MATHEMATICS SOLUTIONS Junior Certificate Higher Level Contents

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1 MATHEMATICS SOLUTIONS Junior Certificate Higher Level Contents Paper SEC Sample Paper 1 (Phase )... Sample Paper 1 Educate.ie Paper 1 (Phase )... 7 Sample Paper Educate.ie Paper 1 (Phase ) Sample Paper Educate.ie Paper 1 (Phase )... Sample Paper 4 Educate.ie Paper 1 (Phase )... 8 Sample Paper 5 Educate.ie Paper 1 (Phase )... 8 Sample Paper 6 Educate.ie Paper 1 (Phase ) Sample Paper 7 Educate.ie Paper 1 (Phase ) SEC Examination Paper 1 (Phase ) SEC Examination Paper 1 (Phase ) SEC Examination Paper 1 (Phase ) Paper Sample Paper 1 Educate.ie Paper (Phase )... 8 Sample Paper Educate.ie Paper (Phase ) Sample Paper Educate.ie Paper (Phase )... 9 Sample Paper 4 Educate.ie Paper (Phase ) Sample Paper 5 Educate.ie Paper (Phase ) Sample Paper 6 Educate.ie Paper (Phase ) Sample Paper 7 Educate.ie Paper (Phase ) SEC Examination Paper (Phase ) SEC Sample Paper (Phase ) SEC Examination Paper (Phase ) SEC Examination Paper (Phase ) SEC Examination Paper (Phase 1) SEC Supplementary Questions

2 015 SEC Sample Paper 1 (Phase ) 1. (a) P Q = {1,,, 4, 5, 6} All the numbers in P or Q Q R = {5, 6} Numbers common to both Q and R P (Q R) = {1,, 4, 5, 6} P union with the last answer Use (P Q) (P R) As in 4( + 5) = 4() + 4(5) in real numbers P Q = {1,,, 4, 5, 6} and P R = {1,, 4, 5, 6, 7} (P Q) (P R) = {1,, 4, 5, 6} which is the same as P (Q R). (a) A B C = {,, 4, 5, 6, 8, 9, 10, 1, 14, 15, 16, 18, 0, 1,, 4, 5, 6, 7, 8, 0} Therefore, the cardinal number of the elements not in this set is 9 = 7 i.e. {7, 11, 1, 17, 19,, 9} Two divisors Divisors: numbers which divide in evenly (c) Prime numbers. (a) U T C 4 8 D (c) = (a) % of 5000 = 150 less tax of 41% = = = All people in Tea or Coffee ovals Total No. of people Soft drink oval is excluded gives interest of x at x% interest rate. Less the tax of 41% leaves interest of 0 015x x = x = x = = 4 5% Reducing by 41% = 59% remaining i.e. multiply by x multiplied by (a) Jerry is treating 0 5 as 100% of the cost of the meal rather than as 109% of the cost of the meal. Mathematics Junior Certificate 1

3 0 5 = 109% of the cost of the meal before VAT 0 8 = 1% 8 = the cost of the meal before VAT At VAT rate of 1 5% this meal would cost = Sample SEC P1 6. (a) If she spends 5 5 x 50 5 will allow the voucher to be used. 5 y 60 All the cash she has She will pay 50 and get a 10 discount. 7. (a) Let x units be the side of the square in the lower left corner. You could The other square would then have a side length of (10 x) units. also sketch the The joint areas of the squares can then be given by x + (10 x). graph to see its Expanding and tidying gives an area function A = x 0x minimum. By completing the square this function can be written as A = (x 5) This function has a minimum turning point at (5, 50), and hence the minimum value for area is 50 unit s. d x 8. (a) (10 x) The side lengths of the right-angled triangle in the bottom right corner are d, x and (10 x). Applying Pythagoras s theorem gives d = x + (10 x). Q.E.D. Higher Level, 015 SEC Sample, Paper 1

4 Stage Perimeter Stage 1 = 4 + 0(8) Stage = 4 + 1(8) Stage = 4 + (8).. Stage n = 4 + (n 1)8 = 8n 4 Drawing a table of values can help when searching for a formula. (c) Stage Area Stage 1 = Stage = + 1 Stage = +.. Stage n = n + (n 1) = n n + 1 (d) Quadratic because the second differences are constant (all equal to 4) 9. x + 8x + x + 10x x + x + 8x + x = 14 4 x + 6x + 80 = 14 4 x + 6x 6 = 0 x x 8x x 8 m x Find the area of each individual section. 10x 10 m 80 10x x 8x x 4 Mathematics Junior Certificate

5 (d) (8 + x)(10 + x) = x + 4 x = 14 4x + 6x 6 = Sample SEC P1 Width = 8 + x Add the distances. x Length = 10 + x x 10 m 8 m x Add the distances. (e) 4x + 6x 6 = 0 (x )(x + 1) = 0 x = 0 and x + 1 = 0 x = 1 5 and x = 10 5 x = 1 5 m (f) (g) Let x = 1 : Area is 10 by 1 = 10 m not true (too small) Let x = : Area is 1 by 14 = 168 m not true (too big) Tony would then have to try values between 1 and m etc. Kevin s or Elaine s Their algebraic method is faster and more accurate. Tony may not have found the exact answer using his method. 10. (a) R(, ): = () + a() + b = 4 + a + b a + b = 1 S( 5, 4): 4 = ( 5) + a( 5) + b 4 = 5 5a + b 5a + b = 9 a + b = 1 The points are on the curve so substitute them into the equation. 5a + b = 9 a b = 1 7a = 8 a = 4 b = 9 Solve simultaneous equations. 4 Higher Level, 015 SEC Sample, Paper 1 5

6 (c) (0, 9) (d) x + 4x 9 = 0 x = 4 ± (4) 4(1)( 9) (1) x = 4 ± 5 x = 1 6 or x = 5 6 Quadratic formula See page 0 of Formulae and Tables. 11. f(x) g(x) Roots of f: 0 and 4 Roots of g: 1 and 5 Roots: points where the curve crosses the x-axis (c) 5 h(x) (d) Complete the square on the RHS. x 10x (x 5) Comparing with the LHS gives p = 5 (e) x = 5 Mirror line which would allow the graph to fold onto itself 1. Some x values have two y outputs. OR It fails the Vertical Line Test. 6 Mathematics Junior Certificate 5

7 Educate.ie Sample 1 Paper 1 Sample 1 Educate.ie P1 1. (a) {1,, 4,, 6, 7, 1} A B {, 7, 1} (c) {1, 4} (d) {6} This can also be written as (A C ) c (e) {, 7, 1,, 5, 14} C. (a) 75 x is a common factor. (5 x 1) (5x + 1)(5x 1) Difference of two squares 14 x x 5 Quadratic factors (7x + 1)(x 5) (c) 4p q 1pq + 5p p is a common factor. p(4 q 1q + 5) Quadratic factors p(4q 1)(q 5). (a) Even number > Sum of primes = 7 + also: 14 = 11 + also: (c) = 100, + 97 = 100 are two possible ways. Are there any more? Other ones are , , Higher Level, Educate.ie Sample 1, Paper 1 7

