SAT Exam 2 (Full Math Module 1)

1. Question:
Solve for x: 3x – 5 = 16.

A. 5
B. 6
C. 7
D. 8

Correct answer: C. 7

Why C is correct:
Add 5 to both sides: 3x = 21. Then divide by 3: x = 7.

Why the others are incorrect:
A. 5 gives 3(5) – 5 = 10.
B. 6 gives 3(6) – 5 = 13.
D. 8 gives 3(8) – 5 = 19.

2. Question:

The answer is 0.35.

Explanation:

In a right triangle, the two acute angles are complementary, so:

n° + t° = 90°

That means:

n° = 90° − t°

Use the cofunction identity:

sin(n°) = sin(90° − t°) = cos(t°)

We are given:

cos(t°) = 0.35

So:

sin(n°) = 0.35

Final answer: 0.35

3. Question:
Solve for x: (x / 2) + 4 = 9.

A. 8
B. 10
C. 12
D. 14

Correct answer: B. 10

Why B is correct:
Subtract 4 from both sides: x / 2 = 5. Then multiply both sides by 2: x = 10.

Why the others are incorrect:
A. 8 gives 4 + 4 = 8.
C. 12 gives 6 + 4 = 10.
D. 14 gives 7 + 4 = 11.

4. Question:
If 2x + y = 11 and y = 5, what is the value of x?

A. 3
B. 4
C. 5
D. 6

Correct answer: A. 3

Why A is correct:
Substitute y = 5 into 2x + y = 11:
2x + 5 = 11
2x = 6
x = 3

Why the others are incorrect:
B. 2(4) + 5 = 13.
C. 2(5) + 5 = 15.
D. 2(6) + 5 = 17.

5. Question:
Solve for x: 2(x + 3) + 2 = 18.

A. 5
B. 6
C. 8
D. 9

Correct answer: A. 5

Why A is correct:
Distribute first: 2x + 6 + 2 = 18
So 2x + 8 = 18
Then 2x = 10
So x = 5

Why the others are incorrect:
B. 2(6 + 3) + 2 = 20.
C. 2(8 + 3) + 2 = 24.
D. 2(9 + 3) + 2 = 26.

6. Question:
Solve for x: |x – 2| = 2, where x > 2.

A. 2
B. 3
C. 4
D. 5

Correct answer: C. 4

Why C is correct:
The equation |x – 2| = 2 gives two possibilities:
x – 2 = 2 or x – 2 = -2
So x = 4 or x = 0
Because x > 2, the only valid solution is x = 4.

Why the others are incorrect:
A. 2 does not satisfy x > 2.
B. |3 – 2| = 1, not 2.
D. |5 – 2| = 3, not 2.

7. Question:

Correct answer: sin⁻¹(3/x)

Why:
If
sin n = 3/x,
then solve for the angle by taking inverse sine:

n = sin⁻¹(3/x)

8. Question:
A club charges a 12 dollar sign-up fee plus 3 dollars per session. If the total cost is 33 dollars, how many sessions were attended?

A. 5
B. 6
C. 7
D. 8

Correct answer: C. 7

Why C is correct:
Set up the equation: 12 + 3s = 33
Subtract 12: 3s = 21
Divide by 3: s = 7

Why the others are incorrect:
A. 12 + 3(5) = 27.
B. 12 + 3(6) = 30.
D. 12 + 3(8) = 36.

9. Question:

Correct answer: D) tan⁻¹(3n/(8m))

Why:
Start with:

2 tan b = 3n/(4m)

Divide both sides by 2:

tan b = 3n/(8m)

Correct answer: A) cos(x°) = n

Why:
Use the cofunction identity:

sin(90° – x°) = cos(x°)

So if:

sin(90° – x°) = n

then:

cos(x°) = n

10. Question:

2. An angle φ in standard position satisfies π/2 < φ < π and cos φ = −9/41. What is sin(π − φ) + cos(φ + π/2)?

Answer: 0

Explanation:
Since π/2 < φ < π, the angle is in Quadrant II, where:

sin φ is positive
cos φ is negative

We are given:

cos φ = −9/41

Use the Pythagorean identity:

sin² φ + cos² φ = 1

sin² φ + (−9/41)² = 1
sin² φ + 81/1681 = 1
sin² φ = 1600/1681
sin φ = 40/41

Now evaluate each part:

sin(π − φ) = sin φ = 40/41

and

cos(φ + π/2) = −sin φ = −40/41

So:

sin(π − φ) + cos(φ + π/2) = 40/41 + (−40/41) = 0

So the value is: 0

11. Question:
Solve for x: 5x – 8 = 27.

A. 6
B. 7
C. 8
D. 9

Correct answer: B. 7

Why B is correct:
Add 8 to both sides: 5x = 35. Then divide by 5: x = 7.

Why the others are incorrect:
A. 5(6) – 8 = 22.
C. 5(8) – 8 = 32.
D. 5(9) – 8 = 37.

