THE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL OF TANZANIA CERTIFICATE OF SECONDARY EDUCATION EXAMINATION
041 BASIC MATHEMATICS
(For Both School and Private Candidates)
Time: 3 Hours Tuesday, 06^{th} November 2018 a.m.
Instructions
SECTION A (60 Marks)
Answer all questions in this section.
1.If and find the fraction of in its simplest form
(b).Find the GCF of 210, 357 and 252.
2. (a). Evaluate log_{10} 40,500 given that log_{10} 2 = 0.3010 , log_{10} 3 = 0.4771 and log_{10} 5 = 0.6990.
(b). Find the values of x and y if
3. (a). In a school of 60 teachers, some drink Fanta and some drink CocaCola. If 46 drink Fanta, 18 drink CocaCola and 14 drink both CocaCola and Fanta. How many teachers drink neither Fanta nor CocaCola? (Use Venn diagram)
(b).Use the figure below to answer the following questions:
(i). Write the expression for the total area of rectangles A and B.
(ii). If the total area of rectangles A and B is 98 square centimeters, find the value of x
4. (a).
Find x and y given that
(b). Find the point of intersection of the lines x − 2y = −5 and 2x + 7y − 34 = 0
5. In the following figure, a regular hexagon is inscribed in a circle. If the perimeter of the hexagon is 42 cm, find:
(a) The radius of the circle.
(b) The area of the circle and the regular polygon.
(c)The area of the shaded region.
6. (a)Mukasa received Ushs 1,000,000 from his sister in Uganda. How much did he get in Tanzanian currency (Tshs) if one Ugandan shilling was equivalent to 0.65 Tanzanian shilling?
(b) The energy (E) stored in an elastic band varies as the square of the extension (x). When the elastic band is extended by 4 cm, the energy stored is 240 joules. What is the energy stored when the extension is 6 cm? What is the extension when the stored energy is 60 joules?
7. (a) Three relatives shared Tshs 140,000 so that the first one got twice as much as the second, and the second got twice as much as the third. How much money did the first relative get?
(b) Kitwana paid Tshs 900,000 for a desktop computer and sold it the following year for Tshs 720,000. Find:
8. (a) If an arithmetic progression has A_{1} as the first term and d as the common difference,
(b) The first and second terms of a geometric progression are 3 and 9 respectively.
by using the results in part (i).
9. (a) Find the distance PR in the following figure if the lines PR and RQ are perpendicular.
(b) A flagpole is 5 meters high. Find to the nearest cm, the length of its shadow when the elevation of the sun is 60^{0}.
10.(a) Use factorization method to solve the quadratic equation x^{2} − 9x + 14 = 0.
(b) Find the values of x that satisfies the equation
SECTION B (40 Marks)
Answer four (4) questions from this section.
11. A farmer needs to buy up to 25 cows for a new herd. He can buy either brown cows at 50,000/= each or black cows at 80,000/= each and he can spend a total of not more than 1,580,000/=. He must have at least 9 cows of each type. On selling the cows he will make a profit of 5,000/= on each brown cow and 6,000/= on each black cow. How many of each type he should buy to maximize profit?
12. The scores of 45 pupils in a Civics test were recorded as follows:
30  65  50  62  40  35  64  32  28  59  60  82  24  35  63 
68  46  48  73  92  54  46  63  75  58  43  71  72  27  28 
61  71  36  64  80  61  64  76  64  35  76  73  70  64  46 
13.(a) In the following cuboid, AB = 5 cm, BC = 12 cm and BG = 10 cm. Calculate:
(i) The length of AH (give your answer correct to one decimal place).
(ii) The angle CAH.
(b) In the following figure A, B, C and D lie on the circle; O is the centre of the circle, BD is its diameter and PAT is the tangent of the circle at A.
If angle ABD = 59 , CDB = 35, find ACD, ADB, DAT and CAO.
14. Mwanne commenced business on 1^{st} April, 2015 with capital in cash 200,000/=
April 2 bought goods for cash 100,000/=
3 bought goods for cash 300,000/=
4 purchased shelves for cash 230,000/=
5 sold goods for cash 400,000/=
9 paid wages for cash 50,000/=
12 purchased goods for cash 70,000/=
13 sold goods for cash 600,000/=
16 paid rent for cash 100,000/=
20 bought goods for cash 60,000/=
25 sold goods for cash 300,000/=
27 paid salary for cash 70,000/=
Prepare the following:
15. (a) Find the point P(x, y)
(b)A translation T maps point P(x, y) in part (a) into (3,2). Find where it takes the point (7,4).
