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

80 W = 1 CV W = CV W C = V W C = V 9. (a) w + d = 1 Substitute d = 1 Multiply by. Divide both sides by C. Take the square root of both sides. 1w + 5d = 10 w = 1 Use Simultaneous Equations w + d = 1 w = 10 w 10d = 0 W = 5 points 8d = 8 8d = 8 D = 1 point Trial and error with verification of both solutions is awarded full marks. With the new system, the ratio of win:draw is higher which rewards a victory more and might encourage a team to go for a win (x)(x + ) = 10 x + x 10 = 0 (x + 5)(x ) = 0 x = 5 (not possible) and x = cm Area of a Triangle Formula: A = 1 b h See page 9 of Formulae and Tables. Note: x must be positive (length). 11. (i) 5x (x ) H.C.F (ii) (x 9y)(x + 9y) (iii) a(a b) + (a b) (a + )(a b) Difference of two squares Factor by grouping terms 1. (a) (i) (x 6)(x + 1) = 0 a = 1, b = 5, c = 6 x = 6 or x = 1 x = 5 ± ( 5) 4(1)( 6) (1) 5 ± (5 + 4) x = x = 5 ± 7 x = 6, x = 1 ( 14) 4(8)() (ii) (4x 1)(x ) = 0 x = 14 ± (8) x = 1 or x = x = 14 ± (8) Quadratic equation Quadratic equation x = 1 4, x = 80 Mathematics Junior Certificate

81 (iii) (x + 5) (4x 1) = ( 1) 4x x + = 8x = 16 x = x = 7 ± 49 4()( 6) 4 x = 7 ± x = 7 ± 97 4 x = 7 ± x = 4 1 or x = 0 71 Multiply across by 6 Roots are found by solving the equation. Two decimal places is a hint to use the quadratic formula (page 0, Formulae and Tables) 1. Statement Always true Never true Sometimes true If a > b and b > c, then a > c ü If a < 4 and b < 4, then a < b ü If a > b, then a > b ü If a > b and b < c, then a < c ü If a + 1 >, then a > 0 ü If b 4 < b 8, then b > 4 ü If a and b are both positive and a < b, then 1 a < 1 b ü 14. (a) g() = 0 = 1 Replace x with (i) h(t) = t t h(t + 1) = (t + 1) (t + 1) = 4t + 4t + 1 6t Examine each statement carefully. Substitution of real numbers could help to understand what is being asked each time. Replace x with t and (t + 1) (ii) 4t t = t t t + t = 0 (t )(t + 1) = 0 t =, t = 1 Equate 01 SEC P1 Higher Level, 01 SEC, Paper 1 81

82 (c) A B C (i) C A, B Sub x = 0 f (x) = 0 y = 8 (x 4)(x + ) = 0 C(0, 8) x = 4 or x = (ii) x 4 A and B are roots which are found by solving the equation. C is the y-intercept which is the constant term (x = 0). Where the function is on or below the x-axis 8 Mathematics Junior Certificate

83 Educate.ie Sample 1 Paper Sample 1 Educate.ie P 1. (a) 1 pr h = 1 p()()(4) = 1p cm Volume of a cone: See page 10 of Formulae and Tables. (i) 1 5 H = 90 60H = 90 H = H = 1 5 cm (ii) (L W) + (L H) + (W H) (1 5) + (1 1 5) + (5 1 5) = = 171 cm 1cm (iii) 1 Block 90 cm = = 756 g = 150 blocks Surface Area (SA) of capsule = SA of (c) (i) prh + 4pr cylinder + SA of two hemispheres = ( 14)(4)(170) + 4( 14)(4)(4) = 44,89 +, = 66, = 67,000 cm (ii) pr h + 4 pr Volume of capsule = Volume of cylinder + Volume of two hemispheres = ( 14)(0 4)(0 4)(1 70) + (1 )( 14)(0 4)(0 4)(0 4) = = = 1 5 m 1 5 cm 5 cm. (a) x y = y = x + y = 6 + = y x = 4 4 = y x = = y Point (, ) Solving simultaneously x y + = 0 let x = 0 0 y + = 0 = y 1 = y C = (0, 1) Higher Level, Educate.ie Sample 1, Paper 8

84 (c) (0, ) x + y 6 = () 6 = 0 0 = 0 (d) Slope of l = 1 Slope of m = Point (4, ) y + = (x 4) y + = x + 8 x + y = 8 x + y = 6 Equation of a line formula: y y 1 = m(x x 1 ). See page 18 of Formulae and Tables.. A = Axial symmetry in the y axis B = Central symmetry C = Translation 4. (a) A = ( 5, ) F y 4 D 4 4 x B C 4 E (c) ( 4 + 0, 4 + (d) = 6 = 4 (e) y 4 = (x 0) y 8 = x x y + 8 = 0 ) = ( 4 5. (a) Approx. 5 times, 4 ) = (, ) Slope formula: See page 18 of Formulae and Tables. Equation of a line formula: y y 1 = m(x x 1 ). See page 18 of Formulae and Tables. Midpoint formula: See page 18 of Formulae and Tables. (c) Approx. 105 times Approx. 105 times Probability = Favourable outcomes Total outcomes 84 Mathematics Junior Certificate

85 6. (a) 8 Class A Class B (c) The lowest score was 76% Key 9 = 9 Median is the middle value if the results are arranged in ascending/descending order. Sample 1 Educate.ie P (d) One student scored 100%. 7. (a) H H T H T H T T H T H T H T Tree diagram Sample space (HHH), (HHT), (HTH), (HTT) (THH), (THT), (TTH), (TTT) 1 8 (c) Probability = Favourable outcomes 8 Total outcomes (d) 1 8 (e) (a) Y = 9x + 10 line, as the slope is 9. Lines 1 and 4 both have the same slope. The lines are in the form of y = mx + c. m is the slope of the line. (c) Lines 5 and 6 since 1. The slopes are 1 and. As these multiply to give 1, the lines are perpendicular. Higher Level, Educate.ie Sample 1, Paper 85

86 9. (a) A(, ) + 4 B(5, 1) P(6, 7) Q(9, ) Q = (9, ) (5 ) + ( 1 ) = (9 6) + ( 7) () + ( 4) = ( ) + ( 4) 5 = (a) Tan A = Opposite Adjacent = 1 4 Hypotenuse = = 17 Adjacent Cos A = Hypotenuse = 1 17 A H 1 Distance formula: (x x 1 ) + (y y 1 ) See page 18 of Formulae and Tables. 4 (c) 60 mins = 84 km 1 mins = = 6 4 km 5 mins = = 160 km sin 6 = AB 160 AB = 160 sin 6 = 94 km OR Speed = Distance Time 84 = Distance Distance = (84)( ) = 160 km 11. Diagram A 4 1 B D C Given: A triangle ABC in which A is 90 To prove: BC = AB + AC Construction: Draw AD BC and mark in the angles 1,, and 4. Proof: Consider the triangles ABC and ABD Standard proof A A 1 B C 1 B D 1 is common to both triangles. BAC = ADB = 90 ΔABC and ΔABD are similar. BC AB = AB BD AB = BC BD (equation 1) 86 Mathematics Junior Certificate

87 Now consider the triangles ABC and ADC. Sample 1 Educate.ie P A A 4 1 B C D C is common to both. BAC = ADC = 90 so ΔABC and ΔADC are similar. AC DC = BC AC AC = BC DC (equation ) Adding equation 1 and equation we get AB + AC = BC BD + BC DC = BC ( BD + DC ) = BC BC AB + AC = BC BC = AB + AC 1. A 5 B 5 C 1. (a) interval Steps: 1. Place the compass on the point B and draw two arcs on the arms of the angle.. Place the compass point where the arcs cut the arms and draw two arcs to cut each other.. Join B to the point where the arcs cut. 4. This line is the bisector of the angle ABC km/h (c) (10 8) + (0 4) + (50 40) + (70 18) + (90 10) = = = = 49 6 km/hr 100 (d) = 8 Higher Level, Educate.ie Sample 1, Paper 87

88 14. Z 40 V 40 O W t Y X (a) 40 Z V Y W Y Both triangles are congruent since YZW = YVW 40 ZY = VW diameter ZYW = VWY 50 W 15. (10) = (6) + (h) 100 = 6 + h 64 = h 8 cm = h 16. 4, 4, 6, 7, 9 Pythagoras s Theorem If the mode is 4 then we must have at least two 4s. As the median is 6 and there are five numbers, 6 is the middle number. As the range is 5 and I am assuming that 4 is the lowest, the maximum number is 9. As the mean of the five numbers is 6 then the total must be 0. So the final number must be (a) (1, 0) (, 0) (, 0) (0, 1) 5 10 (0, ) (0, ) (c) Looking for greater than 9, 5 9 All have a negative slope 1 Use Pythagoras s Theorem. Example (0, ) and (, 0) () + () = 1 Use diagram to find slopes of the line segments. They all have negative slopes. 88 Mathematics Junior Certificate

89 1. (a) 6 cm = 60 mm Educate.ie Sample Paper Sample Educate.ie P 6 cm = 60 mm Area = 6 cm = 6 cm 6 cm Each side is 6 cm or 60 mm. Perimeter = 4(60) = 40 mm (i) (ii) Area of 1 4 of circle = 1 4 p r 1 4 p(14) = 49p cm Length of 1 4 of a circle = 1 4 pr = 1 98 cm Perimeter of shaded region = = cm (c) (i) Volume of hemisphere = pr = p ( 1 ) ( 1 ) ( 1 ) = p 14 cm = radius Volume of a sphere = 4 pr See page 10 of Formulae and Tables. 1 cm (ii) Volume of disk: pr x = p Volume of a cylinder = pr h 1 See page 10 of Formulae and Tables. x = 1 1 x = = cm = 0 09 mm = 4 choices This is the Fundamental Principle of Counting. options and then 8 options give a total of 8 outcomes.. (a) 18 outcomes A B C D E F 1 1, A 1, B 1, C 1, D 1, E 1, F, A, B, C, D, E, F, A, B, C, D, E, F Area of a disc = pr See page 8 of Formulae and Tables. Length of a circle = pr See page 8 of Formulae and Tables. This is the Fundamental Principle of Counting. options and then 6 options give a total of 6 outcomes. (c) outcomes Higher Level, Educate.ie Sample, Paper 89