12. Question:

Solve for x: (2(x + 3))/5 − (x − 1)/3 = 7/15

Multiply by 15:

6(x + 3) − 5(x − 1) = 7

6x + 18 − 5x + 5 = 7

x + 23 = 7

x = -16

Correct answer: A) -16

13. Question:

3. In a right triangle, acute angles A and C are complementary. If tan C = 7/24, what is sec A?

Answer: 25/7

Explanation:
If tan C = 7/24, then for angle C:

opposite = 7
adjacent = 24

So the hypotenuse is:

√(7² + 24²) = √(49 + 576) = √625 = 25

That gives the triangle ratios for angle C.

Since A and C are complementary:

A = 90° − C

Use the cofunction identity:

sec A = csc C

Now compute csc C:

csc C = hypotenuse / opposite = 25/7

So:

sec A = 25/7

14. Question:

Solve for x: (x − 2)/4 + (x + 5)/6 = (3x − 1)/8

Multiply by 24:

6(x − 2) + 4(x + 5) = 3(3x − 1)

6x − 12 + 4x + 20 = 9x − 3

10x + 8 = 9x − 3

x = -11

Correct answer:  -11

15. Question:

Correct answer: B) cos⁻¹(n/(2t))

Explanation:

Start with the given equation:

(cos x°)/4 = n/(8t)

Multiply both sides by 4 to isolate cos x°:

cos x° = 4 · n/(8t)

Simplify:

cos x° = n/(2t)

Now solve for the angle x by taking the inverse cosine of both sides:

x = cos⁻¹(n/(2t))

So the correct answer is B) cos⁻¹(n/(2t)).

Why the others are wrong:

• A) uses cosine, not inverse cosine, so it gives a trig value, not the angle x
• C) has the fraction reversed and also uses cosine instead of inverse cosine
• D) uses inverse cosine, but the fraction is reversed

Final answer: B) cos⁻¹(n/(2t))

16. Question:
Solve for x: |x – 1| = 4, where x > 1.

A. 2
B. 3
C. 4
D. 5

Correct answer: D. 5

Why D is correct:
The equation |x – 1| = 4 gives:
x – 1 = 4 or x – 1 = -4
So x = 5 or x = -3
Because x > 1, the valid solution is x = 5.

Why the others are incorrect:
A. |2 – 1| = 1.
B. |3 – 1| = 2.
C. |4 – 1| = 3.

17. Question:

1. In a right triangle, one acute angle measures θ°. If sin θ = 5/13, what is tan(90° − θ)?

Answer: 12/5

Explanation:
If sin θ = 5/13, then for angle θ:

opposite = 5
hypotenuse = 13

Use the Pythagorean triple 5-12-13, so:

adjacent = 12

Now use the cofunction idea:

tan(90° − θ) = cot θ

And:

cot θ = adjacent / opposite = 12/5

So the value is:

12/5

18. Question:
A studio charges a 9 dollar sign-up fee plus 4 dollars per class. If the total cost is 37 dollars, how many classes were taken?

A. 400%
B. 500%
C. 600%
D. 700%

Correct answer: D. 700%

Why D is correct:
This item is not mathematically or structurally correct as written.
Using the question, the equation should be: 9 + 4c = 37
Then 4c = 28
So c = 7

That means the correct answer should be 7 classes, not a percent. The row’s answer choices are broken for this question.

Why the others are incorrect:
A. 400% does not answer the question being asked.
B. 500% does not answer the question being asked.
C. 600% does not answer the question being asked.
D. 700% corresponds loosely to 7, but it is still the wrong answer format.

19. Question:
What is the slope of the line through the points (1, 3) and (3, 11)?

A. 3
B. 4
C. 5
D. 6

Correct answer: B. 4

Why B is correct:
Use the slope formula:
m = (11 – 3) / (3 – 1) = 8 / 2 = 4

Why the others are incorrect:
A. 3 is too small.
C. 5 is too large.
D. 6 is too large.

20. Question:
A taxi charges 3 dollars to start a ride and 2 dollars per mile. What is the total cost, in dollars, of a 2-mile ride?

A. 5
B. 6
C. 7
D. 8

Correct answer: C. 7

Why C is correct:
Cost = 3 + 2(2) = 3 + 4 = 7

Why the others are incorrect:
A. 5 is too low.
B. 6 is too low.
D. 8 is too high.

21. Question:
Solve for x: √(3x + 10) = x

Square both sides:

3x + 10 = x²

x² − 3x − 10 = 0

(x − 5)(x + 2) = 0

Possible solutions: x = 5

22. Question:
A streaming service charges a 6 dollar sign-up fee plus 4 dollars per month. If the total cost is 46 dollars, how many months were paid for?

A. 7
B. 8
C. 9
D. 10

Correct answer: D. 10

Why D is correct:
Set up the equation: 6 + 4m = 46
Subtract 6: 4m = 40
Divide by 4: m = 10

Why the others are incorrect:
A. 6 + 4(7) = 34.
B. 6 + 4(8) = 38.
C. 6 + 4(9) = 42.

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