SOLUTIONS 1. Is '7 divides 56'a true or false statement? 7 divides 56 is a true statement. We say that x divides y if y x is a whole number. Here, 56 7 = 8 which is a whole number. 2. Is 627,080 divisible by 2? A number is said to be divisible by 2 if the last digit is even (0,2,4,6,8), here the last digit is 0 which is even. Thus, 627,080 is divisible by 2. Is 627,080 divisible by 3? A number is said to be divisible by 3 if the sum of the digits is divisible by 3, here the sum of the digits is 6+2+7+0+8+0 = 23. But, 23 is not divisible by 3. Thus, 627,080 is not divisible by 3. Is 627,080 divisible by 4? A number is said to be divisible by 4 if the last 2 digits are divisible by 4, here the last two digits are 80 which is divisible by 4 (80 4=20 ). Thus, 627,080 is divisible by 4. Is 627,080 divisible by 5? A number is said to be divisible by 5 if the last digit is 0 or 5, here the last digit is 0. Thus, 627,080 is divisible by 5. Is 627,080 divisible by 6? A number is said to be divisible by 6 if the number is divisible by both 2 and 3. From the above work we can see is that this number is divisible by 2 but not by 3. Thus, 627,080 is not divisible by 6. Is 627,080 divisible by 8? A number is said to be divisible by 8 if the last 3 digits are divisible by 8, here the last three digits are 080 which is divisible by 8 (80 8=10 ). Thus, 627,080 is divisible by 8. Is 627,080 divisible by 9? A number is said to be divisible by 9 if the sum of the digits is divisible by 9, here the sum of the digits is 6+2+7+0+8+0 = 23. But, 23 is not divisible by 9. Thus, 627,080 is not divisible by 9. Is 627,080 divisible by 10? A number is said to be divisible by 10 if the last digit is 0, here the last digit is 0. Thus, 627,080 is divisible by 10. Thus, the numbers form the list 2,3,4,5,6,8,9 and 10 that divide 627,080 are 2,4,5,8 and 10. 3. Find the GCD of 45 and 297. Using the prime factorisation method: To find the GCD of 45 and 297, we will first find the factors of 45 and 297. Divisors of 45: 1,3,5,9,15,45. Divisors of 297: 1,3,9,11,27,33,99,297. The common divisors are: 1,3,9. And the greatest common divisor (GCD) is 9. Using the prime factorisation method: To find the LCM of 45 and 297, we will first find the factors of 45 and 297. Prime factors of 45: 3,3,5. Prime factors of 297: 3,3,3,11. Multiplying all the prime factors together, multiplying each number once for each of its occurrences, 3: three occurrences 5: one occurrence 11: one occurrence, So, we get, 3x3x3x5x11 = 1485 Thus, the lowest common multiple (LCM) is 1485. 4. (-3) (+9) + (-7) (+4) = (-27) + (-28) [(-a)(+b) = - (ab)] = -55 [(-a) + (-b) = - (a+b)] So, (-3) (+9) + (-7) (+4) = -55 5. Is the following pair of rational numbers equal? 1 6 5 30 To compare two rational numbers we first make their denominators equal. For this we will take the LCM of 5 and 30. Prime factor of 5: 5 and Prime factors of 30: 2, 3, 5 LCM of 5 and 30: 2 x3 x5 = 30 1 1 6 6 So that 5 = 5 6 = 30 1 6 Thus, 5 = 30 the two rational numbers are equal. B is the correct option. 6. Simplify: 46 1 7 ( ) 105 21 138 We will solve this step by step: Solving the brackets: 1 7 21 138 Making the denominators equal: Prime factors of 21= 3,7 and prime factors of 138= 2,3,23. LCM of 21 and 138 = 2x3x7x23 = 966. 1 46 7 7 4649 3 1 = = = 21 46 138 7 966 966 322 Now, 46 1 7 ( ) 105 21 138 = 46 1 ( ) 105 322 = 46 322 ( ) = 105 1 2116 15 So, 46 1 7 2116 ( ) . 