90 (d) (e) 18 or or 1 6 Probability = 4. (a) Generally this type of data is ignored. Examples: Row 1, Row 4, Row 1, Row Favourable outcomes Total outcomes Boys: Mean = = cm Girls: Mean = = cm According to this data the average height of the boys in this class seems to be greater than that of the girls. (c) Girls Boys Key 1 means 1 cm (d) Any two such as: The girls have a smaller range of heights. The boys have a bigger range of heights. There is one boy who is very tall compared with the rest of the boys. 5. Area of shaded triangle = 1 (8) 8 = cm Area of shaded rectangle = 4 8 = cm Area of triangle = 1 base perp. height Area of rectangle = base height 6. Equation A B C D Graph Proof: AOC = DOB Vertically opposite AO = OB Radii CO = OD Radii Δ AOC and Δ DBO are congruent SAS = SAS 90 Mathematics Junior Certificate

91 8. (i) (ii) 6 or 6 or 1 1 Probability = Probability = Favourable outcomes = Three 1s = = 1 Total outcomes 6 6 Favourable outcomes = Two s = = 1 Total outcomes 6 6 Sample Educate.ie P (iii) 1 6 Probability = Favourable outcomes Total outcomes = One = (i) Chimney A: tan 64 = Approximate height 100 Approximate height = 100 tan 64 = 05 m Chimney B: tan 5 = Approximate height 75 Approximate height = 75 tan 5 = 5 m Ratio of the heights = 05 5 = 41 : 7 This is approximately 6 : 1 (ii) Any two ways such as: Taking her own height into account. Taking the diameters of the bases of the chimneys into account. Any other reasonable way. 10. (a) 1 and 4: and 6: and 5: 6 y Line 1 1 Line (4, ) x (1, ) No Higher Level, Educate.ie Sample, Paper 91

92 (c) On the axes above, plot the points represented by the couples (1, ) and (4, ). y y 1 (d) x x = = 4 Point (4, ) y y 1 = m(x x 1 ) y = 4 (x 4) y 6 = 4x 16 4x y 10 = 0 (e) No Because the slope of this line is 4 and none of the lines 1 to 6 have a slope of (a) cos ABC = 1 15 = 0 8 sin BCA = 1 15 = 0 8 ABC = 7 BCA = 5 1. (a) A C D B AC = Depends on individual diagram. CD = Depends on individual diagram. DB = Depends on individual diagram. Should all be equal Calculator: Calculator: 5 1 Steps: 1. Draw another line from A at an angle to the first line.. With a compass, draw three arcs of equal length to cut this line.. Join the outside arc to B. 4. Through the other arc, draw a line parallel to the line going through the first arc. AB will now be divided into three equal segments. 5. Label the points C and D on the line segment AB. 1. A = ( 5, ) B = (5, ) C = (5, ) D = ( 5, ) B is the image of A under C is the image of A under D is the image of A under Axial symmetry in the y axis. Central symmetry in the origin. Axial symmetry in the x axis. 14. If the sides of two triangles are proportional then the two triangles are similar. 15. D A 70 O B This question is based on the theorem: The angle at the centre of a circle is twice the angle at the circle standing on the same arc. C 9 Mathematics Junior Certificate

93 Educate.ie Sample Paper 1. (a) (i) pr = ( 14)(11)(11) = = 80 cm (ii) 10 ( 14)(11)(11) = 17 cm 60 (i) prh = (p)(7)(0) = 80p cm (ii) prh + pr + 1 (4pr ) = (p)(7)(0) + p(7)(7) + p(7)(7) = 80p + 49p + 98p = 47p cm Area of a disc: = pr See page 8 of Formulae and Tables. Length of a circle: = pr See page 8 of Formulae and Tables. Sample Educate.ie P (c) (i) (ii) 1 p(x)(x)(x) = x p units Volume of nd cone = 1 (p)(x)(x)(1 5x) Ratio = x p:x p = :1 = x p units Volume of a cone: = 1 pr h See page 10 of Formulae and Tables.. (a) Categorical Numerical (c) Categorical (d) Numerical (e) Categorical. (a) H = + 7 H = H = 5 Opposite sin A = Hypotenuse 7 = 5 Adjacent (c) cos A = Hypotenuse = 5 Use Pythagoras s Theorem Higher Level, Educate.ie Sample, Paper 9

94 4. (d) cos A = 5 cos A = A = cos 1 ( ) A = 74 Calculator: D 9 cm C 5 cm 5 cm A 9 cm B 5. (a) 6 Steps: 1. Draw a line 9 cm in length with a ruler. Label as AB.. With the centre of the protractor on A, mark out an angle of 90 o.. With a compass, mark off 5 cm along the 90 o line. Label as D where the arc cuts the line. 4. With the centre of the protractor on B, mark out an angle of 90 o. 5. With a compass, mark off 5 cm along this 90 o line. Label as C where the arc cuts the line. 6. Join D to C. 1 C B D O A 4 E F 6 5 (c) A translation A to C or A translation F to D. 94 Mathematics Junior Certificate

95 6. Diagram A 1 B 4 C Given: A triangle ABC with the side [BC] extended to the point D. The angles marked 1,, and 4 are also given. D Sample Educate.ie P To prove: 4 = 1 + Construction: None Proof: + 4 = 180 Straight angle = 180 Angles in a triangle + 4 = Both equal = 1 + Subtract Standard proof 7. (a) BCA = 90 since the angle at the centre of the circle is a straight angle of 180 and is twice the size of BCA. (c) (d) 8. (a) Since BCA = 90 then CAB = 45 which, by theorem, is equal to CDB. Answer CDB = 45 By Pythagoras s Theorem AB = AC + BC (1) = (BC) 144 = (BC) 7 = BC 7 = BC Area = 7 7 = 7 cm = 5 = x = x = 6(14) 65 + x = 84 x = x = 19 Higher Level, Educate.ie Sample, Paper 95

96 9. (a) Number of log-ons Duration (minutes) A histogram is chosen as it is suitable to display intervals in time = 84 (c) Median value = 4 which is in the interval 9 1 The median line will divide the area of the histogram into two equal parts. 10. Male Female Total Wearing glasses Not wearing glasses Total (a) 5 50 = = 8 5 (c) = (a) tan 56 1 = XY 50 XY = 50 tan 56 1 Calculator: Key as you see: Answer = XY = 75 m (100) = (75) + (KX) Use Pythagoras s Theorem (KX) = 10,000 5,65 KX = 475 KX = 66 m KT = = 16 m 1. (a) BCD = = 17 1h = 90 h = 90 1 = 7 5 cm Probability = Probability = Probability = Favourable outcomes = 5 males = 1 Total outcomes 50 Favourable outcomes = Total outcomes Favourable outcomes = Total outcomes 16 not wearing glasses males wear glasses 50 = 8 5 = Mathematics Junior Certificate

97 1. (a) Line 4: the slope is 1 4 or 0 5 Line and line 5. Both slopes are (c) Line 1 and line 4 Since 4 1 = 1 4 (d) Cuts x-axis at y = 0 0 = x + 1 x = 1 x = 4 (4, 0) Cuts y-axis at x = 0 y = (0) + 1 y = 1 (0, 1) Sample Educate.ie P 14. (a) (1, 1) (1, ) (1, ) (1, 4) (1, 5) (1, 6) (, 1) (, ) (, ) (, 4) (, 5) (, 6) (, 1) (, ) (, ) (, 4) (, 5) (, 6) (4, 1) (4, ) (4, ) (4, 4) (4, 5) (4, 6) (5, 1) (5, ) (5, ) (5, 4) (5, 5) (5, 6) (6, 1) (6, ) (6, ) (6, 4) (6, 5) (6, 6) Scores that add up to 9 are (6, ) (4, 5) (5, 4) (, 6) 4 = 6 = 1 9 (c) Scores that add up to 10 are (4, 6) (6, 4) (5, 5) = 6 = 1 1 (d) 4 or less = (1, 1) (1, ) (1, ) (, 1) (, ) (, 1) = 6 6 = 6 1 (e) = (a) (, 1) y E D F C A 4 B x 5 Higher Level, Educate.ie Sample, Paper 97

98 (c) ( 4, 1 + 1, 0 ) = (, 0) (d) = = or 1 1 (e) = = 1 1 Yes [AD] is parallel to [BC]. ) = ( 6 Midpoint formula: See page 18 of Formulae and Tables. Slope formula: See page 18 of Formulae and Tables. They have the same slope. 16. y l P l 1 Slope of l 1 is 1 = Rise and it is in the positive Run direction. Slope of l 1 is so it is perpendicular to the line l 1. x 17. (a) P(May) = 1 65 = P(June) = 0 65 = 0 08 (c) P(May or June) = = = Mathematics Junior Certificate

99 Educate.ie Sample 4 Paper 1. (a) pr h = ( 14)(4 5)(4 5)(9) = 57 cm (i) 4x = 96 (c) x = 96 4 = 4 m Area = x = (4)(4) = 576 m (ii) prh = ( 14)(0 75)(1) = 4 m (iii) 9 4 = = 75% 576 (i) 1 pr h = (0 )p()()(6) = 8p cm (ii) Amount of sand = 4p cm 4 Rate of flow 45 p cm per second Time = 4p = 45 seconds 4 45 p. A = Central symmetry B = Translation C = Axial symmetry in the y-axis. (a) Line 6 has slope of 11 Volume of a cylinder: = pr h See page 10 of Formulae and Tables. Curved Surface Area of cylinder: = prh See page 10 of Formulae and Tables. Volume of a cone: = 1 pr h See page 10 of Formulae and Tables. 4p 4p 45 4p 4p 45 = 4p 45 4p = 45 = 4p 45 4p = 45 Reason: 6 is the largest coefficient of x when the equations are written as y = mx + c. Sample 4 Educate.ie P Line and line 5 have slopes of. Parallel lines have the same slope. 4. (c) Lines and 4 since 1 = 1 C 9 cm A 7 cm B For perpendicular lines the product of the slopes should equal 1. (m 1 m = 1) Steps: 1. Draw a line 7 cm in length with a ruler. Label as AB.. With the centre of the protractor on A, mark out an angle of 90. Draw a line which will be 90 to the line AB.. With a compass point on B, draw an arc of length 9 cm to cut the 90 line from A. Label as C. 4. Join B to C. Higher Level, Educate.ie Sample 4, Paper 99