8 (d) 1,000,000,000 = ,000,000,000,000 = (e) = 100 trillion 4. (a) squares Explanation: Shape 1: 1 = Shape : = 6 Shape : 4 = 1 Shape 4: 5 4 = 0 Shape 5: 6 5 = 0 Shape n: (n + 1) n = n + n (c) n + n = 90 n + n 90 = 0 (n + 10)(n 9) = 0 n = 9 5. (a) x x x 1 Get the common denominator. (x 1)(x ) 5(x + 1) (x + 1)(x 1) x x + 5x 5 x x 8x 1 x 1 x x x 1 = 1 x 8x x = 1 1 ( x 8x )( x 1) x 1 = 1( x 1) x 8x = 1( x 1) ( x 8x ) = x 1 x 4x 9 = x 1 x 4x 8 = 0 x 1x 4 = 0 a = 1, b = 1, c = 4 x = b ± b 4ac a 8 Mathematics Junior Certificate

9 x = 1 ± 144 4(1)( 4) x = 1 ± 160 x = 1 ± 4 10 x = 6 ± 10 Use a calculator to get the square root of 160. Sample 1 Educate.ie P1 6. (a) John s Wages Hours worked Wages Orla s Wages Hours worked Wages Wages ( ) Orla John Hours worked (c) (d) John: y = 7x Orla: y = 6x Wages ( ) Orla 0 John Hours worked (e) 6 = 18 6 per hour = = 18 (f) From the graph, they earn the same for 0 hours work. Also: John 7(0) = 140 Orla: 6(0) + 0 = 140 Higher Level, Educate.ie Sample 1, Paper 1 9

10 (g) For 5 hours, John earns 5 7 = 175 Gross tax on r % = (r 100) 175 = 1 75r Tax payable = gross tax tax credit = 1 75r 0 = r = r = = 0% 7. (a) is not in the solution so the clear circle indicates this. (c) is in the solution so the shaded circle indicates this. (d) (a) ( ) + ( ) + ( ) (c) = times heavier 10 Mathematics Junior Certificate

11 9. Distance from Limerick (km) Sample 1 Educate.ie P1 11:00 1:00 1:00 14:00 15:00 16:00 Time 17:00 18:00 19:00 0:00 1:00 (a) (i) 1:00 (ii) 15 minutes (c) (d) Speed = distance time = 100 km/h 0 km See graph (e) 100 km = 6 miles As 1 km = 0 6 miles gallons in 6 miles This is gallons in 6 miles. So it is = 5 miles/gallon. 10. a is as then both lines will be parallel. b can have any value provided a = for both lines to be parallel (x + ) + (x) = x + (x) + (x + 1) + x + 5 = 8x + 5x = x = 0 4 cm 1. (a) T = π L g T = ( 14) 9 8 T = 8 seconds Higher Level, Educate.ie Sample 1, Paper 1 11

12 T = π L g T = 4 π L g Square both sides to get rid of the square root. Multiply both sides by g. g T = 4 π L Divide both sides by 4 π. L = g T 4 π 1. (a) A B C D E F G H Number 47 7 o 1 5sin % (1 54) Decimal number F H G A C B E D (c) (i) C H = 5 5 (ii) 5 5 = 5 5 = 7 5 = 1 4 C H = 7 15 = = (a) f (1) = (1 ) = 1 and f ( 1) = [ ( 1) ] = f (1) + f ( 1) = 1 + = 4 f ( ) = ( ) = 5 : f ( ) = [ ( ) ] = 15 Is f ( ) > f ( ) 5 > 15 True as shown 15. (a) Yes Couples on the graph are (0, 0), (, 10), (4, 16), (6, 18), (8, 16). First differences between 0, 10, 16, 18 and 16 are 10, 6,,. The second difference between these are 4, 4, 4 which is a constant. 1 Mathematics Junior Certificate

13 (c) Two points on the graph are (, 10) and (4, 16). f(t) = at + bt (, 10) f() = a() + b() = 10 4a + b = 10 (4, 16) f(4) = a(4) + b(4) = 16 16a + 4b = 16 4a + b = 10 Sample 1 Educate.ie P1 16a + 4b = 16 4a + b = 10 4a + b = 4 b = 6 a = 1 f(t) = 1 t + 6t (d) Temperature C g(t) = x mins mins 4 mins 6 mins 8 mins 10 mins minutes and 8 minutes Higher Level, Educate.ie Sample 1, Paper 1 1

14 Educate.ie Sample Paper 1 1. Description of number Number Square Number 5 5 Reciprocal of a Whole Number 1 1 over Prime Number 1 Irrational Number A number with itself and 1 as the only factors A number that cannot be written as a simple fraction Cubed Number 8 Negative Integer 7 A negative whole number Index Form of a Number 4 A number to a power Estimate for Number pi 14. F = P(1 + i) t See page 0 of Formulae and Tables. F = 5000(1 05) 5 F = Interest = Tax on interest = 5% of = Value of investment after tax = = (a) Points on p Points on k R S D C B A F 14 Mathematics Junior Certificate

15 Points on p Points on k R D A B S C F Points on w Sample Educate.ie P1 4. (a) 5(x + ) [4 (x )] + 5x + 15 [4 x + 6] + 5x x x 1 5 x + 4 x 5(x) (x + 4) (x + 4)(x) 5x 6x 1 x + 4x x 1 x + 4x When x = 1 x 1 x + 4x, becomes Get the common denominator and be careful with the minus between the two fractions. ( 1 ) 1 ( 1_ ) + 4 ( 1_ ) Use brackets when substituting. 1 1 ( 1_ 4 ) + 4 ( 1_ ) 1 1 1_ Higher Level, Educate.ie Sample, Paper 1 15

16 5. (a) X Factor Britain s Got Talent 15 [(9 x) + (6 x) + x] 6 x 9 x x 0 [(9 x) + (7 x) + x] 7 x 10 [(6 x) + (7 x) + x] The Voice 0 [(9 x) + (7 x) + x] = 0 [16 x] = 4 + x 15 [(9 x) + (6 x) + x] = 15 [15 x] = x 10 [(6 x) + (7 x) + x] = 10 [1 x] = + x (4 + x) + (x) + ( + x) + x + (6 x) + (9 x) + (7 x) + = x + x + = 50 x + 46 = 50 x = 4 6. (a) Salary = % = 0% of 800 = % = 41% of ( ) = 1 0 Gross tax = = Net tax = gross tax tax credit = = Salary = % = 7 5% of = 4875 Net salary = gross salary (net tax + PRSI) Net salary = ( ) Net salary = ( 87) Net salary = (a) 5 x 5 x Subtracting from each part x 5 x 5 When multiplying by minus, don t forget to change the inequality signs Mathematics Junior Certificate

17 + x < 9 + x < 9 Subtracting from each part 6 x < 6 x < Divide each part by Sample Educate.ie P1 8. (a) Floor 1 st nd rd 4 th 5 th 6 th Cost ( millions) Cost ( m) Floor number (c) Estimation: 4 million euro (d) 1 st Floor = = (0 1) nd Floor = = (0 1) rd Floor = = (0 1) 4 th Floor = = (0 1) 5 th Floor = = (0 1) n th Floor = (n 1)0 1 Cost for n th Floor = 0 1n Higher Level, Educate.ie Sample, Paper 1 17

