1
Grade 12 tasks:
Grade 12 learners must complete 7 formal assessment tasks.
Term 1: an assisgment; a project or investigation and a test = 3 SBA tasks
Term 2: a test and the midyear examination = 2 SBA tasks
Term 3: a test and a midyear examination. = 2 SBA tasks
These exemplars will not deal with the two examinations as we are still waiting for the national examination guidelines. The test for each term may be compiled from the topical assignments.
Term 1
Topical Assignments
Topic 1; Sequences and series
Question 1
1.1Consider the sequence:6 ; 10 ; 16 ; 24 ; 34 ; …
1.1.1If the sequence behaves consistently, determine the next two terms of
the sequence.(2)
1.1.2Calculate a formula for the nth term of the sequence. (5)
1.1.3Use your formula to calculate n if the nth term in the sequence is 1264.(3)
1.2Consider the following sequence: 6; 18; 54; 162 ……..
1.2.1If the formula for the general term of the sequence is , give the
value of a and r.(2)
1.2.2Which term of the sequence is equal to 1458?(4)
Given the geometric series: 5. + 5. + 5. + ……..
1.3.1Explain why the series converges.(2)
1.3.2Calculate the sum to infinity of the series.(3)
1.3.3Calculate the sum of the first 9 terms of the series, correct to TWO
decimal places.(4)
1.3.4 Use your answers to QUESTION 1.3.2 and QUESTION 1.3.3 to determine
(correct to TWO decimal places)(2)
Question 2
2.1 / 3x + 1 ; 2x ; 3x – 7 are the first three terms of an arithmetic sequence.Calculate the value of x. / (2)
2.2 / The first and second terms of an arithmetic sequence are 10 and 6 respectively.
2.2.1 / Calculate the 11th term of the sequence. / (2)
2.2.2 / The sum of the first n terms of this sequence is –560. Calculate n. / (6)
2.3In the diagram below, the first (outer) triangle is equilateral triangle with sides of 8 units. Another equilateral triangle in drawn within this triangle, by joining the midpoints of the sides of the outer triangle.
This process is continued without end.
2.3.1Calculate the perimeter of the fourth triangle.(2)
2.3.2Show that the sum of the perimeters of all the
inner triangles will never exceed the perimeter
of the outer triangle.(6)
Question 3
3.1 / Given the sequence: 4 ; x ; 32Determine the value(s) of x if the sequence is:
3.1.1 / Arithmetic / (2)
3.1.2 / Geometric / (3)
3.2 / The following sequence is a combination of an arithmetic and a geometric sequence:
3 ; 3 ; 9 ; 6 ; 15 ; 12 ; …
3.2.1 / Write down the next TWO terms. / (4)
3.2.2 / Calculate . / (5)
3.3.3 / Prove that ALL the terms of this infinite sequence will be divisible by 3. / (4)
QUESTION 4
4.1 Given the arithmetic series: – 7 – 3 + 1 + … + 1734.1.1 / How many terms are there in the series? / (4)
4.1.2 / Calculate the sum of the series. / (3)
4.1.3 / Write the series in sigma notation. / (2)
4.2 / Consider the geometric series: 4 + – 2 + 1 + …
4.2.1 / Write the series in sigma notation / (3)
4.2.2 / Determine n if the nth term is . / (4)
4.2.3 / Calculate the sum to infinity of the series … / (2)
4.3 / If x is a REAL number, show that the following sequence can NOT be geometric:
1 ; x + 1 ; x – 3 … / (4)
Topic 2: Inverse functions
Question 1
1.1Say whether the statements that follow are true or false. Give a reason for your answer.
1.1.1The inverse of is (2)
1.1.2 is a many to one function(2)
1.1.3The inverse of 1.1.2 is a function(2)
1.1.4The domain of 1.1.2 is (2)
1.1.5The function (2)
1.2Given
1.2.1Determine (3)
1.2.2Is a function or not? Give a reason for your answer.(2)
1.2.3How will you restrict the domain of the original function to ensure (1)
that will be a function?
1.2.4Draw on the same set of axes(3)
1.2.5Determine the point(s) of intersection of (4)
Question 2
2.1Determine the inverses of the following functions:
2.1.1
2.1.2
1.4Determine the point(s) at which in question 1.3 will intersect.
1.5Given the function .
5.1Determine .(3)
5.2Sketch the graph of .(2)
5.3Explain why is not a function.(1)
5.4Explain how you would restrict the domain of
so that is a function.(2)
6.1The figure represents the graph of .
