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ZIMSEC O Level · 4008/2, 4028/2 · J2000

Mathematics Paper 2 June 2000

Questions
65
Total marks
138
Time allowed
150 min
Syllabus code
4008/2, 4028/2

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Questions
65
Pass mark
39
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Answer every question in the printed order, get marked at the end, then see the answers.

The questions

Question 101

[2 marks]Algebra, functions
Given that y=a+bby = \frac{a+b}{b}, calculate the value of yy when a=40a = 40 and b=0,1b = 0,1.

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Question 102

[3 marks]Algebra, functions
Given that y=a+bby = \frac{a+b}{b}, express bb in terms of yy and aa.

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Question 103

[2 marks]Algebra, functions
Given that f(x)=2x2+7xf(x) = 2x^2 + 7x, calculate f(3)f(3).

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Question 104

[2 marks]Algebra, functions
Given that f(x)=2x2+7xf(x) = 2x^2 + 7x, the equation f(x)=4f(x)=4 has two solutions for xx. Find the positive value of xx.

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Question 105

[2 marks]Algebra, functions
Given that f(x)=2x2+7xf(x) = 2x^2 + 7x, the equation f(x)=4f(x)=4 has two solutions for xx. Find the negative value of xx.

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Question 201

[2 marks]Matrices, sets
Given that P=(−4−713)P = \begin{pmatrix} -4 & -7 \\ 1 & 3 \end{pmatrix} and Q=(2−34−1)Q = \begin{pmatrix} 2 & -3 \\ 4 & -1 \end{pmatrix}, calculate the matrix P+3QP+3Q. Give your answer as (top-left, top-right, bottom-left, bottom-right).

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Question 202

[2 marks]Matrices, sets
Given that Q=(2−34−1)Q = \begin{pmatrix} 2 & -3 \\ 4 & -1 \end{pmatrix}, calculate the matrix Q2Q^2. Give your answer as (top-left, top-right, bottom-left, bottom-right).

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Question 203

[3 marks]Matrices, sets
Given that P=(−4−713)P = \begin{pmatrix} -4 & -7 \\ 1 & 3 \end{pmatrix}, calculate the inverse of PP. Give your answer as decimals (top-left, top-right, bottom-left, bottom-right).

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Question 204

[1 marks]Matrices, sets
The universal set is ξ={x:10<x≤31}\xi = \{x : 10 < x \leq 31\} where xx is an integer, and A={x:x is prime}A = \{x : x \text{ is prime}\}. List all the elements of set AA.

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Question 205

[1 marks]Matrices, sets
The universal set is ξ={x:10<x≤31}\xi = \{x : 10 < x \leq 31\} where xx is an integer, and C={x:x is a multiple of 5}C = \{x : x \text{ is a multiple of 5}\}. Find n(C)n(C).

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Question 206

[2 marks]Matrices, sets
The universal set is ξ={x:10<x≤31}\xi = \{x : 10 < x \leq 31\} where xx is an integer, B={x:x is even}B = \{x : x \text{ is even}\} and C={x:x is a multiple of 5}C = \{x : x \text{ is a multiple of 5}\}. Find n(B∪C)n(B \cup C).

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Question 207

[2 marks]Matrices, sets
The universal set is ξ={x:10<x≤31}\xi = \{x : 10 < x \leq 31\} where xx is an integer, with A={x:x is prime}A = \{x : x \text{ is prime}\}, B={x:x is even}B = \{x : x \text{ is even}\} and C={x:x is a multiple of 5}C = \{x : x \text{ is a multiple of 5}\}. Express the set {15,25}\{15, 25\} in set notation, using some or all of AA, BB and CC.

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Question 301

[2 marks]Algebra, factorising, variation
Factorise completely 8p3−18p8p^3 - 18p.

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Question 302

[2 marks]Algebra, factorising, variation
Factorise completely as+2at−3s−6tas + 2at - 3s - 6t.

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Question 303

[3 marks]Algebra, factorising, variation
yy varies directly as the cube of xx, and y=40y=40 when x=2x=2. Find the value of yy when x=3x=3.

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Question 401

[3 marks]Trigonometry, Bearing & Distances
In triangle ABCABC, AB=12AB=12 cm, BC=15BC=15 cm and CA^B=58°C\hat{A}B=58°. The triangle is acute-angled. Calculate AC^BA\hat{C}B.

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Question 402

[3 marks]Trigonometry, Bearing & Distances
In triangle ABCABC, AB=12AB=12 cm, BC=15BC=15 cm, CA^B=58°C\hat{A}B=58° and AC^B=42.7°A\hat{C}B=42.7°. Calculate the perpendicular distance from CC to ABAB.

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Question 403

[2 marks]Trigonometry, Bearing & Distances
In triangle ABCABC, CA^B=58°C\hat{A}B=58° and the perpendicular distance from CC to line ABAB is 14.7414.74 cm. Calculate ACAC.