105 21 138 = 15 7. Convert the mixed number to an improper fraction. 1 8 4 1 ( 8 4 ) +1 32+1 33 8 = = = 4 4 4 4 b [Since, a c = ( a c ) +b c 1 33 Thus, 8 4 = 4 . 8. Determine whether the number is rational or irrational? 8 8 . 8= 2 2 2=2 2 We can simplify Where 2 is rational and 2 is irrational. Further, product of a rational and an irrational number is again irrational. Thus, 8 is irrational. 9. Determine whether the number is rational or irrational. 2.678667866678... We know that every non-terminating non-repeating number is irrational. The above number is non- terminating and non-repeating. Thus, 2.678667866678... is irrational. 10. Simplify by rationalizing the denominator. 5 7 To rationalize the denominator, we multiply and divide the fraction by 5 7 5 7 5 7 = = 7 7 7 7 7 5 5 7 = 7 7 Thus, 11. Identify the property that the equation illustrates. 8 + (-8) = 0 The equation illustrates 'inverse property'. It says that a + (-a) = 0 D is the correct option. 12. Evaluate the following expression. 3 2 5 5 3 2 5 5 =5 3+( 2 ) m n m +n [Since a a =a 32 = 5 1 = 5 =5 3 2 Thus, 5 5 =5 . 13. Divide 5 3 32 3 5 52 =3 [Since, 32 am =a mn ] n a 3 = 3 = 27 14. Express the number 0.000067 in scientific notation. 7 . 5 0.000067 = 6.7 10 [decimal shifts 5 steps to the right]. 15. Expressing 200,000,000,000,000,000,000 in scientific notation 20 200,000,000,000,000,000,000 = 2 10 [decimal shifts 20 steps to the left]. 16. Determine the next term in the sequence and determine if the sequence is arithmetic, geometric or neither. 1,3,9,27,81 C is the correct option. The given sequence is geometric [a geometric progression is the sequence in which the ratio between any two successive terms is the same]. The common ratio is 3 So, the next term is 81 3 = 243. 17. For the following geometric sequence, find a7 5 5 5 5, 3 , 9 , 27 a7 = 5 81 [We observe that the denominators follow geometric sequence with common ratio 3, and the numerator is the same in each term i.e., 5] 18. Write the decimal 7.76 as a percent. To convert a decimal to a percent, multiply the decimal by 100. 7.76 = 7.76 100 = 776% 19. 17.2 is what percent of 80? Let 17.2 be x 80 . Then, x 80=17.2 x 80=17.2 100 x= 17.2 100 80 = 21.5% Thus, 17.2 is 21.5% of 80. 20. The number 28.35 is 20.25% of what number? Let 28.35 be 20.25% of x. Then, 20.25 x=28.35 20.25 x=28.35 100 x= 28.35 100 20.25 = 140 Thus, 28.23 is 20.25% of 140. 21. Future value, A = P(1 + r t) Given: A = $2,304 r = 7% t = 4 years To find: P 2,304 = P (1+ (0.07 4)) = P (1.28) 2,304 P = 1.28 =1800 So P = $1,800. 22. Formula for future value using compound interest, A = P(1+r)n Given: P = $7000 r = 7% t = 8 years n = number of compounding periods = 8 12 = 96 To find: A 0.07 A = 7000(1+ 12 )96 = 7000 (1.0058)96 = 7000 (1.75) = 12,334.78 Thus, the amount is $12,334.78. 1 23. Finance charge on last month's balance = $440 0.171 12 =$ 6.27 P = Previous month's balance + finance charge + purchases made - returns - payments = $(440 + 6.27 -340 + (140+115+190)) = $541.27 1 Finance charge for next month = $541.27 0.171 12 = $7.71 So, the finance charge is $7.71. 24. Finance charge per $100 = = Finance charge $ 100 Loan $ 600 $ 100 $ 3000 = $20 Thus, the finance charge per $100 = $20. 25. Angle j and t are alternate interior angles. So, E is the correct option. 26. Given: x = 40 To find: 1,23 Solution: We know that corresponding angles are equal. x and 1 are corresponding angles. Therefore, x = 1 = 40 Also, 1+ 2=90 40 +2=90 2=90 40 =50 1 and 3 are linear pairs. 