Vector Functions MCQs in Calculus

Question 1:

\[ \text{Which of the following is a vector function?} \] \[ \text{(a) } f(t) = t^2 + 3t, \quad \text{(b) } \mathbf{r}(t) = (t, t^2, t^3), \quad \text{(c) } f(t) = \sin(t), \quad \text{(d) } g(t) = \cos(t) \] Answer: B

Question 2:

\[ \text{If } \mathbf{r}(t) = (t^2, 2t, 3t), \text{ what is } \mathbf{r}'(t)? \] \[ \text{(a) } (2t, 2, 3), \quad \text{(b) } (2t, 2t, 3), \quad \text{(c) } (2, 2, 3), \quad \text{(d) } (t, 2, 3) \] Answer: A

Question 3:

\[ \text{Which of the following is the equation of the tangent vector to the vector function } \mathbf{r}(t) = (t^3, \sin(t), e^t) \text{ at } t = 0? \] \[ \text{(a) } (0, 1, 1), \quad \text{(b) } (0, 0, 1), \quad \text{(c) } (1, 0, 1), \quad \text{(d) } (0, 0, 0) \] Answer: B

Question 4:

\[ \text{If } \mathbf{r}(t) = (t^2, 3t, e^t), \text{ what is the velocity vector } \mathbf{v}(t)? \] \[ \text{(a) } (2t, 3, e^t), \quad \text{(b) } (t, 3, e^t), \quad \text{(c) } (2t, 3t, e^t), \quad \text{(d) } (0, 3, 0) \] Answer: A

Question 5:

\[ \text{For the vector function } \mathbf{r}(t) = (t^2, \cos(t), t), \text{ the acceleration vector is } \mathbf{a}(t) = ? \] \[ \text{(a) } (2, -\sin(t), 0), \quad \text{(b) } (2t, -\sin(t), 1), \quad \text{(c) } (0, -\cos(t), 0), \quad \text{(d) } (2, \sin(t), 0) \] Answer: B

Question 6:

\[ \text{If } \mathbf{r}(t) = (t^2, 2t, t^3), \text{ what is the speed of the curve at } t = 1? \] \[ \text{(a) } \sqrt{14}, \quad \text{(b) } \sqrt{6}, \quad \text{(c) } 6, \quad \text{(d) } 5 \] Answer: A

Question 7:

\[ \text{What is the position vector at } t = 1 \text{ for the vector function } \mathbf{r}(t) = (e^t, \sin(t), t^2)? \] \[ \text{(a) } (e, \sin(1), 1), \quad \text{(b) } (e, \sin(1), 0), \quad \text{(c) } (1, \sin(1), 1), \quad \text{(d) } (1, \sin(1), 0) \] Answer: A

Question 8:

\[ \text{Which of the following is the arc length of the curve defined by the vector function } \mathbf{r}(t) = (t, t^2) \text{ from } t = 0 \text{ to } t = 1? \] \[ \text{(a) } \sqrt{2}, \quad \text{(b) } 2, \quad \text{(c) } \frac{3}{2}, \quad \text{(d) } 1 \] Answer: A

Question 9:

\[ \text{If the vector function } \mathbf{r}(t) = (e^t, \cos(t), t), \text{ what is the derivative of the position vector at } t = 0? \] \[ \text{(a) } (1, -1, 1), \quad \text{(b) } (1, 0, 1), \quad \text{(c) } (1, 1, 1), \quad \text{(d) } (0, -1, 0) \] Answer: B

Question 10:

\[ \text{The vector function } \mathbf{r}(t) = (t, \cos(t), e^t) \text{ describes a curve. What is the direction of motion at } t = 0? \] \[ \text{(a) } (0, 1, 1), \quad \text{(b) } (1, 0, 1), \quad \text{(c) } (0, 0, 1), \quad \text{(d) } (1, -1, 0) \] Answer: C

Question 11:

\[ \text{What is the derivative of the vector function } \mathbf{r}(t) = (3t^2, t^3, \sin(t)) \text{ with respect to } t? \] \[ \text{(a) } (6t, 3t^2, \cos(t)), \quad \text{(b) } (6t^2, 3t^2, \cos(t)), \quad \text{(c) } (6t, 3t^3, \cos(t)), \quad \text{(d) } (3t^2, 3t^3, \cos(t)) \] Answer: A

Question 12:

\[ \text{For the vector function } \mathbf{r}(t) = (2t, 3t^2, t), \text{ what is the velocity vector at } t = 2? \] \[ \text{(a) } (2, 12, 2), \quad \text{(b) } (4, 12, 2), \quad \text{(c) } (6, 6, 2), \quad \text{(d) } (6, 12, 2) \] Answer: B

Question 13:

\[ \text{If the vector function is } \mathbf{r}(t) = (t^2 + 1, \cos(t), e^{-t}), \text{ what is the acceleration vector at } t = 0? \] \[ \text{(a) } (2, -1, -1), \quad \text{(b) } (0, -1, 1), \quad \text{(c) } (2, 1, -1), \quad \text{(d) } (0, 1, -1) \] Answer: A

Question 14:

\[ \text{What is the equation of the velocity vector for the vector function } \mathbf{r}(t) = (e^t, t, \sin(t))? \] \[ \text{(a) } (e^t, 1, \cos(t)), \quad \text{(b) } (e^t, 0, \cos(t)), \quad \text{(c) } (e^t, t, \cos(t)), \quad \text{(d) } (e^t, 1, \sin(t)) \] Answer: A

Question 15:

\[ \text{What is the position vector at } t = \pi \text{ for the vector function } \mathbf{r}(t) = (t, \sin(t), \cos(t))? \] \[ \text{(a) } (\pi, 0, -1), \quad \text{(b) } (\pi, 0, 1), \quad \text{(c) } (\pi, 1, 0), \quad \text{(d) } (\pi, -1, 0) \] Answer: A

Vector Functions MCQs in Calculus
By: Prof. Dr. Fazal Rehman Shamil | Last updated: February 4, 2025