15. Ans: (A)

The general forms of finding all the polar coordinates are:

1) When r >= 0(meaning positive):(r, θ + 2nπ) where, n = integer

2) When r < 0(meaning negative): (-r, θ + (2n+1)π) where, n = integer

Since r = +1, -1(ordered pair)

θ(given) =

When r = +1(r>0):(1,

+ 2nπ)

When r = -1(r<0):(-1,

+ (2n+1)π)

Therefore, the correct option is (A) (1, pi divided by 3 + 2nπ) or (-1, pi divided by 3 + (2n + 1)π)

16. Ans: (A)

In polar coordinates,

Since x = 3, y=-3; therefore,

To find the angle,

tanθ = y/x = -3/3 = -1

=> θ = -45°

=> θ = -45°+360° = 315° (when

)

If r = -r =

, then,

θ = -45° + 180° = 135°

Therefore, the correct option is (A) (3 square root of 2 , 315°), (-3 square root of 2 , 135°)

17. Ans: (A)

(Question-17 missing Image is attached below) The general form of the limacon curve is:

r = b + a cosθ

If b < a, the curve would have inner loop.

As you can see in the image attached(labeled Question-17), the limacon curve graph has the inner loop. Therefore, the correct option is (A) r = 2 + 3 cosθ, since b = 2, and a = 3; and the condition b < a (2 < 3) is met.

18. Ans: (B)

Let's find out!

1. If we replace θ with -θ, we would get:

r = 4 - 4*cos(-θ )

Since, cos(-θ) = +cosθ, therefore,

r = 4 - 4*cos(θ)

Same as the original, therefore, graph is symmetric to x-axis.

2. If we replace r with -r, we would get:

-r = 4 - 4*cos(θ )

r = -4 + 4*cos(θ)

NOT same as original, therefore, graph is NOT symmetric to its origin.

3. If we replace θ with -θ and r with -r, we would get:

-r = 4 - 4*cos(-θ )

Since, cos(-θ) = +cosθ, therefore,

r = -4 + 4*cos(θ)

NOT same as original, therefore, graph is NOT symmetric to y-axis.

Ans: The graph is symmetric to: x-axis only!

19. Ans:

Explanation:

As the question suggests that it is a horizontal ellipse, therefore, the equation for the horizontal ellipse is:

-- (A)

Since,

x = 21ft,

y = 29ft,

b = 58ft,

= ?

Plug-in the values in equation (A),

(A)=>

=>

= 588

Therefore, the equation becomes,Ans:

20. Ans: x-axis only

Let's find out!

1. If we replace θ with -θ, we would get:

r = 4*cos(-5θ )

Since, cos(-θ) = +cosθ, therefore,

r = +4*cos(5θ) = Same as original

Therefore, graph is symmetric to x-axis.

2. If we replace r with -r, we would get:

-r = 4*cos(5θ )

r = -4*cos(5θ) = Not same

NOT same as original, therefore, graph is NOT symmetric to its origin.

3. If we replace θ with -θ and r with -r, we would get:

-r = 4*cos(-5θ )

Since, cos(-θ) = +cosθ, therefore,

r = -4*cos(5θ) = Not Same

NOT same as original, therefore, graph is NOT symmetric to y-axis.

Ans: The graph is symmetric to: x-axis only!