Parametric Equations and Polar Coordinates MCQs in Calculus

By: Prof. Dr. Fazal Rehman Shamil | Last updated: February 12, 2025

Question 1:

\[
\text{What is the parametric equation of a circle with radius } r \text{ centered at the origin?}
\]
\[
\text{(a) } x = r \cos(t), \quad y = \cos(t), \quad \text{(b) } x = t, \quad y = r \sin(t), \quad \text{(c) } x = t, \quad y = t, \quad \text{(d) } x = r \cos(t), \quad y = r \sin(t)
\]
Answer: D

Step by Step Solution

Solution:

The parametric equation of a circle with radius \( r \) centered at the origin is given by:
\[
x = r \cos(t), \quad y = r \sin(t)
\]
\[
\text{Thus, the correct answer is } \boxed{x = r \cos(t), \quad y = r \sin(t)}.
\]

Question 2:

\[
\text{Which of the following is the correct parametric equation for a line with slope } m \text{ passing through the point } (x_0, y_0)?
\]
\[
\text{(a) } x = t, \quad y = mt + y_0, \quad \text{(b) } x = x_0 + t, \quad y = y_0 + mt, \quad \text{(c) } x = x_0 + mt, \quad y = y_0 + t, \quad \text{(d) } x = t, \quad y = m + t
\]
Answer: B

Step by Step Solution

Solution:

The parametric equation of a line with slope \( m \) passing through \( (x_0, y_0) \) is:
\[
x = x_0 + t, \quad y = y_0 + mt
\]
\[
\text{Thus, the correct answer is } \boxed{x = x_0 + t, \quad y = y_0 + mt}.
\]

Question 3:

\[
\text{What is the polar coordinate representation of the point } (x, y) = (3, 4)?
\]
\[
\text{(a) } r = 7, \quad \theta = \tan^{-1}\left(\frac{3}{4}\right), \quad \text{(b) } r = 4, \quad \theta = \frac{\pi}{4}, \quad \text{(c) } r = 5, \quad \theta = \tan^{-1}\left(\frac{4}{3}\right), \quad \text{(d) } r = 5, \quad \theta = \frac{\pi}{3}
\]
Answer: C

Step by Step Solution

Solution:

The polar representation of a Cartesian point \((x, y)\) is:
\[
r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1} \left(\frac{y}{x}\right)
\]
Substituting \( x = 3 \) and \( y = 4 \):
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
\]
\[
\theta = \tan^{-1} \left(\frac{4}{3}\right)
\]
\[
\text{Thus, the correct answer is } \boxed{r = 5, \quad \theta = \tan^{-1} \left(\frac{4}{3}\right)}.
\]

Question 4:

\[
\text{Which of the following is the equation for a circle in polar coordinates with radius } r = 2 \text{ and centered at the origin?}
\]
\[
\text{(a) } r = 4, \quad \text{(b) } r = 2 \cos(\theta), \quad \text{(c) } r = 2 \sin(\theta), \quad \text{(d) } r = 2
\]
Answer: D

Step by Step Solution

Solution:

A circle centered at the origin in polar coordinates is given by:
\[
r = \text{constant}
\]
For \( r = 2 \), the equation remains:
\[
r = 2
\]
\[
\text{Thus, the correct answer is } \boxed{r = 2}.
\]

Question 5:

\[
\text{The parametric equations } x = 3t, \quad y = 4t \text{ represent which type of curve?}
\]
\[
\text{(a) } \text{Circle}, \quad \text{(b) } \text{Parabola}, \quad \text{(c) } \text{Straight line}, \quad \text{(d) } \text{Elliptical curve}
\]
Answer: C

Step by Step Solution

Solution:

Rewriting the parametric equations:
\[
x = 3t, \quad y = 4t
\]
Solving for \( t \) in terms of \( x \):
\[
t = \frac{x}{3}
\]
Substituting into \( y \):
\[
y = 4 \left(\frac{x}{3}\right) = \frac{4}{3} x
\]
This is a linear equation of the form \( y = mx \), confirming the curve is a straight line.
\[
\text{Thus, the correct answer is } \boxed{\text{Straight line}}.
\]

