1. Find the value of p x +√
14 − x when x = −2.
2. Find the area of the triangle formed by the coordinate axes and the line 3x + 2y = 6.
3. A sequence is defined by an = 3(an−1+ 2) for n ≥ 2, where a1 = 1. What is a4? 4. Christa leaves Town A. After traveling 12 km, she reaches Town B at 2:00 P.M. Then
she drove at a constant speed and passes Town C, 40 km from Town B, at 2:50 P.M.
Find the function d(t) that models the distance (in km) she has traveled from Town A t minutes after 2:00 P.M.
5. What is the largest negative integer that satisfies the inequality |3x + 2| > 4?
6. A person has two parents, four grandparents, eight grand-parents, and so on. How many ancestors does a person have 10 generations back?
7. In 4ABC,6 B is twice 6 A, and 6 C is three times as large as 6 B. Find6 C.
8. If −3 ≤ x ≤ 0, find the minimum value of f (x) = x2+ 4x.
9. The 9th and the 11th term of an arithmetic sequence are 28 and 45, respectively. What is the 12th term?
10. Perform the indicated operation, and simplify: 2
x + 3
x − 1− 4 x2− x. 11. Find the range of the function f (x) = |2x + 1|.
12. Find the value ofh
x − (x − x−1)−1i−1
when x = 2.
13. Find the equation (in the form ax + by = c) of the line through the point (5, 2) that is parallel to the line 4x + 6y + 5 = 0.
14. In 4ABC,6 B = 90◦,6 ACB = 28◦, and D is the midpoint of AC. What is6 BDC?
15. Find all solutions of the system:
2x + y = 1 3x + 4y = 14. 16. If f (2x − 1) = x, what is f (2)?
17. What is the quotient when 3x5+ 5x4− 4x3+ 7x + 3 is divided by x + 2?
18. A man is walking away from a lamppost with a light source 6 meters above the ground.
The man is 2 meters tall. How long is his shadow when he is 110 meters from the lamppost?
19. Find the length of the shorter segment made on side AB of 4ABC by the bisector of
6 C, if AB = 20, AC = 12, and BC = 18 cm.
20. How many different three-digit numbers less than 300 can be formed with the digits 1,2,3 and 5 if repetition of digits is not allowed?
21. Factor completely the expression x3− 7x + 6.
22. An integer between 1 and 10000 inclusive is selected at random. What is the probability that it is a perfect square?
23. If a chord 24 cm long is 5 cm from the center of a circle, how long is a chord 10 cm from the center?
24. Pipes are being stored in a pile with 25 pipes in the first layer, 24 in the second, and so on. If there are 12 layers, how many pipes does the pile contain?
25. Find the solution set of the inequality x2 < 5x − 6.
26. Suppose an object is dropped from the roof of a very tall building. After t seconds, its height from the ground is given by h = −16t2+ 625, where h is measured in feet. How long does it take to reach ground level?
27. In 4ABC, AB||DE, AC = 10 cm, CD = 7 cm, and CE = 9 cm. Find BC.
28. Find a polynomial P (x) of degree 3 that has zeroes −2, 0, and 7, and the coefficient of x2 is 10.
29. Find the equation (in the form ax + by = c) of the perpendicular bisector of the line segment joining the points (1, 4) and (7, −2).
30. The sides of a polygon are 3,4,5,6, and 7cm. Find the perimeter of a similar polygon, if the side corresponding to 5 cm is 9 cm long.
31. If tan θ = 2
3 and θ is in Quadrant III, find cos θ.
32. The first term of a geometric sequence is 1536, and the common ratio is 1
2. Which term of the sequence is 6?
33. Inf 4ABC,6 B = 26 A. The bisectors of theses angles meet at D, while BD extended meet AC at E. If6 CEB = 70◦, find 6 BAC.
34. Find the sum of the series: 1 3+ 21
32 +22 33 +23
34 + · · · .