^{(c) Find the image of the point obtained in part (b) under a rotation of 900 followed by another rotation of 180}^{0}^{ anticlockwise. }
16. (a) A bag contains 6 white shirts and 3 blue shirts. Three shirts are picked at random one after another with replacement. Determine the probability that:
(b) The function f is defined by
(i) Sketch the graph of f.
(ii) Use the graph to determine the domain and range of f.
CERTIFICATE OF SECONDARY EDUCATION EXAMINATION
041 BASIC MATHEMATICS
(For Both School and Private Candidates)
Instructions
This paper consists of sections A and B with a total of sixteen (16) questions.
Answer all questions in sections A and four (4) questions from section B. Each question in section A carries six (06) marks while each question in section B carries ten (10) marks.
All necessary working and answers for each question done must be shown clearly.
Mathematical tables may be used.
Calculators, cellular phones and any unauthorised materials are not allowed in the examination room.
Write your Examination Number on every page of your answer booklet(s).
Answer all questions in this section.
(a) Round off:
9.67 to ones,
0.205 to one decimal place
0.0197 to two decimal places.
Hence, estimate the value of
Simplify the expressions
Express 0.3636… in the form a/b where b≠0
(b) If nlog_{5}125 = log_{2}64, find the value of n.
(a) Factorize the following expressions: (i) 16y^{2 }+ xy − 15x^{2},
(ii) 4 − (3x − 1)^{2}.
(b) At Moiva’s graduation ceremony 45 people drank PepsiCola, 80 drank CocaCola and 35 drank both PepsiCola and CocaCola. By using a Venn diagram, found out how many people were at the ceremony if each person drank PepsiCola or CocaCola.
4(a) Given the three vectors a = 4i + 6j, b = 4i + 10j and c = 2i + 4j determine the magnitude of their resultant.
(b) Camilla walks 5 km northeast, then 3 km due east and afterwards 2 km due south. Represent these displacements together with the resultant displacement graphically using the scale 1 unit = 1 km.
(c) Show that triangle ABC is rightangled where A = (2,1), B = (2,1) and C = (1,3).
5 (a) In the figure below, AB = 10 cm, AX = 6 cm, CX = 8 cm and AB is parallel to DC.
Show whether triangles AXB and CXD are similar or not.
Find the length of CD.
Find the ratio of the areas of triangles AXB and CXD.
(b) Using a ruler and compass, construct an angle of 90°.
6 (a) In the preparation of fanta orange drink, a bottling filling machine can fill 1,500 bottles in 45 minutes. How many bottles will it fill in 4^{1}_{2 }hours?
(b) If X varies directly as Y and inversely as W, find the values of a and b in the table below.
X  8  6  b 
Y  4  a  2 
W  2  3  4 
7. A computer is advertised in a shop as having a list price of sh. 2,500,000 plus value added tax (VAT) of 20%. The sales manager offers a discount of 25% before adding the VAT. Calculate:
The list price including VAT.
The amount of discount before VAT is added.
The reduced final price of the computer.
8. (a) If the sum of n terms of a geometric progression with first term 1 and common ratio 1/2 is 31/16 find the number of terms.
(b) How many integers are there between 14 and 1,000 which are divisible by 17?
9. In the figure below, AE = 20 m, EB = 20 ^{√}2 m and DAE = 45°.
Find:
The length: DE, AD and AB.
The area of triangle ABE, leaving the answer in surd form.
10(a) Solve the equation 4x^{2 }− 32x + 12 = 0 by using the quadratic formula.
(b) Anna is 6 years younger than her brother Jerry. If the product of their ages is 135, find how old is Anna and Jerry.
Answer four (4) questions from this section.
11. Zelda wants to buy oranges and mangoes for her children. The oranges are sold at sh. 150 each and mangoes at sh. 200 each. She must buy at least two of each kind of fruit but her shopping bag cannot hold more than 10 fruits. If the owner of the shop makes a profit of sh. 40 on each orange and sh. 60 on each mango, determine how many fruits of each kind Zelda must buy for the shop owner to realise maximum profit.
12. The heights of 50 plants recorded by a certain researcher are given below:
56  82  70  69  72  37  28  96  52  88  41  42 
50  40  51  56  48  79  29  30  66  90  99  49 
77  66  61  64  97  84  72  43  73  76  76  22 
46  49  48  53  98  45  87  88  27  48  80  73 
54  79 