Calculate the value of a.(2)
6.2.2Draw a sketch graph of if k is the inverse
of f . Indicate the intercept(s), the coordinates
of one other point and the asymptote(s).
Question 2
2.1The diagram alongside represents the functions:
The minimum value of the function
equals 1 when .
The turning point of the parabola is
at the point F.
EF is drawn parallel to the y-axis.
2.1.1Destermine the value of a, b, c, m and k.(5)
2.1.2Calculate the length of EF and GH correct to two decimal places(3)
2.1.3Determine the equations of (4)
2.1.4Hence explain why of why not are symmetrical; with respect to the line (2)
2.2The graph alongside shows the functions g, f and h.
f and g are symmetrical with respect to the y-axis
f and h are symmetrical with respect to the line
y = x. If and the point (1 ; 4) lies on
2.2.1determine the value of a(2)
2.2.2write down the coordinates of P and Q (2)
2.2.3write down the equations of (6)
2.3Given:
2.3.1Explain why, unless the domain of this function is restricted, its inverse is
not a function. (2)
2.3.2Write down the equation of inverse, of for in the
form =… (3)
2.3.3Write down the domain of (1)
2.3.4Draw graphs of both for and on the
same system of axes(4)
Topic 3: Present and Future value formulae
Question 1
1.Felicity purchases a new house. The price of the house is R 825 000. She pays 20% deposit and takes out a bond over 20 years to pay the balance.
1.1Calculate her monthly instalments if the bank charges 11,5% p.a. compounded monthly (4)
1.2Calculate the total amount she repays on the house.(3)
1.3If she increases his monthly repayments by R500, how long will he take to repay the loan. (4)
1.4How much will he save by increasing his monthly instalment by R 500(2)
1.5If she wins the lotto and wishes to settle her bond after 12 years, calculate the outstanding balance (4)
Question 2
2The Mathematics, Science and Technology Academy has just bought a luxury school bus R 750 000. The intention is to replace this bus after 8 years. The bus depreciates at a rate of 15% p.a. calculated on a reducing balance.
2.1What is the trade-in value of the bus after 8 years.(3)
2.2Due to inflation the purchase price of a new bus in 8 years’ time will increase by 25% p.a. simple interest. Calculate the purchase price of the new bus (3)
2.3The school wants to pay cash for the new bus after trading in the old bus.
Calculate the monthly deposits into a sinking fund order to pay cash pay for the new bus after 8 years. This investment renders interest at 8,5% p.a. compounded monthly. (4)
2.4Suppose that 12 months after the purchase of the present bus and every 12 months thereafter, the school withdraws R5 000 from his account, to pay for the maintenance of the new bus. They makes 5 such withdrawals, what will the new monthly deposit be? (6)
Question 3
A construction worker wants to save a monthly amount towards his retirement. His bank offers 11,5% p.a. compounded monthly.
3.1If he pays R250 per month for the next 10 years, how much will he receive at the end of this period. (4)
3.2If he wants to save R100 000 over this period, how much must he deposit monthly? (4)
3.3How long will it take him to save R100 000 if he saves R1000 monthly.(4)
Topic 4 Compound Angles
Question 1
1.1If 4 tanθ = 3 and 1800<θ<360determine using a diagram:
1.1.1 sinθ + cosθ
1.1.2sin 2θ and cos 2θ
1.1.3tan 2θ
1.2Given: and where . Calculate the value of the following without using a calculator:
1.2.1(6)
1.2.2(4)
1.3If , Express each of the following in terms of :
1.3.11.3.21.3.31.3.4 (10)
1.4If sin 360.cos 12= p and cos 36.sin 12= q, determine in terms of p and q the value of:
1.4.1 sin 48°1.4.2sin 24° 1.4.3cos 24° (8)
Question 2
2.1Solve: waar
2.2Determine the general solution:
2.2.1(6)
2.2.2, correct to 1 decimal place if necessary.(6)
2.2.1Solve the equation for (6)
2.2.2Sketch graphs of and on the same system of
axes for . Show the co-ordinates of all points of intersection with
the axes, all turning points and all points at which (8)
2.3Prove that :
2.3.1
2.3.2
Term 2
Topic 1: heights and distances
Question 1
1.1A, B and L are points in the same horizontal plane.
HL is is a vertical pole with lenght 3 meter,
AL = 5,2 m, and AB = 113°
The angle of elevation from B to H is 40°.
1.1.1 Calculate the length of LB. (2)
1.1.2 Hence calculate the lenght of AB. (4)
1.2The diagram below represents the course of a swimming race in a bay on the coast.
