Special Topics 1 – Pre-calculus
Part 1: Algebra and Trigonometry
1. What number has four significant figures?
a. 0.0014
b. 0.01414
c. 0.141
d. 1.4140
2. Simplify the following equation: i3217 – i427 – i18
a. 1 + 2i
b. 1 – I
c. -1 + 2i
d. 1 + i
3. If z = �1 − �1 − √1 − ⋯, what is the value of z?
a. 0.453
b. 0.618
c. 0.816
d. 0.681
4. Determine the absolute value/modulus/magnitude of the complex number 3 + 4i.
a. 4
b. 8
c. 5
d. 6
5. What is the value of k to make the expression kx2 – 3kx + 9 a perfect square?
a. 2
b. 3
c. 4
d. 5
6. If the roots of the equation are -1, 2 and 4, what is the equation?
a. 3 – 5x2 + 2x + 8 = 0
c. x3 – 4x2 – 3x + 8 = 0
d. x3 – 4x2 + 2x + 6 = 0|
b. X3 – 4x2 + 3x + 8 = 0
7. The equivalent whose roots are the reciprocal of the roots of the equation 2x2 – 3x – 5 = 0 is:
a. 3x2 + 5x – 5 = 0
c. 2x2 – 5x + 3 = 0
2
d. 5x2 – 3x + 2 = 0
b. 5x + 3x – 2 = 0
8. The remainder when 2x4 - kx – 15x2 – 3x – 2 is divided by (x – 3) is 4. What is the value of k?
a. 4
c. 3
b. 7
d. 5
9. What is the middle term (2x + 1/x)8
a. 1120
b.2240
c.70
d. None of the above
10. Factor of the said expression (1-a)8 as completely.*
a. (1+a)4(1+a)(1-a)
c. (1+a)4(1-a)4
b. (1+a)4(1+a)(1+a)(1+a)(1-a)
d. (1-a)(1+a)(1-a)4
11. If log 2 = x and log 3 = y, find log 1.2 in terms of x and y.
a. 2x + y – 1
c. 2x – y + 1
b. 3x + 2y – 1
d. 2x – 3y – 1
12. Suppose 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 3 = 𝑝𝑝 𝑎𝑎𝑎𝑎𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 5 = r, then 𝑙𝑙𝑙𝑙𝑙𝑙√𝑎𝑎 25 + 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎2 25 in terms of p and r is:
a. 4r + (3/2)p
b. 4r + (2/3)p
c. 4r – (2/3)p
d. 4r – (3/2)p
13. The value of (3 to 2.5 power) square is equal to:*
a. 730
b. 150
c. 93
d. 243
14. The geometric mean of numbers 3 and 12 is:*
a. 15
c. 6
b. 7.5
d. 8
15. There are 6 geometric means between 4 and 8748. Find the sum of all terms.*
a. 13210
c. 12310
b. 13120
d. 12130
16. A rubber ball is made to fall from a height of 50 ft and is observed to rebound 2/3 the distance it
falls. How far will the ball travel before coming to rest if the ball continues to fall in this manner?
a. 420 ft
b. 343 ft
c.271 ft
d. 250 ft
17. How many terms must the progression 3,5,7… have in order for the sum to be 2600?
a. 47
b. 48
c. 50
d. 52
18. Find the root mean square (RMS) of 11, 23 and 35.
a. 25
b. 27
c. 26
d. 24
19. If x varies inversely as y and when x = 2, y = 5, What is the value of y2 when x = 1/3?
a. 900
b. 1/900
c. 30
d. 1/30
20. If x varies directly as y and inversely as z, and x = 14 when y = 7 and z = 2, find the value of x
when z = 4 and y = 16.
a. 12
b. 18
c. 14
d. 16
21. Given that w varies directly as the product of x and y and inversely as the square of z and that w
= 4 when x = 2, y = 6 and z = 3. Find the value of w when x = 1, y = 4.
a. 2
b. 3
c. 4
d. 5
22. A man left Sta. Rosa City to drive to Lopez, Quezon at 6:15pm and arrived at 11:45 pm. If he
averaged 50 kph and stopped 1 hour for dinner, how far is Lopez, Quezon from Sta. Rosa City?
a. 225 km
b. 252 km
c. 522 km
d. 215 km
23. A jogger starts a course at a steady rate of 8 kph. Five mins later, the second jogger starts the
same course at 10 kph. How long will it take for the second jogger to catch the first?
a. 20 mins.
b. 21 mins.
c. 22 mins.
d. 18 mins.