18 (e) (i) T = n [ a + (n 1)d ] n = 8: a = 0 5: d = 0 1 T = n [a + (n 1)d ] T = 8 [(0 5) + (8 1)(0 1)] T = 14[1 + 7] T = 14[ 7] T = 51 8 million euro (ii) T = n [a + (n 1)d ] = 0 n =?: a = 0 5: d = 0 1 n [(0 5) + (n 1)(0 1)] = 0 n [ n 0 1] = 0 n[ n 0 1] = n n = 40 n + 9n 400 = 0 (n + 5)(n 16) = 0 n = Floors (f) Total cost = 51 8 million 1 = 0 87 x = 51 8 million x = 51 8 million 0 87 x = million euro 9. (a) 4 x 9x = 0 x(4x 9) = 0 x = 0 or 4x 9 = 0 x = 0 or x = x 9 = 0 (x + )(x ) = 0 x + = 0 or x = 0 x = or x = (c) 4 x 16x 9 = 0 (x + 1)(x 9) = 0 x + 1 = 0 or x 9 = 0 x = 1 or x = 9 18 Mathematics Junior Certificate

19 10. Graph A B C D E F Story (a) x + y = 50 x + y = 410 -cent coins Sample Educate.ie P cent coins 90 one-cent coins and 160 two-cent coins (c) x + y = 50 x + y = 410 y = 160 y = 160 x = 50 x = (a) (i) (ii) Given that (4 x y ) (xy) = p Simplifying we get 4 x y x y = p 4x = p The question asks to show that x y 4 (xy) is the reciprocal of p x y Simplifying 4 x y 1 4x = 1 p which is the reciprocal of p Higher Level, Educate.ie Sample, Paper 1 19

20 1. (a) a a Common factor a a (a ) a a b b (a b ) + ( a b) Grouping (a + b)(a b) (a + b) (a + b)(a b 1) 14. x +4 x 6x + +1 x + 4 or Use long division or Factorise 6 x + 17x (a) {, 4, 6, 8} {4, 6, 8, 10} (c) {, 4, 6, 8, 10} 16. (a) g(x) = 1 x g(x 1) = 1 (x 1) = 1 x + = 4 x g(x) g(x 1) When graphed they form parallel lines. 0 Mathematics Junior Certificate

21 (c) (i) g(x) = 1 x + k = 1 x k = 5 g(x) + k 4 Sample Educate.ie P (ii) 5 g(x) + k 4 y = 4 x = Higher Level, Educate.ie Sample, Paper 1 1

22 Educate.ie Sample Paper 1 1. (a) 4 = 5 = 9 Highest common factor =. (a) If you have a Casio Calculator (NATURAL V.P.A.M) to find the prime factors of 4. Type in 4 then press the equals button. Then press and the prime factors will show up as 4 x. 1 = means of 1 of the full circle and this is, from the diagram, 1 of the circle (a) Total paid = $9 Use ratio 1 : $1 8 = x : $9 1 = $1 8 1 x = $9 1 8 = x 9 Therefore x = (9 1) 1 8 = 8 41 Number of months = May, June, July = F = P(1 + i ) t F = 8 41(1 019 ) F = See page 0 of Formulae and Tables. Mathematics Junior Certificate

23 4. (a) 8 + x x + x = 00 4x = 00 4x + 7 = 00 4x = 18 x = = 8 A 8 x x x 1 C B 8 U 5. (a) 16 6 (c) (d) 6 Sample Educate.ie P1 (e) 17 (f) Copper: 60% of 4 5 g = 55 g Copper: 0% of 4 5 g = 0 85 g Copper: 0% of 4 5 g = 0 85 g Statement Always True Never True Example If a, b Z with both a and b < 0, then (a + b) N If a = and b = : ( ) = 5 N If a, b Z with both a and b > 0, then (a + b) N If a = and b = : ( + ) = 5 N If a, b Z with a < b, then (a b) < 0 If a = and b = 1: ( + 1) = 1 True. If a, b Z with both a and b < 0, then a b N If a = and b = : ( ) = 6 N If a, b Z with both a and b < 0, then ( a + b ) < 0 If a = and b = : ( + ) = 1 is not less than zero. 8. (a) Time (sec) Volume (l ) First change 4 4 First change (difference) is a constant therefore linear. Higher Level, Educate.ie Sample, Paper 1

24 Volume (litres) Time (seconds) (c) 1 litres See broken lines on graph above. (d) litres/sec Rate of change = Slope of Line = Rise Run = 1 = (e) (f) Volume = 0 (number of seconds) V = 0 s Height (cm) Time (seconds) (g) Volume of cone = 1 p r h See page 10 of Formulae and Tables. 1 ()h = 90% of 0 litres h = 7 litres h = 7 = metres 9. (a) n + (n + 1) + (n + ) n + n n + n + This expression is divisible by. (n + ) = n + 1 (n ) + (n 1) + n n + n 1 + n n This expression is divisible by. (n ) = n 1 (c) n + (n + 1) + (n + ) + (n + ) n + n n + + n + 4n + 6 This expression is not evenly divisible by 4. 4 Mathematics Junior Certificate

25 10. (a) (i) a ac ab + bc Grouping ( a ac) + ( ab + bc) a(a c) b(a c) (a b)(a c) (ii) 5 x + 5x 0 Quadratic factors 5( x + x 6) 5(x + )(x ) (iii) 4 x y Difference of two squares (x + y)(x y) x y = 4 Difference of two squares (x + y)(x y) = 4 As x + y = ()(x y) = 4 x y = 8 x y = 16 Sample Educate.ie P1 11. (a) (i) x = 5x x 5x = 0 x(x 5) = 0 x = 0 or x 5 = 0 x = 0 or x = 5 (ii) x + 4 = 8x 8 x 8x + 1 = 0 (x )(x 6) = 0 x = 0 or x 6 = 0 x = or x = 6 (iii) 4x 7 x = 0 7 x + 4x = 0 (7x )(x + 1) = 0 7x = 0 or x + 1 = 0 x = or x = 1 7 Higher Level, Educate.ie Sample, Paper 1 5

26 x + x 15 = 0 a =, b =, c = 15 x = b ± b 4ac a x = ± () 4()( 15) () x = ± x = ± 14 4 x = ± 1 4 x = 1 ± 1 1 ± x = x = or x = x = 88 or x = 88 x = 8 or x = (a) (c) 1000 cm x 1000 x x = 1000 x x = 1000 x (x + 0) x(x + 0) = 10 m = 1000 cm 1000x + 75(x)(x + 0) x(x + 0) 1000x = 1000x + 75 x + 50x 75 x + 50x = 0 x + 0x 400 = 0 (x + 40)(x 10) = 0 x + 40 = 0 or x 10 = 0 x = 40 or x = cm 6 Mathematics Junior Certificate

27 1. (a) = = = 5 million (c) Day 1 4 = ( ) = 6 Day 6 = 7 Day 7 = 8 Day 4 8 = 9 Day 5 9 = 10 Critical Value 14. f (x) = x 1 = 54 x 1 = 7 x 1 = x 1 = x = 4 Sample Educate.ie P1 15. Functions x x + 5 x x x x + Graph Higher Level, Educate.ie Sample, Paper 1 7

28 Educate.ie Sample 4 Paper 1 1. U U U A B A B A B A/B B/A A B U U U A B A B A B A B A or A c (A B)' or (A B) c. (a) 6 See diagram for explanation. A B See diagram for explanation. A B (c) See diagram in part (a). 8 Mathematics Junior Certificate