- P is the starting and finishing point;
- A is a buoy, 300 metres from P on a bearing of
- B is a buoy, 500 metres from P on a bearing of .
Calculate:
1.2.1the bearing of the point P from the buoy, B and(1)
1.2.2the distance competitors must swim (from P to A, then to B and then back to P).(6)
1.3In , and
1.3.1In , express in terms of and (2)
1.3.2In , express in terms of and (3)
1.3.3Hence show that (3)
Question 2
2.1.is a vertical tower in a horizontal plane BCD. The angle of elevation of A from C is and the distance
2.1.1Prove that the height of the tower AB = (5)
2.1.2Calculate the height of the tower if = 40 , and .(2)
2.1.3Calculate the area of (3)
2.2.1Prove that p = (6)
2.2.2Now calculate the value of p if h = 50 m; x = 32,3 and y = 25,8.
Give the answer to one decimal place.(3)
Topic 2: Calculus
Question 1
1.1Given , determine from first principles.(5)
1.2Determine the derivative of:
1.2.1(3)
1.2.2(4)
Question 2
Given:
2.1Show that () is a factor of f (x).(2)
2.2Hence factorise f (x) completely.(2)
2.3Determine the co-ordinates of the turning points of f.(4)
2.4Draw a neat sketch graph of f indicating the co-ordinates of the turning points as well as the x-intercepts. (4)
2.5For which value of x will f have a point of inflection?(4)
Question 3
A cubic function has the following properties:- f decreases for x only
Draw a possible sketch graph of f, clearly indicating the x-coordinates of the turning points and ALL the x-intercepts. (5)
Question 4
The graph of the function is sketched below.
4.1 / Calculate the x-coordinates of the turning points of f. / (4)
4.2 / Calculate the x-coordinate of the point at which is a maximum. / (3)
Question 5
The graph of , where is a cubic function, is sketched below.Use the graph to answer the following questions:
5.1 / For which values of x is the graph of decreasing? / (1)
5.2 / At which value of x does the graph of f have a local minimum? Give reasons for your answer. / (3)
Question 6
6.1A pasta company has packaged their spaghetti in a box that has the shape of a rectangular prism as shown in the diagram below. The box has a volume of 540 cm3, a breadth of 4 cm and a length of x cm.
6.1.1Express h in terms of x.(2)
6.1.2Hence show that the total surface area
of the box (in cm2 ) is given by:
(3)
6.1.3Determine the value of x for which the total
surface area is a minimum. Round the
answer off to the nearest cm.(4)
6.2Water is flowing into a tank at a rate of 5 litres per minute. At the same time water flows out of the tank at a rate of k litres per minute. The volume (in litres) of water in the tank at time t (in minutes) is given by the formula .6.2.1 / What is the initial volume of the water in the tank? / (1)
6.2.2 / Write down TWO different expressions for the rate of change of the volume of water in the tank. / (3)
6.2.3 / Determine the value of k (that is, the rate at which water flows out of the tank). / (2)
6.3A particle moves along a straight line. The distance, s, (in metres) of the particle from a fixed point on the line at time t seconds () is given by .
6.3.1 / Calculate the particle's initial velocity. (Velocity is the rate of change of distance.) / (3)
6.3.2 / Determine the rate at which the velocity of the particle is changing at t seconds. / (1)
6.3.3 / After how many seconds will the particle be closest to the fixed point? / (2)
Topic 3: Analytical geometry
Question 1
1.1Show that is a right angled isosceles triangle.(6)
1.2Determine the area of (3)
1.3Given that BC is a diameter of the circumscribed circle of , show that the centre of
this circle is M, the point (2)
1.4Calculate the equation of the circumscribed circle of (4)
1.5Determine the equation of the tangent to the circle at C and show that this tangent is
parallel to MA.(8)
Question 2
2.1Calculate the equation of the circle with centre M (2;3) which passes through the point P(6;1). (5)
2.2Show that if Q is the point Q( the perpendicular bisector of PQ passes
Through the centre of the circle. (6)
2.3Does the point lie on the circle, inside the circle or outside the circle?
Justify our answer (3)
Question 3
3.1Determine the equation of the tangent to the circle x2 + y2 = 10 at the point P(–1 ; 3). (4)
3.2Determine the equation of the tangent to the circle x2 + y2 + 2x – 4y = 20 at the
point (–4 ; –2).(7)
3.3Determine the coordinates of S, the fourth vertex of the parallelogram PQRS,
if P(–2 ; –1), Q(1 ; 6) and R(3 ; 4).(4)
3.4ABCD is a rhombus with diagonals meeting in K. A(0 ; 4), B(– 3 ; – 2)
and K(1½ ; 2½)
3.4.1Determine the co-ordinates of C.(2)
3.4.2Show by analytical means that .(4)
Question 4
4.1The circle with centre O and the straight line intersect at the points A(0 ; 5) and B(4 ; 3).
