dfrac{3}{4}\right)\)
- B) \(\ln(2)\)
- C) 1
- D) \(-\ln(2)\)
- E) \(\ln(2)\) (Note: duplicate option text preserved from source)
-
What is true about the graph of \(f(x) = \ln(x^2 + 2x + 1)\)?
- A) It has no x-intercept
- B) It has a horizontal asymptote
- C) It is exactly the same as the graph of \(g(x) = 2 \ln|x + 1|\)
- D) It is exactly the same as the graph of \(g(x) = 2 \ln(x + 1)\)
- E) It has no y-intercept
-
For what value of \(c\) do we have the following?
\[ \int_{0}^{c} \ln(x+1) \, dx = 1 \]
- A) \(e\)
- B) \(e - 1\)
- C) \(1 - e\)
- D) 1
- E) \(2e\)
-
Find constant \(K\), \(K > 0\), so that the volume of the solid of revolution determined by rotating the area bounded by \(f(x) = x^2\) and \(g(x) = -x(x - K)\) is equal to \(200\pi\).
- A) \(4 \cdot 5^{2/3}\)
- B) \(\dfrac{1}{2}\)
- C) 120
- D) \(1200^{1/5}\)
- E) \(2 \cdot 600^{1/5}\)
Answers to the Above Questions
1. E
2. A
3. B
4. C
5. D
6. D
7. C
8. B
9. E