100 5. (a) ( 1, h) is on ( 1) 4(h) + 7 = = 4h 4 = 4h 1 = h (k, 0) is on 4x + y 4 = 0 4(k) + (0) 4 = 0 4k = 4 k = 6 x 4y = 7 (mult. ) 4x + y = 4 (mult. 4) 9x 1y = 1 16x + 1y = 96 5x = 75 x = Substitute into top equation () 4y = 7 9 4y = = 4y 16 = 4y 4 = y Point of intersection (, 4) = R Substitute the point ( 1, h) into the line x 4y + 7 = 0 to find h. Substitute the point (k, 0) into the line 4x + y 4 = 0 to find k. Solving simultaneously 6. (a) Spinner Die (1, 1) (, 1) (, 1) (4, 1) (5, 1) (6, 1) (1, ) (, ) (, ) (4, ) (5, ) (6, ) (1, ) (, ) (, ) (4, ) (5, ) (6, ) Less than = 1 (c) P(8) = 18 = (a) AOB = 100 or twice ACB When the outcomes are added the only outcome giving less than is (1, 1). There are two outcomes that, when added, give a total of 8 and they are (5, ) and (6, ). There are 18 outcomes in total. ADB = 10 or half AOB = 60 (c) OAB = 40 BAD = 5 so OAD = 65 Probability = 18 = Mathematics Junior Certificate

101 8. (a) Discrete Discrete (c) Continuous (d) Continuous (e) Discrete 9. (a) P let y = 0 x = 0 x = 9 x = Q let x = 0 (0) y + 9 = 0 (, 0) = P y = 9 y = 4 1 ( 0, 4 1 ) = Q P Q (?,?) (, 0) ( 0, 4 1 ) (, 9) (, 9) Sample 4 Educate.ie P 10. Diagram A B D 1 4 C Given: A parallelogram ABCD To prove: AB = DC and AD = BC Construction: Draw the diagonal [BD] and mark in the angles 1,, and 4. Proof: This is a standard proof of this theorem In the triangles ABD and BCD 1 = Alternate = 4 Alternate DB = DB Common Hence ΔABD ΔBCD Therefore AB = DC AD = BC Corresponding side Also DAB = DCB Corresponding angles and ABC = ADC Higher Level, Educate.ie Sample 4, Paper 101

102 11. (a) A Tan A = 1 6 Tan A = 0 5 A = Calculator: (a) 8 Tan 8 = Tree 1 Tree = 1 tan 8 = 6 9 m x + 5 = x = 0 x = 1 Calculator: Key as you see: Answer = 6 91 Marks No. of students ( 0) + (9 50) + (9 70) + (4 90) (c) Mean = = = 5 = 60 0, 50, 70 and 90 are the mid-interval values 1. (a) A = (, ) (, 1) (0, 1) (, 1) B = (, 1) 10 Mathematics Junior Certificate

103 (c) ( 5, ) ( 5, 0) ( 5, ) C = ( 5, ) The answer to part and (c) are plotted on the Cartesian plane. y 4 C 1 0 x B 4 5 Sample 4 Educate.ie P (d) Slope of BC = = 4 7 y + 1 = 4 7 (x ) 7y + 7 = 4x + 8 4x + 7y = 1 7y = 4x + 1 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. y = 4 7 x (a) = x 9 x = 18 x = 6 cos C = Adjacent Hypotenuse = x 9 cos C = 6 9 C = cos 1 ( ) C = (a) = Since = A B C D The data is skewed to the right. The data is skewed to the left. There is a single mode. Higher Level, Educate.ie Sample 4, Paper 10

104 17. (a) (7x 8) + (5x + 8) + (10x 10) + (x 5) = 60 5x 15 = 60 5x = 75 x = 75 5 x = 15 ABC = 60 (7(15) 8) = = Mathematics Junior Certificate

105 Educate.ie Sample 5 Paper 1. (a) ( 14)(8) = 90 cm (i) pr h = p(14)(14)(57) = 11 17p cm (c) (ii) 1 pr h = 1 p(14)(14)(57) = 74p cm (iii) 11,17p 74p = (i) pr 4 = p(80)(80) p 111,17p = = 1600( 14) m = 504 m (ii) pr = ( 14)(1 )(1 ) = 4 5 m (iii) (4 5) m 4 = = m The lengths of two semicircles added give the length of one circle: Length of a circle = p r See page 8 of Formulae and Tables. Volume of a cylinder: = pr h See page 10 of Formulae and Tables. Volume of a cone: = 1 pr h See page 10 of Formulae and Tables. Sample 5 Educate.ie P. (a) 110 since the sum of the angles in the polygon ABED is since BAC = CAD = 5 and ADC = 55 (c) Since BE is to AB and DE then ABC = DEC both 90 BAC = DCE both 5 BCA = CDE both 55 (d) AC = AB + BC AC = (5) + ( 5) AC = AC = 7 5 AC = 7 5 = 6 1 Use Pythagoras s Theorem Higher Level, Educate.ie Sample 5, Paper 105

106 . (a) A = ( 5, 0) y D 4 E 1 C F B 4 5 x (c) ( 1 + 1, (d) = = 1 6 (e) y 0 = 1 (x + 5) y = x + 5 y = 1 x (f) 1 = 4 = 1 (g) y = 1 (x 1) y 4 = x + 1 y = x y = x + 5 x + y 5 = 0 + ) = ( 0, 0 ) = (0, 0) Midpoint formula: See page 18 of Formulae and Tables. 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. 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. (h) The two lines are perpendicular, meeting at 90 angles to each other. This can be seen in part when the points are plotted on the Cartesian plane. 4. (a) The word implies means that when one statement is made, another statement follows on logically from the first statement. (c) (d) (e) A corollary is a statement that follows readily from a previous theorem. Multiple possible answers. For example: A diagonal divides a parallelogram into congruent triangles. An axiom is a rule or statement that we accept without any proof. Multiple possible answers. For example: There is exactly one line through any two given points. 106 Mathematics Junior Certificate

107 5. (a) No. of shots No. of rounds (7 67 5) + ( ) + (15 7 5) + (9 78) = = 50 = , 70 5, 7 5 and 78 are the mid-interval values. 6. (a) Line 5 because it has a slope of 8 Lines and 6 since both have slopes of 4 (c) Lines and or lines 6 and Since 4x 1 = 1 4 The product of their slopes equals 1. Sample 5 Educate.ie P (d) Cuts the x axis at y = 0 0 = 7x 14 7x = 14 x = (, 0) Cuts the y axis at x = 0 y = 7(0) 14 y = 14 (0, 14) 7. (a) ACB = 75 since = 150 = 75 ACD = 105 since it makes a straight angle 8. (a) BOC = 80 since OBC = 50 also since Δ is isosceles (c) (d) BAC = 40 since it is equal to angle OCA Since they are both standing on the same arc BD BAE = EDC Alternate AB = CD Told ABE = ECD Alternate Hence the two triangles are congruent by ASA. Higher Level, Educate.ie Sample 5, Paper 107

108 9. A 7 cm C 11 cm 10. (a) 1st spin Black White Grey Black BB WB GB White BW WW GW Grey BG WG GG nd spin 8 cm B Steps: 1. With a ruler, draw a line segment 11 cm in length. Label as AB.. Place the compass on the point A and mark off an arc that is 7 cm from A.. Place the compass on the point B and mark off an arc that is 8 cm from B. 4. Label where the two arcs meet as C. (c) 6 9 = BB, WW, GG Probability = 11. (a) A pie chart (or a bar chart or a line plot) as one can quickly identify all the different segments. Public transport = 5 60 = Car = = Walk = = Favourable outcomes = 6 with different colours Total outcomes 9 total outcomes Walk Car Public transport 1. (a) ADB = 45 since BAD = 90 and ABD is also equal to 45 DAC = 45 since ΔACD is an isosceles triangle and we previously know ADB is also Mathematics Junior Certificate

109 1. (a) If one event has m possible outcomes and a second event has n possible outcomes, then the total number of possible outcomes is m n. 4 6 = 48 (c) 14. (a) 4 6 = 4, hence 10 4 = 5. So there are 5 desserts Key 4 0 = = 51 = 5 5 (c) 4 (d) 4 11 = 1 Sample 5 Educate.ie P (e) (f) 15. (a) (c) He was best in the class. (0 7) + (1 9) + ( 11) + ( 1) + (4 7) + (5 4) = = = days 100 = 100 = 46% 50 The most common number of days missed 16. (a) = = = = = = = 0 18 Higher Level, Educate.ie Sample 5, Paper 109

110 (c) The die may be slightly biased, but as the die is thrown more often the relative frequency of all the numbers should be 1 = Not enough trials were done to decide = = Mathematics Junior Certificate

111 Educate.ie Sample 6 Paper 1. (a) ( 14)(4)(8) = 01 cm (i) L + (p) ( 100 p = 400 L + 00 = 400 L = 100 m Curved surface area of cylinder = prh See page 10 of Formulae and Tables. (ii) Each lap = 400 m so after 100 m you are at point A, hence the finishing line is at D. (c) (iii) (i) 1500 m = mins 6 sec 1500 m = 06 sec 7 m = 1 sec Speed = 7 m/s 4 (p)()()() = 10 p cm (ii) pr h = p(5)(5)(6) = 150p cm (iii) pr h = 10 p 5ph = 10 p 10 _ h = 5 h = 0 4 cm Volume of a sphere: = 4 pr See page 10 of Formulae and Tables. Volume of a cylinder = pr h See page 10 of Formulae and Tables. Sample 6 Educate.ie P. (a) = 96 Opposite angles of a cyclic quadrilateral add to = 107. (a) sin A = 0 5 A = sin 1 (0 5) A = 0 0 cos 0 = x x cos 0 = 0 0 x = cos 0 x = 09 m You could also use trigonometric ratios. See page 1 of Formulae and Tables. Higher Level, Educate.ie Sample 6, Paper 111