29 . (a) x 10x 4 (x 1)(x + ) Quadratic factors abx + y by ax abx ax + y by ax(b ) y( + b) (b )(ax y) Grouping (rearrange first) (c) x 4 16 Difference of two squares (x ) (4) (x + 4) (x 4) Difference of two squares (x + 4)(x + )(x ) 4. (a) p + 1 p, p True False Reason: When you add 1 to any integer you will always get a larger integer. p + 1 p, p True False Reason: p is always greater than p for p because when you square a positive or negative number you will always get a larger positive number. Sample 4 Educate.ie P1 (c) p + 1 p, p True False Reason: When you add 1 to any real number (including a fraction) you will always get a larger real number. (d) p p, p True False Reason: If p is negative, this statement is not true. (e) p p, p True False Reason: p can t be negative so this statement is always true. 5. (a) % of = 450 Balance = = % of = 5000 Total Levy = = 5450 % of = % of 5980 = 9 0 Balance = ( ) = % of = Total USC = = Higher Level, Educate.ie Sample 4, Paper 1 9

30 (c) 0% of = 860 Balance = = 00 41% of 00 = 951 Total gross tax = = Net tax = gross tax tax credits Net tax = = 1 9 (d) Net salary = (pension levy + USC + income tax) Net salary = ( ) Net salary = Length (cm) Width (cm) Area (cm ) (a) (i) Length against width: Prediction: Linear graph (ii) Length against area: Prediction: Quadratic graph Explanation for (i) The first difference (change) is a constant. Explanation for (ii) The second difference (change) is a constant. 0 Mathematics Junior Certificate

31 Width (cm) Length (cm) (c) (d) Width + length = 5 cm Area (cm ) Sample 4 Educate.ie P Length (cm) (e) Maximum area = 156 cm Higher Level, Educate.ie Sample 4, Paper 1 1

32 See page of Formulae and Tables. 7. (a) Number/Set Natural Integers Rational Irrational \ Real π Area = 7 = 1 8. Distance (km) Graph 1 (a) Distance 8 km Time 8 minutes (c) Speed 8 km in 8 minutes = 60 km/h Time (minutes) Mathematics Junior Certificate

33 Distance (m) Graph (a) Distance 0 m Time 6 seconds (c) Speed 0 metres in 6 seconds = 18 km/h Time (secs) Graph 16 Distance (km) (a) Distance = 4 km Time 1 minutes (c) Speed 4 km in 1 minutes = 10 km/h Sample 4 Educate.ie P1 Time (minutes) 18 Graph 4 16 Distance (m) (a) Distance 0 m Time 10 seconds (c) Speed 0 km/h (at rest) Time (secs) (a) Motor tax = 0 Insurance = 908 Petrol (1 cent)( km) = 080 Oil (0 16 cent)( km) = 5 60 Tyres (1 8 cent)( km) = 88 Servicing (1 8 cent)( km) = 88 Repairs (6 7 cent)( km) = 107 Total = Higher Level, Educate.ie Sample 4, Paper 1

34 Motor tax = 90 Insurance = 940 Petrol (1 cent)( km) = 40 Oil (0 16 cent)( km) = 8 80 Tyres (1 8 cent)( km) = 7 60 Servicing ( cent)( km) = 96 Repairs (6 6 cent)( km) = 1188 Total = (c) Cost: 01 Cost: 014 Motor tax 0 Motor tax 90 Insurance 908 Insurance 940 Total 110 Total (a) Increase = = 10 % Increase = (10 110) 100 = 9.9% = 10% x 5 y = Simplify 6x 10y = x y 1 = 4 6x 10y = 45 x 4y = 1 6x 10y = 45 6x 8y = 6 Simplify y = 19 y = 9 5 6x 10y = 45 6x 10( 9 5) = x = 45 x = 5 x 5 y = When x = 5 and y = 9 5 ( 5 ) ( 9 5) = = 9 = = ( x 9 ) ( 4y 4 ) = x 10y = 45 x 4y = 1 4 Mathematics Junior Certificate

35 11. (a) Area = (x + 1)(x) = 90 x + x = 90 x + x 90 = 0 (x + 10)(x 9) = 0 x = 9 10 cm 9 cm (Diagonal) = Diagonal = 181 cm Area = 1 (4x + ) ( x ) = 18 x + x = 6 x + x 6 = 0 (x + 9)(x 4) = 0 x = 4 Sample 4 Educate.ie P1 p cm cm 18 cm 9 cm p = 9 + p = 85 Perimeter = Perimeter = ( 85 = 40 ) b ± b 1. (a) x = 4ac a See page 0 in Formulae and Tables. a = 1, b =, c = 8 x = ( ) ± ( ) 4(1)( 8) (1) x = ± 9 + x = ± 41 x = ± Higher Level, Educate.ie Sample 4, Paper 1 5

36 x = or x = x = or x = x = 4 70 or x = 1 70 x = or x = x = 4 70 or x = 1 70 (t + ) (t + ) 8 = 0 x = t + t = x t = or t = t = 0 9 or t = 4 1. (a) (i) Drawn below (ii) Drawn below f(x) 4 5 g(x) Where the two graphs intersect: x = 1 and x = (c) x 7 = x 4 x x = 0 (x + 1) (x ) x = 1 x = 6 Mathematics Junior Certificate

37 14. (a) A and B are the roots of x 8x + 1 = 0 (x 6)(x ) = 0 A(, 0) B(6, 0) A(, 0) is on the function k (x) y = x + b 0 = + b b = The function k(x) is x + Sample 4 Educate.ie P1 Higher Level, Educate.ie Sample 4, Paper 1 7

38 Educate.ie Sample 5 Paper 1 1. (a) 0 6. is a rational number because it can be expressed as a fraction. π = Real Numbers, (c), 0 7., (144) , 5 0 1, 5, 1, π, (144), (a) : 5 : 1 means 0 parts in total 0, of = of = 650 = of = = of = = and 0 5% on 8450 = 8450(0 05) = 11 5 After tax the person had = See page of Formulae and Tables.. (a) U U U P Q P Q P Q R R R (P Q) / R (P Q R) R / (P Q) 8 Mathematics Junior Certificate 1

39 U U U P Q P Q P Q R R R (P Q R) P / (Q R) (Q R) / P P Q U R 4. (a) Pattern Rows of disks Columns of disks Number of disks 1 st 1 1 = nd 4 4 = 8 rd 6 6 = 18 4 th = 5 th = n th n n n n = n Sample 5 Educate.ie P Disks Pattern Higher Level, Educate.ie Sample 5, Paper 1 9

40 (c) Explanation on diagram above (i) (ii) 18 disks 10 th Pattern (d) (i) (8) = 18 (ii) n = 00 n = 100 n = 10 (e) n n = n 5. (a) (i) 4ab A = 0ab 5b (ii) 0ab 4ab = 5b x + 1 A = x + 16x + 48 x + 4 x + 16x + 48 (x + 1) (x + 4) (i) x + 5 4x + Area = (x + 5)(4x + ) = 8x + 6x + 0x + 15 = 8x + 6x + 15 (ii) x + 5x + x + Area = (x + 5x + )(x + ) = x + x + 5x + 15x + x + 6 = x + 8x + 17x Mathematics Junior Certificate

41 6. (a) (i) x > 1 (ii) x < and x (i) 0x < 10x + 60 (ii) 0x < 10x x 10x < x < 10 x < 1 0 days 7. (a) Athlete A Athlete C overtook Athlete B approximately 0 km from the start. Then Athlete B overtook Athlete C at about the 56 km mark. (c) Athlete A: Distance = 6 km: Time = 00 minutes: Speed = Distance Time = 18 6 km/h (d) (e) 8. (a) Athlete B: Distance = 6 km: Time = 5 minutes: Speed = Distance Time = 16 5 km/h Athlete C: Distance = 6 km: Time = 5 minutes: Speed = Distance Time = 15 8 km/h 14 minutes The cycle, because they finished last in the other two legs. x + 4 x 1 x + 5 x + 1 (x + 4)(x + 1) (x + 5)(x 1) (x 1)(x + 1) x + 5x + 4 x 4x + 5 (x 1)(x + 1) x + 9 x 1 x + 4 x 1 x + 5 x + 1 = x x 1 x + 9 x 1 = x x 1 x + 9 = x x x 9 = 0 a = 1, b = 1, c = 9 x = b ± b 4ac a x = ( 1) ± ( 1) 4(1)( 9) (1) See page 0 of Formulae and Tables. Sample 5 Educate.ie P1 Higher Level, Educate.ie Sample 5, Paper 1 41