24. A father, Max is 24 years older than Joey. In 8 years, he will be twice as old as his son joey. What
is the present age?*
a. 35
b. 40
c. 50
d. 45
25. Two times the mothers age is 8 more than six times her daughters age. Ten years ago, the sum
of their ages was 44. What is the daughter’s age?
a. 15
b. 12
c. 18
d. 16
26. Douglas can paint a fence 50% faster than Nonoy and 20% faster than Jerome. Together, they
can paint the fence in 4 hours. How long will it take Douglas to paint the same fence if he had to
work alone?*
a. 9
b. 8
c. 10
d. 11
27. A piece of work can be done by Ador and Marco in 10 hours, by Ador and July in 12 hours and by
Marco and July in 20 hours. Find how long would it take by Ador to do the job alone?*
a. 60 hours
b. 25 hours
c. 15 hours
d. 30 hours
28. How much of a 40% solution of a denatured alcohol should be mixed with an 80% to give 150
liters of a 50% solution?*
a. 112.5 Liters
b. 122.5 Liters
c. 37.5 Liters
d. 127.5 Liters
29. 10L if 25% salt solution and 15L of 35% salt solution are poured into a drum originally containing
30 liters of 10% salt solution. What is the percent concentration of the mixture?
a. 0.1955
b. 0.1530
c. 0.1140
d. 0.05
30. The measure of 2.25 revolutions counter-clockwise is:
a. -835o
b. 805o
c. 810o
d. -810o
31. The supplement of an angle is 5/2 of its complement. Find the angle.
a. 50
b. 30
c. 25
d. 45
32. The sides of the triangle are 5,8 and 12. Find the angle opposite of the longest side in degrees.
a. 96
b. 44
c. 133
d. 113
33. In triangle ABC, A = 45 degrees, a = 7 sqrt 2 and b = 7. Find c.
a. 23.8
b. 13.5
c. 78.9
d. 44.7
34. One leg of a right triangle is 20cm and the hypotenuse is 10cm longer than the other leg. Find
the length of the hypotenuse.
a. 10cm
b. 15cm
c. 25cm
d. 20cm
35. A man finds the angle of elevation of the top of the tower to be 30 degrees. He walks 85m
nearer the tower and finds its angle of elevation to be 60 degrees. What is the height of the
tower?
a. 76.31m
b.76.61m
c. 73.31m
d. 73.61m
36. Solve for x in the equation arctan (x + 1) + arctan (x-1) = arctan 12
a. 1.5
b. 1.33
c. 1.20
d. 1.25
37. In the triangle, find the side c if angle C = 100o, side b = 20 and side a = 15.
a. 28
b. 27
c. 29
d. 26
38. A man who can row 6 km/hr in still water heads directly across the current of a river flowing at a
rate of 4.5 kph. Find the actual rate and direction of his motion with respect to the current.
a. 5.2 kph at 50.75 deg.
c. 7.5 kph at 53.12 deg.
b. 3.5 kph and 36.87 deg.
d. 8.5 kph at 30.43 deg.
39. Given a right triangle ABC. Angle B is right angle and has legs of 5m and 12m. Find the length of
a line drawn from B perpendicular to the hypotenuse.
a. 2.46m
b. 4.62m
c. 6.24m
d. 4.26m
40. Simplify the expression 4 cos y sin y [1 – 2(siny)2].
a. Sec 2y
b. cos 2y
c. tan 4y
d. sin 4y
41. Solve for A in the given equation cos2A = 1 – cos2A
a. 45,125,225,335 degrees
c. 45,135,225,315 degrees
b. 45,125,225,315 degrees
d. 45,150,220,315 degrees
42. If arctan 2x + arctan 3x = 45o what is the value of x?
a. 1/6
c. 1/5
b. 1/3
d. 1/4
43. If sin 3A = cos 6B, then,
a. A+B = 90
b. A+2B = 30
c. A+B = 180
d. none of these
44. What is the sum of the squares of the sine and cosine of an angle?
a. 0
c. 2
b. 1
d. sqrt 3
45. In the equation below, solve for X:
a. 1
b. sin ∅
𝑋𝑋 = (tan ∅ + cot ∅)2 𝑠𝑠𝑠𝑠𝑠𝑠2 ∅ − 𝑡𝑡𝑡𝑡𝑡𝑡2 ∅
c. 0
d. cos ∅