112 (c) tan 0 = h 0 h = 0 tan 0 h = m 4. (a) A theorem is a rule or statement that you can prove by following a certain number of logical steps or by using a previous theorem or axiom that you already know. (c) (d) Vertically opposite angles are equal in measure (multiple possible answers). The converse of a theorem is the reverse of the theorem. If two angles in a triangle are equal then the triangle is isosceles (multiple possible examples). 5. A 55 C 8 cm 40 B Steps: 1. With a ruler, draw a line segment 8 cm in length. Label as AB.. Place the centre of a protractor at the point A and mark off an angle of 55.. Draw a line from A at Place the centre of a protractor at the point B and mark off an angle of Draw a line from B at 40 until it crosses the 55 line. Label this C. 6. (a) 4 F B 1 x C A D 4 E y (c) Area = 1 base height = 1 () 4 = 4 units See diagram above. Area of a triangle = 1 (base)(perp. height) 11 Mathematics Junior Certificate

113 (d) Area = 1 base height = 1 (7) 6 = 5 6 = 1 units (e) 4:1 (f) Answer: No The triangles are not equal in all respects so they are not congruent. Reason: The lengths of the sides of Δ ABC are not equal to Δ DEF. 7. (a) Line 5, as the slope is equal to. Lines 1 and, as both slopes are. (c) Lines 1 and or lines and since 1 = 1 (d) Cuts x axis at y = 0 4x (0) = 1 4x = 1 x = (, 0) Cuts y axis at x = 0 4(0) y = 1 y = 1 y = 4 (0, 4) Sample 6 Educate.ie P 8. (a) FBO HDO (c) FBO FAO FBO HCO 9. (a) A population is when everybody is used to collect data, for example, the entire school. A sample is when only a part of the population is used to collect data, for example, the rd year students in a school. (i) Face-to-face interview (ii) (iii) (iv) Postal Telephone Online (c) Questionnaire should (i) Be relevant to the survey you re undertaking Higher Level, Educate.ie Sample 6, Paper 11

114 (ii) (iii) (iv) Use clear, simple language Be as brief as possible Have no leading questions 10. (a) 8 0 = 840 m sin 0 = Height 840 Height = 840 sin 0 Height = 87 m Speed = Distance Time: Distance = Speed Time Calculator: Key as you see. 11. (a) X = 11 Y = 4 ABC = AMN = = 65 since both are corresponding angles Based on the facts that: (a) three angles of a triangle add to 180. in a triangle, angles across from equal sides are equal in measure. (c) = 50 since AB = AC makes Δ ABC isosceles 1. (a) = 18 = 6 years = 10 (c) 150 = 10 students 10 0 = = students (d) 0 = students = = 4 students 0 (e) Spanish = 60 = 4 students. Hence the number who do not study Spanish is 4 4 = 0 students. (f) 0 4 = x = x = 8 x = Mathematics Junior Certificate

115 14. (a) Charolais Limousin Key 8 1 = 81 kg Charolais is better as they are generally the heavier animals. The heaviest Charolais is 847 kg compared to the heaviest Limousin is 87 kg. The heavier the animal, the better the price a farmer gets from the butcher. 15. (a) (6 4) + ( 6) = () + (4) = 0 = 5 M = ( 1 +, (c) = = ) = (, 6 ) = (1, ) Distance formula: (x x 1 ) + (y y 1 ) See page 18 of Formulae and Tables. Slope formula: See page 18 of Formulae and Tables. (d) Slope (m) = Point (1, ) y = (x 1) y = x + x + y = 0 x + y 5 = 0 Equation of a line formula: y y 1 = m(x x 1 ) See page 18 of Formulae and Tables. Sample 6 Educate.ie P (e) x y = 0 y = 0 x + y = 5 = y x y = 0 1 = y 4x + y = 10 5x = 10 x = Point (, 1) 16. (a) 4 = 4 4 = 1 1 Solving simultaneously This is the Fundamental Principle of Counting. Choosing from and then choosing from and then choosing from 4 gives a total of 4 choices. Higher Level, Educate.ie Sample 6, Paper 115

116 Educate.ie Sample 7 Paper 1. (a) x + x + x + x = 00 10x = 00 x = 0 Length = (0) = 60 cm Breadth = (0) = 40 cm Area = = 400 cm Ratio : means a total of 5 parts, and 5 5 Half the perimeter = of 100 = 0: of 100 = 60: of 100 = Length = 60 cm Breadth = 40 cm (i) (7 5) = (6) + (r) = r 0 5 = r 4 5 cm = r (ii) prl + pr = ( 14)(4 5)(7 5) + ( 14)(4 5)(4 5) = 170 cm (c) (i) Volume = pr h + pr = p(6)(6)(14) + (0 66)p(6)(6)(6) = 504p + 144p = 648p cm Total surface area of a cone = Curved surface area of cone + Area of the circle at bottom (ii) Volume of water = 648p = 16p 144p + p(6)(6)h = 16p 6H = 7 H = cm Depth = 6 cm + cm = 8 cm. (a) XYZ, XZY, ZXY, ZYX, YZX, YXZ (c) (d) = 1 Probability = Favourable outcomes Total outcomes In the question, all horses are said to finish the race.. BAC = 180 BAC = BAC = 19 Based on the fact that, given parallel lines with a transversal, alternate angles are equal. 116 Mathematics Junior Certificate

117 4. (a) (, 4) y 4 D F 1 E B C 4 5 x (c) 5. (a) Right-angled isosceles triangle since BC = BD = 4 units and BD BC. Male Female Key 59 0 = 590 Sample 7 Educate.ie P Clearly the male circumferences are much larger than the female circumferences, hence male heads are generally larger. 6. A pie chart could be used as there would only be four sectors easily identified. 45 Pop: 60 = 180 Classical Rock: 60 = 100 Rock 90 Other Classical: Other: 5 60 = = Pop 100 Higher Level, Educate.ie Sample 7, Paper 117

118 7. (4 5) + ( 7 5) + (14 1 5) + (x 17 5) + (6 5) x x + 15 = 46 + x x = x x = x 17 5x 11 1x = x = x = 6 4 x = 4 5, 7 5, 1 5, 17 5 and 5 are the mid-interval values. 8. (a) Line 5 because the slope is 11 Line and line 6, both slopes are 7 (c) Cuts x axis at y = 0 0 = x + 6 x = 6 x = (, 0) Cuts y axis at x = 0 9. (a) x y = 0 y = x + y = 1 x y = (0) + 6 y = 6 (0, 6) Rewriting the equation as y = mx + c m will be the slope of the line. Slope = 1 Slope Point (, 5) y 5 = (x + ) y 5 = x 4 x + y = 1: K Equation formula: y y 1 = m(x x 1 ). See page 18 of Formulae and Tables. (c) x y = 1 y = 1 = y x + y = 1 = y x y = 1 = y 4x + y = 5x = 5 x = 1 Point of intersection (1, 1) 118 Mathematics Junior Certificate Solving simultaneously

119 (d) (, 5) (1, 1) (4, 7) + 6 Answer (4, 7) 10. (a) ΔPQT ΔSWR By ASA Since PQT = WSR Told PQ = SR By theorem QPT = WRS Alternate Therefore PT = WR ΔPSW ΔQTR Since PS = QR By theorem SPW = QRT Alternate PW = TR Since PT = WR in part (a) Therefore they are congruent by SAS. 11. Diagram A L B C Given: The triangle ABC with angles marked 1, and Sample 7 Educate.ie P To prove: = 180 Construction: Draw a line L through A parallel to BC. Mark the angles 4 and 5. Proof: = 4 and = 5 Alternate angle Standard proof = 180 Straight angle For 4 and 5 substitute and = = 180 Steps: 1. Draw a line 11 cm long.. Draw another line at an angle to the first line.. With a compass, draw three arcs of equal length to cut this line. 4. Join the outside arc to the end point of the 11 cm line. 5. Through the other arcs, draw lines parallel to the line going through the first arc. The 11 cm line will now be divided into three equal segments cm Higher Level, Educate.ie Sample 7, Paper 119

120 1. x = 54 alternate angle y = 180 (sum of angles in Δ) y = y = 88 Adjacent 14. cos A = Hypotenuse = x A () = x + ( ) 4 = x + 1 = x tan = Opposite Adjacent = (a) sin CBD = 4 6 CBD = sin 1 (0 666) CBD = 4 tan CAD = 4 9 CAD = tan 1 (0 444) CAD = 4 (c) sin x = 1 x = sin 1 (0 5) = 0 cos 0 = 16. (a) 8x 8 = 5x x 5x = x = 18 x = 6 You could also use trigonometric ratios. See page 1 of Formulae and Tables. The two angles at Q marked by the dots are equal. Angle PQR (8(6) 8) + (5(6) + 10) (48 8) + (0 + 10) = Mathematics Junior Certificate

121 1. (i) H H H T H H H H T T H T H T H T T H H T T T T T (ii) Pr( H, 1 T ) = 8 (iii) 014 SEC Paper (Phase ) Answer: No Reason: Pr ( H) = 1 8 but Pr ( H, 1 T ) = 8 Or: Try to list the outcomes systematically. Favourable outcomes Probability = Total outcomes Count your answers from the table above. Again, use the table above to count the events. Reason: There is only 1 way to get three heads. There are ways to get two heads and one tail. (iv) Answer: Yes Reason: Pr (H H H) = 1 and Pr (H T H) = Or: Reason: There is only one way to get each event.. (i) Volume = = 64 cm The volume of a cuboid is given by Length Width Height (ii) Box A: Box B: Volume = 0 78 = cm Volume = = cm Box A Box B cm 0 cm 40 cm 9 cm 014 SEC P 78 cm 48 cm V = L W H (iii) Box A: 8 = 4; 0 6 = 5; 78 1 = 6; so Box A can be filled completely. Box B: 48 6 = 8; 40 8 = 5; 9 1 = ; so Box B can be filled completely. Total: = 10 individual phone boxes. Also 8 5 = 10 (Box B) Higher Level, 014 SEC, Paper 11

122 (iv) Box A: Box B: Surface Area = ( ) = cm Surface Area = ( ) = cm S.A. = (Front + Top + Side) (v) Use Box B. The cost is given per m, so convert surface area to m (or cost to per cm ). 1 cm = 0 01 m, so 1 cm = 0 01 m = m. Surface area = cm = m = m Cost of box = = = 0 7, to the nearest cent (vi) Cost of Box A = ( ) = = 0 78, to the nearest cent Saving per annum = ( ) = (0 06) 1680 = Or: Difference in area = ( ) cm = 888 cm = m Saving per annum = = IQ Test IQ Test (i) IQ Test 1 IQ Test Key: 9 7 = a score of 97 Make sure to order the scores in the plot. (ii) Range of IQ Test 1 = = 40 Range of IQ Test = 10 8 = 7 Range = Highest Score Lowest Score 1 Mathematics Junior Certificate