42 x = 1 ± x = 1 ± 7 x = or x = 1 7 x = 54 or x = (a) Equation 1: 5x + 4y = 9 Equation : x + 6y = 8 4 5x + 4y = 9 x + 6y = x + 1y = 7 6 6x + 1y = x = 10 8 x = 1 0 5(1 0) + 4y = y = 9 4y = y = 0 8 Ice creams cost 1 0 and smoothies cost (a) V i = R = R R = V i = R V = Ri V R = i 11. See how much interest 100 would earn. F = P(1 + i) t See page 0 of Formulae and Tables. F = 100(1 1) 5 F = Interest = = x x = = Mathematics Junior Certificate

43 (a) a = 4, b = 1, p = x =, x = 1 (c) In grey on graph above (d) x = 4 and x = 1 x + 4 = 0 and x 1 = 0 (x + 4)(x 1) = 0 x + x 4 = 0 p = 4 (e) x + x = 0 x(x + ) = 0 x = 0 or x = Sample 5 Educate.ie P1 1. (a) f (0) = 41 f () = 47 f (6) = 71 41, 47, 71 are all prime numbers. f (40) = 1601 f (41) = is also a prime number but 1681 is not as it is a square number. (c) f() + f(6) = = 118 f( + 6) = f(9) = 11 f() + f(6) f( + 6) Higher Level, Educate.ie Sample 5, Paper 1 4

44 14. Shown on diagram below Axis of symmetry (a) x = and x = 1 7 (c) Axis of symmetry: x = Mathematics Junior Certificate

45 Educate.ie Sample 6 Paper 1 1. (a) U Facebook Twitter (15 + x) x 1 11 ( + x) 4 Google Plus x x + 11 x + 4 = x x + 8 x + 4 = x + 4 = x = 40 x = 6. (a) 50x is a common factor. (5x 1) Difference of two squares also (5x + 1)(5x 1) Sample 6 Educate.ie P1 p 9x 6x p Grouping (p 9x ) + ( 6x p) (p + x)(p x) (x + p) (p + x)(p x ) (p + x)(p x ) (c) x + 4x 4x x is a common factor. x(x + 4x 4) Quadratic factors x(x )(x + ) Higher Level, Educate.ie Sample 6, Paper 1 45

46 . (a) A 5 k y p B m 8 (i) 4 (ii) 5 (iii) 1 (iv) 8 Statement True or False Reason #(A/B) = #(B/A) False #{5, k, y} #{m, 8,, } (A B) (A B) True {p} is a subset of {5, k, y, p, m, 8,, } #(A B) = #B False 1 5 #[(A/B) (B/A)] = 7 True [(A/B) (B/A)] = {5, k, y, m, 8,, } which has 7 elements. 4. Choice (4, 1) Reason: It satisfies both equations: x y = 11: (4) ( 1) = = 11 x + y = 10: (4) + ( 1) = 10 1 = (a) Number ( 1 1_ 4 ) A B C D E F G H 1 tan 45 5% (1 5) Decimal Number (c) 1 = 1 1 = 4 = (d) (i) (ii) = = 19 1 Using calculator: = Mathematics Junior Certificate

47 6. (a) Day Brían Máire Free texts Brían Máire Day 1 (c) (d) (e) Brían: Total free texts = 0 + times the number of days Máire: Total free texts = times the number of days Brían: T = 0 + d: d = number of days, T = total number of free texts Máire: T = d: d = number of days, T = total number of free texts Free texts Máire Brían Sample 6 Educate.ie P Week Higher Level, Educate.ie Sample 6, Paper 1 47

48 (f) (i) Day 10 (ii) 66 texts (g) 50 (iii) = 0 7. (a) = x + 5 x x x = x + 5 x x = x + 5 x x 5 = 0 (x 5)(x + 1) = 0 x = 5 or x = 1 x 1 x = (x ) (x 1) = (x 1)(x ) (x 1)(x ) (x 1)(x ) x 4 x + = (x x + ) x 1 = x + 6x 4 x 7x + = 0 (x 1)(x ) = 0 x = 1 or x = 8. Depth Time Depth Time 48 Mathematics Junior Certificate

49 Depth Time Depth Time 9. (a) (i) (ii) (iii) (iv) < x < x Solution set: { 4,,, 1, 0, 1, } Sample 6 Educate.ie P1 10. (a) 1 yottagram = 10 4 grams = tonne = 10 kilograms 104 kg = 10 = 101 kg 0 tonne = 0 10 kilograms = decagrams = 10 6 decagrams (c) (d) 1 cow = 0 5 tonne 4 cow = 1 tonnes kg kg = kg tonnes Higher Level, Educate.ie Sample 6, Paper 1 49

50 11. (a) z = xy z y z = y = xy z y z y = xy z z y xy = z y(z x) = z z y = z x z y = x z y = z x z ( 1 4 ) ( ) ( 1 4 ) y = ( 1 16 ) ( y = ( 1 y = ( 1 y = ( 1 y = 1 16 ) ( 4 16 ) ( 16 ) ( ) ) 16 ) ) 1. Joint salary before tax = % of = % of ( ) = 41% of = Total gross tax = = Net tax = gross tax tax credits Net tax = = Net salary = gross salary net tax Net salary = = (a) x = x = 0 x = 1 x + 1 = 0 (x )(x + 1) = 0 6x x = 0 a = 6, b = 1, c = 50 Mathematics Junior Certificate

51 14. x = x = 0 x = x + = 0 [ x ][ x + ] = 0 x + x x = 0 x = 0 a = 1, b = 0, c = (a) 6 metres 6 5 metres 15. Functions x + x + x x (x ) x Graph Sample 6 Educate.ie P1 Higher Level, Educate.ie Sample 6, Paper 1 51

52 Educate.ie Sample 7 Paper 1 1. Description of number Numbers Natural Numbers 7, 5 0, 144 Integers 9, 7, 5 0, 144 Prime Numbers 7 Irrational Numbers, π Squared Number 144 Negative Integer 9 Reciprocal of a, where a 1 5, 6 1, 5 0 Recurring Decimal. See page of Formulae and Tables.. (a) 5 (c) 15 A B U x 17 x 0 x 5 17 x + x + 0 x = 45 x =. (a) (A B C) A B U C 5 Mathematics Junior Certificate

53 (A B C) c A B U C (c) (B C) \ A A B U C (d) (A B) C A B U C 4. (a) 4xy y n = (x)(y 6 ) 4xy y n = 4xy 6 4xy n = 4xy 6 y n = y 6 n = 6 n = 4 Sample 7 Educate.ie P1 x 7x 6 x x 7x 6x 1 x 7x 6 x + x + 7x + 7x + 6x + 6 = x + 8x + 1x + 6 Higher Level, Educate.ie Sample 7, Paper 1 5

