21. suppose points a and b lie on the outside of a circle, separated by an angle of 45°. suppose the arc length between those two points is π meters long. which of the following statements is true?
a) the circumference of the circle is 4π meters.
b) the diameter of the circle is 8 meters.
c) the radius of the circle is 16 meters.
d) the area of the circle is 64π square meters.

22. consider the following system of equations. what value of x is a solution to the system?
x + y + z = 10
x − y − z = 2
x − 2y = 4
a) -10
b) 0
c) 6
d) 10

23. for which of these integers does the total number of its unique factors equal the sum of its digits?
a) 5
b) 12
c) 13
d) 22

24. which of the following is equivalent to the expression cos(x)csc(x)cos(2x) + sin(2x)?
a) cos(x)
b) sin²(x)
c) cot(x)
d) sin(x)cos(x)

25. consider the parabolas described by the equations y = x² + x − 2, and y = 2x² + 2x − 4. for what values of x do these two parabolas intersect?
a) -2 and 1
b) -2 and -4
c) 1 and 4
d) these parabolas do not intersect.

Question

21. suppose points a and b lie on the outside of a circle, separated by an angle of 45°. suppose the arc length between those two points is π meters long. which of the following statements is true?
a) the circumference of the circle is 4π meters.
b) the diameter of the circle is 8 meters.
c) the radius of the circle is 16 meters.
d) the area of the circle is 64π square meters.

22. consider the following system of equations. what value of x is a solution to the system?
x + y + z = 10
x − y − z = 2
x − 2y = 4
a) -10
b) 0
c) 6
d) 10

23. for which of these integers does the total number of its unique factors equal the sum of its digits?
a) 5
b) 12
c) 13
d) 22

24. which of the following is equivalent to the expression cos(x)csc(x)cos(2x) + sin(2x)?
a) cos(x)
b) sin²(x)
c) cot(x)
d) sin(x)cos(x)

25. consider the parabolas described by the equations y = x² + x − 2, and y = 2x² + 2x − 4. for what values of x do these two parabolas intersect?
a) -2 and 1
b) -2 and -4
c) 1 and 4
d) these parabolas do not intersect.

Accepted Answer

### Question 21
# Explanation:
## Step1: Recall arc length formula
The arc length $ s $ of a circle with radius $ r $ and central angle $ \theta $ (in degrees) is given by $ s=\frac{\theta}{360^\circ}\times2\pi r $. Here, $ \theta = 45^\circ $ and $ s=\pi $.
## Step2: Substitute values into formula
Substitute $ s = \pi $ and $ \theta=45^\circ $ into the formula: $ \pi=\frac{45^\circ}{360^\circ}\times2\pi r $.
Simplify $ \frac{45}{360}=\frac{1}{8} $, so the equation becomes $ \pi=\frac{1}{8}\times2\pi r $.
Simplify $ \frac{1}{8}\times2\pi r=\frac{\pi r}{4} $, so $ \pi=\frac{\pi r}{4} $.
Divide both sides by $ \pi $ (assuming $ \pi
eq0 $): $ 1 = \frac{r}{4} $, so $ r = 4 $ meters.
## Step3: Check each option
- Option A: Circumference $ C = 2\pi r=2\pi\times4 = 8\pi $, not $ 4\pi $. So A is false.
- Option B: Diameter $ d = 2r=2\times4 = 8 $ meters. So B is true.
- Option C: Radius is 4, not 16. So C is false.
- Option D: Area $ A=\pi r^2=\pi\times4^2 = 16\pi $, not $ 64\pi $. So D is false.
# Answer: B) The diameter of the circle is 8 meters.

