**THE UNITED REPUBLIC OF TANZANIA ****NATIONAL EXAMINATIONS COUNCIL **

**CERTIFICATE OF SECONDARY EDUCATION EXAMINATION**

**041** **BASIC MATHEMATICS**

(For School Candidates Only)

__Time: 3 Hours____ __**Monday, 4**^{th}* October 2010 a.m.*

**Instructions**

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 ****shown clearly.**

4. Mathematical tables may be used unless otherwise stated.

5. Calculators and cellular phones are **not **allowed in the examination room.

8. You are advised to spend not more than **two (2) **hours on section **A **and the remaining time on section B.

9. 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
- three (3) decimal places.

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(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)*

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2. (a) Evaluate without using mathematical tables 2 log 5 + log 36 — log 9.

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(b) Simplify

*(6 marks)*

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3. (a) Given that A= {x : 0 ≤ x ≤8}

*B *= {x : 3 ≤ x ≤8}

where x is an integer, in the same form, represent in a Venn diagram

- A u
*B* - A n
*B*

and hence find the elements in each set.

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(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)*

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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^{°} to the positive direction of the x-axis.

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(b) The position vectors of the points *A, B *and *C *are 4i - 3j , i +3j and -5i + *j *respectively. Find the vectors *AB, BC *and *AC* hence verify that *AB + BC = AC** ***(6 marks)**

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5. (a) The volume of two similar cylinders is 125 cm^{3} and 512 cm^{3}. If the radius of the larger cylinder is 8cm, find the radius of the smaller cylinder.

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(b) In the diagram below, show that

*(6 marks)*

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6. (a) Juma bought motor vehicle spare parts from Japan worth 5,900,000 Japanese Yen. When he arrived 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.

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(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)**

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7. (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?

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(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:

- The salary of this worker.
- The amount of money which he/she spent on food.
**(6 marks)**

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8. (a) Find the general term and hence the 30^{th} term of the sequence

1, -2, 4, -8, ......

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(b) Given the series 100 + 92 + 84 +.....

Find

- the 20
^{th} term - the sum of the first 20 terms. (6 marks)

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9. (a) If tan A = ¾ and A is acute, find cos A, sin A and hence verify the identity cos^{2} A + sin^{2} A =1

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(b)

Given the right angled triangle above whose sides are measured in centimeter determine:

- the value of
*x* - the area of the triangle
*(6 marks)*

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10. (a) Factorize each of the following expressions:

*3a*^{2}c -5a ^{2}d - 3b^{2} c + 5b^{2}*d* *3(2 - y*^{2})-17y

(b) Find the value of y which satisfies the equation *3(2 - y*^{2}) - 17y = 0 **(6 marks)**

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**SECTION B ( 40 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:

*x ≥ *0

y ≥ 0

2x+y ≤ 6

x+2y ≤ 6

(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)*

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12. The data below represent masses in kg of 36 men.

51 | 61 | 60 | 70 | 75 | 71 | 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 |

- 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.
- 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)*

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13. (a) Below is a circle with centre **0 **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:

- arc length AB
- area of sector AOB.

(b) Find (i) the length of arc *AB*

(ii) the area of the sector

If θ=57^{0} and r=5.4cm (Use π =22/7)

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14. From 1^{st} January to 29^{th} 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 | 320,000.00 |

You are required to:

- prepare the trial balance
- open capital and cash account.

**N.B **All payments and receipts were made in cash. *(10 marks)*

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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.*

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(b) For what values of *n *will the matrix

be non-singular? *(10 **marks)*

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16. (a) If *f (x) = -2x + *3 find *f *^{-1} (3)

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(b) Draw the graph of** ****f (x) = | x —1| **for -4 ≤ x ≤ 4

(c) State the domain and range of* ***f (x)=|x-1|**

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(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.
- either Rose or Juma will be selected.
*(10 marks)*

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