35. What is the remainder when x3− x + 1 is divided by 2x − 1?
36. Triangle ABC has right angle at B. If6 C = 30◦ and BC = 12 cm, find AB.
37. What is the largest root of the equation 2x3+ x2− 13x + 6 = 0?
38. Find the equation (in the form x2+ y2+ cx + dy = e) of the circle that has the points (1, 8) and (5, −6) as the endpoints of the diameter.
39. The distance from the midpoint of a chord 10 cm long to the midpoint of its minor arc is 3 cm. What is the radius of the circle?
40. Find the equation (in the form ax+by = c) of the line tangent to the circle x2+y2 = 25 at the point (3, −4).
41. How many triangles can be formed by 8 points, no three of which are collinear?
42. Two tangents to a circle form an angle of 80◦. How many degrees is the larger inter-cepted arc?
43. Find all real solutions of the equation x12 + x16 − 2 = 0.
44. The sides of a triangle are 6, 8, and 10, cm, respectively. What is the length of its shortest altitude?
45. In a circle of radius 10 cm, arc AB measures 120◦. Find the distance from B to the diameter through A.
46. Find the remainder when x100 is divided by x2− x.
47. Find the area (in terms of π) of the circle inscribed in the triangle enclosed by the line 3x + 4y = 24 and the coordinate axes.
48. In how many different ways can 5 persons be seated in an automobile having places for 2 in the front seat and 3 in the back if only 2 of them can drive and one of the others insists on riding in the back?
49. Factor completely the expression x4+ 3x2+ 4.
50. A bag is filled with red and blue balls. Before drawing a ball, there is a 1
4 chance of drawing a blue ball. After drawing out a ball, there is now a 1
5 chance of drawing a blue ball. How many red balls are originally in the bag?
2 Answers with Solutions
3 greater than 1 2− 2?
250000 in scientific notation.
Answer: 4.4 × 10−7
4. The product of two prime numbers is 302. What is the sum of the two numbers?
Answer: 153
Since 302 is even, one of the factors is 2. Thus the other prime factor is 151. Which gives a sum of 153 .
5. A shirt is marked Php 315 after a discount of 10% and value added tax of 12%. What was the price of the shirt before the tax and discount?
Answer: Php 312.5
Let p be the original price of the shirt. So we have 0.9p after the 10 % discount and (0.9)(1.12)p after the 12% added tax. Thus,
(0.9)(1.12)p = 315
6. How many different lengths of diagonals does a regular octagon have?
Answer: 3
Draw an octagon and count the number of diagonals from a vertex with different lengths.
7. What number is midway between 3 + 1
3 and 2 − 1
10. The sum of the measures of the interior angles of a polygon is 1980◦. How many sides has the polygon?
Answer: 13
180(n − 2) = 1980 n − 2 = 11
n = 13
11. The average of three numbers is 20. Two numbers are added to the set and the average of the five numbers becomes 42. If one of the added numbers is twice the other, what
are the two numbers added to the data set?
Answer: 50 and 100
Let S be the sum of the three numbers. So S = 3(20) = 60. Let a and b be the two numbers added to the set. Now we have S + a + b = 5(42) = 210. Subtracting the two equations, a + b = 150. Also, a = 2b, gives 2b + b = 150. Hence, b = 50. The numbers added are 50 and 100 .
12. Compute 24 ÷1 + 15 2 − 13. Answer: 12
24 ÷ 1 + 1
5 2 −1 3
= 24 ÷ 6 5 5 3
= 24 ÷ 2
= 12
13. Subtract 5a − 2b + c from the sum of 3a + b − 2c and a − b + 3c.
Answer: −a + 2b
[(3a + b − 2c) + (a − b + 3c)] − (5a − 2b + c) = (4a + c) − (5a − 2b + c)
= 4a + c − 5a + 2b − c
= −a + 2b
14. Alex, Beth, and Carla play a game in which the losing player in each round gives each of the other players as much money as the player has at that time. In Round 1, Alex loses and gives Beth and Carla as much money as they have. In round 2, Beth loses and in Round 3, Carla loses. After 3 rounds, they find that they each have Php 40.
How much money did Alex have at the start of the game?