Question 6:

\[
\text{Which of the following represents the polar form of the equation of a line passing through the origin with slope } m?
\]
\[
\text{(a) } r = \frac{\theta}{m}, \quad \text{(b) } r = \frac{1}{\cos(\theta – m)}, \quad \text{(c) } r = \frac{1}{\sin(\theta)}, \quad \text{(d) } r = \frac{\theta}{\cos(\theta – m)}
\]
Answer: D

Step by Step Solution

Solution:

A straight line passing through the origin in polar form is given by:
\[
r = \frac{\theta}{\cos(\theta – m)}
\]
\[
\text{Thus, the correct answer is } \boxed{r = \frac{\theta}{\cos(\theta – m)}}.
\]

Question 7:

\[
\text{Which of the following represents the parametric equations of a spiral?}
\]
\[
\text{(a) } x = t \cos(t), \quad y = t \sin(t), \quad \text{(b) } x = \cos(t), \quad y = t, \quad \text{(c) } x = \cos(t), \quad y = t \cos(t), \quad \text{(d) } x = t, \quad y = t^2
\]
Answer: A

Step by Step Solution

Solution:

A spiral is defined by increasing distance from the origin with increasing angle. The standard parametric form of a spiral is:
\[
x = t \cos(t), \quad y = t \sin(t)
\]
\[
\text{Thus, the correct answer is } \boxed{x = t \cos(t), \quad y = t \sin(t)}.
\]

Question 8:

\[
\text{Which of the following is the correct conversion formula from polar coordinates to Cartesian coordinates?}
\]
\[
\text{(a) } x = r \cos(\theta), \quad y = r \sin(\theta), \quad \text{(b) } x = r \sin(\theta), \quad y = r \cos(\theta), \quad \text{(c) } x = r \cos(\theta), \quad y = \tan(\theta), \quad \text{(d) } x = r, \quad y = \theta
\]
Answer: A

Step by Step Solution

Solution:

The conversion from polar to Cartesian coordinates is given by:
\[
x = r \cos(\theta), \quad y = r \sin(\theta)
\]
\[
\text{Thus, the correct answer is } \boxed{x = r \cos(\theta), \quad y = r \sin(\theta)}.
\]

Question 9:

\[
\text{Which of the following is the correct parametric form of the equation of an ellipse with semi-major axis } a \text{ and semi-minor axis } b?
\]
\[
\text{(a) } x = a \cos(t), \quad y = b \sin(t), \quad \text{(b) } x = a \sin(t), \quad y = b \cos(t), \quad \text{(c) } x = b \cos(t), \quad y = a \sin(t), \quad \text{(d) } x = a \sin(t), \quad y = a \cos(t)
\]
Answer: A

Step by Step Solution

Solution:

The standard parametric equations of an ellipse centered at the origin with semi-major axis \( a \) and semi-minor axis \( b \) are:
\[
x = a \cos(t), \quad y = b \sin(t)
\]
\[
\text{Thus, the correct answer is } \boxed{x = a \cos(t), \quad y = b \sin(t)}.
\]

Question 10:

\[
\text{The equation } r = 2 \cos(\theta) \text{ represents which type of curve in polar coordinates?}
\]
\[
\text{(a) } \text{Circle}, \quad \text{(b) } \text{Spiral}, \quad \text{(c) } \text{Straight line}, \quad \text{(d) } \text{Cardioid}
\]
Answer: A

Step by Step Solution

Solution:

The general equation of a circle in polar form is:
\[
r = a \cos(\theta) \quad \text{or} \quad r = a \sin(\theta)
\]
Here, \( r = 2 \cos(\theta) \) represents a circle centered at \( (a/2, 0) = (1,0) \) with radius 1.
\[
\text{Thus, the correct answer is } \boxed{\text{Circle}}.
\]

Question 11:

\[
\text{Which of the following is the correct polar form of the equation of a cardioid?}
\]
\[
\text{(a) } r = 1 – \cos(\theta), \quad \text{(b) } r = 1 + \sin(\theta), \quad \text{(c) } r = 1 + \cos(\theta), \quad \text{(d) } r = \cos(\theta)
\]
Answer: A

Step by Step Solution

Solution:

A cardioid has the general polar equation:
\[
r = a(1 \pm \cos(\theta)) \quad \text{or} \quad r = a(1 \pm \sin(\theta))
\]
For \( a = 1 \), we get:
\[
r = 1 – \cos(\theta)
\]
\[
\text{Thus, the correct answer is } \boxed{r = 1 – \cos(\theta)}.
\]

Question 12:

\[
\text{Which of the following parametric equations represents a parametric form of a hyperbola?}
\]
\[
\text{(a) } x = t, \quad y = \frac{1}{t}, \quad \text{(b) } x = t^2, \quad y = t, \quad \text{(c) } x = \cos(t), \quad y = \sin(t), \quad \text{(d) } x = \frac{1}{t}, \quad y = t
\]
Answer: A

Step by Step Solution

Solution:

A hyperbola has the form:
\[
xy = C \quad \text{or} \quad \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1
\]
The parametric equation \( x = t, y = \frac{1}{t} \) satisfies \( xy = 1 \), which is a hyperbola.
\[
\text{Thus, the correct answer is } \boxed{x = t, \quad y = \frac{1}{t}}.
\]

Question 13:

\[
\text{What is the conversion from Cartesian coordinates to polar coordinates?}
\]
\[
\text{
(a) } r = x^2 + y^2, \quad \theta = \tan^{-1}\left(\frac{x}{y}\right), \quad \text{
(b) } r = \sqrt{x + y}, \quad \theta = \tan^{-1}\left(\frac{x}{y}\right), \quad \text{
(c) } r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right), \quad \text{
(d) } r = x + y, \quad \theta = \sin^{-1}\left(\frac{y}{x}\right)
\]
Answer: c

Step by Step Solution

Solution:

The conversion from Cartesian to polar coordinates is:
\[
r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
\[
\text{Thus, the correct answer is } \boxed{r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)}.
\]

Question 14:

\[
\text{Which of the following parametric equations represents the equation of a cycloid?}
\]
\[
\text{(a) } x = t – \sin(t), \quad y = 1 – \cos(t), \quad \text{(b) } x = 1 – \cos(t), \quad y = t – \sin(t), \quad \text{(c) } x = t, \quad y = \sin(t), \quad \text{(d) } x = t, \quad y = \cos(t)
\]
Answer: A

Step by Step Solution

Solution:

A cycloid is the curve traced by a point on the circumference of a rolling circle of radius \( R \). The parametric equations for a cycloid are:
\[
x = R(t – \sin t), \quad y = R(1 – \cos t)
\]
For \( R = 1 \), we get:
\[
x = t – \sin t, \quad y = 1 – \cos t
\]
\[
\text{Thus, the correct answer is } \boxed{x = t – \sin t, \quad y = 1 – \cos t}.
\]

Question 15:

\[
\text{Which of the following represents the polar equation of a straight line passing through the origin at an angle } \theta_0?
\]
\[
\text{(a) } r = \frac{1}{\cos(\theta – \theta_0)}, \quad \text{(b) } r = \cos(\theta – \theta_0), \quad \text{(c) } r = \frac{1}{\sin(\theta – \theta_0)}, \quad \text{(d) } r = \sin(\theta – \theta_0)
\]
Answer: A

Step by Step Solution

Solution:

A straight line passing through the origin in polar form is given by:
\[
r = \frac{1}{\cos(\theta – \theta_0)}
\]
\[
\text{Thus, the correct answer is } \boxed{r = \frac{1}{\cos(\theta – \theta_0)}}.
\]

More MCQs on Calculus

  1. Functions and Models MCQs in Calculus
  2. Limits and Derivatives MCQs in Calculus
  3. Differentiation Rules MCQs in Calculus
  4. Applications of Differentiation MCQs in Calculus
  5. Integrals MCQs in Calculus
  6. Applications of Integration MCQs in Calculus
  7. Techniques of Integration MCQs in Calculus
  8. Differential Equations MCQs in Calculus
  9. Parametric Equations and Polar Coordinates MCQs in Calculus
  10. Infinite Sequences and Series MCQs in Calculus
  11. Vectors and the Geometry of Space MCQs in Calculus
  12. Vector Functions MCQs in Calculus
  13. Partial Derivatives MCQs in Calculus
  14. Multiple Integrals MCQs in Calculus
  15. Vector Calculus MCQs in Calculus
  16. Second-Order Differential Equations MCQs in Calculus

Leave a Reply