54 5x 6x + 1 (c) x + ) 10x + x 16x + 10x + 15x 1x 16x + 1x 18x x + x + or 5x 6x 1 x 10x 1x 6x 15x 7x x 5. (a) F = P(1 + i) t See page 0 of Formulae and Tables. F = 1 (1 09) 6 F = cm F = P(1 + i) t 44 = 1 5(1 + i) 10 (1 + i) 10 = (1 + i) 10 = i = i = i = r i = 100 = r = 4 89 r = 5% 6. (a) Total cards = = 18 Working from the top down 1 st Row (1) = nd Row () = 6 rd Row () = 9 10 th Row (10) = = 165 cards 54 Mathematics Junior Certificate

55 (c) (1) + () + () (65) [ ] [ ] [145] (x )(x + 4) = 0 x + x 1 = 0 p = 1, w = 1 Use calculator and patterns. Look up Gauss on Google. n(n+1) In this case it is 65(65+1) = 145 (145) = Statement Always Never Sometimes Example true true true x 1 < x, x, and x If x = : 4 1 <, False If x = 1: 1 1 < 1, True x + < x, x, and 0 < x 5 If x = 0: 0 + < 0, False If x = 4: < 4, False x < x, x, and 0 x 5 If x = 0: 0 < 0, True If x = 5: 5 < 0, True x < x, x and 5 x < 0 If x = 5: 5 < 5, False If x = 1: 1 < 1, False 9. (a) x y + y x x + y yx (x ) (x 6) (c) (4x + 9 1x) (x + 6 1x) 4x + 9 1x x 6 + 1x x 7 [= (x 9) = (x + )(x )] 6x x + 0 x 5 (x 4)(x 5) x 5 x 4 Sample 7 Educate.ie P1 10. (a) Area of A = (6x + )(x + 1) Area of B = 5x(4x + ) Area of A = 18x + 15x + Area of B = 0x + 10x 18x + 15x + = 0x + 10x x 5x = 0 (x + 1)(x ) = 0 x = 1 or x = x = Higher Level, Educate.ie Sample 7, Paper 1 55

56 Perimeter of A = (9x + 4) Perimeter of B = (9x + ) Perimeter of A = 18x + 8 Perimeter of B = 18x + 4 Rectangle A has the longest perimeter. Perimeter of A = 18() + 8 Perimeter of B = 18() + 4 Perimeter of A = 6 Perimeter of B = Dublin (Heuston) km from Cork Stop 1 Stop 4 Stop Cork (Kent) Stop 5 Stop 7:00 7:15 7:0 7:45 8:00 8:15 8:0 8:45 9:00 9:15 9:0 9:45 Time (a) = 50 km Dublin/Cork express: stops Cork/Dublin express: stops (c) Stop 1: 5 minutes Stop : 15 minutes Stop : 5 minutes Stop 4: 5 minutes Stop 5: 10 minutes (d) Dublin/Cork express: 9:5 am Cork/Dublin express: 9:0 am (e) (f) Speed = Distance Time Dublin/Cork express: 70 0 min = 140 km/h Cork/Dublin express: 50 0 min = 100 km/h Where: 10 km from Dublin or 140 km from Cork: Time: 8:1 am 1. (a) 5 5x 15 1 x 1 x Mathematics Junior Certificate

57 6(x + 5) > (7 x) 6x + 0 > 14 x 8x > 16 x > (c) 5 x 8 16 x 4 1 x (a) x 5 60 = 41 x 5 = x 5 = 101 x = % of % of % of (c) 4% of 505= 0 0 (d) 505 ( ) = f (x) = 5 ax f (1) = 5 a(1) = 1 5 a = 1 a = 6 a = f (x) = 5 ()x = 5 6x f () = 5 6() = 7 f (x) = 5 6(x) = 5 6x = 0 x = 5 Sample 7 Educate.ie P1 Higher Level, Educate.ie Sample 7, Paper 1 57

58 15. (a) 10 x x Area = x(10 x) = 0x 4x (c) (i) 5 m (ii) Maximum area is a square of sides 5 m. 58 Mathematics Junior Certificate

59 1. (a) 1 4,, ( = , = 1 5 ) (c) Answer: π 014 SEC Paper 1 (Phase ) Reason: It cannot be written as a fraction. (i) n n (17) + 1 = = 89 4 (19) + 1 = or (1) + 1 = or (ii) Answer: 89 The other answers are non-whole numbers. Reason: It is a positive whole number. Change and into decimals with your calculator if you need to. Rational numbers can be expressed as the ratio of two integers i.e. as a fraction. Replace n with 17, 19, 1 in turn in the formula.. (a) (i) p 6p + 1 6p Replace p in the formula with the values under p in the first column (ii) There are a number of different reasons any two will suffice. Reasons related to all prime numbers : The formulas do not generate, which is prime. The formulas do not generate, which is prime. Reasons related to only prime numbers : The formulas generate 1, which is not prime. The formulas generate 5, which is not prime. The formulas generate 5, which is not prime. Note: 1 is non-prime as it doesn t have two factors. 014 SEC P = 41, which has 41 as a factor. 41 = 1681 which obviously has 41 as a factor along with 1 and Higher Level, 014 SEC, Paper 1 59

60 . (a) (i) A B C (ii) A B = {, 4, 6} A B: means what is in both A and B. B\(A C) = {, 6, 8, 10} (B\A) (B\C) = {, 6, 8, 10} B\(A C): B less (A and C). (B\A) (B\C): (B less A) united with (B less C) (iii) (A C)\B or equivalent A null set contains no elements. (i) U M N = 110 OR maximise M N = 10 Minimum = 10 To make #(M N) as small as possible, make # (M N) = 0. (ii) U M N Maximum = 8 OR minimise M N To make M N as big as possible, make the smaller set a subset of the larger set. 60 Mathematics Junior Certificate

61 4. (a) 9a 6ab + 1ac 8bc = a(a b) + 4c(a b) = (a b)(a + 4c) Factors by grouping 9x 16y = (x 4y)(x + 4y) The difference of two squares x (c) + 4x x + x 6 = x(x + ) (x + )(x ) Factorise numerator and denominator x = Note: x + x + = 1 x 5. One method: Dots on the number-line as the question involves integers Or: 17 1 x < 1 1: 18 x < 1 ( ): 6 x > 4 i.e. 4 < x x and 1 x < 1 x 18 and x < 1 x 6 and x > 4 i.e. 4 < x (a) Container 1 Graph C A B h Reason your way through this type of question e.g. container has two sections so the graph will also (B) The rate of change of container is constant and so must be graph (A), etc. t The container fills quickly at first and then slows down. 7. (i) %: = %: = 5980, and = 9 0 7%: = 0 944, and = Total USC = SEC P1 (ii) 0%: = %: = 4160, and = Gross Tax: Tax Credits: = 00 (iii) Total Deductions: = Deductions 100% Gross Higher Level, 014 SEC, Paper 1 61

62 Total Deductions as % of Gross Income: = = 19%, correct to the nearest percent 6, (i) First difference: Second difference: Answer: Quadratic Examine the differences. Reason: The first differences are not all the same, but the second differences are. (ii) (iii) 5 metres Use the differences to approximate the heights. Second difference: First difference: Height (m): Time (s): 5 5 Continuing the method for (ii): Second difference: First difference: Height (m): Time (s): Answer: The ball spends roughly 4 4 seconds in the air. Its height is 0 just before 4 5 seconds. Or, graphically: From the graph, the ball spends roughly 4 4 seconds in the air. 8 Height (m) Time (s) 6 Mathematics Junior Certificate