Answer: 65
Let the starting amounts of Alex, Beth, and Carla be a, b and c.
The table below shows the amounts of they have every after round
Round Alex Beth Carla
1 a − b − c 2b 2c
2 2a − 2b − 2c −a + 3b − c 4c 3 4a − 4b − 4c −2a + 6b − 2c −a − b + 7c Then,
4a − 4b − 4c = 40 → a − b − c = 10 (1)
−2a + 6b − 2c = 40 → a − 3b + c = −20 (2)
−a − b + 7c = 40 → a + b − 7c = −40 (3)
From (1), (2), and (3), a = 65, b = 35, c = 20. Thus, Alex had 65 coins at the start of the game.
15. Three two-digit numbers have consecutive tens digits and have units digit all equal to 5. If the tens digit of the smallest number is n, what is the sum of the three numbers?
Answer: 30n + 45
So n is the smallest tens digit. Also, note that the integer AB can be written as 10A + B. So the numbers are:
10n + 5, 10(n + 1) + 5, 10(n + 2) + 5 or 10n + 5, 10n + 15, 10n + 25 Hence, their sum is 30n + 45 .
16. The length of a rectangle is 8 cm more than its width. If the length is decreased by 9 and the width is tripled, the area is increased by 50%. What was the area of the original rectangle?
Answer: 180 cm2
Let w be the width of the rectangle. So the length is w +8 and the area of the rectangle is w(w + 8). Using the first condition,
[(w + 8) − 9] (3w) = w(w + 8)(1.5) (w − 1)(3w) = (w2+ 8w)(1.5)
2w(w − 1) = w(w + 8)
2(w − 1) = w + 8 (since w 6= 0) 2w − 2 = w + 8
w = 10
Thus, the original dimensions of the rectangle are 10 cm by 18 cm which gives an area of 180 cm2 .
17. Evaluate: 236 × 542 + 458 × 764 + 542 × 764 + 236 × 458.
Answer: 1000000
Let a = 236, b = 542, c = 458, d = 764
We have now ab + cd + bd + ac which can be factored as (a + d)(b + c). Plugging in the respective values,
(a + b)(c + d) = (236 + 764)(542 + 458) = 10002 = 1000000 18. TRUE or FALSE: If n is a real number, then n2 is positive.
Answer: FALSE
When n = 0, n2 = 0 which is neither positive nor negative.
19. If n kilos of rice costs p pesos, how much will x kilos of rice cost?
Answer: px
k pesos Rate = p
k pesos per kilo. thus, x kilos is worth px
k pesos.
20. If the letters of the word MATHEMATICS are repeatedly and consecutively written, what is the 2016th letter?
Answer: T
There are 11 letters on the word so every 11 letters, it repeats the sequence. Thus the remainder when 2016 is divided by 11 gives the 2016th term. 2016 ≡ 3mod11. The third letter of MATHEMATICS is T .
21. Simplify: x −√ 2
x +√ 2 x3− 2x . Answer: 1
x
x −√ 2
x +√ 2
x3− 2x = x2 − 2 x(x2 − 2)
= 1
x
22. If x is three times as far from −5 as it is from 15, what are the possible values of x?
Answer: x = {25, 10, 20}
The equation is |x + 5| = 3|x − 15|.
Case 1: If x < −5,
−x − 5 = 3(−x + 15)
−x − 5 = −3x + 45 2x = 50
x = 25 But x < −5, so the solution is extraneous.
Case 2: If −5 ≤ x < 15,
x + 5 = 3(−x + 15) x + 5 = −3x + 45
4x = 40 x = 10
Case 3: If x ≥ 15,
x + 5 = 3(x − 15) x + 5 = 3x − 45
2x = 50 x = 25
23. If three children eat 4 kilos of rice in 5 days, how long will 12 children eat 48 kilos of rice?
Answer: 15 days
Since the number of children is directly proportional to the number of kilos they eat and inversely proportional to the number of days they take.
