Grade 11 Assessment Booklet - Maths Excellence

1 Grade 11 - 2 - Exemplar Assessments 2008 Grade 11 Assessment Exemplars 1 Learning Outcomes 1 and 2 Assignment : Functions 5 Investigation: Ratios 8 Control Test: Equations, Inequalities, Exponents 10 Project: Finance 12 Exam A: Paper 1 13 Exam B: Paper 1 19 2 Learning Outcomes 3 and 4 Assignment: Analytical and Transformation Geometry 25 Investigation: Shape and Space 27 Control Test: Trigonometry, Mensuration 29 Project: Enlargements 31 Exam A: Paper 2 35 Exam B: Paper 2 46 Grade 11 - 3 - Exemplar Assessments 2008 Information Sheet: Mathematics aacbbx242 = ).

2 1(niPA+= niPA)1(+= niPA)1( = ).1(niPA = ==nin11 () =+=ninni121 ()()()() = += +nidnandia11221 () = =ninirraar1111 ; 1 r = =niiraar111 ; 11<< r iixFn]1)1[( += iixPn])1(1[ + = hxfhxfxfh)()(lim)('0 += 212212)()(yyxxd + = ++2;22121yyxxM cmxy+= ()11xxmyy = 1212xxyym = tan=m ()()222rbyax= + ABCIn ; CcBbAasinsinsin== )sin(+=+ )sin( = += )cos( =+ )cos(+= 22sincos2cos = 2sin212cos = 1cos22cos2 = nxx = nfxx = nxxnii = =12)(var nxxSDni = =12)( )()()(snAnAP= )()()()(BandAPBPAPBorAP += Grade 11 - 4 - Exemplar Assessments 2008 Instructions and Information Read the following instructions carefully before answering this question paper: 1 This question paper consists of.)

3 Questions. Answer ALL questions. 2 Clearly show ALL calculations, diagrams, graphs, et cetera, which you have used in determining the answers. 3 An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 4 If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 5 Number your answers correctly according to the numbering system used in this question paper. 6 Diagrams are not necessarily drawn to scale. 7 It is in your own interest to write legibly and to present your work neatly. Grade 11 - 5 - Exemplar Assessments 2008 Grade 11 Assignment: Functions Marks: 95 Question 1: What do you understand by the asymptote of a function; (2) the axis of symmetry of a function; (2) the zeros of a function?

4 (2) ()()xfxg = and ()()xfxh =. Write down the equation of the line about which: ()xg is a reflection of ()xf (1) ()xh is a reflection of ()xf (1) What does it mean if ()()22gf=? (2) How do you test whether or not the point ()ba; lies on the graph of ()xfy=? (2) Write down a formula for the average gradient of a curve ()xgy= between the points ()11;ba and ()22;ba (2) [14] Question 2: Sketch the graph of xxy24 = showing all axes of symmetry, asymptotes, intersection with the axes and any other critical points.

5 (7) Give the equation of; the horizontal asymptote (1) the axis of symmetry that has a negative gradient (2) the graph that would result if you shifted your sketch up by 4 units (2) [12] Question 3: Sketch the graphs of ()2xxfy == , ()()21 ==xxgy and ()()212 ==xxhy on the same system of axes. Label each graph, any lines of symmetry or asymptotes that may exist as well as at least 2 points on each graph. (11) Describe, in words, the effect on the graph of ()2xxfy == of the parameters a, c and d in the equation ()dcxaxpy++ ==2)(.

6 (8) [19] Grade 11 - 6 - Exemplar Assessments 2008 Question 4: Given 241)(1 = xxh Write down the equation of the asymptote of h (1) Determine the coordinates of the intercepts of h with the x and y axes (6) Write down the equation of the reflection of 241)(1 = xxh in the y axis. (2) [9] Question 5: The sketch below represents the graph of qpxaxf++=)( If the line of g is the vertical asymptote of the above function, determine the value of a, p and qand hence the equation of f. (5) What is the equation of the horizontal asymptote?

7 (1) What is the equation of the axis of symmetry that has a positive gradient? (2) [8] Grade 11 - 7 - Exemplar Assessments 2008 yxfgOSPQREDCBAQ uestion 6: Given dcbxaxf++=o)sin()( and srqxpxg++=o)cos()( Determine the values of a, b, c, d, p, q, r and s and hence the equations of f and g (4) Read from the sketch, the values of xfor which )()(xgxf=for []00180;0 x (2) [6] Question 7 ()1242 +=xxxf 122)(+=xxg Determine the lengths of OA; OB and OC. (4) Determine the coordinates of P.

8 (5) Calculate the length of RS if S is the turning point. (5) Determine the lengths of BD; QB and EC. (6) Write down the equation of: the reflection of ()xf in the yaxis; (2) the parabola with the same zeros as ()xf, which has been stretched through thexaxis by a factor of 2 (2) Calculate the average gradient of ()xf between S and B.

9 (3) [27] Grade 11 - 8 - Exemplar Assessments 2008 A1A2A3A4A4A3A2A1A0 Grade 11 Investigation: Ratios Marks: 100 1. Solve the following equation for x in terms of y: 02322=+ yxyx and hence show that the ratio x : y is 1 : 1 or 2 : 1. 2. Most paper is cut to internationally agreed sizes: A0, A1, A2, ..A7 with the property that the A1 sheet is half the size of the A0 sheet and has the same shape as the A0 sheet, the A2 sheet is half the size of the A1 sheet and has the same shape and so on.

10 Explain what it means that the sheets all have the same shape. Find the ratio of the length to the breadth of each rectangular piece of paper. All sheets have the same shape An A0 sheet folds into two A1 sheets 3 The golden rectangle has been recognised through the ages as being aesthetically pleasing. It can be seen in the architecture of the Greeks, in sculptures and in Renaissance paintings. y x Grade 11 - 9 - Exemplar Assessments 2008 ABBC= Measure x and y and hence estimate the golden ratio x : y The golden rectangle has the property that when a square the length of the shorter side of the rectangle is cut from it, another rectangle with the same shape is left.