Prel Examination 2005 / 06

Transcription

1 Prel Examination 00 / 0 MATHEMATICS National Qualifications - Intermediate Maths and Paper (non-calculator) Time allowed - minutes Read carefully. You may NOT use a calculator.. Full credit will e given only where the solution contains appropriate working.. Square-ruled paper is provided.

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

3 All questions should e attempted Marks. (a) Find the equation of the line shown in the diagram. y - x () Find the point at which the line with equation y = x intersects with the line in the diagram.. At a meeting of a Music Appreciation Society the memers were asked to vote for their favourite musical. The following results were otained: Cats Mamma Mia Les Miserales Miss Saigon If a memer was chosen at random, what is the proaility that Les Miserales was not their favourite musical? Give your answer as a fraction in its simplest terms.

4 . (a) Factorise fully: x 8 () A curve has equation y x x. ( i ) Find the coordinates of the points where the curve crosses the x and y-axis. ( ii ) Sketch the curve, marking clearly the points from part ( i ), and the turning point of the curve.. The numer of missed hospital appointments over a period of days were: (a) Construct an ordered stem-and-leaf diagram to illustrate this information. () Hence, or otherwise, write down the lower quartile, median and upper quartile of the data. (c) Calculate the semi-interquartile range.. The circle in the diagram opposite has centre O and radius cm. P R is the point the point of contact of the tangent PQ. R Given that OQ = 0cm calculate the length of RQ. cm O 0cm Q

5 . The square and rectangle shown elow have the same perimeter. (x + ) cm (x + ) cm Find an expression for the length of the rectangle. END OF QUESTION PAPER

6 Prelim Examination 00 / 0 MATHEMATICS National Qualifications - Intermediate Maths and Paper Time allowed - hour 0 minutes Read carefully. Calculators may e used in this paper.. Full credit will e given only where the solution contains appropriate working.. Square-ruled paper is provided.

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

8 All questions should e attempted Marks. Mr. Jones ought a rand new car which cost The car depreciated in value y 0% during its first year, y % in its second year and y 8% in its third year, Calculate, to the nearest 00, the value of the car after three years.. Solve the equation x 7x 0, giving your answer correct to two decimal places.. A group of students were asked to investigate the effect on the Ph value of adding a certain chemical to soil. They egan y testing the Ph value of six samples taken from various regions in the country. The results are given here: 9 9 (a) Calculate the mean and Standard Deviation of the results. () After adding the chemical the mean Ph value was and the Standard Deviation. Comment on the effect of the chemical on the PH value of the soil samples, y making reference to oth the mean and the standard deviation.

9 . Pot pourri is packaged in a large cylindrical container which has diameter cm and height 0cm. cm 0cm (a) Calculate the volume of the cylindrical container For the Christmas Gift market the pot pourri is packaged into glass spheres. () If 7 spheres can e filled from one cylinder of pot pourri, calculate the radius of each sphere. Give your answer correct to significant figures.. Two golfers teed off from the same spot, marked T on the diagram elow. Harry hit his shot on a earing of 0 o and his all travelled 87 metres. David hit his a distance of 7 metres on a earing of 0 o. How far apart were the golf alls after this first drive? 87m T o 7m

10 . A child s toy is in the shape of a sphere with a duck and some water inside. As the all rolls around the water remains at the same level. The diagram opposite shows the cross section when the sphere has een halved. w cm Given that the radius of the sphere is cm, and the depth of the water is cm, calculate w, the width, cm in centimetres, of the water surface. 7. A charity produced this adge to promote itself in the community. It is made up from a circular hoop of metal with an isosceles triangle set into it as shown in the diagram elow. C is the centre of the circular hoop. The radii of the circles are cm and cm. C The dimensions of the isosceles triangle are as shown in this diagram. cm cm 0 o Calculate the area of metal required to make the adge. 7