C = kK
D → k = CD K where k is the constant of proportion.
k = 3 · 5
4 = 12 ∗ D 48 D = 15 days
24. A conical tank full of water is emptied into an empty cylindrical tank of the same height. If the base radius of the cylinder is twice that of the cone, what fraction of the cylinder will be filled with water?
Answer: 81 3 %
Let the common height be h and the radius of the cone is r so the radius of the cylinder is 2r.
The volume of the cone therefore is 1
3πr2h while the cylinder has a volume π(2r)2h = 4πr2h. The required answer is the ratio of the volumes of the volume of the cone to the volume of the cylinder.
Vcone
Vcylinder = 1
12 = 81 3% 25. Find the minimum integer n for which 18
n + 1 is an integer.
Answer: -19
n is minimized when n + 1 = −18. Hence, n = −19 .
26. The sum of the square roots of two positive integers is 5. If the two integers differ by 5, what are the integers?
Answer: 9 and 4
If a is the square root of the larger integer, the square root of the smaller integer is 5 − a.
Expressing the second condition,
a2− (5 − a)2 = 5 10a − 25 = 5 10a = 30
a = 3
So the square roots of the two integers are 3 and 2. Thus the integers are 9, 4 27. A worm crawls 7.5 inches in 80 seconds. What is its speed in feet per hour?
Answer: 28.125
speed = 7.5 in 80 s
= 7.5 in 80 s · 1 ft
12 in · 3600 s 1 hr
= 28.125 ft per hr
28. By how much is (3x − 5)(x + 2) greater than (x + 4)(2x − 1)?
Answer: x2− 6x − 6
(3x − 5)(x + 2) − (x + 4)(2x − 1) = (3x2+ x − 10) − (2x2+ 7x − 4)
= 3x2+ x − 10 − 2x2− 7x + 4
= x2− 6x − 6
29. A game consists of drawing a number from 1 − 20. A player wins if the number drawn is either a prime number or a perfect square. What is the probability of winning in this game?
Answer: 3 5
The prime numbers in that range are: 2,3,5,7,11,13,17, and 19 while the perfect squares are: 1,4,9, and 16. Which means there are 12 numbers that give a player the win. Thus, the probability of winning the game is 12
20 = 3 5 .
30. The number of boys in a class is equal to the number of girls. Nine boys are absent today, and this leaves twice as many girls as boys in the classroom. How many students belong to the class?
Answer: 36
Let n be the number of girls and boys in the class. So, n = 2(n − 9) → n = 18
So the total number of students in the class is 2n which is 36 .
31. A square region is removed from a rectangular region. Which of the following can be true?
(a) The perimeter is decreased.
(b) The perimeter is not changed.
(c) The perimeter is increased.
Answer: B and C
When the square region has a common vertex to the rectangle, the perimeter won’t change. Otherwise, the perimeter increases.
32. A bag contains 5 black, 5 red, 6 blue, 7 white, 8 yellow, and 10 orange beads. At least how many beads must be drawn from the bag to ensure that at least 3 beads of the same color are chosen?
Answer: 13
The worst case scenario occurs when you have already picked 2 beads from each color and surely after picking the next bead, it will have already three beads of the same color. Thus there should be 2(6) + 1 = 13 beads that will be drawn.
33. How many positive numbers less than 1000 are divisible by 6 but not by 5?
Answer: 133
Counting the number of integers less than 1000 that are divisible by 6:
1 < 6n < 1000 → 1 ≤ n ≤ 166 So there are 166 − 1 + 1 = 166 numbers that are divisible by 6.
To remove the numbers divisible by 5 out of the 166 multiples of 6, the multiples of 30 on that set should be removed.
1 < 30n < 1000 → 1 ≤ n ≤ 33 Which is 33.
Therefore the answer is 166 − 33 = 133
34. A 4 cm × 5 cm × 7 cm rectangular prism is painted on all faces. If the prism is cut into 1 cm × 1 cm × 1 cm how many cubes do not have paint on any of its sides?
Answer: 30
It is simplfy (4 − 2)(5 − 2)(7 − 2) = 30 cubes.
