1. Which of the following statements are true?
(i) r = 4 – 3 sin θ is the equation for a limaçon rotated 90°.
(ii) r = 3 cos 8θ is the equation for a rose curve with 8 petals.
(iii) rθ = 16 is the equation for a circle.
A. (i) and (iii)
B. (i) only
C. (ii) only
D. (i) and (ii)
2) How many petals do the two rose curves r = 9 cos 61θ and r = 8 sin 60θ have combined?
A. 182
B. 242
C. 181
D. 131
18. What is true of the curve r = ƒ(θ) if its shape is unaffected when r is replaced by –r and θ is replaced by p – θ?
A. It's symmetric with respect to the x-axis.
B. It's symmetric with respect to the origin.
C. It's symmetric with respect to the line y = x.
D. It's symmetric with respect to the y-axis.
1. To answer this question, we need to understand the equations for the given curves.
(i) r = 4 – 3 sin θ represents a limaçon curve. A limaçon is a type of polar curve shaped like a heart or a circular loop with a dent in it. If we rotate it 90°, the shape will change. Therefore, statement (i) is false.
(ii) r = 3 cos 8θ represents a rose curve. A rose curve is a type of polar curve that creates a figure with multiple petals. In this case, it has 8 petals. Therefore, statement (ii) is true.
(iii) rθ = 16 represents a circle in polar coordinates. The equation for a circle in polar form is usually r = constant. In this case, rθ = 16 doesn't represent a circle. Therefore, statement (iii) is false.
Based on our analysis, the correct answer is D. (i) and (ii) are the true statements.
2. To find the number of petals in the combined rose curves r = 9 cos 61θ and r = 8 sin 60θ, we need to look at the coefficient of θ in each equation.
For r = 9 cos 61θ, the coefficient of θ is 61. This means there are 61 petals in this curve.
For r = 8 sin 60θ, the coefficient of θ is 60. This means there are 60 petals in this curve.
To find the total number of petals, we add the number of petals in both curves:
61 + 60 = 121.
Therefore, the correct answer is not provided in the options.
For the third question, let's break down the given transformations:
Replacing r with -r reflects the curve with respect to the origin.
Replacing θ with π - θ reflects the curve with respect to the y-axis.
If the shape of the curve is unaffected by both transformations, it means that the curve is symmetric with respect to the y-axis. Therefore, the correct answer is option D. It's symmetric with respect to the y-axis.