### Question 22
# Explanation:
## Step1: Add first two equations
Add $ x + y+z = 10 $ and $ x - y - z = 2 $: $ (x + y + z)+(x - y - z)=10 + 2 $.
Simplify left side: $ 2x=12 $.
## Step2: Solve for x
Divide both sides by 2: $ x = 6 $.
We can check with the third equation $ x-2y = 4 $, if $ x = 6 $, then $ 6-2y = 4\Rightarrow - 2y=-2\Rightarrow y = 1 $. Then from first equation $ 6 + 1+z = 10\Rightarrow z = 3 $. The values satisfy all equations.
# Answer: C) 6

### Question 23
# Explanation:
## Step1: Analyze Option A: 5
Unique factors of 5: 1, 5. Number of unique factors: 2. Sum of digits: $ 5=5 $. $ 2
eq5 $, so A is false.
## Step2: Analyze Option B: 12
Unique factors of 12: 1, 2, 3, 4, 6, 12. Number of unique factors: 6. Sum of digits: $ 1 + 2=3 $. $ 6
eq3 $, so B is false.
## Step3: Analyze Option C: 13
Unique factors of 13: 1, 13. Number of unique factors: 2. Sum of digits: $ 1+3 = 4 $. $ 2
eq4 $, so C is false.
## Step4: Analyze Option D: 22
Unique factors of 22: 1, 2, 11, 22. Number of unique factors: 4. Sum of digits: $ 2 + 2=4 $. $ 4 = 4 $, so D is true.
# Answer: D) 22

### Question 24
# Explanation:
## Step1: Recall trigonometric identities
Recall that $ \csc(x)=\frac{1}{\sin(x)} $ and $ \cos(2x)=\cos^{2}(x)-\sin^{2}(x) $, $ \sin(2x) = 2\sin(x)\cos(x) $.
Substitute $ \csc(x)=\frac{1}{\sin(x)} $ into the expression: $ \cos(x)\times\frac{1}{\sin(x)}\times\cos(2x)+\sin(2x)=\frac{\cos(x)\cos(2x)}{\sin(x)}+2\sin(x)\cos(x) $.
## Step2: Simplify the expression
First term: $ \frac{\cos(x)\cos(2x)}{\sin(x)} $, second term: $ \frac{2\sin^{2}(x)\cos(x)}{\sin(x)}=\frac{2\sin^{2}(x)\cos(x)}{\sin(x)} = 2\sin(x)\cos(x) $ (wait, better to get a common denominator).
Common denominator is $ \sin(x) $: $ \frac{\cos(x)\cos(2x)+2\sin^{2}(x)\cos(x)}{\sin(x)} $.
Factor out $ \cos(x) $: $ \frac{\cos(x)(\cos(2x)+2\sin^{2}(x))}{\sin(x)} $.
Recall $ \cos(2x)=1 - 2\sin^{2}(x) $, substitute: $ \frac{\cos(x)(1 - 2\sin^{2}(x)+2\sin^{2}(x))}{\sin(x)}=\frac{\cos(x)(1)}{\sin(x)}=\cot(x) $.
# Answer: C) $ \cot(x) $

### Question 25
# Explanation:
## Step1: Set the equations equal
To find intersection, set $ x^{2}+x - 2=2x^{2}+2x - 4 $.
## Step2: Rearrange into standard quadratic form
Subtract $ x^{2}+x - 2 $ from both sides: $ 0=2x^{2}+2x - 4-(x^{2}+x - 2) $.
Simplify right side: $ 2x^{2}+2x - 4 - x^{2}-x + 2=x^{2}+x - 2 $.
So the equation is $ x^{2}+x - 2 = 0 $.
## Step3: Factor the quadratic
Factor $ x^{2}+x - 2=(x + 2)(x - 1)=0 $.
## Step4: Solve for x
Set each factor to zero: $ x+2 = 0\Rightarrow x=-2 $; $ x - 1=0\Rightarrow x = 1 $.
# Answer: A) -2 and 1

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QUESTION IMAGE

Question

Explanation:

Question 21

Step1: Recall arc length formula

Step2: Substitute values into formula

Step3: Check each option

Step1: Add first two equations

Step2: Solve for x

Step1: Analyze Option A: 5

Step2: Analyze Option B: 12

Step3: Analyze Option C: 13

Step4: Analyze Option D: 22

Answer:

Question 22