Copy and complete this tally table for the data given above.
Height (cm)  Tally  Frequency 
2130 


3140 


4150 


5160 


6170 


7180 


8190 


91100 


Use this table to:
Draw a histogram for the height of the plants.
Find the mean height of the plants (do not use the assumed mean method).
Find the median of the heights of the plants.
13. In the figure below, BC is a diameter of the circle, O is the centre of the circle and side CD of the cyclic quadrilateral ABCD is produced to E.
With reasons, name the right angles in this figure.
Show that AD̂E = AB̂C.
If AD̂E = 60 and CÂD = 25, find:
the value of AB̂D,
the lengths AB and BD given that CB = 10cm.
14. (a) What is a trial balance and what is its main purpose.
On January 1^{st }2015 Semolina Women Group started a business with a capital in cash of 2,000,000/=
January 2 Purchased goods for cash 1,400,000/=
3 Sold goods for cash 1,000,000/=
6 Purchased goods for cash 600,000/=
15 Paid rent for cash 220,000/=
26 Paid wages for cash 220,000/=
15 Sold goods for cash 620,000/=
Prepare:
The cash account and balance it.
The Trial Balance.
15. (a) Find the inverse and identity matrix of A = .
Triangle OAB has vertices at O(0,0), A(2,1) and B(1,3). If the triangle is enlarged by
E = and then translated by T = , find the vertices of the triangle.
Draw on the same x  y plane triangle OAB and the images after being:
enlarged
translated
16. (a) A function f is defined on the set of integers as follows:
Draw a pictorial diagram for f (x) .
Find the domain and range of f (x) .
(b) Given that
(c)In a yard there are 500 vehicles, of which 160 are cars, 130 are vans and the remaining are lorries. If every vehicle has an equal chance to leave, find the probability of:
A van leaving first,
A lorry leaving first,
A car leaving second if either a lorry or van had left first.
CERTIFICATE OF SECONDARY EDUCATION EXAMINATION
(For Both School and Private Candidates)
Instructions
This paper consists of sections A and B.
Answer all questions in section A and four (4) questions from section B. Each question in section A carries 6 marks while each question in section B carries ten (10) marks.
All necessary working and answers for each question done must be shown clearly.
Mathematical tables may be used.
Calculators and cellular phones are not allowed in the examination room.
Write your Examination Number on every page of your answer booklet(s).
Answer all questions in this section.
1. (a) From the set of numbers {1, 3, 4, 5, 6, 8, 10, 15, 17, 21, 27} ; write down:
the prime numbers,
the multiples of 3,
the factors of 60.
(b) Four wooden rods with lengths of 70 cm, 119 cm, 84 cm and 105 cm are cut into pieces of the same length. Find the greatest possible length for these pieces if no wood is left over.
(a) Solve for x in the equation
b) Show that
(a) By using a=1/x and b= 1/y in the following system of equation
Find the solution set (x,y)
(b) Let U be a universal set and A and B
be the subsets of U where: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , A = { odd numbers } and B = { prime numbers }
Represent this information in a venn diagram.
^{Find }^{A }^{⋂ }^{B}′ ^{and (}^{A }^{⋃}^{ }^{B}^{) }′
(a) Given vectors a = 6i + 12j, b = 17i + 18j:
Find the vector c = 2a – b and its magnitude correctly to 3 significant figures.
Represent vector c in part (a)(i) on the x  y plane.
(b) Find the equation of the line passing passing through the midpoint of the points A(− 3, 2) and B(1,− 4) and which is perpendicular to line AB .
(a) In triangle ABC , X ,Y and Z are the midpoints of sides AB , AC and BC respectively. If ZX = ZYand ZX^{ˆ}^{ }B = ZY^{ˆ}^{ }C = 90° ;
Represent this information diagrammatically,
Show that AB^{ˆ}^{ }Z = AC^{ˆ}^{ }Z .
(b) The areas of two similar polygons are 27 and 48 square metres. If the length of one side of the smaller polygon is 4.5 cm, find the length of the corresponding side of the larger polygon.
6. (a) The number of tablets given to a patient was found to be directly proportional to the weight of the patient. If a patient with 36 kg was given 9 tablets, find how many tablets would be given to a patient whose weight is 48 kg.
(b) Four people can eat 2 bags of rice each weighing 10 kg in 12 days. How many people can eat 6 bags of rice of the same weight in 18 days?
7. (a) Mariam, Selina and Moses contributed 800,000, 1,200,000 and 850,000 shillings respectively while starting their business.