123 (iii) data points in each set, so the median is the = 8th data point. Median of IQ Test 1 = 10 Median: Middle value when the scores are arranged Median of IQ Test = 106 in ascending/descending order (iv) Mean of IQ Test 1 = = (v) (vi) Mean of IQ Test = = In general, the scores in IQ Test are slightly higher than in IQ Test 1, as both the mean and median are higher for IQ Test. The scores are slightly more spread out in IQ Test 1 than in IQ Test, as the range is bigger for IQ Test 1; or The spread of scores is very similar, as the two ranges are almost the same. Answer: No. Mean: Sum of the scores divided by the number of scores Measures of Central Tendency: Mean or Median Measure of Variability: Range Explanation: The person who got 119 on IQ Test 1 could have got less, e.g. 94, on IQ Test. Or: Explanation: The maximum score on IQ Test 1 is greater than the minimum score on IQ Test. 4. (a) Diagram: B s r t O p w 014 SEC P A D C Given: A circle with centre O. Points A, B, and C on the circle. Angles p and r, as shown. To Prove: p = r. Construction: Join B to O, and extend to D. Mark the angles s, t, and w. Proof: OA = OB Radii of circle Step 1 s = t Isosceles triangle Step Standard Proof w = s + t Exterior angle Step w = t Step 4 Higher Level, 014 SEC, Paper 1

124 Similarly, (p w) = (r t). So p = (p w) + w = (r t) + t = r Step 5 (i) R (i) Angle at the centre theorem (above in part (a)) (ii) Angles standing on the same arc are equal. Q O P S SOP = = 64 (ii) SQP = SRP = (c) A Try to show that the opposite angles are right angles. B D C We just need to prove that the four angles are 90. BAD = BCD = 90, as [BD] is a diameter. Similarly, CBA = CDA = 90 So ABCD is a rectangle. 14 Mathematics Junior Certificate

125 5. (a) Credit Card Debit Card Debit Card Cash Debit Card Credit Card Cash Cash Credit Card Debit Card Debit Card Debit Card Cheque Cash Cash Cash Cash Debit Card Cash Credit Card (i) Categorical Nominal (ii) Method of Payment Credit Card Debit Card Cash Cheque Frequency (iii) Mode = Cash (iv) He cannot add up his values and divide by 0. (v) Credit Card: 4 60 = 7 0 Debit Card: 7 60 = Cash: 60 = Cheque: 60 = Mode: Which appears most often? The Mode is the only measure of centre for this type of data. Cheque 18 John's Data Credit Card 7 Use proportions of 60 to decide on the angle measures. Cash 144 Debit Card 16 Margaret s data may be biased because her sample is probably not representative. She will probably have a lot more people answering Lidl than she should. (c) (i) Answer: Pie chart Reason: It s easy to see where half the pie chart is (180 ). 014 SEC P Money Spent on Weekly Shopping ( ) Number of People Money Spent on Weekly Shopping ( ) Amount of Money ( ) Higher Level, 014 SEC, Paper 15

126 (ii) Answer: Bar chart/histogram Reason: It s easy to see which bar is highest. 6. (i) C 10 cm Standard construction A 6 cm B Note: It is also possible to work out the length of the third side, [BC], using the Theorem of Pythagoras, and then construct [BC] and [AC]. (ii) CAB = 5 Use your protractor. (iii) 7. (i) cos(5 ) = = 0 60, correct to three decimal places (iv) They are not the same because CAB = cos 1 ( 6 10 ) = So if X is a whole number then cos X can never be exactly y A (0, 5) 4 (ii) B ( 4, 0) AB = = 41 Similarly, AC = 41-1 O C (4, 0) Or use the Length formula. See page 18 of Formulae and Tables. Total length of metal bar = 41 = 1 80 = 1 8 m, correct to one decimal place. x 16 Mathematics Junior Certificate

127 (iii) AB: AC: Slope = Rise Run = 5 or Slope = = 5 or (iv) Answer: No Reason: Product of slopes = 5 5 = Or: Or use the Slope formula. See page 18 of Formulae and Tables. Perpendicular: Product of the slopes is 1. Reason: When you invert one slope and change the sign, you don t get the other slope. (v) tan X = 5 4 X = tan 1 ( 5 4 ) = = 51 4, correct to two decimal places. A Use the inverse (shift) key on your calculator to get tan 1. 5 cm (vi) B X 4 cm Recall from (ii) that AB = 41 m. Increase X by 0% to get X ' : X ' = = From the diagram, sin X' = sin = h = 41 sin = 5 6 = 5 6 m, correct to one decimal place. A O h SEC P 41 h B X Higher Level, 014 SEC, Paper 17

128 8. (i) Method 1: Method : y = x + 6 Step 1 y = x 6 y = 1 x Step Slope = a b = 1 Use y = mx + c Slope = 1 Step = 1 (ii) Substitute in (1, ) to l: LHS = 1 ( ) 6 = 1 0 = RHS. Point not on l. (iii) Slope of k = 1 Point on k = (1, ) Equation of k Or: Equation of k: y ( ) = 1 (x 1) x y + c = 0 y = x 7 1 ( ) + c = 0 or x y 7 = 0 c = 7 x y 7 = 0 If a point is on a line, when you substitute it into the equation of the line it will satisfy the equation (LHS = RHS) Equation of a Line formula. See page 18 of Formulae and Tables 9. 6 N 4 P M 9 10 Q R (i) Proof: MNP = PRQ (given) NPM = QPR (vertically opposite) NMP = PQR (third angle) The triangles are similar. Similar Triangles: They have the same angles. (ii) Answer: Yes Reason: MNP = PQR or NMP = PQR or alternate angles are equal. (iii) Given MN = 6, NP = 4, QP = 9, and PR = 10, find: QR By similar triangles ΔMNP and ΔQRP. QR 6 = 10 4 QR = = Mathematics Junior Certificate 8 In Similar Triangles, the corresponding sides are in proportion.

129 (iv) QM By similar triangles ΔMNP and ΔQRP: PM 9 = 6 15 or 4 10 PM = 18 5 QM = = (i) Volume = 1 π(6) 14 = 168π cm Or: PM 4 = or 6 PM = 4 10 = 18 or or 1 6 QM = = or Volume of a Cone formula: V = 1 πr h See page 10 of Formulae and Tables. 14 cm 6 cm (ii) Volume of upper (removed) portion: 1 π() 7 = 1π cm Volume of frustum: 168π 1π = 147π cm Subtract the volume of the smaller (missing) cone from your answer in part (i) to get the volume of the frustum. Ratio = Fraction or proportion Or: Volume of frustum = 1 πh[r + Rr + r ] = 1 π 7( ) = 147π cm Required ratio: 147π 168π = 7 or 7:8 or SEC P cm 7 cm 6 cm Higher Level, 014 SEC, Paper 19

130 014 SEC Sample Paper (Phase ) 1. (i) Area = Length Width = 8 m 4 m = m Area of rectangle = Length Width (ii) Total length needed = 5(semicircles) + length = 5( 14 ) + 8 = = 9 4 metres (iii) Length: = 1 5 m since (0 5) Length + Diameter of circle + buried pieces Area = Length Width = = m Width 6 78 m since πr + (0 5) ( 14)() (iv) Volume = Cylinder = πr h Volume of Cylinder = πr h. See page 10 of Formulae and Tables. ( 14)()()(8) = = 50 4 m (v) Area of bed = (8 0 4) (4 0 4) = = 7 6 m Volume of topsoil = = 6 84 m Cost of topsoil = = (i) 600 ( ) = 89 Total frequency must add to 600. (ii) Based on the above results, I disagree. Each number should have a 1 6 or 16 7% chance of occurring. Currently, 5 has a 14 8% chance of occurring, whereas has a 19 % chance of occurring. (iii) Answer: 15 Reason: the probability of getting an even number from six hundred throws is = = 15 Favourable outcomes 00 Probability = Total outcomes 10 Mathematics Junior Certificate

131 . (i) Example 1: Black jeans, White shirt, Black jumper and Boots Example : Black jeans, Red shirt, Black jumper and Flip-flops (ii) 4 = 7 outfits This is the Fundamental Principle of Counting. 4. (i) U NY B S.F (ii) (iii) (iv) 5 = = = 5 7 Probability = Favourable outcomes Total outcomes Always simplify. 5. (i) No. of hours No. of students (ii) 4 hours (iii) (1 11) + ( 1) + (5 18) + (7 1) + (9 11) + (11 ) + (1 1) + (15 1) + (17 6) + (19 1) + (1 4) = = 6 5 hours (iv) (1) He is using a much smaller sample. The mode is the most common interval: 4 occurs 1 times. () His sample consists of First Year boys only and ignores other years and the opinions of girls. () His survey is being conducted after the mid-term break, when students would have had more free time and probably spent many more hours on social networking sites. 014 Sample SEC P 6. (i) No. of jelly beans Brand A Brand B Brand C Higher Level, 014 SEC Sample, Paper 11

132 5 4 Brand A Brand B Brand C Frequency Number of jelly beans One could have used separate bar-charts or dot plots. (ii) 7. (i) Example answer: If I had to choose, I would buy Brand C. In Brand C, the mean number of sweets is = 7 6 sweets, compared with a mean of 4 5 sweets for Brand B and 9 1 for Brand A. The reason I didn t pick Brand A, even though it has a greater mean, is because Brand C has a range of 6 (1 5) unlike Brand A (5 ) and Brand B (9 17), which both have a range of 1, double that of Brand C. This means there is a greater difference in the number of sweets between the biggest and smallest packages. When buying sweets, I d expect a consistent number of sweets in any brand package I buy. I would choose Brand C = 84 schools 60 Angle 60 Total no. of schools (ii) Example answer: I disagree, as the first pie chart represents 165 primary schools, but the second chart only represents 79 post-primary schools. Both angles are approximately Primary: = 96 schools 45 Post-primary: = 91 schools 8. (i) Answer: No Reason: No two lengths are equal in measure, hence, an isosceles triangle with two sides of equal measure cannot be constructed. (ii) In a parallelogram, opposite sides are of equal length, but no two strips are of equal measure. Hence, they cannot be used to form a parallelogram. 1 Mathematics Junior Certificate

