Mathematics 17

Third Long Exam 11 September 2007

TOTAL : 50 POINTS

“A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself.

The larger the denominator, the smaller the fraction.”

- Tolstoy

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This exam is good for 1 hour and 15 minutes only. Use black or blue ballpen – no pencils. For

computations, show all your solutions and BOX your final answers.

I. Write TRUE if the statement is true. Otherwise, write FALSE. (1 pt each)

1. For any real number x, logx x = 1.

2. The sequence { 3x, 9

x, 27

x, … } forms a geometric progression.

3. If f(x) varies inversely as the square root of x, then �(�)�(�) = 2.

4. The sequence {log 6, log 12, log 24, … } forms an arithmetic progression.

5. Any polynomial of degree n > 1 has at least 1 real zero.

II. Do as indicated.

1. Given P(x) = 6x4 – 13x

3 + 13x

2 – 2

(a) Using factor / remainder theorem, show that (3x +1) is a factor of P(x) (2 pts)

(b) Find the solution set of the equation P(x) = 0. (3 pts)

2. Given loga 3 = 0.14 and loga 5 = 0.42, find:

(a) loga 0.36 (b) log0.2

�� (4 pts each)

3. Find k such that – k + 1, 2k + 1, and 10k + 3 form a geometric progression (4 pts)

4. Find the range of � = ����� + 2. (4 pts)

III. Find the solution set of the following equations: (4 pts each)

1. 3x – 3

xe

4x + 2 = 0

2. 2 log√� = log(4x – 1) – log 2

3. � ����� = 3√���

4. log(2x + 6) + log(x +2) = log(ln e) + log(-x2 – 2x)

IV. Word Problems (4 pts each)

1. For a vibrating string, the rate of vibration is directly proportional to the square root of the

tension on the string. Given that a particular string vibrates 864 times per second under a tension

of 24 kg, find the tension for the rate of 432 vibrations per second.

2. In an amateur singing contest of six contestants, the cash prize given forms an arithmetic

progression. If the sixth placer got 500 pesos and the third got 875 pesos, how much did the

champion got?