35. Andrew is 5 years old and Charlie is 26. In how many years will Charlie be 21 2 times as old as Andrew?
Answer: 9 years
26 + n = 5
2(5 + n) 52 + 2n = 25 + 5n
n = 9
36. A car is driving along a highway at 55 kph. The driver notices a bus, 1
2 km behind.
The bus passes the car one minute later. What was the speed of the bus?
Answer: 85 kph
Using the formula rate · time = distance,and letting r be the speed of the bus,
r 1 60
− 55 1 60
= 1 2 r − 55 = 30
r = 85 So the speed of the bus is 85 kph .
37. With what polynomial must 8x5− 10x3+ 2x + 5 be divided to get a quotient of 4x2− 3 and a remainder of 5 − x?
Answer: 2x3− x
8x5− 10x3 + 2x + 5 = P (4x2− 3) + 5 − x P = 8x5 − 10x3+ 2x + 5
4x2− 3
2x3 − x 4x2− 3
8x5− 10x3+ 3x
− 8x5 + 6x3
− 4x3+ 3x 4x3− 3x 0
Thus, P = 2x3− x
38. The volume of a sphere is equal to its surface area. What is the diameter of the sphere?
Answer: 6
4
3πr3 = 4πr2 r = 3 So the radius is 3 which means the diameter is 6 .
39. If 3A54B10 is divisible by 330, what are the values of A and B?
Answer: A = 4, B = 1
Since 330 = 3 ∗ 10 ∗ 11, 3A54B10 should be divisible by 3, and 11 to make it divisible by 330 (10 is not needed already since the last digit is 0).
Divisible by 3:
A + B + 22 = 27, 36 → A + B = 5, 14 Divisible by 11:
3 − A + 5 − 4 + B − 1 + 0 = −A + B + 3 = 0, 11 → −A + B = −3, 8
It forms 4 systems of of equations but solving the 4 systems of 2 equations each, there will only be one pair that is acceptable which is A = 4, B = 1 .
40. Find the solution set: |5 − 2x| < 19.
Answer: −7 < x < 12
|5 − 2x| < 19
−19 < 5 − 2x < 19
−24 < −2x < 14 12 > x > −7
−7 < x < 12
41. If 47A2969 is the square of 3(721 + A), find the digit A.
Answer: 8
The square of 3(721+A) is 9(721+A)2. Which means 47A2969 should be divisible by 9.
4 + 7 + A + 2 + 9 + 6 + 9 = A + 37 = 45 → A = 8 It only has one possible value which is A = 8 . Verifying it,
[3(721 + 8)]2 = 4782969
42. Consider the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, . . . . How many numbers in the sequence are needed so that the sum of the reciprocals is 100?
Answer: 5050
Getting the sum of reciprocals, it simplifies to
1 + 1 + 1 + ... + 1
| {z }
n times
= 100 n = 100
So we need one hundred 1’s which means we need the sequence to be untill the 100th 100. Which, in total, has 100(101)
2 = 5050 numbers.
43. If n is a positive odd integer, which of the following is a perfect square: 2n2, 25(3n+2), 7n(n+1)? Answer: 7n(n+1)
The exponent with an even exponent will be a perfect square. Since n is odd, n2, n + 3 will also be odd. So we only need to prove that n(n + 1) is even.
Let n = 2k + 1
n(n + 1) = (2k + 1)(2k + 2) → n(n + 1) = 2(2k + 1)(k + 1)
Since there is a actor of 2, n(n + 1) is surely an even number. Thus the only perfect square in that set is 7n(n+1).
44. The sum of the first 50 odd integers is 2500. What is the sum of the next 50 odd integers?
Answer: 7500
The sum of the first n odd integers is actually n2. So to get the required sum, we need to get the sum of the first 100 odd integers, which is 10000 and then subtracting it to 2500 which gives 7500 .
45. Felix has an average of 90 in five tests, each test is with 100 points. What is the lowest possible score Felix could have gotten in a test?
Answer: 50
The sum of his scores in his 5 tests is 5(90) = 450. To minimize his score in one of the tests, we need to maximize the other 4 tests. Since the maximum value of the score in one test is 100, the 4 tests have a maximum total of 400 points. Thus the minimum possible score in one of Felix’s tests is 50 .
