Elgin Academy. MATHEMATICS National Qualifications - National 5 Paper 1 (non-calculator)

Transcription

1 N5 Prelim Elgin Academy Examination 01 / 14 MATHEMATICS National Qualifications - National 5 Paper 1 (non-calculator) Covering all three Units Time allowed - 1 hour Read carefully what is printed elow Total marks You may NOT use a calculator.. Use lue or lack ink. Pencil may e used for graphs and diagrams only.. Write your working and answers on the lank paper provided. Write clearly the numer of the question you are attempting. Extra paper may e requested at any time from the invigilator. 4. Square ruled paper is also provided. 5. Full credit will e given only where the solution contains appropriate working. 6. State the units for your answer where appropriate.

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

3 All questions should e attempted 1. Calculate Find the gradient of the line which has equation 5x + 7y + 5 = 0. Change the suject of formula to r V = π r h 4. (a) Factorise 5x 49 () Factorise fully 10x + 9x 7 (c) Simplify 5x 49 10x + 9x 7 1

4 5. Vectors a and have components as follows: a = 4 4 and = (a) Find the components of the vector represented y a. 1 () Calculate the magnitude of the vector represented y a. 6. Multiply the rackets and simplify ( x )(5x 4x ) The area of rectangle is 4 cm². It has length5. Calculate its readth,, leaving your answer as a surd in its simplest form with a rational denominator. Dimensions are in centimetres. 5

5 9. Write as a single fraction in its simplest terms 5 x x + 1 x ; x Simplify x 4 y 6 x y expressing your answer with positive indices. 11. AB is a diameter and O is the centre of the circle shown elow. BD is a tangent to the circle with B the point of contact. Triangle BCD is isosceles. B O A 50 o D C Given that angle BAC = 50 o, find the size of angle BDC.

6 1. The graph shows the height aove sea level, in metres, of eight places in Scotland and the corresponding mean temperature in degrees Celsius. The line of est fit has also een drawn on the graph Temperature ( o C) Height (metres) Determine the equation of line of est fit.

7 1. The graph shown has equation y = (x + )(x 4). y 0 4 x C (a) Find the coordinates of point C, where the graph cuts the y-axis. () Find the coordinates of the turning point. (c) State the equation of the axis of symmetry of the paraola. 1 1 End of Question Paper

8 National 5 Paper 1 ~ 01/14 Marking Scheme Qn Give one mark for each Illustrations for awarding mark 1 1 ans: marks 15 1 changes to improper fraction and simplifies 6 5 multiplies ans: m = marks 7 5 y = x 5 rearranges equation to y = mx + c 7 states gradient of line 5 m = 7 Note: Candidates may multiply then simplify. ans: V r = marks πh 4a divides oth sides y πh takes square root of oth sides ans: (5x 7)(5x + 7) marks r r = = V πh V πh 1 one racket correct second racket correct 1 (5x 7)...(5x + 7) ans: (5x + 7)(x 1) marks 1 one racket correct second racket correct 1 (5x + 7)(...(x 1) c ans: 5x 7 (x 1) 1 mark 1 simplifies 1 (5x 7) (x 1) 5a ans: 1 mark 6 states components ans: 7 marks 6 knows how to find magnitude evaluates + ( ) = 7

9 Qn Give one mark for each Illustrations for awarding mark 6 ans: 5x 14x + 6x + 4 marks starts to multiply completes multiplication simplifies 7 ans: 1 (Source: Pegasys Credit prelim 00 ) 4 sets up equation equates to zero factorises correctly selects correct answer 4 marks N.B. AWARD /4 for 1 with no working and AWARD 4/4 for 1 with evidence of trial and improvement. 5x 4x x. 10x² + 8x + 4 5x 14x + 6x + 4 [must have x term] 1 x ( x ) = 54 x x 108 = 0 ( x 1 )( x + 9) = 0 4 x = 1 8 ans: 1 5 marks 9 knows how to find readth knows how to rationalize denominator simplifies 1 - x ans: marks ( x - )( x + 1) 4 5 / (1 ) / 5 10 correct denominator correct numerator simplifies numerator y ans: x marks (x )(x + 1) (x + 1) 5(x ) 1 x starts to simplify writes with positive powers 11 ans: BDC = 80 o marks y² x y x realises ACB is right and finds ABC finds CBD and BCD finds BDC 1 ans: T = 1/100H + 15 marks ACB = 90 o ; ABC = 40 o CBD = 50 o ; BCD = 50 o BDC = 80 o Angle must e stated explicitly 1 finds gradient identifies y intercept states equation 1 m = 1/100 c = 15 T = 1/100H + 15 [or equivalent]

10 Qn Give one mark for each Illustrations for awarding mark 1a ans: C(0, 8) marks knows to su x = 0 states point C y = (0 + )(0 4) C(0, 8) ans: (1, 9) marks knows to sustitute x = 1 into equation states coordinates of T.P. y = (1 + )(1 4) (1, 9) c ans: x = 1 1 mark states equation of axis of symmetry x = 1 Total 40 marks

11 N5 Prelim Elgin Academy Examination 01 / 14 MATHEMATICS National Qualifications - National 5 Paper (Calculator) Covering all three Units Time allowed - 1 hour and 0 minutes Read carefully what is printed elow Total marks You may use a calculator.. Use lue or lack ink. Pencil may e used for graphs and diagrams only.. Write your working and answers on the lank paper provided. Write clearly the numer of the question you are attempting. Extra paper may e requested at any time from the invigilator. 4. Square ruled paper is also provided. 5. Full credit will e given only where the solution contains appropriate working. 6. State the units for your answer where appropriate.

