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- Which one of the following equations has no real solution?
a) 2 x^2 − 8 x + 5 = 0 b) 4 x^2 − 4 x + 1 = 0 c) 3 x^2 + 4x − 5 = 0 d) 5 x^2 − 2 x + 1 = 0
- Find all solutions of the equation x^2 x^ + 1 4 = 2.
a) x = 1 or x = − (^12) b) x = ±1 or x = ± √^12
c) x = ±1 or x = ± √i 2
d) x = ±i or x = ± √^12
- Solve the inequality (^) x −^3 1 ≥ (^4) x.
a) (−∞, 0) ∪ (1, 4] b) (0, 1) ∪ [4, ∞) c) (−∞, 4] d) [4, ∞)
- A circle has center (1, −3), and contains the point (− 2 , 0). Find the equation of this circle.
a) (x − 1)^2 + (y + 3)^2 = 18 b) (x + 1)^2 + (y − 3)^2 = 3√ 2 c) (x − 1)^2 + (y + 3)^2 = 3√ 2 d) (x + 1)^2 + (y − 3)^2 = 18
- Find the equation of the line that passes through the points (5, −1) and (2, 4).
a) 3 y − 5 x + 28 = 0 b) 2 y + 3x + 25 = 0 c) 5 x + 3y − 22 = 0 d) 3 x + 5y − 10 = 0
- Find the range of the function f (x) = − 2 x^2 + 12x − 13.
a) (−∞, 3] b) (−∞, 5] c) [5, ∞) d) [3, ∞)
- Let f (x) = √x − 3. Find the inverse function f −^1.
a) f −^1 (x) = x^2 + 3, x ≥ 0 b) f −^1 (x) = x^2 + 3, x ≥ 3 c) f −^1 (x) = (x − 3)^2 , x ≥ 3 d) f −^1 (x) = (x + 3)^2 , x ≥ 0
- Write the complex number^3 1 +^ −^ ii in standard form (a + bi with a, b real numbers) a) 2 − 1 i b) 1 − 2 i c) 12 +^13 i d) 23 +^12 i
- Which of the following functions is 1-1?
a) f (x) = −(x + 5)^2 + 13 b) f (x) = 2 tan(x − π) c) f (x) = e^2 −x^ + 1 d) f (x) = x x^5 −+ 1 1
- Find the quotient and remainder for^3 x
(^3) − 2 x (^2) − 4 x − 3 x^2 + 3x + 3. a) quotient 3x − 7, remainder − 7 x − 5 b) quotient 3x − 7, remainder 16x + 18 c) quotient 3x − 11, remainder 20x + 30 d) quotient 3x − 11, remainder 20x − 36
- Find the slant asymptote of the function f (x) = x^2 + 3 x + 5x^ + 2.
a) y = x + 12 b) y = x^2 + 5x + 12 c) y = x + 5 d) y = x − 2
- Find the horizontal and vertical asymptotes of f (x) = 3 x
− 7 x^2 + x. a) horizontal asymptote y = − 37 , vertical asymptotes: x = 0, x =^17 b) horizontal asymptote y =^37 , vertical asymptote: x =^17 c) horizontal asymptote y = 0, vertical asymptotes: x = −√2, x = √ 2 d) no horizontal asymptote, vertical asymptotes: 1 x = 0, x = 7
- Solve the equation 3x^2 −^3 x^ = 81.
a) x = 4 b) x = − 1 , 4 c) x = 3, − 3 d) x = 3
- Solve the equation log(x − 3) + log(x − 2) = log(2x + 24).
a) x = 2, x = 9 b) x = 9 c) x = − 2 , x = 9 d) x = 12
- Solve the equation e^2 x^ + 4ex^ − 21 = 0.
a) x = ln 3 b) x = 3 c) x = ln(−7) and x = ln(3) d) no solution
- Find the area of an equilateral (all sides the same length) triangle with side length of 10 m.