1 + 3 = 180 40 + 3 = 180 3 = 180 40 3 = 140 So, m 1=40 , m 2=50 m 3=140 27. Given: Circumference of the circle = 42 metres Arc length = 7 metres. To find: Central angle. Solution: Circumference of the circle = 42 metres 2r = 42 metres r = 42/2 metres r = 21/ metres We know that central angle = Arc length/ Radius 7 = 21 / = 7 21 = 3 Thus, m ACB= 3 28. The measure of the interior angle of a regular polygon is given by 180 (n2) n where n is the number of sides. Here, n = 6 So, measure of interior angle = = 180 (62) 6 180 (4) 6 = 120 Thus, the measure of an interior angle of a regular polygon with 6 sides is 120 . 29. We know that in similar triangles: Corresponding sides are in the same ratio and the measure of corresponding angles are equal. From figure ABCDE we get that A=90 , E=90 , C=90 . We join DB and observe that CDB=54 . Then by angle sum property DBC=36 . So that, CBA=54 Measure of F= A=90 , G= B=54 , H = C=90 , I = D=36 , J = E=90 Length of FG = 195, GH = ED 195 ED 3 = , JF = 260 4 CB 195 CB 3 = , HI = 200 4 EA 195 EA 3 = 260 4 180 195 =135 , IJ = 260 1 (base )(height ) 2 30. Area of a triangle = Here, base = 22 cm and height = 18 cm 1 So, Area of the triangle = 2 (22)(18) = 198 cm2 Thus, the area of the triangle = 198 cm2. 31. Area of the shaded region = Area of the square - Area of the two semicircles Area of the square = a2 [where is the length of one side] = (16 m)2 = 256 m2 r2 Area of a semicircle = [where r is the radius of the circle] 2 = ( 8)2 2 = (3.14)(8)2 2 [as diameter= 16m, so radius= diameter/2= 8m] = 100.48 m2 Area of shaded region = [256 - (2 100.48)] m2 = (256 - 200.96) m2 = 55.04 m2 Thus, the area of the shaded region is 55.04 m2. 32. The distance from the home plate to diamond base will be calculated by Pythagoras theorem. Pythagoras Theorem states that in a right angled triangle: (Hypotenuse)2 = (Base)2 + (Perpendicular)2 Here we have to find the hypotenuse. Base = 79 ft and Perpendicular = 79 ft. 2 2 So that, (Hypotenuse)2 = 79 +79 = 12482 Hypotenuse = 12482 = 111.72 Thus, the distance from the home plate to diamond base = 111.72 33. Area to be tiled = Area of desk + Area of the pond Desk: length = 10ft, breadth = 100ft Area of desk = Length Breadth [as desk is a rectangle] = (10 100)ft2 = 1000ft2 Pond: Area of pond = Area of rectangle (length=80ft, breadth=10ft) + Area of rectangle (length=42ft, breadth=(14-10)ft) = [ ( 80 10 ) + ( 42 4 ) ] ft2 = (800 + 320)ft2 = 968ft2 Area to be tiled = (1000+968)ft2 = 1968ft2 Thus, the area to be tiled is 1968ft2. 34. Area of court = n r2 360 = where n is the angle an r is the radius. 45 2 (3.14)(18) 360 = 127.17ft2 Thus, area of the court is 127.2ft2. 35. C = 5 ( F32) 9 Here, F =199 5 So, C = 9 (19932) 5 = 9 (167) = 92.78 Thus, 199 F = 92.78 . 36. Area of the rectangular piece = 3.4 4.1 ) km2 = 13.94 km2 1 km2= 100 ha 13.94 km2= (13.94 100) ha = 1394 ha So, the measure of the rectangular piece is 1394 ha. 37. A is the correct option. We have to select two different members from S without repetition. So the sequence is MN, MP, MQ, MR, NM, NP, NQ, NR, PM, PN, PQ, PR, QM, QN, QP, QR, RM, RN, RP, RQ 38. B is the correct option. For the 1st place there are just 2 options Y and N, when the 1st place is occupied, we are left with one option. For 3rd place there are 5 options 6, 0, 2, 9 and 7, for the 4th place there are 4 options, for 5th 3 options, for 6th 3 options, for 7th 2 options and for 8th 1 option. So, number of licence plates that the police must investigate to find the car = 2 1 5 4 3 2 1 = 240 39. Since there are 8 members, so there are 8 options for the in charge of marketing, once the position is left there are 7 members left so 7 options for the in charge of research. Number of ways in which these two positions can be filled = 8 7 = 56. 