133 (iii) Use Pythagoras s theorem: Is (5) = (4) + (7)? 65 = = 65 It is a right-angled triangle. (iv) (Missing side) = (0) + (4) (Missing side) = Missing side = 976 = = 1 4 cm 7 cm Red 0 cm White 5 cm Yellow 4 cm Blue? 4 cm Blue 9. (i) The sin of EAB = opposite hypotenuse which can be written as BE = 80 AE 10 or CD AD = ED 80 (ii) sin EAB = 10 = DE 80(10 + DE ) = 00(10) DE = 4000 E 80 DE = = m DE = 180 m 10. (i) ADB is similar to ΔAPC: BAP = PAC told. ABD = APC (x + y) ADB = ACP (180 x y) Similar by A, A, A ( angles the same) (ii) P x + y B x + y 10 m A B C x A x 180 x y x + y y P B x x y D C D 180 x y 00 m 180 x y 014 Sample SEC P 180 x y C x A D 180 x y x A Small Δ Big Δ AC AD = EC BD = AP AB Cross multiply AC BD = AD PC Proportional sides in similar triangles 4 Higher Level, 014 SEC Sample, Paper 1

134 11. A F C D E B (i) (ii) DE = EF because the arc BE has the same radius as the arc EF. F is a point on two circles of equal radii. In the triangle BDF and the triangle BEF DE = EF..Part (i) BF = BF..Same line BD = BE..On same arc Therefore the triangle BDF and the triangle BEF.. SSS = SSS Therefore DBF = EBF Therefore BF bisects the angle ABC. 1. (a) axes of symmetry x = y A B C D A Central symmetry B Axial symmetry (through the x-axis) C Translation D Axial symmetry (through the line x = y) 14 Mathematics Junior Certificate

135 1. (i) 10 8 D C 6 4 A B (ii) AD = BC (, )(4, 8) = (10, 4)(1, 9) (4 ) + (8 ) = (1 10) + (9 4) () + (5) = () + (5) = = 9 (iii) E = midpoint of (, )(1, 9) = ( , ) = (7, 6) F = midpoint of (10, 4) (4, 8) = ( , ) = (7, 6) Since the midpoint of both diagonals AC and BD is (7, 6) the diagonals bisect each other. Distan ce Formula: (x x 1 ) + (y y) See page 18 of Formulae and Tables. Mid-point Formula: See page 18 of Formulae and Tables. (iv) No, we cannot. We would have to prove that opposite sides and opposite angles are equal. 14. (i) and (ii) l Sample SEC P A l Higher Level, 014 SEC Sample, Paper 15

136 15. (a) Answer: False/incorrect Reason: tan 60 = 1 7, which is greater than 1. Answer: True/correct Reason: sin 0 = 0 5, but sin 60 = 0 8. (c) False/incorrect Reason: cos 0 = 0 866, but cos 60 = 0 5. (d) () = l + h = h = h sin 60 = Theorem of Pythagoras Opposite Hypotenuse = cm 60 1 cm cm sin 60 = cm 16. The Leaning Tower of Pisa Tower of Suurhusen Church A 9 A 47 9 cos A = cos A = cos A = cos A = A = cos 1 ( ) A = cos 1 ( ) A = 86 A = 84 8 The most tilted tower is the tower of Suurhusen Church, which makes an angle of (i) x + 4x + 90 = 180 5x = 90 x = = 18 Hence, the other angle = 4(18) = 7 Answer: 18, 7 (ii) Slope = tan x = tan 18 = = Mathematics Junior Certificate

137 01 SEC Paper (Phase ) 1. (a) h = a + b Length = = 1 m Area = 1 (1)(5) = 0 m (c) Area of patio = 1 = 169 m Apply Pythagoras s Theorem. or Area of patio = 17 (4 0) = 169 m Area of a Triangle Formula A = 1 b h See page 9 of Formulae and Tables. (d) Area of flagstone = = 0 4 m Number of flagstones = 169 = 4 5 (or 4) 0 4 Extra 0% = = 84 5 or 10% = Total number of flagstones = 507 or Extra 0% = 4 0 = 84 6 or 10% = 4 1 Total number of flagstones = or Area to cover = = 0 8 Total number of flagstones = = 507. (a) Aerobics class Swimming class Key: 1 5 means 15 5 m 1 m Area of a Square formula A = l l = l Don t forget to fill in the key Aerobic median: = Swimming median: = 0 (c) Mean or Mode Median is the middle value if the results are arranged in ascending/descending order. Mean and mode are also considered measures of centre. 01 SEC P (d) An older age group take Aerobics class. Higher Level, 01 SEC, Paper 17

138 or A younger age group take Swimming class. or Similar. (a) A 1 1 B 1 1 C D E F G (i) 1. A B E. A B D E ways. A C D E Count all the individual paths from A to E. (ii) = = 1 Probability = = 1 = (a) Different strips: 5 = 0 Different strips: 4 = (a) Girls phone under pillow = 5% of 4171 = = Use the females pie chart. = 1460 (or or 1459) Calculate the probabilities of the different paths separately e.g. A-B-E means A to B and B to E = 1 1 = 1 4 ( and means multiply), etc. This is the Fundamental Principle of Counting. Choosing from and then choosing from and then choosing from 5 gives a total of 5 choices. Total number of students = 7150 Boys phone under pillow = % of 979 Overall : get the number from both pie charts to calculate the percentage. 18 Mathematics Junior Certificate

139 = = 685 (or or 686) Total = = 145 (or 145 0) Percentage = ( or ) = 0% (or 0 000%). (c) Angle = 0% of 60 = 60 0 Use the solution from part. Remember there are 60 in a pie-chart. = 108 (or ) 6. (a) Salary ( 1000) No. of Employees [Note: 10 0 means or more but less than 0 000, etc.] = The mid-interval values are 5000, 15000, 5000, 5000, 45000, 55000, Mean 5, 15, 5, 5, 45, 55 and 65 are the mid-interval values. (5000 1) + ( ) + (5000 1) + (5000 9) + (45000 ) + ( ) + ( ) = = = 8,750 (c) (i) Add up all the individual salaries and divide by. 7. (a) U (ii) Answer: Adding up individual salaries and dividing by Reason: This gives the actual mean as estimates (mid-intervals) are not used. C L Fill in the intersection of all sets first. 01 SEC P 10 R 4 Probability person did not vote = 4 54 or 7 Probability = 4 (did not vote) 54 (total) = 7 Higher Level, 01 SEC, Paper 19

140 (c) Probability person voted for at least two parties = = Probability = (d) Probability person voted for the same party = 19 (voted for two or three parties) 54 (total) = Probability = 1 (voted for C only or L only or R only) 54 (total) 8. C 1. Place the tip of the compass on the vertex of the angle and draw an arc which cuts each arm of the angle at two different points.. Place the tip of the compass on one of the points where the fi rst arc cuts an arm of the angle. B A. Place the compass tip on the point of intersection of the fi rst arc and the other arm, without changing the width of the compass α β l 1 l 4. Draw a line connecting the vertex of the angle and the point of intersection of the second and the third arcs. 5. This line segment bisects the angle. 115 γ l α = 180 ( ) = 5 β = = 140 γ = 40 A straight angle measures 180. Alternate angles are equal. The three angles in a triangle sum to Mathematics Junior Certificate

141 10. (a) h = a + b (c) (d) = AC + 1 AC = 1 = cos BAC = BAC = 0 Hypotenuse = = cos 45 = 1 cos 75 = 6 = cos 45 + cos 0 = 1 + ( ) Apply Pythagoras s Theorem. cos A = Adjacent Hypotenuse = = A B C 11. x = x = 10 Similar triangles have corresponding sides in proportion. x SEC P Higher Level, 01 SEC, Paper 141

142 1. A B C (a) A(, 6) B( 6, 0) C(4, ) D = ( 6, ) = ( (c) Slope AB = =, ) Equation AB: y 0 = (x + 6) or y 6 = (x ) or y = x + 4 x y + 1 = 0 (d) Perpendicular slope = Line through C: y + = (x 4) x + y 8 = 0 or The line is of the form x + y + c = 0 (4, ): (4) + ( ) + c = 0 c = 8 x + y 8 = 0 Midpoint formula: See page 18 of Formulae and Tables. Equation formula: y y 1 = m(x x 1 ) See page 18 of Formulae and Tables. Remember: to find a perpendicular slope, invert and change the sign. (e) E the point of intersection of two lines x y + 1 = 0 (i) x + y 8 = 0 (ii) x + 1 (i) 4x 6y = 4 or y = + (ii) 9x + 6y = 4 x + x + 1 ( ) 8 = 0 9x + 4x = 0 x = 0 and y = 4 Use simultaneous equations to find E. 14 Mathematics Junior Certificate

143 (f) CD = ( 4 + ) + ( ) = 55 5 or 7 4 CE = (4 0) + ( 4) = 5 or 7 11 CE is the shorter distance. or CE (is the perpendicular distance and therefore is the shorter distance.) CE = (4 0) + ( 4) = 5 or 7 11 Length formula: See page 18 of Formulae and Tables. 1. θ tan θ = or tan α = 6 6 θ = α = 6 07 θ = θ = = 6 56 = = = m tan A = Opposite Adjacent 14. (a) Volume of hemisphere = πr = π 1 = 115π cm Note: This formula is found by halving the volume of a sphere. See page 10 of Formulae and Tables. 01 SEC P (c) Volume of cone = 1 πr h = 1 π 4 1 = 64π cm Remaining metal = 115π 64π = 1088π Volume of a cone: = 1 π h See page 10 of Formulae and Tables. Higher Level, 01 SEC, Paper 14

144 Volume of cone = 64π Number of cones = 1088π 64π = 17 or Number of cones = 115π 64π 1 = 18 1 = h h r r (a) Curved surface area of a cylinder = πrh Curved surface area of small cylinder = π r h = πrh Curved surface area of large cylinder = π (r) (h) = 8πrh Ratio = πrh : 8πrh = 1 : 4 Curved surface area of cylinder = πrh See page 10 of Formulae and Tables. Volume of a cylinder = πr h Volume of a cylinder V = πr h Volume of small cylinder = π r h = πr h See page 10 of Formulae and Tables. Volume of large cylinder = π (r) (h) = 8πr h Ratio = πr h : 8πr h = 1 : Mathematics Junior Certificate