a) 30 m b) 50 √3 m c) 25 √3 m d) 25 m
- In a triangle ABC, if b = 10, ∠A = 45◦^ and ∠B = 30◦, what is a?
a) a =^10
b) a = 10√ 2 c) a =^10
d) a cannot be determined
- In a triangle ABC, if a = 2, b = 3 and c = √7, what is C?
a) C = 60◦ b) C = 30◦ c) C = 75◦ d) C cannot be determined
- Find the amplitude and period of the function f (x) = 2 + 4 sin(3x + π).
a) amplitude: 2, period: π b) amplitude: 4, period: 2π c) amplitude: 2, period: 23 π d) amplitude: 4, period: 23 π
- Determine which of the following functions is even.
a) f (x) = sin x b) f (x) = sin x cos x c) f (x) = x sin x d) f (x) = x cos x
- Calculate sin(240◦).
a) (^12) b) − (^12)
c)
d) −
- Simplify the expression sec
(^2) x csc^2 x + 1. a) csc^2 x b) sin^2 x c) sec^2 x d) cos^2 x
- Find the exact value of sin 10 πcos^25 π + cos 10 πsin^25 π
a)
b)
c) 0 d) 1
- Simplify sin 2x cot x − 1
a) cos 2x b) tan 2x c) cot 2x d) sec 2x
- A 50 ft pole casts a shadow as shown. Express the angle of elevation θ of the sun as a function of the length s of the shadow.
a) θ = cos−^1 (^50 s ) b) θ = cot−^1 ( 50 s) c) θ = tan−^1 (^50 s ) d) θ = tan−^1 ( 50 s)
- Find all solutions of the equation 2 sin x + √3 = 0 in the interval [0, 2 π)
a) 56 π,^53 π b) 56 π,^116 π c) 43 π,^53 π d) 43 π,^116 π
- What is the length of the circular arc subtended by an angle of arccos( −
2 ) with a radius of 2? a)
√ 3 π 2 b) 53 π c) 35 π d) π 6
Addition/Subtraction Formulas cos(A + B) = cos(A) cos(B) − sin(A) sin(B) cos(A − B) = cos(A) cos(B) + sin(A) sin(B)
sin(A + B) = sin(A) cos(B) + cos(A) sin(B) sin(A − B) = sin(A) cos(B) − cos(A) sin(B)
Half Angle Formulas sin
( A
√ (^1) − cos(A) 2
cos
( A
√ (^) 1 + cos(A) 2 Product-to-Sum Formulas cos(A) cos(B) = 12 (^ cos(A − B) + cos(A + B)) sin(A) sin(B) = 12 (^ cos(A − B) − cos(A + B)) cos(A) sin(B) = 12 (^ sin(A + B) − sin(A − B)) sin(A) cos(B) = 12 (^ sin(A + B) + sin(A − B))
Sum-to-Product Formulas sin(α) + sin(β) = 2 sin
( (^) α + β 2
cos
( (^) α − β 2
cos(α) + cos(β) = 2 cos^ (^ α^ + 2 β^ ) cos^ (^ α^ − 2 β^ )
sin(α) − sin(β) = 2 cos^ (^ α^ + 2 β^ ) sin^ (^ α^ − 2 β^ ) cos(α) − cos(β) = −2 sin
( (^) α + β 2
sin
( (^) α − β 2
Law of Cosines: a^2 = b^2 + c^2 − 2 bc cos A b^2 = a^2 + c^2 − 2 ac cos B c^2 = a^2 + b^2 − 2 ab cos C
FINAL EXAM- VERSION A
1. D 2. C 3. A 4. A 5. C 6. B 7. A 8. B 9. C 10. C 11. D 12. A
13. B 14. B 15. A 16. C 17. B 18. A 19. D 20. C 21. D 22. B 23.
B 24. A 25. C 26. D 27. A 28. C 29. C 30. B
