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:
\[
r = \frac{1}{\cos(\theta – \theta_0)}
\]
\[
\text{Thus, the correct answer is } \boxed{r = \frac{1}{\cos(\theta – \theta_0)}}.
\]
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