145 01 SEC Paper (Phase ) = m = = 7 8 m = = 7 8 m 10 mm = 1 cm 1000 mm = 1 m mm = 1 m. (a) πr h = V 1 litre = 1000 cm π r 70 = m = 70 cm r = Volume of a Cylinder formula: π 70 r V = πr = 7 6 h See page 10 of Formulae and Tables. r = 15 Diameter = 0 cm. (a) = cm or 1 5 L (c) π h = h = π h = 44 6 cm = 6% 50 6% of the height of 70 cm is 44 1 cm 4 4 = 6 cm (6 6 6) + ( 4 π ) = (c) = 658 cm The height consists of four side lengths of the cube = = = 8% = 16 4 π = = = = 8% 864 Change all to centimetres. Remember 1 m = 100 cm. Method 1: Compare volumes Method : Compare percentages The volume is made up of spheres and cubes. Volume of a Sphere formula: V = 4 πr See page 10 of Formulae and Tables. Remaining 100% Total 01 SEC P Higher Level, 01 SEC, Paper 145

146 4. (a) Answer = Diagram D πr = π 5 = cm Need a piece 10 cm to make this cylinder. Only D has this. Area of sheet: 18 = 414 Surface area of cylinder: 10 + (π 5 ) = = 97 Metal remaining: = 117 cm (c) V = π 5 10 = cm Surface Area of Cylinder = Rectangle + Two Circular ends Volume of a Cylinder formula: V = πr h See page 10 of Formulae and Tables. 5. (a) Event Probability A club is selected in a random draw from a pack of A playing cards. 1 4 A tossed fair coin shows a tail on landing. B 1 OR evens OR The sun will rise in the east tomorrow. C 1 OR certain May will follow directly after June. D 0 OR impossible A randomly selected person was born on a Thursday. E 1 7 Probability = Favourable outcomes Total outcomes 0 E A B 1 D C 6. (a) Male actors Female actors Key: 5 is 5 years old Don t forget to fill in the key Same shape of distribution No one is over 61. No one is under 4. The range is similar in both. Discuss shape/range/unusual numbers etc. 146 Mathematics Junior Certificate

147 There is an outlier in the female winners. No female is in her 50s. The females are younger. (c) (i) Male Female Sum = 887 Sum = 714 Mean = 44 5 Mean = 5 7 (d) 7. (a) (c) 8. (a) The mean age of women is lower so the statement is true for mean age. (ii) Male Female ( ) ( + ) Median = 45 Median = The median age of women is lower so the statement is true for median age. Male Female = = 11 5 or = = ( ) = 86 (c) (d) = = = = = = V 0 14 Experimental error Probability = (blue) (total) Use numbers to back up your arguments. I.Q.R. = Upper Quartile Lower Quartile and : Multiply the individual probabilities. and : Multiply the individual probabilities. Probability = 1 (one shows on the die) 6 (total) Relative Frequency: How often something happens divided by all the outcomes 01 SEC P Higher Level, 01 SEC, Paper 147

148 9. (a) and 4 D (0, 4) A (1, 4) E ( 6, 0) 1 B (, 0) C ( 4, 4) 4 5 F (4, 4) (c) ( 0 + 4, 4 4 ) = (, 0) 4 0 (d) 4 = 4 (e) y = x + 4 y 0 = (x ) (f) 0 ( 4) 6 ( 4) = 4 = y = x + 4 Midpoint formula: See page 18 of Formulae and Tables. 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. Slope formula: See page 18 of Formulae and Tables. (g) y 0 = (x + 6) y = x 1 x + y + 1 = 0 Equation formula: y y 1 = m(x x 1 ) See page 18 of Formulae and Tables. (h) Area of ΔBCE Area of ΔBCF 1 (8)(4) 1 (8)(4) Ratio: 1:1 or 1/1 Area of a Triangle Formula A = 1 b h See page 9 of Formulae and Tables. (i) Answer: Yes Congruent: identical in all respects Reason: CFBE is a parallelogram and CB is a diagonal which divides the parallelogram into two congruent triangles. or SSS or SAS or ASA argument 148 Mathematics Junior Certificate

149 10. (a) Line OR y = 5x + 0 Use y = mx + c 5 is the biggest number in front of x for any of the lines. Line 1 and Line y = x 6 and y = x + 1 They have the same slope (). Parallel lines have equal slopes. (c) y = x 6 x = 0, y = 6 y = 0, x = (, 0), (0, 6) (d) slope = = y-intercept = 4 equation y = x + 4 Answer = Line 5 (y = x + 4) Equation formula: y y 1 = m(x x 1 ) See page 18 of Formulae and Tables SEC P Higher Level, 01 SEC, Paper 149

150 (e) y = 4x m = 9 7 = 4 y = 4(7) 16 = 1 y 1 = 4(x 7) y = 4(9) 16 = 0 y = 4x 16 y = 4(10) 16 = 4 Answer = Line 6 Find the equation which is satisfied by all the points. (f) y = x 7 y = 4x 16 0 = x 9 x = y = 7 = 4 (, 4) Answer: x = Simultaneous Equations 11. (a) 1. (g) Line 4 y = () 7 = 4 Line 6 y = 4() 16 = 4 A A = 84 cos A = Draw a rough diagram. Opposite. Remember that sin A = Hypotenuse = Draw a base of any length. 4. Use your protractor/set square to draw a perpendicular line to the base. 5. Mark off 4 cm on this line using your compass. 6. Use the compass to draw a 10 cm line segment from the top of the 4 cm line segment to the base. 7. Label angle A and the side lengths 4 cm and 10 cm. cos A = Adjacent Hypotenuse 5 cm 10 cm 0 cm ramp Apply Pythagoras s Theorem 105 cm 150 Mathematics Junior Certificate

151 Ramp = OR = 109 cm H = H = H = 15 Ramp = H Ramp = 15 Ramp = 109 cm 1. A B h h 0 P 60 4 m x (a) tan 60 = h x tan 60 = h h x tan 0 = 4 x = h x 1 7 = h 1 x = h 4 x h = x h =1 7 x h = 4 x tan A = Opposite Adjacent tan 0 = 1 = h 4 x x 4 x h tan 0 = 4 x = 1 7x 4 x x = 4 x x = 4 x 4x = 4 4x = 4 x = 6 x = 6 h = 6 m h = 6 m or 6(1 7) m = 10 9 m 01 SEC P Higher Level, 01 SEC, Paper 151

152 14. Given: A circle with centre O, with points A, B and C on the circle To Prove: BOC = BAC Construction: Join A to O and extend to R Proof: In the triangle AOB AO = OB Radii OBA = OAB Theorem (isosceles Δ) BOR = OBA + OAB Theorem 6 (exterior angle) BOR = OAB + OAB BOR = OAB Similarly ROC = OAC BOC = BAC Standard Proof 15. (a) z b Label clearly any angles you add in. x a c y x = b + c (external angle) a + b + c = 180 (Triangle) y = a + b (external angle) x + y + z = 60 z = a + c (external angle) x + y + z = (a + b + c) a 5b Alternate angles are equal b = 150(alt) b = 0 a + 5b = 180 a = 0 a = Mathematics Junior Certificate

153 011 SEC Paper (Phase 1) 1. (a) Semicircular lengths = πr = 14π Step 1 Straight lengths = (8) or 56 Step Total length = 14π + 56 = (i) Volume of cylinder = πr h (ii) Volume of cone = 1 πr h = 100 cm Step = π(7) (70) = 5100π cm = 1 π(7) (70) = 17010π cm or Volume of cone = 1 (5100π) = 17010π cm Both semicircular ends together form a full circle. Don t forget to give your answer correct to the nearest whole number. Volume of a Cylinder formula: See page 10 of Formulae and Tables. Leave your answer in terms of π. This means leave π in your answer. Volume of a Cone formula: See page 10 of Formulae and Tables. (c) (iii) Remainder = 5100π 17010π = 400π Fraction = 400 π 5100π 400 = 5100 or or 1 1 = E Volume of Cylinder Volume of Cone C A B D 011 SEC P Higher Level, 011 SEC, Paper 15

154 (i) Area of CDE = 1 πr = 1 π(100) 4 4 = 500π = = m (ii) Area of throwing zone = πr = π(1) = 1π = 1416 = 14 m CDE is one quarter of the disc. Circle of radius 1 m (iii) Area of the shot-put zone = 4 (Area of throwing zone) + Area CDE 4 (Area of throwing zone) = 4 ( 14) = 55 Area of shot-put zone = = = or 4 (Area of throwing zone) = 0 75π Area of shot put zone = 0 75π + 500π = π = = m or 1 4 (Area of throwing zone) = 1 4 ( 14) = Area of shot put zone = = = or 1 4 (Area of throwing zone) = 0 5π. Blood Group Percentage in Irish population Area of shot put zone = 500π + 1π 0 5π = π = = m Blood groups to which transfusions can be safely given 154 Mathematics Junior Certificate Blood groups from which transfusions can be safely received O 8 All O O+ 47 O+, AB+, A+, B+ O+ and O A 5 A, A+, AB+, AB A and O A+ 6 A+ and AB+ A+, O, O+, A B B, B+, AB, AB+ B and O B+ 9 B+ and AB+ B+, B, O, O+ AB 1 AB and AB+ AB, O, A, B AB+ AB+ All Source: Irish Blood Transfusion Service

155 (a) B (%) + O (8%) = 10% OR B ( 100 ) + O ( ) = 100 (c) O+(47%) + AB+(%) + A+(6%) + B+(9%) = 84% = (d) 1% = 5 O can only receive blood from other O people. This is only 8% of the population, therefore this category needs to be encouraged to donate blood. or O can safely give blood to all other groups and so is the best to have if there is any shortage of blood.. Colour Frequency Relative frequency Daily frequency (Part (e) below) = (reduce to simplest form where possible in 10 probability answers) = 1 Red 70 Blue 100 Yellow 45 White 55 Black 90 Silver 140 Total or or or or or = or (a) 500 ( ) = = 90 black cars Done in table Relative Frequency: How often something happens divided by all the outcomes (c) Method: The sum of the relative frequencies should total to 1. OR The percentages should sum to 100%. Check: Candidate to show his/her check (d) 70 = 0 14 = 14% 500 No. of red cars Total No. of cars 011 SEC P Higher Level, 011 SEC, Paper 155