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Question 404

[2 marks]Trigonometry, Bearing & Distances
In triangle ABLABL, AB=12AB=12 cm, AL=6AL=6 cm and angle A=58°A=58° (the angle CA^BC\hat{A}B shared with triangle ABCABC). Using the cosine rule, calculate BL2BL^2.

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Question 405

[2 marks]Trigonometry, Bearing & Distances
In triangle ABLABL, it is found that BL2=103.69BL^2 = 103.69 cm2^2. Calculate BLBL.

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Question 501

[1 marks]Percentages, income
A salesman's annual salary of $39\,000 is paid in twelve equal monthly amounts. Calculate his monthly salary.

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Question 502

[2 marks]Percentages, income
A salesman's annual salary was 39 000.Inaddition,attheendoftheyearhewaspaidabonusof39\,000. In addition, at the end of the year he was paid a bonus of 3\frac{1}{2}\%$ of his total annual sales, which amounted to $100\,000 that year. Calculate his total income in the first year.

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Question 503

[2 marks]Percentages, income
In the second year, a salesman's annual salary remained 39 000,buthistotalincomefortheyearamountedto39\,000, but his total income for the year amounted to 44\,145, the difference being his bonus. Calculate the value of his bonus for the second year.

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Question 504

[2 marks]Percentages, income
In the second year, a salesman was paid a bonus of $5145, calculated as 312%3\frac{1}{2}\% of his total sales. Calculate his total sales during the second year.

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Question 505

[2 marks]Percentages, income
A salesman's annual salary was increased from 39 000to39\,000 to 46\,800 in the third year. Calculate the percentage increase in his annual salary.

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Question 506

[3 marks]Percentages, income
In the third year, a salesman's annual salary was 46 800andhistotalincomefortheyearwas46\,800 and his total income for the year was 50\,000, with sales totalling $80\,000. Calculate his new percentage bonus payment.

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Question 601

[1 marks]Constructions & Loci
A field is drawn to a scale of 1 cm to represent 10 m. One side of the field measures 70 m on the ground. How long is that side on the drawing?

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Question 602

[2 marks]Constructions & Loci
A field ABCD is bounded by two straight roads AB and BC. Name the locus of the points inside the field that are the same distance from road AB as from road BC.
  1. Athe arc of a circle centred at B
  2. Bthe bisector of angle ABC
  3. Cthe line joining the midpoints of AB and BC
  4. Dthe perpendicular bisector of AB

Question 603

[2 marks]Constructions & Loci
A field ABCD has corners A and B. Name the locus of the points inside the field that are the same distance from A as from B.
  1. Athe perpendicular bisector of AB
  2. Bthe arc of a circle centred at the midpoint of AB
  3. Cthe bisector of angle ABC
  4. Dthe line through A parallel to BC

Question 604

[2 marks]Constructions & Loci
In quadrilateral ABCD, angle ABC is 90 degrees and angle BCD is 120 degrees. Angle ADC is measured as 73 degrees. Calculate angle BAD.

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Question 701

[1 marks]Algebra, speed-distance-time
A car travels from Bulawayo to Harare by a route 440 km long, at an average speed of xx km/h. Write down an expression, in terms of xx, for the time, in hours, needed for the journey.

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Question 702

[1 marks]Algebra, speed-distance-time
A car travels at an average speed of xx km/h from Bulawayo to Harare. A mini-bus travelling a different, longer route has an average speed 4 km/h less than the car. Write down an expression, in terms of xx, for the average speed of the mini-bus.

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Question 703

[1 marks]Algebra, speed-distance-time
A mini-bus travels from Bulawayo to Harare by a route 448 km long (8 km longer than the car's 440 km route), at an average speed of (x−4)(x-4) km/h. Write down an expression, in terms of xx, for the time, in hours, needed for the mini-bus's journey.

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Question 704

[2 marks]Algebra, speed-distance-time
A car travels 440 km at xx km/h, taking 440x\frac{440}{x} hours. A mini-bus travels 448 km at (x−4)(x-4) km/h, taking 448x−4\frac{448}{x-4} hours, and this is 12\frac{1}{2} an hour longer than the car's journey time. Write down the (unsimplified) equation that xx must satisfy.

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Question 705

[3 marks]Algebra, speed-distance-time
Solving 448x−4−440x=12\frac{448}{x-4}-\frac{440}{x}=\frac{1}{2} leads to the equation x2−20x−3520=0x^2-20x-3520=0. Solve this equation and give the value of xx (the car's average speed in km/h) correct to 3 significant figures, rejecting any negative solution.

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Question 706

[3 marks]Algebra, speed-distance-time
The car's average speed was found to be x=70.2x=70.2 km/h. Hence find the average speed of the mini-bus, correct to 3 significant figures.