46. In square ABCD, P is the midpoint of AB and Q is the midpoint of BC. What percent of the area of ABCD is the area of 4P QD ?
Answer: 37.5%
Denote [XY Z] be the area of a polygon XYZ. Now, note that [P QD] = [ABCD] − [P BQ] − [P AD] − [QCD]
Getting the areas needed in the RHS by letting one side of the square be n, [ABCD] = n2
Thus LHS is greater than RHS. Since √3
3x is originally at the LHS, √3
3x >√ 2x in the interval 1 < x < 9
8.
48. Find the least positive integer that leaves remainders of 1, 2, and 3 when divided by 3, 5, and 7, repsectively.
Answer: 52
Let n ≡ 1 mod 3 ≡ 2 mod 5 ≡ 3 mod 7 Since n ≡ 1 mod 3, n = 3a + 1 for any positive integer a.
3a + 1 ≡ 2 mod 5 3a ≡ 1 mod 5 6a ≡ 2 mod 5
Since 5a ≡ 0 mod 5, (6a − 5a) ≡ (2 − 0) mod 5 or a ≡ 2 mod 5.
So, a = 5b + 2 or n = 3(5b + 2) + 1 = 15b + 7.
Finally,
15b + 7 ≡ 3 mod 7 15b ≡ −4 mod 7
b ≡ 3 mod 7
Therefore, b = 7c + 3, i.e, n = 15(7c + 3) + 7 = 105c + 52. Hence the smallest positive integer n is when c = 0 which is 52 .
49. Find all points x on the real number line such that the sum of the distances from x to 4 and from x to −4 is 12.
Answer: 6, −6
The equation needed is |x − 4| + |x + 4| = 12.
(a) When x < −4
−x + 4 − x − 4 = 12
−2x = 12 x = −6 (b) When −4 < x ≤ 4
−x + 4 + x + 4 = 12 8 6= 12 Gives no solution.
(c) When x ≥ 4
x − 4 + x + 4 = 12 2x = 12 x = 6 Thus, the points are located at 6, and − 6 .
50. If 11n leaves a remainder of 6 when divided by 7, what is the remainder when 5n is divided by 7?
Answer: 4
11n ≡ 6 mod 7 4n ≡ 6 mod 7 2n ≡ 3 mod 7 2n ≡ −4 mod 7
n ≡ −2 mod 7 n ≡ 5 mod 7
5n ≡ 25 ≡ 4 mod 7
2.2 Grade 8
1. Simplify (a − b)2(a + b)2 + 2a2b2. Answer: a4+ b4
(a − b)2(a + b)2+ 2a2b2 = [(a + b)(a − b)]2+ 2a2b2
= a2 − b22
+ 2a2b2
= (a4− 2a2b2+ b4) + 2a2b2
= a4+ b4
2. Simplify 125x4y3 27x−2y6
1 3.
Answer: 5x2 3y
125x4y3 27x−2y6
1
3 = 53x6 33y3
1 3
= 5x2 3y
3. Solve for x in the equation x4− 5x2+ 4 = 0.
Answer: ±1, ±2
x4− 5x2+ 4 = 0 (x2− 1)(x2− 4) = 0 (x + 1)(x − 1)(x + 2)(x − 2) = 0
x = ±1, ±2 4. In the arithmetic sequence 10 + 10√
3 , 11 + 9√
3 , 12 + 8√
3 , · · · , what term has no
√3 ?
Answer: 20
an = a1+ (n − 1)d → an= 10 + 10√
3 + (n − 1) 1 −√
3 = (n + 9) + (11 − n)√ 3 Thus, the term that has no √
3 is when n = 11, i.e, a11= 20
5. If x + y = 12 and xy = 50, what is x2+ y2? Answer: 44
Note that
x2+ y2 = (x + y)2− 2xy Plugging in the values of x + y and xy,
x2+ y2 = 122− 2(50) = 44
6. What is the sum of the first ten terms of the geometric sequence 4, 8, 16, · · · ,?
Answer: 8188
Sn = a rn− 1 r − 1
S11= 4 211− 1 2 − 1
= 8188
7. If the product of two consecutive odd integers is 783, what is the sum of their squares?
Answer: 1570
• Solution 1
By letting x and y be the two integers where x > y,we have x − y = 2 and xy = 783.