4.1.1Determine the equation of the circle.(5)
4.1.2Calculate the length of chord AB.(3)
4.2In the figure B(1 ; 1) is the centre of the circle. CA is a tangent at A. C is the point (1 ; 6).. CA = 20 units.
Calculate:
4.2.1the length of the radius of the circle (2)
4.2.2the equation of the circle (3)
4.2.3the equation of the tangent CR (4)
4.2.4the equation of the radius AB
(4)
4.2.8the co-ordinates of A (4)
Term 3
Topic 1 Geometry
Question 1
1.1In the diagram two circles intersect in A and C. BA is a tangent to the larger circle at point A. Straight lines ATD and BCD intersect the circles in T and D, and C and D respectively. The larger circle passes through centre O of the smaller circle.
Let = x.
1.1.1Prove that = 180° – 2x.(4)
1.1.2Prove that AD = BD.(5)
1.1.3Prove that TC AB.(2)
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1.2 / In the diagram below, two circles have a common tangent TAB. PT is a tangent to the smaller circle. PAQ, QRT and NAR are straight lines.Let
1.2.1 / Name, with reasons, THREE other angles equal to x. / (5)
1.2.2 / Prove that APTR is a cyclic quadrilateral. / (5)
Question 2
2.1In PQW, S is a point on PW and R is a point on QW such that SR || PQ.T is a point on QW such that ST || PR.RT = 6 cmWS : SP = 3 : 2
Calculate:
2.1.1 / WT / (3)
2.1.2 / WQ / (4)
2.2In the diagram PQRS is a parallelogram. Side RS is produced to W. WQ intersect PS in X. M is a point on XQ so that MX = XW.
Given MT // XS, PQ = 12 cm, WS = 4 cm
2.2.2Determine the length of TR(2)
2.2.3Determine the value of (4)
Question 3
3.1In the figure, AB is a tangent to the circle with centre O. AC = AO and BA // CE. DC produced, cuts tangent BA at B.
3.1.1Show that (4)
3.1.2Prove that ACF /// ADC(5)
3.1.3Prove that AD = 4 AF(5)
3.2In the diagram DA is a tangent to the circle ACBT at A. CT and AD are produced to meet at P. BT is produced to cut PA at D. AC, CB, AB and AT are joined.
AC // BD. Let .
3.2.1Prove that ABC /// ADT.(5)
3.2.2Prove that PT is a tangent to the circle ADT at T(3)
3.2.3Prove that APT /// TPD.(5)
3.2.4If , show that (4)
Topic 2: Statistics
QUESTION 4As part of an environmental awareness initiative, learners of Greenside High School were requested to collect newspapers for recycling. The cumulative frequency graph (ogive) below shows the total weight of the newspapers (in kilograms) collected over a period of 6 months by 30 learners.
4.1 / Determine the modal class of the weight of the newspapers collected. / (1)
4.2 / Determine the median weight of the newspapers collected by this group of learners. / (1)
4.3 / How many learners collected more than 60 kilograms of newspaper? / (2)
QUESTION 3
The length of time, in minutes, of a certain number of telephone calls was recorded. No call lasted 25 minutes or longer. A cumulative frequency diagram of this data is shown below.3.1 / Determine the total number of calls recorded. / (1)
3.2 / Complete the frequency table for the data / (3)
3.3 / Hence, draw a histogram for the data / (3)
3.4 / Draw a box and whisker plot using the data on the ogive / (4)
QUESTION 4
In the grid below a, b, c, d, e,f and g represent values in a data set written in an increasing order. No value in the data set is repeated.a / b / c / d / e / f / g
Determine the value of a, b, c, d, e, f and g if:
- The maximum value is 42
- The range is 35
- The median is 23
- The difference between the median and the upper quartile is 14
- The interquartile range is 22
- The mean is 25
Topic 3: probability
QUESTION 1
1.1Every client of CASHSAVE Bank has a personal identity number (PIN) which is made up of 5digits chosen from the digits 0 to 9.1.1.1 / How many personal identity numbers (PINs) can be made if:
(a) / Digits can be repeated / (2)
(b) / Digits cannot be repeated / (2)
1.1.2 / Suppose that a PIN can be made up by selecting digits at random and that the digits can be repeated. What is the probability that such a PIN will contain at least one 9? / (4)
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1.2Consider the digits 1, 2, 3, 4, 5, 6, 7 and 8 and answer the following questions:
1.2.1 / How many 2-digit numbers can be formed if repetition is allowed? / (2)
1.2.2 / How many 4-digit numbers can be formed if repetition is NOT allowed? / (3)
1.2.3 / Determine the probability that numbers between 4 000 and 5 000 can be formed if repetition is not allowed? / (4)
[8]
1.3Consider the word PRODUCT
1.3.1How many different letter arrangements are possible (3)
1.3.2How many different letter arrangements are possible if the first letter is T (3)
and the fifth letter is C
1.3.3What is the probability that different letter arrangements are formed with the letters R, O and Dfollowing each other in any order. (4)
Question 2
2.1M1, M 2 and M 3 are 3 machines in a factory that manufactures nuts and bolts. They produce 25%, 30% and 45% of the total production of the factory. Of the products of M1, M 2 and M 3 is 18%, 5% en 2% are defective. A random sample is drawn of the products
2.1.1Represents this data in a tree diagram.(6)
2.1.2Determine the probability that
(i).The defective products are manufactures by machine M1(2)
(ii)The products are defective(3)
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2.2A survey of 200 learners regarding their preferences for chicken, lamb or beef
yielded the following data:
90 prefers chicken as first choice.
