MATH 115 FINAL EXAM, Spring 2000

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1. a) Solve: x^{2} − 2x − 7 = 0  b) Solve: |x − 2| ≤ 3  1  11
2  12
3  13
4  14
5  15
6  16
7  17
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2. Given the points P (1, 2), Q(4, 6), find the equations of:  9  19

a) The line through P and Q b) The circle through Q centered at P.

10

20 Tot

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3. a) Find the slope of the line ⊥ to 2x − 3y = 6 b) Find the minimum value of f(x) = 3x^{2} − 12x + 5.

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4. a) Let f(x) = 3x^{3} − 2, Find (f ∘ f)(1). b) let f(x) = 3x^{3} − 2. Find f^{−1}(x).

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5. a) Factor f(x) = x^{3} − 8x^{2} + 17x − 10, given that x = −1 is a zero.

b) Find the other zeros.

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6. a) Find the domain of f(x) = (x^{2} + 1)/x. b) Find the asymptotes of f(x) = (x^{2} + 1)/x.

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7. a) Solve for x: 9^{x} = 3^{1−x} b) Simplify: (b^{2}b^{−4})^{−2}.

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8. a) Let f(x) = 12^{x}, Find k so that f(x) = e^{kx} b) Evaluate log_{2}(1/8).

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9. a) Solve for t: 4e^{−3t} = 5. b) Solve for x: ln(x + 2) = 1 + ln(2x − 2).

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10. How long does it take for a population to double if after 1 year it has increased by 4%.

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11. a) Evaluate exactly: cos 150^{o}. b) Evaluate exactly: tan(4π/3)

12. a) Find sin t, given cos t = 5/13, 0 ≤ t ≤ π/2 b) Find the amplitude and period of f(x) = −2 cos 3t

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13. Compute the exact value of cos 75^{o} using:

a) The sum formula. b) The half angle formula.

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14. a) Simplify: (1 − sin^{2} t)(sec^{2} t − 1) b) Prove that: (sin t + cos t)^{2} = 1 + sin 2t

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15. a) Simplify: sin(t − 3π/2) b) Simplify: cos 75^{o} cos 15^{o} + sin 15^{o} sin 75^{o}

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16. In a triangle ABC, a = 4, b = 7, c = 9

a) Find angle α. b) Find the area

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17. a) Compute: 1 + 2 + 3 + ⋯ + 1000 b) Write as a fraction: 1.233333 …

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18. a) Evaluate the sum: 12 + 9 + 27/4 + · · ·.

b) Find a_{4}, if a_{n+2} = 2a_{n+1} + 3a_{n}, a_{0} = 0, a_{1} = 1

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19. a) Evaluate:( \begin{matrix}200 \\ 198 \end{matrix} ) b) Write the first 3 terms of (a + b)^{30}

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20. Solve by Gaussian elimination:

3x − 2y + 4z = 0 x − y + 3z = 1 4x + 2y − z = 3

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