63 9. (i) Amount of Euro ( ) Jack Sarah Amount of Sterling ( ) 56 (ii) Slope = 40 0 =, or 1 15 Slope formula. See page 18 of Formulae and Tables. 0 Explanation: Each extra 1 costs Jack an extra Or: Explanation: Each 1 costs Jack 1 15, after an initial fee of 10. (iii) e = 1 15s + 10, where s is the amount in sterling, and e is the amount in euro. (iv) 48 4 Slope = 40 0 = 5 6, or 1, y-intercept = 0 e = 1 s, where s is the amount in sterling, and e is the amount in euro. (v) Using formulas: Solve simultaneously e = 1 15s + 10 and e = 1 s, so 1 15s + 10 = 1 s, i.e. s = 00 and e = 40. Amount of sterling: 00 From table: Slope formula. See page 18 of Formulae and Tables. Each time the amount of sterling goes up by 0, the difference between the costs decreases by 1. This difference is 9 for 0. So after 9 increases, i.e. increase of 9 0 = 180, the costs are the same, i.e. for SEC P1 Higher Level, 014 SEC, Paper 1 6

64 10. (a) 6 y 4 y 4 x x x will have shape. x will have shape. Answer: f(x) Answer: g(x) 8 y 6 4 x y 6 4 x Roots : points of intersection of the graph and x-axis If x = is a root, then (x ) is a factor. 6 6 h(x) k(x) Roots of h(x): x = and x = Equation: h(x) = (x + )(x ), or h(x) = x x 6 Check y-intercept is correct, i.e. co-efficient of x is correct: h(0) = 6, which corresponds to the graph. Roots of k(x): x = and x = Equation: k(x) = (x + )(x ), or k(x) = x + x 6 Check y-intercept is correct, i.e. co-efficient of x is correct: k(0) = 6, which corresponds to the graph. 11. (i) Increase x by 1: x + 1 Decrease x by : x (ii) (x + 1)(x ) = 1 or equivalent. Product means multiply. (iii) (x + 1)(x ) = 1 x x = 0 x = ( 1) ± ( 1) 4(1)( ) (1) x = 08 and x = 1 08 decimal places is a hint to use the quadratic formula. See page 0 of Formulae and Tables. x = 0 and x = 1 0, correct to three decimal places. 64 Mathematics Junior Certificate

65 1. (a) (6x )(x 1) = 1x 1x + Multiply carefully! x + x x 1) x x x + x x x x + x x x + x + Answer: x + x. (c) (i) 1 : 6x 9y = 54 : 5x + 9y = 10 11x = 44 (ii) 1. (i) x = : x = 4 Substitute in x = 4 in: (4) y = 18 8 y = 18 y = 18 8 y = 10 ( 1): y = 10 : y = 10 = 10 or equivalent Answer: x = 4 and y = 10. Note: You only need to check the equation that wasn t used to find the second variable. In this case, we only need to use. Verify using substitution. 10 5(4) + 9 ( ) = 0 0 = 10. = 18 or Or: sin 45 = x 1 = x x = Long division Or: x x x x x x 1 x x Answer = x + x Choose the letter to get rid of carefully. (y here) Use Pythagoras s Theorem or sin A = Opposite Hypotenuse 014 SEC P1 x Higher Level, 014 SEC, Paper 1 65

66 (ii) (iii) y = 1 = 8 or y x 1 Apply Pythagoras s Theorem. Note: 8 = 4 = 18 = 9 = y Perimeter = x + y = = ( ) + ( ) = Mathematics Junior Certificate

67 14. (i) Number of Bacteria, in thousands 4 16 g(t) f(t) Time in Days, t f(0) = 1 g(0) = 1 f(1) = g(1) = 4 f() = 4 g() = 9 f() = 8 g() = 16 f(4) = 16 g(4) = 5 f(5) = g(5) = 6 g(t) = t + t + 1 Note: f is an exponential function. g is a quadratic function. t t +t +1 y Show workings in the grid after the question. Use your calculator to verify that the points are correct. 014 SEC P1 Higher Level, 014 SEC, Paper 1 67

68 (ii) g(t) = t + t + 1 g(0) = (0) + (0) + 1 = = 1 g(1) = (1) + (1) + 1 = = 4 g() = () + () + 1 = = 9 g() = () + () + 1 = = 16 g(4) = (4) + (4) + 1 = = 5 g(5) = (5) + (5) + 1 = = 6 Marie after 5 days: bacteria, approximately Paul after 5 days: 6000 bacteria, approximately Difference: = 6000 bacteria Read each separately from the graph and subtract. (iii) (iv) t 4 days t = 5 days Range implies that your answer will involve an inequality. Read your answer from the graph. (v) Answer: Paul, i.e. f(t) Substitute t =14 into both formulae and compare. Reason: f(14) = = , so Paul predicts = g(14) = 5 = 10, so Marie predicts = Mathematics Junior Certificate

69 1. (a) (i) (ii). (a) U 01 SEC Paper 1 (Phase ) Number/Set N Z Q (R/Q) R 5 No No No Yes Yes 8 Yes Yes Yes No Yes 4 No Yes Yes No Yes 1 No No Yes No Yes π 4 5 cannot be written as a fraction. 7 + = 5 P No No No Yes Yes P\(E O), P E O, E O, (E O)\P (c) 1 5. (a) (i) U O E 1. Rational numbers (Q) can be expressed as the ratio of two integers (i.e. as a fraction). π is not an element of 4 Q (rationals) since π is not an integer. Also, 1 = 7. You can also use your calculator to check whether a number is rational or not. Prime numbers have only themselves and 1 as factors. A null set doesn t contain any elements. Probability = 1 (the number ) 5 (primes less than 1) A B C means what is common to all sets A B C 01 SEC P1 Higher Level, 01 SEC, Paper 1 69

70 (ii) U A B (A B ) \ C means is common to (A and B) less C C (A B) \ C (iv) A\B = B\A or (iii) (A\B)\C = A\(B\C) Diagram or explanation (c) F S Examine each statement separately, preferably by drawing diagrams. Let x represent the number of people who had been to both countries. (x + ) x x x + 4 (x + ) x + x + x + = 4 x = 5 4. (a) = which is 6 06 correct to two decimal places or any other check = 8 1 Or find 6% of 8 65 and subtract (c) No = 8 6 This is not as high as the original starting point. 5. (a) Start at the corner flag. Use the tape measure to measure a certain distance out along the side-line. e.g. 5 m. Then measure a certain distance out along the goal-line. e.g. 4 m. Then measure the distance between these two end points Using Pythagoras s Theorem, see if the calculated distance is the same as the measured distance. 70 Mathematics Junior Certificate

71 Use the trundle wheel to measure the radius, i.e. the distance from the centre spot to anywhere on the circumference. Use circumference = πr to calculate the circumference. Then use the trundle wheel to measure the circumference on the circle and see if the two match. 6. Car A: (Time to reach D) T = D = 70 S 50 = 1 4 h Car B: Distance travelled = 6 km Car B: Distance = Speed Time 7. (a) Story Angela walks at a constant pace and stops at 5.08 for four minutes. She then walks at a slower pace and arrives at practice at Angela walks at a constant pace and stops at 5 1 for four minutes. She then walks at a faster pace and arrives at practice at Angela walks at a constant pace and stops at 5.08 for five minutes. She then walks at a faster pace and arrives at practice at Angela walks at a constant pace and stops at 5.08 for four minutes. She then walks at a faster pace and arrives at practice at Angela walks at a constant pace and stops at 5.08 for four minutes. She then walks at the same pace and arrives at practice at Tick one story (ü) ü Mary s journey Try to understand WHY each of the other stories is incorrect. Distance (metres) Constant pace : Mary s journey will have to be represented by a straight line segment. 100 H Time (minutes) 01 SEC P1 Higher Level, 01 SEC, Paper 1 71