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Question 801

[2 marks]Mensuration (sectors and prisms)
In sector AOBAOB of a circle, centre OO, the radius is 6.56.5 cm and AO^B=120°A\hat{O}B=120°. Taking π=3.142\pi = 3.142, calculate the length of arc ABAB.

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Question 802

[1 marks]Mensuration (sectors and prisms)
Sector AOBAOB of a circle, centre OO, has radius 6.56.5 cm, AO^B=120°A\hat{O}B=120°, and arc AB=13.61AB=13.61 cm. Calculate the perimeter of the sector AOBAOB.

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Question 803

[2 marks]Mensuration (sectors and prisms)
In sector AOBAOB of a circle, centre OO, the radius is 6.56.5 cm and AO^B=120°A\hat{O}B=120°. Taking π=3.142\pi = 3.142, calculate the area of the sector AOBAOB.

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Question 804

[2 marks]Mensuration (sectors and prisms)
Sector AOBAOB (area 44.2444.24 cm2^2) is the cross-section of a solid of length 1212 cm. Calculate the volume of the solid.

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Question 805

[2 marks]Mensuration (sectors and prisms)
A solid with sector cross-section has volume 530.9530.9 cm3^3 and mass 41924192 g. Calculate the density of the solid.

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Question 806

[3 marks]Mensuration (sectors and prisms)
A solid has sector cross-section AOBAOB (radius 6.56.5 cm, AO^B=120°A\hat{O}B=120°, arc AB=13.61AB=13.61 cm, sector area 44.2444.24 cm2^2) and length 1212 cm. Calculate the total surface area of the solid.

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Question 901

[1 marks]Functional Graphs
The curve y=12x2−2x+3y=\frac{1}{2}x^2-2x+3 is tabulated for −1≤x≤6-1 \leq x \leq 6. Calculate the value of aa, the yy-value when x=−1x=-1.

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Question 902

[1 marks]Functional Graphs
The curve y=12x2−2x+3y=\frac{1}{2}x^2-2x+3 is tabulated for −1≤x≤6-1 \leq x \leq 6. Calculate the value of bb, the yy-value when x=6x=6.

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Question 903

[2 marks]Functional Graphs
For the curve y=12x2−2x+3y = \frac{1}{2}x^2-2x+3, calculate the gradient of the curve at x=3x=3.

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Question 904

[2 marks]Functional Graphs
Calculate the area of the region between the curve y=12x2−2x+3y=\frac{1}{2}x^2-2x+3, the xx-axis, and the lines x=0x=0 and x=4x=4.

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Question 905

[2 marks]Functional Graphs
For the curve y=12x2−2x+3y=\frac{1}{2}x^2-2x+3, the region between the curve, the xx-axis and the lines x=0x=0 and x=4x=4 has area 6.676.67 square units. Hence find the area of the region enclosed by the curve and the line y=3y=3 (between x=0x=0 and x=4x=4).

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Question 1001

[1 marks]Statistics & Probability
Cumulative frequency pp represents the number of pigs with mass m≤80m\leq80 kg. Given cumulative frequencies of 26 pigs at m≤70m\leq70 kg and 18 pigs in the class 70<m≤8070<m\leq80 kg, calculate pp.

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Question 1002

[1 marks]Statistics & Probability
Cumulative frequency qq represents the number of pigs with mass m≤100m\leq100 kg. Given cumulative frequencies of 66 pigs at m≤90m\leq90 kg and 14 pigs in the class 90<m≤10090<m\leq100 kg, calculate qq.

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Question 1003

[2 marks]Statistics & Probability
A farmer's 100 pigs have a cumulative mass distribution where 44 pigs have mass m≤80m\leq80 kg and 66 pigs have mass m≤90m\leq90 kg. Using linear interpolation between these two points, estimate the median mass of the pigs (the mass below which half the pigs fall).

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Question 1004

[1 marks]Statistics & Probability
A histogram is drawn for pig masses using class widths of 20 kg, where the column for 40<m≤6040<m\leq60 kg (12 pigs) has height 2.42.4 cm. Calculate the height of the column representing 60<m≤8060<m\leq80 kg (32 pigs), also of class width 20 kg.

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Question 1005

[1 marks]Statistics & Probability
A histogram is drawn for pig masses using class widths of 20 kg, where the column for 40<m≤6040<m\leq60 kg (12 pigs) has height 2.42.4 cm, i.e. 0.20.2 cm per pig per 20 kg of width. Calculate the height of the column representing 100<m≤140100<m\leq140 kg (20 pigs, class width 40 kg).

