有幾道 F.5 數學不懂，數學高手快入!!

1. Use a calculator to solve the equation on the interval 0~360. round the ans to 2 decimal places
cos θ= 0.67
a. 0.84,2.41
b. 0.84,3.98
c. 0.84,2.31
d. 0.84,5.45

θ = cos ^-1(0.67) = 0.8366
Only in the 1st quadrant and 4th quadrant, cos θ is positive
In the 4th quadrant, θ = 2pi - 0.8366 = 2(3.14159) – 0.8366 = 5.44658
Θ = 0.84 or 5.45 (angle in radians)

2. Solve the equation for solutions in the interval (0,360)

square root 2 cos 2θ =1
2θ = cos^-1(1/2^0.5) = 45 deg, θ = 22.5 deg
2θ = 360 deg – 45 deg = 315 deg = 157.5 deg
2θ = 360 deg + 45 deg = 405 deg = 202.5 deg
2θ = 360 deg + 315 deg = 675 deg = 337.5 deg

θ = 22.5 deg, 157.5 deg, 202.5 deg or 337.5 deg (angle in degree)

3. In a shipment of 20 Harry Potter books, 7 have a missing
cover, out of a sample of 4, what is the probability that at least 3 have the cover missing?
% defective = (7/20)(100) = 35%
n = 4 Bernoulli trial, p = 0.65, q = 0.35
In Bernoulli trial the probability of exactly x success or failure is given by
n
p(x) = ( ) p^x q^(n-x) where q = 1 - p
x

n n!
( ) = -----------------------
x x! (n - x)!

Let X be the number of defective samples
You have either 3 or 4 defective samples out of 4 samples

P(X>= 3) = p(3) + p(4)

4
p(3) = ( )(0.35)^3(0.65)^1= [4!/(3!)(4-3)!] (0.35)^3(0.65)^1 = 4(0.0428)(0.65) = 0.11147
3

4
p(4) = ( )(0.35)^4(0.65)^0= [4!/(4!)(4-4)!] (0.35)^4(0.65)^0 = 1(0.0428) = 0.015
4

P(X>= 3) = 0.11147+ 0.015 = 0.12647
Out of a sample of 4, the probability that at least 3 have the cover missing is 0.12647

4. Find an expression for the slope of the line tangent to the curve at x, and determine the slope of tangent at the point
y=x^2 + 11x - 15 at x=1
Differentiate y with respect to x [Calculus]
dy/dx = 2x^(2-1) + 11 = 2x + 11
The slope of the line tangent at x = 1 = 2x + 11 = 2(1) + 11= 13

a. 11x,11
b. 2x-11,-9
c. 2x+11,13
d.11x+15,26

Comment: I did probability very long time ago. I'm not so sure about the answer. Let someone do it.