Polar Coordinates

Step 1: Understanding the Concept:
This question tests the rules for symmetry of curves given in polar coordinates .
Symmetry about the initial line ( , the x-axis): Occurs if replacing with leaves the equation unchanged.
Symmetry about the line (the y-axis): Occurs if replacing with leaves the equation unchanged.
Symmetry about the pole (origin): Occurs if replacing with OR replacing with leaves the equation unchanged.
Step 2: Detailed Explanation:

(A) r = a(1 + cos ): Replace with . Since , the equation becomes , which is unchanged. So, it is symmetrical about the initial line. Statement (A) is true.
(B) r = 2(1 - 2 sin ): Replace with . Since , the equation becomes , which is different. So, it is not symmetrical about the initial line. Statement (B) is false.
(C) r = a(1 + sin ): Replace with . Since , the equation becomes , which is unchanged. So, it is symmetrical about the line . Statement (C) is true.
(D) r = a sin(3 ): Replace with . Since , the equation becomes , which is different. So, it is not symmetrical about the initial line. Statement (D) is false. (It is symmetric about ).
(E) r² = a²cos(2 ): To test for symmetry about the pole, replace with . The equation becomes , which is , unchanged. So, it is symmetrical about the pole. Statement (E) is true.
Step 3: Final Answer:
The true statements are (A), (C), and (E). This corresponds to option (D).