40. We have to construct a 7-digit number. So, there are 7 places to be filled. Taking from left to right, for the 1st place there are 5 options viz. 5,6,7,8,9 as the number should be greater than 5. For the 2nd, 3rd, 4th, 5th and 6th place there are 10 options viz. 0,1,2,3,4,5,6,7,8,9. Finally for the 7th place there are 5 options viz. 1,3,5,7,9 as the number must be odd. Thus, the number of 7-digit numbers that can be constructed = 5 10 10 10 10 10 5 = 2500000 41. 12 ! 12 11 10 ! 12 11 = = =132 10 ! 10! 1 Thus, 12 ! =132 10 ! 42. There are total 36 outcomes. (a) To get a sum of 10 roll there are 3 outcomes: (4,6), (5,5), (6,4) number of favourable outcomes So, the probability of rolling a sum of 10 = total number of outcomes = 3 36 = (b) There are 33 way to fail. Odds against rolling sum of 10 = = 1 12 33 /36 3 /36 33 3 = 11 to 1 So, A is the correct option. 43. P(A B ) = 0.8, P(A) = 0.3 and P(A B ) = 0.15 To find: P(B) We know that P(A B ) = P(A) + P(B) - P(A B ) 0.8 = 0.3 + P(B) - 0.15 0.8 = 0.15 + P(B) 0.65 = P(B) Thus, P(B) = 0.65 44. From the table, Probability that the randomly selected consumer spends 3-9 hours = 572 1616 Probability that the randomly selected consumer has an annual income $40,000 to 521 $60,000 = 1616 So, Probability that the randomly selected consumer spends 3-9 hours or has an 572 521 1093 annual income $40,000 to $60,000 = 1616 + 1616 = 1616 45. There are total 52 cards in a deck and there are 4 ten cards. 4 3 12 3 Probability of drawing two tens without replacement = 52 52 = 52 = 13 So, Probability of drawing two tens without replacement = Probability of drawing two tens with replacement = 3 13 4 4 4 = 52 52 13 So, Probability of drawing two tens with replacement = 4 13 46. From the tree diagram P(N R) = 0.4 0.6=0.24 47. A is the correct option. This histogram corresponds to the given data. 48. The five-number summary of a data set consists of the five numbers determined by computing the minimum, Q1 , median, Q3 , and maximum of the data set. The data set in increasing order is 9,14,15,22,27,29,29,35,35,37,42,42,45,47,53 Minimum - 9 Q1 - The lower half of the data is 9,14,15,22,27,29,29 so the middle value of this data is Q1 that is 22. Median - There are total 15 values, so the middle data value is the 8th value that is 35. Q2 - The upper half of the data is 35,37,42,42,45,47,53, so the middle value of this data is Q2 that is 42. Maximum - 53 Thus, the five number summary is 9,22,35,42,53. For box-whisker plot B is the correct option. 49. Standard deviation = Variance Variance = Average of squared differences of the mean 47+51+ 47+48+ 49+ 48+ 45+ 49 =48 Mean = 8 Variance = ( 4847 )2 + ( 4851 )2 +(4847)2+(4848)2 +(4849)2+(4848)2 +( 4845)2+(4849)2 8 = 1+ 9+ 1+ 0+1+0+9+ 1 11 = 8 4 Standard deviation= 11 =1.658 4 50. Arranging the data in increasing order 4,5,5,7,8,10,14,18 Range = 18-4 = 14 4+5+ 5+7+8+10+14 +18 71 = =8.875 Mean = 8 8 Variance = 1351 8 Standard deviation= 1351 =12.99 8