MATHEMATICS National Qualifications - National 5 Paper 1 (Non Calculator) Testing EF and REL

Transcription

1 N5 Prelim Practice Paper B MATHEMATICS National Qualifications - National 5 Paper 1 (Non Calculator) Testing EF and REL Time allowed - 1 hour Fill in these boxes and read carefully what is printed below Full name of centre Town Forename(s) Surname Date of birth Day Month Year Candidate number Seat number Total marks You may NOT use a calculator.. Use blue or black ink. Pencil may be used for graphs and diagrams only.. Write your working and answers in the spaces provided. Additional space for answers is provided at the end of the booklet. If you use this space, write clearly the number of the question you are attempting. 4. Square ruled paper is provided. 5. Full credit will be given only where the solution contains appropriate working. 6. State the units for your answer where appropriate. 7. Before leaving the examination room you must give up this booklet to the invigilator. If you do not, you may lose all the marks for this paper.

2 FORMULAE LIST The roots of ax + bx + c = 0 are x = b ( b 4ac) a Sine rule: a sina b c sinb sinc Cosine rule: a = b + c bc cos A or cos A = b c a bc Area of a triangle: Area = ½ ab sin C Volume of a sphere: Volume = 4 r 1 Volume of a cone: Volume = r h 1 Volume of a Pyramid: Volume = Ah Standard deviation: s = ( x x) x ( x) n 1 n 1 / n, where n is the sample size.

3 Marks Do not write in this margin. All questions should be attempted 1. Factorise fully 0x² + 6x 8. Find the gradient of the line joining the points (4, 6) and (, 8).. Evaluate 9x² 100y² when x = 1 5 and y = 0 55.

4 Marks Do not write in this margin. 4. Change the subject of this formula to h. r V h Find the value of (a) Express with a rational denominator. 5 (b) Simplify

5 7. (a) State the coordinates and nature of the turning point of the graph of the parabola with equation. Marks Do not write in this margin. y = (x + 1) 1 (b) Find the coordinates of the point where the graph crosses the y axis. (c) Make a sketch of the parabola on the grid below showing clearly its turning point and y axis intercept. y O x

6 Marks Do not write in this margin. 8. (a) If f(x) = x + x 5, find the value of f( ). (b) Given that f(a) =, find the value of a given that a > The graph in the diagram has equation of the form y acos bx c. y O 1 10 o x Write down the values of a, b and c.

7 Marks Do not write in this margin. 10. Write as a single fraction in its simplest form: 7 5 ; k (k ) k 0; k 11. The logo for Cyril's Cars is shown below. The logo is a sector of a circle of radius 9 cm. The reflex angle at the centre is 40 o. 40 o Taking π = 14, calculate the perimeter of the logo. End of Question Paper

8 ADDITIONAL SPACE FOR ANSWERS

9 ADDITIONAL SPACE FOR ANSWERS

10 National 5 EF and REL Paper 1 Marking Scheme Qu Give one mark for each Illustrations for awarding mark 1 ans: (5x + 4)(x 1) marks takes out common factor one bracket correct second bracket correct ans: m marks 7 knows how to find gradient states gradient (10x + x 4) (5x + 4). (x 1) 8 6 m = 4 m = 7 ans: 10 marks 4 recognises difference of two squares substitutes values evaluates V ans: h marks r squares both sides (x + 10y)(x 10y) [(1 5) + 10(0 55)][(1 5) 10(0 55)] (10)( 1) = 10 r V h multiplies through by h divides by r 5 ans: 1/ marks r h V V h r 6a evaluates 7 1/ evaluates 81 /4 evaluates ans: knows to multiply by simplifies 5 5 marks 7 1/ = 81 /4 = 1/7 1/ b ans: 6 marks 1 simplifies surd simplifies

11 Qu Give one mark for each Illustrations for awarding mark 7a ans: ( 1, 1); minimum marks correct x coordinate correct y coordinate states nature ( 1,.. 1) minimum b ans: (0, 0) marks knows to substitute 0 for x states coordinates y = (0 + 1)² 1 (0, 0) [accept origin] c ans: graph drawn marks 8a parabola with minimum TP turning point and origin marked ans: 5 marks correct shape both points marked clearly knows to substitute value evaluates f ( ) = ( ) + ( ) 5 5 b ans: a = 4 marks knows to substitute a for x and equates brings all to LHS and factorises solves discards one value 9 ans: a = ; b = ; c = 1 marks 4 a + a 5 = a + a 8 = 0; (a + 4)(a ) = 0 a = 4 or 4 a = 10 states value of a states value of b states value of c 9k 1 ans: k( k ) marks a = b = c = 1 correct denominator correct numerator simplifies numerator 11 ans: 55 68cm marks k (k ) 7(k ) 5k 9k 1 knows how to find perimeter simplifies calculation evaluates 40 P [+ 18] 60 P cm Total 40 marks

12 N5 Prelim Practice Paper B MATHEMATICS National Qualifications - National 5 Paper (Calculator) Testing EF and REL Time allowed - 1 hour and 0 minutes Fill in these boxes and read carefully what is printed below Full name of centre Town Forename(s) Surname Date of birth Day Month Year Candidate number Seat number Total marks You may use a calculator.. Use blue or black ink. Pencil may be used for graphs and diagrams only.. Write your working and answers in the spaces provided. Additional space for answers If you use this space, write clearly the number of the question you are attempting. is provided at the end of the booklet. 4. Square ruled paper is provided. 5. Full credit will be given only where the solution contains appropriate working. 6. State the units for your answer where appropriate. 7. Before leaving the examination room you must give up this booklet to the invigilator. If you do not, you may lose all the marks for this paper.