Find the ratio of their contributions in simplest form.
If the business made a profit of 1,900,000 shillings; find how much each got if the profit was shared in the same ratio as their contributions.
(b) A dealer bought 10 books for 200,000. He sold^{ }^{2}^{ }of them at 30,000 shillings each and the remaining at 25,000 shillings each. What was his percentage profit?
8.(a) The 8^{th }term of an arithmetic progression is 9 greater than the 5^{th }term and the 10^{th }term is 10 times the 2^{nd }term. Find the common difference and the first term of the arithmetic progression.
(b) The sum of the first two terms of a geometric progression is 18 whereas the sum of the second and third term is 54, find the first term and the common ratio.
9 (a) A river with parallel banks is 20 m wide. If P and Q are two points on either side of the river, as shown in the figure below, find the distance PQ
(b) In the triangle LMN , LM = 5 m, LN = 6 m and angle M LN = 66° . Find MN .
10. (a) If one of the roots of the quadratic equation x^{2 }+ bx + 24 = 0 is 1 1/2, find the value of b
(b) Two numbers differ by 3. If the sum of their reciprocals is 7/10 , find the numbers.
Answer four (4) questions from this section.
A shopkeeper sells refrigerators and washing machines. Each refrigerator takes up 1.8 m^{2 }of space and costs 300,000 shillings; whereas each washing machine takes up 1.5 m^{2 }of space and costs 500,000 shillings. The owner of the shop has 6,000,000 shillings to spend and has 27 m^{2 }of space.
Write down all the inequalities which represent the given information.
If he makes a profit of 30,000 shillings on each refrigerator and 40,000 shillings on each washing machine, find how many refrigerators and washing machines he should sell for maximum profit.
07, 19, 78, 53, 43, 67, 12, 54, 27, 22, 33, 80, 25, 58, 50, 36, 65, 33, 16, 19, 34, 20, 55, 27, 37, 41, 04, 32,
48, 28, 70, 31, 61, 08, 35
Prepare the frequency distribution table using the class intervals: 0–9, 10–19, 20–29, etc.
Which class interval has more students?
Represent the information in a histogram and a frequency polygon and then find the mode.
Calculate the median mark.
(a) In the figure below, BD is a tangent to the circle having the centre O .
Given that angle OEC = 28° , find the values of angles marked X , Y and Z .
(b)Calculate the distance from Chagwe (5°S, 39°E) to Minga (12°S, 39°E) in kilometres. Use π = 3.14 , the radius of the earth R = 6370 km and write the answer correct to 1 decimal place.
(c)If a bus leaves Chagwe at 8:00 am on Monday and travels at 40 km/hour, at what time will it reach Minga?
(a) Given:
Opening stock 01012012 34,430/=
Closing stock 31122012 26,720/=
Net purchases during 2012 212,290/=
Expenses for the year 45,880/= Gross Profit is 50% of cost of goods sold
Find: (i) Cost of goods sold (ii) The gross profit
(b) On 1^{st }June, 2013 Mrs. Lemisha started business with capital of 100,000/= and mad ehte following transactions
June 2 bought furniture 40,000/=
7 bought goods 70,000/=
11 sold goods 65,000/=
16 paid Sundry expenses 30,000/=
19 cash sales 80,000/=
24 paid wages 50,000/=
26 withdraw cash 30,000/=
Prepare the cash account
Prepare the balance sheet as at 30/06/2013
Explain the importance of the balance sheet you have prepared in part (b)(ii) above.
(a) Given matrices :
Find A^{2} + 2A
Find t and y such that B^{2} = tB + yI where I is an identity matrix.
Find the value of k if the determinant of C is 5.
(b) A linear translation Q carried point (x, y)into
Determine the transformation matrix Q .
Find Q(3, 3)
Find the image of the point obtained in part (b)(ii) under Q .
(a) The function f is defined as follows:
Sketch the graph of f (x) ,
Use the graph to determine the domain and range of f (x) .
(b) (i) Two numbers are chosen at random from 1, 2 and 3. What is the probability that their sum is an odd number if repetition is not allowed?
(c) If A and B are two events such that, P(A)=1/4, P(B)=1/2 and P(AnB)=1/8. Find ^{P}^{ }^{(}^{A}^{ }^{⋃}^{ }^{B}^{)}′
CERTIFICATE OF SECONDARY EDUCATION EXAMINATION
(For Both School and Private Candidates)
Instructions
This paper consists of sections A and B.
Answer all questions in sections A and four (4) questions from section B. Each question in section A carries 6 marks while each question in section B carries 10 marks.