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Question 1101

[3 marks]Geometrical Transformation
Triangle AA has vertices (1,2)(1,2), (2,2)(2,2) and (1,4)(1,4). Triangle BB, the image of AA under a certain single transformation, has vertices (−2,−1)(-2,-1), (−4,−1)(-4,-1) and (−2,−5)(-2,-5). Describe fully this single transformation.
  1. AEnlargement, centre (0,1), scale factor 2.
  2. BEnlargement, centre the origin, scale factor -2.
  3. CEnlargement, centre (0,1), scale factor -2.
  4. DRotation, centre (0,1), through 180°.

Question 1102

[2 marks]Geometrical Transformation
Triangle AA has vertices (1,2)(1,2), (2,2)(2,2) and (1,4)(1,4). Triangle CC is the image of triangle AA under a two-way stretch in which the origin is invariant, with scale factor 33 parallel to the xx-axis and scale factor −2-2 parallel to the yy-axis. Find the coordinates of the vertices of triangle CC.

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Question 1103

[2 marks]Geometrical Transformation
Triangle AA has vertices (1,2)(1,2), (2,2)(2,2) and (1,4)(1,4). The transformation XX is represented by the matrix (0−110)\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix} and maps triangle AA onto triangle DD. Find the coordinates of the vertices of DD.

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Question 1104

[2 marks]Geometrical Transformation
The transformation XX is represented by the matrix (0−110)\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix} and maps triangle AA (vertices (1,2)(1,2), (2,2)(2,2), (1,4)(1,4)) onto triangle DD (vertices (−2,1)(-2,1), (−2,2)(-2,2), (−4,1)(-4,1)). Describe fully the single transformation XX.
  1. AReflection in the line y = x, swapping x and y values.
  2. BEnlargement, centre the origin, scale factor negative 1.
  3. CRotation, centre the origin, through 90° clockwise.
  4. DRotation, centre the origin, through 90° anticlockwise.

Question 1201

[1 marks]Vector Geometry
In a vector diagram, OA⃗=a\vec{OA}=\mathbf{a} and OB⃗=b\vec{OB}=\mathbf{b}. Express, in terms of a\mathbf{a} and/or b\mathbf{b}, the vector AB⃗\vec{AB}.

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Question 1202

[1 marks]Vector Geometry
In a vector diagram, OA⃗=a\vec{OA}=\mathbf{a}. The point DD is such that OD=DAOD=DA. Express OD⃗\vec{OD} in terms of a\mathbf{a}.

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Question 1203

[1 marks]Vector Geometry
In a vector diagram, OA⃗=a\vec{OA}=\mathbf{a}, OB⃗=b\vec{OB}=\mathbf{b}, and AB⃗=b−a\vec{AB}=\mathbf{b}-\mathbf{a}. The point CC is such that AC=3CBAC=3CB. Express AC⃗\vec{AC} in terms of a\mathbf{a} and/or b\mathbf{b}.

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Question 1204

[1 marks]Vector Geometry
In a vector diagram, OA⃗=a\vec{OA}=\mathbf{a}, OB⃗=b\vec{OB}=\mathbf{b}, and AC⃗=34(b−a)\vec{AC}=\frac{3}{4}(\mathbf{b}-\mathbf{a}). Express OC⃗\vec{OC} in terms of a\mathbf{a} and/or b\mathbf{b}.

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Question 1205

[2 marks]Vector Geometry
In a vector diagram, OA⃗=a\vec{OA}=\mathbf{a}, OB⃗=b\vec{OB}=\mathbf{b}, DD is the midpoint of OAOA (since OD=DAOD=DA), and XX lies on BDBD such that BX⃗=kBD⃗\vec{BX}=k\vec{BD}. Express BX⃗\vec{BX} in terms of a\mathbf{a}, b\mathbf{b} and kk.

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Question 1206

[1 marks]Vector Geometry
In a vector diagram, OA⃗=a\vec{OA}=\mathbf{a}, OB⃗=b\vec{OB}=\mathbf{b}, OC⃗=14a+34b\vec{OC}=\frac{1}{4}\mathbf{a}+\frac{3}{4}\mathbf{b}, and OX⃗=hOC⃗\vec{OX}=h\vec{OC}. Express OX⃗\vec{OX} in terms of a\mathbf{a}, b\mathbf{b} and hh.

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Question 1207

[2 marks]Vector Geometry
Equating two expressions found for OX⃗\vec{OX} (one in terms of kk, one in terms of hh) gives h=2kh=2k and 1−k=32k1-k=\frac{3}{2}k. Solve for kk.

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Question 1208

[1 marks]Vector Geometry
Given that h=2kh=2k and k=25k=\frac{2}{5}, find the value of hh.

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Question 1209

[1 marks]Vector Geometry
Given that BX⃗=kBD⃗\vec{BX}=k\vec{BD} with k=25k=\frac{2}{5}, find the numerical value of the ratio BX:XDBX:XD.

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