Math help please!!!?

Question

1. The number of solutions to the equation sin (2000x) = 3/7 in the interval [0, 2π ] is

A. 1000
B. 2000
C. 4000
D. 6000
E. 8000

2. If x=a is an asymptote of the secant function, then cot(a) = 0.
True or false? and why?

3. The graph of y =sec x never intersects the graph of y=
A. x
B. x^2
C. csc x
D. cos x
E. sin x

Answer 1

1. If x is in [0, 2π] then 2000x is in [0, 4000π]. In each interval of width 2π there are two solutions, so overall there are 2×2000 = 4000 solutions.

2. x=a is an asymptote of the secant function <=> cos a = 0 <=> cot a = cos a / sin a = 0 / ± 1 = 0. So it is true.

3. E is the correct answer. Note that |sec x| >= 1 where it is defined, |sin x| <= 1 where it is defined, and where |sec x| = 1, |cos x| = 1 and so |sin x| = 0. So they cannot intersect.

Answer 2

1) Two solutions only
2) since at x=a tan(a) is undefined, so cot(a) = 0, the statement is true
3) E. sin x
for if sec x = sin x then sinx cos x = 1
i.e. 1/2 sin 2x = 1
So sin 2x = 2 impossible