All necessary working and answers for each question done must be shown clearly.
Mathematical tables may be used.
Calculators and cellular phones are not allowed in the examination room.
Write your Examination Number on every page of your answer booklet(s).
Answer all questions in this section.
(a) If p = 6.4 × 10^{4} and q = 3.2 × 10^{5} , find the values of:
p × q ,
p + q .
Write the answers in standard form.
(b) Evaluate the following using mathematical tables and write the answer correctly to 3 significant figures
(a) Solve for x in the equation 4^{−2}^{x} × 8^{2} = 4 × 16^{x}
(b) Find the value of log 900 given that log 3 = 0.4771 .
(a) Find the solution set of the inequality and indicate it on a number line.
(b) The Venn diagram below shows the universal set U universal set U and its two subsets A and B.
Write down the elements of:
A′ ,
B′ ,
A ⋃ B ,
A′ ⋃ B′ .
(c)Verify that n(A ⋃ B) = n(A) + n(B) − n(A ⋂ B) where A and B are the sets given in part 3(b).
(a) Given vectors a = 3i + 2j, b = 8i 3j and c = 2i + 4j find:
a unit vector in the direction of d.
(b) Find the equation of the line passing at point (6, 2) and it is perpendicular to the line that crosses the xaxis at 3 and the yaxis at 4.
(a) Two triangles are similar. A side of one triangle is 10 cm long while the length of the corresponding side of the other triangle is 18 cm. If the given sides are the bases of the triangles and the area of the smaller triangle is 40 cm^{2}, find the area and the height of the larger triangle.
In the figure below CB = BD = DA and angle ACD = x .
Show that angle ADE = 3x ,
Calculate the measure of angle CDA if x = 39° .
(a) The variable varies directly as the square of x and inversely as y . Find v when x = 5 and y = 2 ; given that when v = 18 and x = 3 the value of y = 4 .
(b) The temperature (T _{i}) inside a house is directly proportional to the temperature (T_{o})
outside the house and is inversely proportional to the thickness (t) of the house wall. If T _{i }= 32°C when T_{o} = 24°Cand t = 9cm , find the value of twhen T _{i }= 36°C and T_{o} = 18°C(a) A shopkeeper makes a 20% profit by selling a radio for sh. 480,000.
Find the ratio of the buying price to the selling price.
If the radio would be sold at 360,000, what would be the percentage loss?
(b) A farmer sold a quarter of his maize harvest and gave one third of the remaining to his relatives. If the farmer remained with 25 bags of maize find how many bags of maize did the farmer harvest.
(a) How many terms of the series 3 + 6 + 9 + 12 + . . . are needed for the sum to be 630?
(b) Jennifer saved sh. 6 million in a Savings Bank whose interest rate was 10% compounded annually. Find the amount in Jennifer’s savings account after 5 years.
(a)Find the value of the following^{ }without using mathematical tables.
(b) Calculate the angles of a triangle which has sides 4m , 5m and 7m .
(a) Factorize completely 2x^{2 }+ x − 10 by splitting the middle term.
(b) Solve the equation
Answer four (4) questions from this section.
A small industry makes two types of clothes namely type A and type B. Each type A takes 3 hours to produce and uses 6 metres of material and each type B takes 6 hours to produce and uses 7 metres of material. The workers can work for a total of 60 hours and there is 90 metres of material available. If the profit on a type A cloth is 4,000 shillings and on a type B is 6,000 shillings, find how many of each type should be made for maximum profit.
The following marks were obtained by 32 students in a physics examination:
32, 35, 42, 50, 46, 29, 39, 38, 45, 37, 48, 52, 37, 58, 52, 48, 36, 54, 37, 42, 64, 37, 34, 28, 58, 64, 34, 57,
54, 62, 48, 67.
Prepare a frequency distribution table using the class intervals: 24 29, 30 35 etc.
Draw the histogram.
Draw the cumulative frequency curve and use it to estimate the median.
Find the mean mark.
(a) Find the value of the angles a and b in the figure below.
A rectangular box with top WXY Z and base ABCD hasAB = 9cm , BC = 12cm and WA = 3cm .
Calculate:
The length of AC ,
The angle between WC and AC .
Two places P and Q both on the parallel of latitude 26°N differ in longitude by 40° . Find the distance between them along their parallel of latitude.
The following trial balance was extracted from the businessman books’ of Chericho Ramaji, at 31^{st }December 2006.
S/N  Details  Dr (Tshs.)  Cr (Tshs.) 
1.  Capital 
 830,000 
2.  Purchases  1,200,000 

