THE UNITED REPUBLIC OF TANZANIA, PRIME MINISTER’S OFFICE REGIONAL ADMINISTRATION AND LOCAL GOVERNMENT

FORM THREE MIDTERM-2 EXAMINATION

041 BASIC MATHEMATICS

TIME 3.00 HOURS AUG./SEPT. 2026

INSTRUCTIONS

1. This paper consists of sections A and B with a total of 14 questions
2. Answer all questions in all sections
3. Each question in section A carries six (06) marks while each question in section B carries ten (10) marks

SECTION A (60 MARKS)

1. (a) There are 150 people at a cricket match, 20% are children, one half are men and the rest are women. How many women are at the cricket match?

(b) The LCM and GCF of y, 18 and 60 are 360 and 2 respectively. Find the value of y

2. (a) Determine the value of x if

 (b) Rationalize the denominator in

3. (a) If n(A) = 17, n(A ∪ B') = 8 and n(A ∪ B) = 31.

Find   (i) n(A ∩ B)    (ii) n(B)

(b) In a class of 30 students, 17 participate in English debate, 12 participate in English and Sports. If every student is required to participate in at least one of these two events, find the number of students who participated in:-

1. English debate only
2. Sports only

4. (a) What is the value of P which makes a line passing through the points A(−5, P) and (1, 2) having equal slope to another line whose equation is x = 2y + 3?

(b) If A = {(x, y): x + y = 8} and B = {(x, y): 2x + 3y = 20}. Find A ∩ B

5. (a) Two similar rectangles have areas 80cm² and 60cm². The length of the smaller rectangle is 12cm, what is the length of the larger rectangle?

(b) The area of the rectangle shown is 125cm². Find the value of x

(c) Each of the angles of a polygon is 140°. Find the number of sides of the polygon.

6. (a) If (x + 1) is directly proportional to y² and inversely proportional to z and x = 10 when y = 8, z = 9. Find the value of x when y = 10 and z = 12.

(b) 60 people working 8 hours a day take 2 days to cultivate a maize farm. How long will it take 20 people to cultivate the same farm if they work 16 hours a day?

7. (a) A book is bought for Tsh. 15,000 and sold at a percentage profit of 10%. Find its selling price

(b) A factory employs skilled, semi – skilled and office staff workers in the ratio of 6: 5: 4 respectively. If there are 120 semi – skilled workers, how many skilled workers are there?

8. Kabaka borrowed a loan from a CRDB bank under the condition that he is required to repay the loan on a monthly installments basis. He repaid 22,000, 24,000, and 26,000 Shillings for the first, second and third installments respectively until the loan was fully repaid;

(a) If the final monthly installment was 86,000 Shillings, how long did he take to repay the whole loan?

(b) Calculate the total amount of loan borrowed

9. (a) A rectangular field is 42m long and 40m wide. The soldiers were arranged along the diagonal of the field at equal intervals of two metres. How many soldiers were arranged on the diagonal of the rectangular field?

(b) Given that Sin B = 4/5. Find the value of

without using tables or calculator

10. (a) Solve the quadratic equation by completing the square:

2x² + 3x − 2 = 0

(b) Factorize x² − y² and use it to find the exact value of 73865² − 26135²

SECTION B (40 MARKS)

11. The following marks were obtained by 50 students in a mathematics examination.

43, 60, 65, 80, 50, 55, 40, 45, 70, 50, 50, 68, 55, 50, 60, 45, 70, 65, 45, 50, 65, 48, 45, 80, 56, 65, 80, 43, 55, 50, 65, 50, 82, 60, 40, 43, 55, 45, 50, 50, 65, 44, 60, 44, 82, 70, 55, 40, 50, 45. 

1. Construct a frequency distribution table with classes starting from 39 – 49, 50 – 60, 61 – 71, etc
2. Calculate the mean score and use it to interpret the results
3. Calculate the median score and use it to interpret the results
4. Draw a histogram and use it to estimate the mode.

12. (a) The domain of the relation R = {(x, y): y = x² − 1} is {0, 2, 4, 6} What is the range of the relation R?

(b) A relation R is such that, R = {(x, y): 3x + 2y ≥ 6}

1. Draws the graph of this relation
2. Write the inverse of this relation.

(c) Find the domain and range of R⁻¹ given that R = {(x, y): y = x² + 4}

13. (a) (b) A translation T takes the origin to (2,5). Find where T taken the point (7,0).