13 FORMULAE LIST The roots of ax + bx + c = 0 are x = b ( b 4ac) a Sine rule: a sina b c sinb sinc Cosine rule: a = b + c bc cos A or cos A = b c a bc Area of a triangle: Area = ½ ab sin C Volume of a sphere: Volume = 4 r 1 Volume of a cone: Volume = r h 1 Volume of a Pyramid: Volume = Ah Standard deviation: s = ( x x) x ( x) n 1 n 1 / n, where n is the sample size.

14 All questions should be attempted Marks Do not write in this margin. 1. When x 11x 7 is expressed in the form (x a) + b, what are the values of a and b?. Simplify x x 1 6x x

15 . The circle in the diagram below has centre O. Points P, Q and R lie on its circumference. Angle RPO = 70 o. Marks Do not write in this margin. P R 70 o x o O Q (a) Calculate the size of angle ROQ. (b) Given that the diameter of the circle above is 0 cm, calculate the area of minor sector ROQ giving your answer correct to significant figures. 4

16 Marks Do not write in this margin. 4. A quadratic equation is given as x² px + 4 = 0. Given that this quadratic has equal roots, find the positive value of p giving your answer as a surd in its simplest form In a human being weighing 150 pounds there are 10 4 pounds of copper. Calculate how much copper there would be in 100 such people. Give your answer in Scientific Notation. 6. Multiply the brackets and simplify (x )(x² 7x + )

17 Marks Do not write in this margin. 7. During one Saturday morning the charity car wash raised by washing 0 cars and 6 vans. (a) Write an equation to illustrate this information. 1 The following Saturday they washed 1 cars and vans and raised 105. (b) Write an equation to illustrate this information. 1 On the third Saturday they hoped to raise at least 10 by washing 5 vans and some cars. (c) What is the minimum number of cars that would have to be washed to raise this amount of money? 4

18 8. A Christmas decoration is formed from a cone and a hemisphere as shown in the diagram below. The diameter of both is 0cm. Marks Do not write in this margin. h cm The volume of the decoration is 49 5 cm. Calculate the height, h cm, of the decoration. 6 0 cm 9. Find the roots of the quadratic equation 4x + 7x 5 = 0 Give your answers correct to 1 decimal place. 4

19 Marks Do not write in this margin. 10. Shown is a children s play tunnel. The end of the tunnel consists of part of a circle, centre C, with diameter 1 metres. The width of the floor of the tunnel is 1 metre. 1 m C h m Would a toddler of height 80cm be able to walk through the tunnel? 5 You must show all your working and give a reason for your answer.

20 Marks Do not write in this margin. 11. Solve the equation 5 sin x sin x 0 x The two boxes below are mathematically similar. 40cm 0cm If it requires 08 m [ignoring overlaps] of card to make the large box, calculate the amount of card needed to make the smaller box. End of Question Paper

21 ADDITIONAL SPACE FOR ANSWERS

22 ADDITIONAL SPACE FOR ANSWERS

23 National 5 EF and REL Paper Marking Scheme Qu Give one mark for each Illustrations for awarding mark 1 ans: a = 5 5; b = 7 5 marks a starts to complete square completes square correct values of a and b x ans : x factorises numerator factorises denominator simplifies ans: 140 o marks marks (x 5 5)². (x 5 5)² = (x 5 5)² 7 5 a = 5 5; b = 7 5 (x 1)(x 1) x(x 1) x 1 x recognises isosceles triangle finds angle POR states size of angle ROQ explicitly angle PRO = 70 o angle POR = 40 o angle ROQ = 140 o b ans: 70cm³ 4 marks 4 correct fraction correct radius in formula answer not rounded correctly rounded answer 4 ans : p = 4 4 marks π 15² cm³ Unrounded answer must be stated and correct units should also be stated. knows condition for equal roots calculates b 4ac solves for p simplifies surd 5 ans : marks 4 uses correct calculation answer in Scientific Notation 6 ans : 6x³ x² + 7x 9 marks three terms correct further three terms correct simplifies b² 4ac = 0 [stated or implied] ( p)² 4 4 = 0; p² = 0 p = 4 p = x³ 14x² + 6x.... 9x² + 1x 9 6x³ x² + 7x 9 [must have x³ term]

24 Qu Give one mark for each Illustrations for awarding mark 7a ans: 0c + 6v = mark constructs equation 0c + 6v = b ans: 1c + v = mark constructs equation 1c + v = 105 c ans: 4 cars minimum 4 marks knows to use system of equation finds correct value for c finds correct value for t finds minimum number of cars 8 ans: 1cm 6 marks knows to find volume of hemisphere finds volume of hemisphere finds volume of cone subs values in formula for cone finds height of cone finds h 9 ans :, marks 4 knows to use quadratic formula calculates b 4ac subs correctly into formula states both roots correctly rounded 10 ans : yes since 0 9m > 0 8m 5 marks assembles facts in right triangle knows to use Pythagoras uses Pythagoras correctly finds height conclusion 11 ans: 8 6 o, 11 4 o 4 marks 4 5 solves for sinx o finds relative angle finds one solution 4 finds second solution 1 ans: 1 17m² marks evidence of scaling equations c = 4 5 v = cars V 10 HS = V cone cm = 1cm evidence , 0 5 x² = 0 6² 0 5² x = h = yes since 0 9m > 0 8m [or equivalent] sinx o = / o 8 6 o o xm 0 5m 0 6m finds linear scale factor for reduction finds area scale factor multiplies by ASF to answer /4 (/4)² (/4)² 08 = 4p Total 50 marks