Since x2+ y2 = (x − y)2+ 2xy, x2+ y2 = 22+ 2(783) = 1570
• Solution 2
Since 783 = 33· 29 = 27 · 29 and luckily 27 and 29 are consecutive odd integers, we have 272 + 292 = 1570 .
8. If r and s are the roots of the equation 2x2− 3x + 4 = 0, what is 4r2+ 7rs + 4s2? Answer: 7
Notice that 4r2 + 7rs + 4s2 = 4(r + s)2 − rs. By Vieta’s Identities, r + s = 3 2 and rs = 2. Thus, 4(r + s)2− rs = 4 3
2
− 2 = 7 .
9. A long wire is cut into three smaller pieces in the ratio of 7 : 3 : 2. If the shortest piece is 16 cm, what is the area of the largest rectangle that can be created using the longest piece?
Answer: 196 sq. cm
The three smaller pieces can be expressed in the lengths 7k, 3k, and 2k, so 2k = 16 or k = 8. The longest piece, therefore, is 56 cm long.
The largest rectangle that can be formed is a square; so its perimeter is 56 cm.
4s = 56 → s = 14cm . Hence, the area is 196 cm2 . 10. A boat takes 2
3 as much time to travel downstream as to its return. If the rate of the river’s current is 8 kph, what is the rate of the boat in still water?
Answer: 40 kph
Let b be the rate of the boat in still water.
b − 8 = 2
3(b + 8) 3b − 24 = 2b + 16
b = 40 kph
11. How many prime numbers between 40 and 240 ends with 4?
Answer: 0
A number is divisible by 2 and is composite if its last digit is even. The only time that an even number is prime when the number is 2. Since 4 is even, there are 0 primes that satisfy the condition.
12. Simplify: x(1 − 6x) − (1 − 2x)(3x − 2).
Answer: −6x + 2
x(1 − 6x) − (1 − 2x)(3x − 2) = (x − 6x2) − (−6x2+ 7x − 2)
= −6x + 2 13. What is the last digit if 72016?
Amswer:
• Solution 1
Since φ(10) = 4, 74 ≡ 1 mod 10. Thus, 72016 = (74)504 ≡ 1504 = 1 mod 10
• Solution 2
Getting the last digits of the powers of 7,
71 → 7, 72 → 9, 73 → 3, 74 → 1, and repeats the pattern, 7,9,3,1, every 4th power.
Since 2016 is divisible by 4, the fourth term will be the last needed digit which is 1 .
14. If a = 3 and b = 7, what is 4a3b + 6a2b2+ 4ab3? Answer: 7518
Note that 4a3b + 6a2b2+ 4ab3 = 2ab(2a2+ 3ab + 2b2) = 4ab(a + b)2− 2(ab)2. Substitutiong the values, 4(3)(7)(10)2− 2(21)2 = 7518
15. What is the median of the numbers a + 1, a + 3, a − 2, a + 5, and a − 4?
Answer: a + 1
Arranging the numbers in increasing order, a − 4, a − 2, a + 1, a + 3, a + 5, the median is a + 1
16. A rectangle is formed by putting two squares side by side. If each square has perimeter 28 cm, what is the perimeter of the rectangle?
Answer: 42 cm
The side of each square is 7 cm. Referring to its figure, the perimeter will be 42 cm . 17. Solve for x: 4(1 − 3x) − 2x(1 − 3x) + 5(1 − 3x) + 3x(1 − 3x) = 0.
The side of each square is 7 cm. Referring to its figure, the perimeter will be 42 cm . 17. Solve for x: 4(1 − 3x) − 2x(1 − 3x) + 5(1 − 3x) + 3x(1 − 3x) = 0.