(b) Find the image of the point Q (1,3) after rotating it through 180° Clockwise.

(c) Find the value of the angles a and b in the figure below.

14. (a) Given that f(x) = 5x² − 3 and g(x) = 6x − 1. Find  

 (b) The function is defined by

1. Sketch the graph of f(x)
2. Determine domain and range of f(x)
3. Calculate f(−2) and f(0.5)
4. Is f(x) one – to – one? Why?

PRESIDENT’S OFFICE

REGIONAL, ADMINISTRATION AND LOCAL GOVERNMENT

SECONDARY EXAM SERIES

FORM THREE MID TERM EXAMINATION

041                                                               BASIC MATHEMATICS

TIME:3 HOURS                                                                                                                                                                                                                    AUG, 2023

INSTRUCTION

1. This paper consist of two sections A and B

2. Answer all questions

3. Calculators and mathematical tables may be used

4. All diagram must be drawn by using pencil

5. All writing must be in a blue or black ink

                                SECTION A (60 MARKS)

1. (a) If a=write in the form of a/b where b≠0 

(b)Three bells are set to ring at intervals of 12 minutes,15 minutes and 24 minutes. If they started together at 2:00 pm, then find at what time will the bells ring together for second time

2. (a)Find the value of X if ()^x+3()^-x=1  

 (b)Solve for X if log₃9^-1=X²-2x-5

3. (a)If  = p+q√r, simplify by rationalizing the denominator and hence find the value of p, q and r

 (b)In a class of 100 students, 38 study mathematics,20 study biology and 45 study neither of two  subjects by using venn diagram, find the number of students who

 (i)Study both subjects

 (ii)Study exactly one subject

4. (a) A straight line has a gradient of  and it passes through the point (-1,2). Find 

 (i)Its equation    

 (ii)The Y-intercept

 (b)Make V the subject from   = 

5. (a) a radio is bought for sh.24,000 a sold for sh. 36,000.  Find the 

1. The profit made
2. The percentage profits

 (b) The operation ab = (a+b)²- ab

Find the value

1. 32
2. 9-

6. (a) The variable V varies directly as the square of X and inversely as Y.  Find V when X =5 and Y=2, When V =18, X=3 and Y =4.

 (b) Water from a tap gets into a tank at a rate of 20 litres per minutes. How long will it take to fill a tank that is 10,000 litres. Give your answer in hours and minutes.

7. (a) Find the first term and common difference of an arithmetic progression whose 5^th term is 21 and 8^th term is 30.

 (b) The products of three terms of a geometric progression (GP) is 8000. If the first term is  4. Find the second term and the third term. 

8. (a) How many grams are there 0.00912 tones?

 (b) A ladder 15cm long leans against of vertical wall so that angles make with the horizontal ground is two times that makes vertical wall. Calculate how far up the wall does the ladder reaches?

9. (a) Basil has to share eighty books with his daughters Rose and Nancy. He decided that for every two books Nancy gets, Rose gets three books and he gets five books. Find the number of books each gets.

 (b) Solve by elimination method          2x – y = 1

                                                               x + y = 1

10. (a) Solve the quadratic equation x²-8x+7=0 by completing the square

 (b) A field is 10m longer than its wide. The area is 7200m². What is the width?

                                           SECTION B (40 MARKS)

11. The table below represents score of 100 students in geography test

1. Determine the value of X

(b)  By using assumed A=74.5, determine mean

(c)  Find mode

 (d)  Find median

12. (a) Kwembe went to the market with 3000Tsh to buy oranges and mangoes. He bought 20 oranges and 5 mangoes. If Grace went to the same market with 2000Tsh and bought 10 oranges and 5 mangoes. What is the price of one mango and one orange? 

 (b) Mchilo invested a certain amount of money in a saving Bank whose interest rate was 10 compound annually. After two years he got 5000 shillings.

(i) How much did he invest at the start?

 (ii) How much did he receive as interest at the end of two years?

