**THE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL CERTIFICATE OF SECONDARY EDUCATION EXAMINATION**

**041 BASIC MATHEMATICS**

(For School Candidates Only)

__Time: 3 Hours Tuesday, 05__^{th}*November 2013 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. Each question in section A carries **six (6)** marks while each question in section B carries **ten (10)** marks.

3. All necessary working and answers for each question attempted must be shown clearly.

4. Mathematical tables may be used.

5. The following constants may be used:

(a) The radius of the earth *R* = 6370*km*

(b) π =

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

7. Write your **Examination Number** on every page of your answer booklet(s).

**SECTION A (60 Marks)**

Answer **all** questions in this section.

1. (a) A shopkeeper sold 292 t-shirts at the price of sh. 6950. Estimate how much money she got.

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(b) Express in the form of a/_{b} where *a* and *b* are integers such that *b* =/ 0 .

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2. (a) If log*y* + 2log(3*x* + 1) = 1, express *y* in terms of *x* .

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(b) Simplify by rationalizing the denominator.

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3. (a) If *m* * *n* = *m* + 4*n*, find *x* given that 3 * (*x* * 1) = 27.

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(b) There are 48 men at a meeting of whom 24 are teachers, 36 are parents and 16 are both teachers and parents. By using a Venn diagram, find the number of men who are neither teachers nor parents.

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4. (a) Find the direction cosines of *C* = 9*i* + 12*j*, hence show that the sum of the squares of these direction cosines is one.

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(b) Find the equation of the line through the points (4,6) and the midpoint of (2,4) and (10,4).

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5. (a) The length of two sides of a triangle are 16*cm* and 20*cm*. Find the area of the triangle if the included angle is 30° .

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(b) In the figure below, calculate the length *BD*:

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6. (a) A bus travels 240*km* using 16 litres of diesel. How many litres of diesel are needed to drive 90*km*?

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(b) If *y*^{2} varies directly to *x* − 1 and inversely to *x* + *d* and if *x* = 2 , *d* = 4 for *y* = 1, then find *x* when *y* = 2 and *d* = 1.

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7. (a) A radio is bought for sh 400,000 and sold for sh 500,000. Find:

(i) The profit made

(ii) The percentage profit.

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(b) Find the time in which sh 300,000 will earn an interest of sh 60,000 if the interest rate is 10% per annum.

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8. (a) The first term of an arithmetic progression is 12 and the common difference is 10. Find the *n*^{th} term.

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(b) Find the amount of money accumulated at the end of 2 years after investing 500,000 shillings at a compound interest rate of 10% annually.

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9. (a) A ladder leans against a wall. If the ladder reaches 12*m* up the wall and its foot is 9*m* from the base of the wall, find the length of the ladder.

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(b) Given that *A* and *B* are complementary angles and sin*A* = , find tan*B* (Leave your answer as an improper fraction).

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10. (a) What must be added to *x*^{2} + 8*x* to make the expression a perfect square?

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(b) Find two consecutive odd numbers whose product is 195.

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**SECTION B (40 Marks)**

Answer any **four (4)** questions from this section.

11. (a) Solve by graphical method the following system of simultaneous equations:

4*x* + *y* = 6

5*x* + 2*y* = 9

(b) A farm is to be planted with sorghum and maize while observing the following constraints:

| **Sorghum** | **Maize** | **Maximum total** |

Days labour per hectare | 4 | 2 | 20 |

Labour cost per hectare | 1400 | 1200 | 8400 |

Cost of fertilizer per hectare (shs) | 600 | 800 | 4800 |

If sorghum yields a profit of 800,000 shillings per hectare while maize yields 600,000 shillings per hectare, how many hectares should be planted with each crop for maximum profit?

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12. Carefully study the frequency distribution table which shows the marks of 100 students in a Physics examination.

**Marks** | 41 - 50 | 51 - 60 | 61 - 70 | 71 - 80 | 81 - 90 | 91 - 100 |

**Number of Students** | 10 | 22 | 34 | 25 | 7 | 2 |

Calculate

(a) the mean given the assumed mean is 75.5,

(b) the median in two decimal places,

(c) the mode in two decimal places.

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13. (a) The figure below shows a rectangular prism in which *AB* = 10*cm* , *BC* = 8*cm* and Calculate the length of *DF* (Leave your answer in surd form)

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(b) If *O* is the centre of a circle given below, find the value of *x*.

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(c) Calculate the circumference of a small circle, in kilometres, along the parallel of latitude 30°*N* (Leave your answer in surd form).

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14. Mr. Chapakazi commenced business on 1^{st} June, 2011 with capital in cash 5,500,000/=

June 2 Bought goods by Cash 2,000,000/=

3 Bought furniture for Cash 600,000/=

7 Sold goods for Cash 2,000,000/=

11 Cash purchases 700,000/=

13 Paid Rent for Cash 2,000,000/=

16 Cash Sales 950,000/=

20 Paid Transport for Cash 80,000/=

22 Sold goods for cash 250,000/=

26 Paid salaries for Cash 160,000=

Record the above transactions in the cash account, balance them and extract the Trial Balance.

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15. (a) Given

find *A*^{−1}.

(b) Use the results obtained in part (a) to find the point of intersection of the following system of simultaneous equations:

2*x* + 6*y* = 22

4*x* + 8*y* = 32

(c) Find the image of the point obtained in part (b) above under a rotation 90° anticlockwise.

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16. (a) A die and a coin are tossed together. What is the probability of getting a tail and an even number?

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(b) Draw the graph of the inverse of *R* = {(*x*,*y*) : *y* ≥ 0 *and* *y* ≤ *x*}. Find its domain and range.

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(c) Without using a table of values, draw the graph of *y* = *x*^{2} − 4*x* + 2 and use it to solve the equation *x*^{2} − 4*x* = 5.

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