72 8. (a) 4(5 x) + 5(x 4) = x 0 0 Common denominator is needed to add these two fractions. x + 11x 4 = 0 x + 11x 4 = 0 (x 1)(x + 4) = 0 x = 11± 11 4()( 4 ) () x = 1, x = 4 x = 11±1 6 x = 1, x = 4 Rearrange to form a quadratic equation before you solve for x. (c) Method A x 5x + x + x + x 1x + 6 Method B x + 6x 5x 1x 5x 15x x + 6 x + 6 Long division: Try to learn the Method B shown in the solutions as it is more convenient. ax bx c x ax bx cx + ax bx c ax = x a = x (a + b) = 5 a + b = 5 6 = b = 5 b = 11 c = 6 c = (d) 5x + 4y = = 10 x + 6y = = 1 60 x = 10 y = (a) 1 ( ) = 1 or 10 or Represent this situation with simultaneous equations. Substitute the values for S and P into the formula. The denominator increases so the value of the fraction decreases. 1 (c) M = Multiply both sides by (S + P) S + P Subtract MS from both sides MS + MP = 1 Divide both sides by M MP = 1 MS P = 1 MS M or P = 1 S M You could check this by substituting values of P (>0.1) in the fraction in the solution to part (a) and taking a look at the effect. 7 Mathematics Junior Certificate

73 10. (a) (7) = 14 (7) + 1 = 15 Substitute the values in. Even numbers differ by (i) is the lowest even number so adding onto an even number will give the next even number. x + (ii) x = 8 NB is subtracted from Multiply each part x = 4 of the inequality by. 11. (a) x < 8 Add 6 to each part of the inequality. (i) x or similar (ii) x 00 x (a) Width 7 mm Height 15 mm Divide both sides by (positive so it won t affect the inequality sign). 1 m = 1000 mm Make height of logo = 1000 mm Make height of logo = 1 m 15 7 = 1000 x 1. (i) x 1 x = 1800 mm (= width) (or 1 8 m) (or 180 cm) Scale Factor = = = mm D F C 15 OR 7 = 1 x x = 1 8 m A E B (ii) Two decimal places is a hint to use the quadratic formula. x 1 = 1 x 1 (See page 0 of Formulae and Tables) x(x 1) = 1 x x 1 = 0 x = 1 ± 5 x = = 1 6 cm (discard neg. value) 14. Term 1 Term Term Term 4 Term 5 a b + c 8a b + c 18a b + c a 4b + c 50a 5b + c 01 SEC P1 Subtract to get the differences. Higher Level, 01 SEC, Paper 1 7

74 a b + c 8a b + c Diff = 6a b 18a b + c Diff = 10a b Diff = 4a a 4b + c Diff = 14a b Diff = 4a 50a 5b + c Diff = 18a b Diff = 4a nd difference is constant therefore the relationship is quadratic. 15. (a) Solve f (x) = 0 Solve g(x) = 0 Solve h(x) = 0 (x )(x + ) = 0 (x )(x ) = 0 x(x ) = 0 x =, x = x = x = 0, x = The solutions to these equations are called the roots of the functions. y axis Diagram 1 x axis Diagram y axis Diagram y axis x axis x axis Diagram 4 y axis h(x) y axis Diagram 5 f(x), Diagram 6 y axis x axis x axis x axis g(x) Roots represent the point(s) of intersection of the function with the x-axis. Use the answers from (a) to decide on the appropriate sketch. 74 Mathematics Junior Certificate

75 01 SEC Paper 1 (Phase ) 1. (a) Reason 1: 7 is not a positive number. OR 7 is a minus number. Reason : 7 is not a whole number. OR 7 is a decimal. Natural numbers: Positive whole numbers excluding 0 R Q Z N 1, 4 5, , π Notes: 4 5 = 9 Rational 7 1 = 1 7 Rational π and can t be written as fractions Irrational 01 SEC P1 Higher Level, 01 SEC, Paper 1 75

76 . (a) +. (a) (i) = $ $1 = (Multiply both sides by 100.) (ii) $ = $ = = R = 060 OR = 968 R = = 1 45 (c) 1 Euro = = anything greater than = Euro 1 = anything less than (a) = 5000 To get each proportion: Minutes played 50 = Total minutes = = (i) Michael Paul John % = = % = (ii) = (one part) Michael = Paul = Or find % and subtract Try to understand both methods in the solutions. Let Michael be represented by 1 when building up the ratio. 76 Mathematics Junior Certificate

77 5. (a) = 00 7 (% USC charge) = 9 (4% USC charge) Finding %: Multiply by 0 0 Finding 4%: Multiply by 0 04 etc. [ ( )] 0 07 = (7% USC charge) = Total USC Charge = 00 (c) = 6560 Tax at lower rate = = 500 Tax at upper rate = Total gross Tax = 86 Total Net Tax Standard Rate Cut Off Point: First 800 is taxed at the lower rate (0%) (d) ( ) = = = 4 69 Net Pay = Take Home Pay 6. (a) Roots and 4 Roots: Points of intersection of the graph and the x-axis. These are found by solving the equation (d) f(x) = x + x + k = + () + k = k k = 1 Substitute (, ) into the function and it will lead to an equation in k. 01 SEC P1 Higher Level, 01 SEC, Paper 1 77

78 (c) (x + t) = x + x + k x + tx + t = x + x + k t = t = 1 (d) Shown in part (a). (e) (x + 5)(x ) = 0 x + x 15 = 0 k = 15 Constant is product of roots 5 = 15 k = 15 Two roots are the same: x-axis is a tangent to the graph of the function. If x = 5 is a root then (x 5) is a factor etc. 7. (a) 56 8 = 18, = 18, 9 74 = 18, = 18, = 18, = 18, = 18 Common first difference of 18 The first difference is constant for all linear functions. 150 A continuous line is needed. Cost in euro Plan B Units Used (c) 0 Point of intersection with the y-axis. 78 Mathematics Junior Certificate

79 (d) Method: When the units used go down by 100 then the cost goes down by = m = = 0 18 ( or 9 50 ) y 8 = 0 18 (x 100) 0 18x y + 0 = 0 sub x = 0 y = 0 Standing charge: 0 Slope formula: See page 18 of Formulae and Tables. Equation formula: y y 1 = m(x x 1 ) See page 18 of Formulae and Tables. (e) Cost = x (f) = = = = = 1 5% VAT = 1 5% VAT 1700 (g) Units Used Plan B Cost in euro Rate = Amount of VAT 100% Total (h) (i) Scenario 1: Concentrates on 650 units [ = 16 75] The cost of Plan A and Plan B are very similar therefore it doesn t really matter which plan Lisa chooses. OR Continuous line again. Scenario : Concentrates on low and/or high usage If Lisa tends to use a low number of units on average, then plan A is better but if she uses a high number of units on average then Plan B is better. Lisa should choose plan B as it is 5c cheaper. See graph in part (c). (j) 640 units Point of Intersection 8. (a) W = 1 CV W = 1 (500)() W = Substitute in the values for C and V. 01 SEC P1 Higher Level, 01 SEC, Paper 1 79