3.  Sales 
 1,750,000 
4.  Return inwards  55,000 

5.  Return outwards 
 64,000 
6.  Plant and machine  240,000 

7.  Furniture and fittings  75,000 

8.  Sundry debtors  137,000 

9.  Sundry creditors 
 86,000 
10.  Wages  228,000 

11.  Bad debts  36,000 

12.  Discount received 
 27,000 
13.  Opening stock  500,000 

14.  Insurance  16,000 

15.  Commission receivable 
 43,000 
16.  Trade expenses  22,000 

17.  Cash in hand  17,000 

18.  Cash at bank  274,000 

Total  2,800,000  2,800,000 
Prepare Trading, Profit and Loss account for the year ended 31^{st }December 2006.
find the elements of matrix P .
(b)Determine the matrix A from the equation
(c) a triangle with vertices A(0, 0) , B(3, 0) and C(3, 1) ; find its image under:
(i) a translation by the vector (2, 3) ,
the enlargement matrix
(d) Sketch the triangle and the images in parts (c)(i) and (ii) on the same pair of axes and comment on their sizes.
(a) The function f is defined as follows:
Sketch the graph of f(x) ,
Determine the domain and range of f(x) .
(c) If a number is to be chosen at random from the integers 1, 2, 3, that: . . . , 11, 12 ; find the probability
It is an even number,
It is divisible by 3.
(d) If in part 16(c) above, E_{1 }is the set of even numbers and E_{2 }the set of numbers that are divisible by 3, show whether E_{1} and E_{2}are mutually exclusive events.
THE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL
CERTIFICATE OF SECONDARY EDUCATION EXAMINATION
BASIC MATHEMATICS
(For School Candidates Only)
Time: 3 Hours Monday, 4^{th} October 2010 a.m.
1. This paper consists of sections A and B.
2. Answer all questions in section A and four (4) questions from section B.
3. All necessary working and answers for each question done must be showmclearly.
4. Mathematical tables may be used unless otherwise stated.
5. Calculators and cellular phones are not allowed in the examination room.
6. You are advised to spend not more than two (2) hours on section A and the remaining time on section B.
7, Write your Examination Number on every page of your answer booklet(s)
SECTION A (60 Marks)
Answer all questions in this section showing all necessary working and answers.
1. (a) Write 624.3278 correct to:
five (5) significant figures (ii) three (3) decimal places.
(b) A mathematics teacher bought 40 expensive calculators at shs.16,400 each and a number of other cheaper calculators costing shs.5,900 each. She spent a total of shs. 774,000. How many of the cheaper calculators did she buy?
(6 marks)
2. (a) Evaluate without using mathematical tables 2 log 5 + log 36 — log 9.
(b) Simplify
(6 marks)
3. (a) Given that A = {x : 0 x 8}
where x is an integer, in the same form, represent in a Venn diagram (i) AuB (ii) An B and hence find the elements in each set.
(b) In a school of 75 pupils, 42% of the pupils take Biology but not Chemistry, 32% take both subjects and 10% of them take Chemistry but not Biology. How many pupils do not take either Biology or Chemistry?
(6 marks)
4. (a) (i) Without using mathematical tables, find the numerical value of
(ii) Write down the equation of the line which passes through (7, 3) and which is inclined at 45^{0} to the positive direction of the x — axis.
(b) The position vectors of the points A, B and C are 4i —3 j , i+3 j and
—51+j respectively. Find the vectors AB, BC and AC hence verify that
AB + BC =AC (6 marks)
5. (a) The volume of two similar cylinders is 125 cm and 512 cm . If the radius of the larger cylinder is 8cm, find the radius of the smaller cylinder.
(b) In the diagram below, show that
c
(6 marks)
6. (a) Juma bought motor vehicle spare parts from Japan worth 5,900,000 Japanese Yen. When he anived in Tanzania he was charged custom duty of 25% on the spare parts. If the exchange rates were as follows:
1 US dollar = 118 Japanese Yen
1 US dollar = 76 Tanzania Shillings
Calculate the duty he paid in Tanzania shillings.
(b) The distance of the horizon d km varies as the square root of the height h m of the observer above sea level. An observer at a height of 100m above sea level sees the horizon at a distance of 35.7 km.
Find (i) the distance of the horizon from an observer 70m above sea level.
(ii) an equation connecting d and h. (6 marks)
(a) An amount of Tshs. 12,000 is to be shared among Ali, Anna and Juma in
the ratio 2:3:5 respectively. How much will each get?
(b) A certain worker used his salary as follows: 20% on house rent, 45% on food, 10% on refreshment and 15% on school fees. If he/she was left with Tsh.22,000, determine:
(i) The salary of this worker.
(ii) The amount of money which he/she spent on food. (6 marks)
(a)Find the general term and hence the 30 term of the sequence
(b) Given the series 100 + 92 + 84 + .
Find
(i) the 20^{th} term
(ii) the sum of the first 20 terms. (6 marks)
9. (a) If tan A =3/4 and A is acute, find cos A, sin A and hence verify the
identity cos^{2} A+sin^{2} A = 1
(b)
Given the right angled triangle above whose sides are measured in centimeter determine:
(i) the value of x
(ii) the area of the triangle (6 marks)
10. (a) Factorize each of the following expressions:
(b) Find the value of y which satisfies the equation 3(2y^{2})17y=0
(6 marks)
Answer four (4) questions from this section. Extra questions will not be marked.
11. (a) Maximize f= 2y — x subject to the following constraints:
2x+y<=6
x+2y <= 6
x≥0
y≥0
(b) Sara had 300 shillings to buy erasers and pencils. An eraser cost 20 shillings while a pencil costs 30 shillings. If the number of erasers bought is at least twice the number of pencils, formulate the inequalities that represent this information. (10 marks)
12. The data below represent masses in kg of 36 men.
51 61 60 70 7571 75 70 74 73 72 82
70 71 76 74 50 68 68 66 65 72 69 64
83 63 83 58 80 90 50 89 55 62 62 61
(i) Prepare a frequency distribution table of class interval of size 5 beginning with the number 50 taking into consideration that both lower limit and upper class limits are inclusive.
(ii) Calculate the mean and mode from the frequency distribution table prepared in (i) above by using assumed mean from the class mark of the modal class. (10 marks)
13. (a) Below is a circle with centre O and radius r units. By considering the circumference of the circle, the area of the circle, the given angle 0 and the degree measures of a circle (360^{0}), develop the formula for finding:
(i) arc length AB
(ii) area of sector AOB.
(b) Find (i) the length of arc AB
(ii) the area of the sector AOB
if 0 is 57^{0} and r is 5.4 cm use pie=22/7 (10 marks)
14. From 1 ^{st} January to 29 January 2006 Mr. Bin decided to keep records of his business as follows:
Jan. 1 Mr. Bin started a business with capital in cash  500,000.00 
5 Purchased goods  254,000.00 
6 Sold goods  290,000.00 
9 Purchased goods  204,000.00 
10 Expenses  24,000.00 
29 Sold goods You are required to: (a) prepare the trial balance (b) open capital and cash account.  320,000.00 
N.B and receipts were made in cash.  (10 marks) 
All payments
15. (a) A transformation T has the matrix
Under the same
transformation T, the point (4, 1) is mapped onto the point (6, 3). Find X and r.
(b) For what values of n will the matrix be nonsingular?
(10 marks)
16. (a) If f(x) = 2x + 3 find f^{1}(3)
(b) Draw the graph of f (x) =  x—l  for —4 ≤ x ≤4
(c) State the domain and range of f (x)= x1 
(d) The probability that Rose and Juma will be selected for A — level studies after completing their O — level studies are 0.4 and 0.7 respectively. Calculate the probability that:
both of them will be selected.
(ii) either Rose or Juma will be selected.
(10 marks)