13. (a) Given a function f(x)=x²-2x-3. Find 

(i) Line of symmetry

 (ii) The turning point

 (iii) Express f(x) in the form of a(x+b)²+c where a,b and c are real numbers

 (b) If f(x) is the function such that

                                                     F(x)=

(i) Sketch the graph of f(x)

(ii) State the domain and range of f(x)

(iii) Find the value of f(100) 

14.  MRS CHENI started a business on 16^thMarch 2023 with capital in cash 2,066,000

March 17 Bought goods for cash 1,000,00/-

19 Bought shelves for cash 110,000/-

20 Sold goods for cash 900,000/-

21 Purchased goods for cash 800,000/-

22 Sold goods for cash 1,400,000/-

26 Paid rent            300,000/-

 (i) Record the above transaction in cash account

 (ii) Extract a trial balance

PRESIDENT’S OFFICE 
REGIONAL ADMINISTRATION AND LOCAL GOVERNMENT AUTHORITY 

BASIC MATHEMATICS EXAMINATIONS- AUG 2021
FORM THREE

TIME: 3:00HRS

INSTRUCTIONS

•         This paper consists of Sections A and II
•         Answer ALL questions in both sections
•         Mathematical table and graph paper may be used unless otherwise stated.
•         Write your Examination Number on every page of your answer sheet(s) provided.

SECTION A (60MARKS)

1.   (a) A school trip of 32 people went to a tour which costs a transport fee of 580/= each person. What was the approximate total transport cost?

b) Rajabu is making some small metal rods.He has three peaces of metals of length 432cm, 648cm,540cm.What is the largest length of a rod he can make if the rods have the same length and no metal is wasted.

2.   (a) Without using mathematical table simplify √63+√72 

√32+√28

(b)Find the value of x in log( - 1) + 2 = log(3 + 2) + log 25

3.   a)(i) List all the subsets of the following set A = {3, 6, 8} 
   ( ii) Work out the number of subsets in a set containing 6 elements.

b) In a class of 45 students, 19 study commerce but not physics, 16 study physics but not commerce and 3 studies neither commerce nor physics, find the number of students who study (i) Physics or commerce (ii) Both subjects

4.   (a) The line joining (2,-3) and (k,5) has a gradient -2, find k

b) ∆PQR is such that PQPR and PQ:QR is equal to 3:4.If the perimeter of the triangle is 60cm. Calculate the value of QR.

5.   (a) Calculate to 1 decimal place the area of an equilateral triangle of side 10cm.

(b) In the figure below find MK and MP

6.        (a) A man is paid 6000/= for 8 hours work (i) What is his rate of pay? (ii) At this rate how long must he work to receive 30,000/=

b) In working for 10 hours a day, 12 men can do a certain piece of work in 6 days. For how many hours a day must 20 men work in order to do the same amount of work in 14 days?

7.     a) By selling an article at Tsh 225,000 shopkeeper makes a loss of 10%.At what price must the shopkeeper sell the article in order to get a profit of 10%?

b) Find the total amount of money received if Tsh 800,000 is deposited in a bank at the rate of 9% compounded semi anually in one year.

(b) Given the angles A^̂ C = 2x+30° and C^̂ D = (x+15)°. If two angles are complementary find the value of x and the size of each angle

9.        (a) Given the right angled triangle below with sides marked x , y , z ,

Prove that cos² + sin²=1

b) A pole 7.45m high casts a Shadow 4.05m long on horizontal ground. Find the angle of elevation of the sun.

10. (a) Expand the expression (ax + c) (bx – d)

(b) A boy bought some packets of biscuits for 120/=.If the biscuits had been 3/= a packet cheaper he would have received 2 more packets for his money. How many packets did he buy?

SECTION B (40MARKS)

11. In a mathematics Examination the following marks were obtained:

27 57 57 40 70 48 59 60 42 44 47 44 44 59 35 48 43 52 36 48 Group the marks in class interval of 20-29, 30 -39 , . . . Then

1. Construct the frequency distribution table
2. Calculate the mean marks by using assumed mean method (Take A =44.5)
3. Calculate the mode
4. Draw the cumulative frequency curve then estimate the median

12. (a)Four positive numbers are consecutive elements of geometric progression (G.P).The product of the first and the third number is 36 while the product of the second and fourth number is 324. Find the sum of nine terms of the G.P

(b) The second , fourth and eighth terms of an Arithmetic Progression form a Geometric progression and the sum of the third and fifth terms of AP is 20. Find the first four terms of the geometric progression.

13. (a) A function is defined as f(x ) = √4 - ² find (i) Domain and Range of f(x) (b) Sketch the graph of function f(x)= { + 3 ℎ  < 1

² ℎ > 1

Hence (i) state its domain and range (ii) Find f(6) , f(-2) (iii) State whether the graph is one to one function ?

14. The relation is defined as R={(, ): ≥ —4, 3 - 4 ≤ 12  5 + 4 ≤ 20}

1. Draw the graph of R
2.  State the domain and range of R