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

13 All questions should e attempted 1. Jasper invests in a ank that pays 6% interest per annum. If Jasper does not withdraw any money, how much will he have in his account after years? runners took part in the 011 New York marathon which set a new world record for the numer of runners finishing a marathon. During the race each runner was supplied with 5 litres of water. How many litres of water were supplied altogether? Give your answer in Scientific Notation correct to significant figures.. In the diagram, M is the mid point of BC. AB represents vector p and AC represents vector q. B p M A q (a) Express BC in terms of p and q. C () Express BM in terms of p and q. 1

14 4. Solve the quadratic equation x² x 4 = 0 giving your answers correct to one decimal place I ought an antique painting a few years ago. It has gained 45% in value since it was ought and is now worth 655. Calculate how much the painting was worth when it was ought.

15 7. A metal ottle stopper is made up from a cone topped with a sphere. The sphere has diameter 1 5cm. Radius = 0 9cm h cm (a) Calculate the volume of the sphere. The total volume of the stopper is 6 cm³ () If the cone has radius 0 9cm, calculate the height, h cm, of the cone. 8. Last year, for a Mathematical Competition the organisers ought 0 medals and 4 trophies for 7.50 (a) Write down an equation in m and t to illustrate information. 1 This year they ought in 50 medals and 8 trophies which cost them () Form another equation in m and t to illustrate information. 1 (c) If the cost of a medal and a trophy remained the same for oth years, find algeraically the cost of 1 medal and 1 trophy. 4

16 9. Over the course of six weeks couple A scored the following marks out of 40 in a dancing competition: (a) Calculate the mean and standard deviation of these scores. 4 Over the same six weeks, couple B s scores were, on average, the same as couple A ut were slightly less consistent. () Write down a possile standard deviation for couple B s scores The diagram shows the cross section of a paper weight. It consists of part of a circle with a horizontal ase. The centre of the circle is O and it has radius 5cm. AB is a chord of the circle and measures 4cm. Calculate the height, hcm, of the paperweight. 4 O hcm A 4cm B

17 11. A metronome is a music tool which helps players with rhythm and tempo. A weight on the pendulum is adjusted so that the metronome swings ack and forth to give the correct tempo for a piece of music. For one particular piece the pendulum is set to a length of 1cm and as it swings it traces out an arc of a circle, AB, of length 15cm. A 15cm B 1cm x o Calculate, to the nearest degree, the angle x o, through which the pendulum swings.

18 1. Chris threw 100 darts at a target and a statistician recorded his results as follows: OUTER out of 5 landed in the inner section 7 out of 5 landed in the outer section 4 out of 50 missed the target completely. INNER Did the statistician record the results correctly? Give a reason for your answer. 1. These two tues of toothpaste are mathematically similar. The cost of the tue depends on its volume. The larger tue is 18cm long and holds 40 ml of toothpaste, the smaller one measures 1 cm. 40 ml 18cm 1cm If the larger tue costs 1.44, how much should the small one cost? End of Question Paper

19 National 5 Paper ~ 01/14 Marking Scheme Qn Give one mark for each Illustrations for awarding mark 1 ans: marks correct multiplier correct method answer ans : marks a knows to multiply answer in Scient. Not. correctly rounded ans : q p marks = chooses correct path answer BC = BA + AC q p ans: ½ (q p) 1 mark answer 4 ans : 4, marks 4 knows to use quadratic formula calculates 4ac sus correctly into formula states oth roots correctly rounded 5 ans : 1.7m/min (Source Pegasys Credit prelim 007) ½ (q p) evidence 41 ± , 0 9 finds the missing angle required sets up sine rule calculation correctly processes sine rule calculation 4 uses result from in susequent calc 5 6 processes calculation correctly uses s = d/t to find speed 6 ans : 4500 marks knows that 145% = 655 knows to divide 655 y 1 45 answer 6 marks 19 o x sin m 80 = sin19 h 4 sin 6 = h = 76. m speed = 76./6 = 1.7m/min 145% = % = a ans: 1 8 cm³ marks sustitutes values in appropriate formula answer ans: 5 cm marks m³ [accept any rounding] π finds volume of cone sustitutes into appropriate formula finds height of cone Pegasys 01 h = 5cm National = 4 m³ 4 = 1 π 0 9 h

20 Qn Give one mark for each Illustrations for awarding mark 8a ans: 0m + 4t = mark constructs equation 0m + 4t = 7 5 ans: 50m + 8t = mark constructs equation 50m + 8t = 66 5 c ans: medal costs 0.85; trophy costs 4 marks 9a 4 scales equations finds value for m finds value for t communicates answer ans: mean = ; S.D. = 8 4 marks finds mean finds ( x) and x sustitutes into formula 4 Or 4 answer finds mean finds deviations squared knows how to find SD answer ans: value > 8 1 mark evidence m = 0 85 t = 4 medal costs 0.85; trophy costs 19/6 = 6864; 618 ( ) [accept any correct rounding] 19/6 = = [accept any correct rounding] states possile value for SD 10 ans: 9.6m 4 marks assemles facts in right-angled triangle knows to use Pythagoras answer finds height of paperweight 11 ans : 7 o marks 4 SD > 8 evidence of R.A.T. 5 1 = 4.6m = 9.6m uses appropriate ratios sustitutes and re-arranges answer x/60 = 15/πD x/60 = 15/(π 4); x = (15 60) / 4π 7 o

21 Qn Give one mark for each Illustrations for awarding mark 1 ans : No, there are 4 unrecorded marks (Source: dept staff) denominator of 100 or % finds total correctly y addition conclusion 1 ans: 4p marks 0%, 68%, 8% 96% No, there are 4 unrecorded finds reduction scale factor finds volume scale factor multiplies y VSF to answer 1/18 = / (/)³ = 8/7 8/ = 4p Total 50 marks