156 (e) (f) = = = OR = = = = = = = = = = 67 No. A test is reliable if repeated runs of the test would give the same results. There is no reason to say that if this test was run again it would be different because of the sample not being random. The colour of a vehicle is random and running the test at different times of the day or on different days would not necessarily make the test any more reliable. 4. (a) 5 9 = 45 OR = 6 OR = = 4 dessert choices 6 9 = 4 dessert choices 6 9 = 4 dessert choices S M D Fundamental Principle of Counting = 18 different -course lunches 5. Airplane A B C D E F G Distance (cm) (a) Median = 50 OR C = 1680 = 40 cm 7 7 (c) 80 7 = 1750 OR = = = 100 cm = 100 cm OR x = x = x = 1750 x = 100 cm Median: Middle value when the values are arranged in order of size Mean: Sum of all values divided by total number of values (d) The minimum distance is anything greater than 50 cm. OR x > 50 cm, x R 156 Mathematics Junior Certificate

157 % % Broadband connection By type of connection DSL (<Mb/S) 1 9 DSL (>Mb/S) Other fixed connection 1 0 Mobile broadband 4 7 (a) Bar Charts DSL (<Mb/S) DSL (>Mb/S) Other fixed connection Mobile broadband 7. (a) The fixed connection went down a lot. The DSL > Mb (faster connection) went up. The DSL < Mb (slower connection) went down. There was no increase in broadband connection. Mobile broadband went up slightly. Test Test The stem-and-leaf diagram need not be sorted. A comparison bar chart is very useful with this type of data. Test 1: Before practice Test : After practice A key should be included with all stem and leaf plots e.g. 5 means 5 sit-ups. 4 (c) Test = Test 6 = 9 The range is the largest value minus the smallest. (d) Yes. Only people did worse after practising. did the same and 19 did better. Yes, there is a higher average. Yes, the median is higher. Yes, there is a general shift of data upwards. Most students did better after the exercise. 011 SEC P Higher Level, 011 SEC, Paper 157

158 (e) Compared Favourably: The class average improvement is 67. John s improvement is. Therefore he improved by more than the average improvement of his classmates. Compared Unfavourably: There were 8 people below him before the practice. There were only 7 people below him after the practice. Therefore he moved down relative to his classmates. 8. Number of days absent None One Two Three Four Five Number of students (a) (c) The 9 students (or none ) who missed no days would not change. The 5 who were absent on the Friday would fall under one of the other five categories, since they had missed at least one day (the Friday). Smallest possible number of days missed Number of days absent None One Two Three Four Five Number of students (d) Largest possible number of days missed Number of days absent None One Two Three Four Five Number of students Number of days absent None One Two Three Four Five Number of students Number of degrees = A = = = 1 7 days Mean: sum of all values divided by total number of values 48 m B 57 m E BE = 57 m BC = 48 m CD = 1 m C 1 m D 158 Mathematics Junior Certificate

159 (a) (c) Peg C must be collinear with the two trees, A and B. Pegs E and D must be collinear with each other and the tree A. Also [BE] must be parallel to [CD]. BA AC = BE 1 AB = AB CD 76 AB = 76 AB 48 + AB = 57 AB = 6 m 1 CD 40 = 45 6 CD = 50 m 6 A Allows us to use similarity to find AB A line drawn parallel to one side of a triangle divides the other two sides in the same proportion. 9 B 40 C E D (d) Create a parallelogram CBEX using strings, where CB = XE = 48 m And BE = CX = 57 m Then extend [CX] until D is collinear with E and A. OR Create a parallelogram BX 1 DC, where BX 1 = CD = 1 m and BC = X 1 D = 48 m 10. (a) Equilateral triangle showing the three axes of symmetry Axis of Symmetry: line along which the shape will fold onto itself 011 SEC P Higher Level, 011 SEC, Paper 159

160 A square showing the four axes of symmetry 11. (a) To Prove: [AE bisects DAC. Proof: Y = W Alternate Y = X Isosceles X = Z Corresponding W = Z Therefore [AE bisects DAC D Z A W T E Make sure to label all angles clearly. You should also justify with a reason, any geometric statement that you make. 1. No, the result in (a) would not still apply. Angle Y would not be equal to angle X. B X Y C (40 40) Distance (km) l1 l (5 75) Fuel (litres) 75 0 (a) 5 0 Slope of l 1 = OR Slope = Rise Run = 75 5 Rise Slope = Run = Slope of l = 8.5 = 11 = 8.5 Slope l 1 = 11 Slope l = 8.5 Slope formula: See page 18 of Formulae and Tables. 160 Mathematics Junior Certificate

161 The higher slope for l 1 indicates that you get more km per litre at the lower speed. OR More fuel is used at the higher speed. Slope is the rate of change of distance with respect to fuel consumption. (c) l 1 : l : = 8.0 OR 80c y = mx y = mx 00 = 11x 00 = 8 5x Calculate the costs separately and subtract. x = litres x = 5 litres = = Museum N North Street You are here Perpendicular Avenue B(, 10) A(1, 8 5) C(10, 9) Tangent Street E(6, 6) D(17, 8) Town Hall Straight Road East Drive Road Path (a) (x x 1 ) + (y y 1 ) (10 ) + (9 10) (7) + ( 1) km Distance formula: See page 18 of Formulae and Tables. You could also use Pythagoras s theorem. 011 SEC P Higher Level, 011 SEC, Paper 161

162 (x x 1 ) + (y y 1 ) = 10 (10 6) + (9 6) = km (4) + () (c) m = y y 1 x x = 1 = 1 5 (d) ( or y y 1 = m(x x 1 ) y 10 = ) (x ) or y 8 5 = 1 5 (x 1) ( or x 4y + 1 = 0 Equation of Tangent Street Perpendicular slope = 1 5 ( or 4 ) y y 1 = m(x x 1 ) y 8 = 1 5 (x 17) 4 ( or used as slope ) 4x + y 9 = 0 4 used as slope ) Distance formula: See page 18 of Formulae and Tables. Slope formula: See page 18 of Formulae and Tables. Perpendicular slopes: invert and change sign Equation of a Line formula: See page 18 of Formulae and Tables. (e) x 4y + 1 = 0 Museum at (11, 16) 4x + y 9 = 0 Solve simultaneous equations (f) North to Tangent Street (7 75 km) and then on to the Museum (1 75 km) i.e. distance from (0, 7 75) to (11, 16) = 1 5 km East for 1 km to Straight Road. Then North to A (8 5 km). Then from A to the Museum i.e. distance from (1, 8 5) to (11, 16) = 1 5 km = km 14. (a) = cm (9x) + (16x) = x + 56x = x = x = 0 6 x = 5 54 length = = = 89 to nearest cm height = = = 50 to nearest cm (c) (4x) + (x) = x + 9x = x = x = x = Mathematics Junior Certificate The ratio is 16:9. Apply Pythagoras s theorem. Make sure to give your answer correct to the nearest cm. Apply Pythagoras s theorem and subtract the answers.

163 length = 4 0 = 81 8 = 81 to nearest cm height = 0 = = 61 to nearest cm (81 61) (89 50) = 491 cm 15. (a) πr = r = 7 07 r = m Height is measured from the centre of the base to the top of the spire. OR: The circumference can be used to calculate the radius, which will give the full distance that Maria is from the centre of the base of the spire. Maria s distance to the centre of the base is ( ) m. Apply the Tan ratio Opposite to solve. Adjacent 1 7 m m 1 1 m x tan 60 = 71 1 x = 1 Spire = = 14 9 = 15 m 011 SEC P Higher Level, 011 SEC, Paper 16

164 015 SEC Supplementary Questions 1. Approximating the island with a conical shape, 916 m is represented with approx. 4 cm in the photograph. 9 m = 1 cm The width of the island is approx. 11 cm. The radius is 5 65 cm = m Volume of a Cone = 1 π r h See page 10 of Formulae and Tables. V = 1 π (19 85) (916) V = m. (a) n + 1, n +, n + (n + 1) + (n + ) + (n + ) = n + 6 n + 6 = n + i.e. divides evenly by (c) (n) + (n + 1) + (n + ) + (n + ) = 4n + 6 Natural numbers differ by 1. Since 4n + 6 is not evenly divisible by 4, the sum of four natural numbers will never be evenly divisible by 4.. (a) 4 Drawing lines can help to see where these numbers come from. Two lines Each square can be reached by at least one vertical and at least one horizontal line. (c) Maximum is 4 for ODD values of n. Maximum is for EVEN values of n. The only squares which belong to 4 lines are the centre squares in any grid. Centre squares will only occur for odd values of n. 4. Let the selling price be 0 and the VAT be 10%. 0 = 110% Divide both sides by = 1% Multiply both sides by = Price before VAT VAT = 0 00 = 0 Draw grids for n an odd number and n an even number to understand this solution. 164 Mathematics Junior Certificate

165 5. (a) If a triangle is right angled, then it has sides cm, 4 cm and 5 cm. False There are right-angled triangles with different side lengths, e.g. 5 cm, 1 cm and 1 cm. Reverse the order of Maisy s statement to build its converse. 6. (a) 5(x ) ( x) H.C.F. Not all converses are true statements. 5(x ) ( x) = 5(x ) (x ) = 5 which is constant proportional (x ) (x ) = 1 (c) x + x + = (x + 1)(x + ) = x + which is a variable x + (x + 1) Not proportional (x + 1) (x + 1) = 1 Factorise above and below 7. Let b = No. of hours in Bob s Bakery. Let c = No. of hours in Ciara s Café. Solve simultaneous equations b + 9 0c = 6 40 b + c = 4 (Multiply by 11. 5) 11 50b + 9 0c = b c = 91 (Subtract) c = 8 6 c = 1 b = 1 hours 8. (a) { 6, 6, 6, 6, 6 } Data sets in which all the elements are identical 9. Supplementary Sample Questions 4 9x Sides are in proportion 16:9. 16x 4 = (9x) + (16x) 1764 = 81 x + 56 x 1764 = 7 x = x 144 x1764 = 144 x 7 = Area of television = 754 inches Higher Level, Supplementary Sample Questions 165

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