64 prefers lamb as first choice
77 prefers beef as first choice
8 prefers all three types as first choice.
18 prefers beef and lamb.
27 prefers chicken and beef.
26 are vegetarians.
number of learners prefers chicken and lamb above beef.
2.2.1Represent this information ion a Venn diagram.(6)
2.2.2How many learners prefers chicken only(4)
2.2.3Determine the probability that a randomly chosen learner prefers chicken or beef(3)
[13]
Question 3
The following contingency table shows the results of a survey amongs male and female drivers stopped along a highway in Cape Town.
Speed fine / No speed fine / totalMale drivers / 45 / 25 / 70
Female drivers / 35 / 45 / 80
total / 80 / 70 / 150
3.1How many persons participated in this study(1)
3.2Calculate the following probabilities
3.2.1 P(male drivers)(2)
3.2.2P(speed fines)(2)
3.3Are the events male driver and speed fine independent. Justify your answer through calculations. (5)
Topic 4: Regression
QUESTION 1A group of students attended a course in Statistics on Saturdays over a period of 10 months. The number of Saturdays on which a student was absent was recorded against the final mark the student obtained. The information is shown in the table below and the scatter plot is drawn for the data.
Number of Saturdays absent / 0 / 1 / 2 / 2 / 3 / 3 / 5 / 6 / 7
Final mark (as %) / 96 / 91 / 78 / 83 / 75 / 62 / 70 / 68 / 56
1.1 / Calculate the equation of the least squares regression line. / (4)
1.2 / Draw the least squares regression line on the grid provided on DIAGRAM SHEET 1. / (2)
1.3 / Calculate the correlation coefficient. / (2)
1.4 / Comment on the trend of the data. / (2)
1.5 / Predict the final mark of a student who was absent for four Saturdays. / (2)
Question 2
A learner conducted an experiment to investigate the relationship between age and resting heart rate (in beats per minute). He sought assistance of the local clinic. The information for 12 people is shon in the table below.
Age / 59 / 32 / 42 / 50 / 22 / 39 / 21 / 20 / 27 / 40 / 29 / 47Resting heart rate (beats per minute) / 88 / 74 / 74 / 93 / 85 / 71 / 78 / 82 / 70 / 75 / 95 / 75
2.1Represent the data in a scatter plot(3)
2.2determine the equation of the least squares line(4)
2.3Draw the least squares line on the scatter plot(2)
2.4Calculate the correlation coefficient for the data(2)
2.5Use the correlation coefficient to comment on the relationship
between age and the resting heart rate(2)
2.6If a learner uses the least squares line to predict the resting
heart rate of a 45 year old person, will his answer be reliable?
Motivate your answer. (2)
Question 3
The student enrolment at an FET college for the past 5 years is represented in the table below: ( NB: we denote 2005 as year 1, 2006 as year 2, 2007 as year 3, 2008 as year 4 and 2009 as year 5, etc for convenience)
Year / 1 / 2 / 3 / 4 / 5Enrolment / 60 / 65 / 145 / 220 / 312
3.1 Represent that data on a scatter plot(4)
3.2 Determine the equation of the “least squares” ( regression) line for the data(4)
3.3Draw the least squares line for the data(1)
3.4 Estimate the enrolment in 2012(3)