11 8. At a party, Peter ought cans of cola and ottles of orange juice costing a total of. (a) Let x e the cost of a can of cola, and let y e the cost of a ottle of orange juice. Write down an equation to illustrate this. Later, Billy ought cans of cola and ottles of orange juice and was charged 70. () Construct a second equation to illustrate this. (c) Calculate the cost of a can of cola and the cost of a ottle of orange juice. 9. The upright posts on a ungee jump are metres apart and 0 metres high. For one jump the initial fall was not exactly half way etween the posts, ut made angles of depression of 0 o and o with the horizontal as shown in the diagram. A metres 0 o o C 0 metres Calculate how far from the ground (h metres) the ungee jumper will e after this initial fall. h metres B END OF QUESTION PAPER

12 Intermediate Paper ~ 00/0 Marking Scheme Qu Give one mark for each Illustration for awarding mark a ans: y x marks finds gradient finds y - intercept states equation ans: (-, ) marks knows to solve system of equations finds value for x and y states point of intersection ans : marks m = c = y x evidence of two equations x = - and y = (-, ) a correct numerator finds total simplifies fraction ans : x x marks 8 / / knows to take common factor recognises difference of two squares ans: ( x )( x ) marks x x x a c sets y = 0 and factorises correctly states points sets x = 0 and finds y-intercept correctly finds turning point sketches curve, annotated correctly ans : stem and leaf drawn marks stem correct ordered leaf key and n shown ans:, 9, marks states lower quartile states median states upper quartile ans: marks uses formula(stated of implied) answer ans : 8cm marks knows that ORQ is right angled assemles facts in right-angled triangle knows to use Pythagoras answer ans : 7x + marks finds expression for two missing sides knows to divide y ( x )( x ) (,0),(,0) ( 0, ) (, 9) see end of marking scheme for graph 9 ( - )/ stated or shown in a diagram diagram - RQ = 0-8 cm x + 8 7x + Total 0 marks

13 Intermediate Paper ~ 00/0 Marking Scheme Qu Give one mark for each Illustrations for awarding mark ans : 00 marks finds value after year finds value after years finds rounded value after years = = = 8.80 ans : x = -.78 or x = -0.7 marks a a knows to use Quadratic Formula sustitutes values into formula simplifies calculates oth roots ans : Mean ; S.D.. marks finds mean finds deviations from the mean squaring deviations sustituting in formula answer ans: mean smaller, more spread out marks statement aout the mean statement aout S.D. ans : 0cm using correct radius sustituting in formula answer ans: cm finds volumes of one sphere equating to formula finds r finds r correct rounding marks marks ans : 7 metres marks calculates angle etween sides knows to use cosine rule sustitutes correctly evaluates to t knows to take square root answer ans : cm marks assemles facts in right angled triangle knows to use Pythagoras finds half width final answer Quadratic Formula stated or implied x 7 7 ()() () x 7 7 x 0.7 or x. 78 7/ = -0, -0,, 0,, 0 09, 0,, 0,, = 8 8 (8 8/) the mean had een reduced SD higher so values are more spread out r = (or su. in formula) V = 0 0cm 0/7 = cm r r / = r... cm 0 o o = o evidence of cosine rule t cos m cm cm - = =. 7 =

14 Qu Give one mark for each Illustrations for awarding mark 7 ans : cm 7 marks 8a c 7 knows to find area of triangle answer uses correct angle for sector finds area of outer circle finds area of inner circle finds area of ring finds total area ans : x y mark states equation ans: x y 70 mark states equation ans : p and 0p marks knows to solve simultaneous eqs. finds a value for x and y finds correct values for x and y states cost of each drink 9 ans : 8 m marks finds third angle in triangle uses sine rule to find AB or BC answer uses SOH CAH TOA to find vertical answer sutracts to find height Area Δ = sin 0 9. cm 0 o = cm x y x y 70 knows to solve simultaneous equations. any values found for x and y x = ; y = 0 coke costs p; orange costs 0p 80 (0 + ) = 8 o /sin8 o = c/sin o (or equivalent) m sin0 o = x/ (or equivalent) x = m 0 = 8 m Total marks