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

041 BASIC MATHEMATICS

(For Both School and Private Candidates)

Time: 3 Hours Tuesday, 03^rdNovember 2015 a.m.

 Instructions

 1.  This paper consists of sections A and B.

2.  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.

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

4.  Mathematical tables may be used.

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

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

SECTION A (60 Marks)

Answer all questions in this section.

1.  (a) If p = 6.4 × 10⁴ and q = 3.2 × 10⁵ , find the values of:

(i)  p × q ,

(ii)  p + q .

Write the answers in standard form.

1.(b) Evaluate

Using mathematical tables and write the answer correctly to 3 significant figures

2.  (a) Solve for x in the equation 4^-2x × 8² = 4 × 16^x .

(b) Find the value of log900 given that log3 = 0.4771 .

 3.(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 and its two subsets A and B.

Write down the elements of:

(i)  A'

(ii)  B',

(iii)  A UB ,

(iv)  A'U B'.

(c) Verify that n(A U B) = n(A) + n(B) - n(A n B) where A and B are the sets given in part 3(b).

4.  (a) Given vectors (i) the vector a = 3i + 2 j , b = 8i +­ 3j and c = 2i + 4 j find:

(ii) 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 ­x-axis at 3 and the y­-axis at ­4.

5.  (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 cm² , find the area and the height of the larger triangle.

(b) In the figure below CB = BD = DA and angle ACD = x .

(i)  Show that angle ADE = 3x ,

(ii)  Calculate the measure of angle CDA if x = 39°.

6. (a) The variable v 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°C and t = 9cm , find the value of t when T_i = 36°C and T_o = 18°C

7.  (a) A shopkeeper makes a 20% profit by selling a radio for sh. 480,000.

(i)  Find the ratio of the buying price to the selling price.

(ii)  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.

8.  (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.

9.(a) Find value of 

Without using mathematical tables

(b)  Calculate the angles of a triangle which has sides 4m, 5m and 7m.

10.  (a) Factorize completely 2x² + x - 10 by splitting the middle term.

10(b) Solve the equation 

SECTION B (40 Marks)

Answer four (4) questions from this section.

11.  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.

12.  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.

(a)  Prepare a frequency distribution table using the class intervals: 24 ­ 29, 30 ­ 35 etc.

(b)  Draw the histogram.

(c)  Draw the cumulative frequency curve and use it to estimate the median.

(d)  Find the mean mark.

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

(b)  A rectangular box with top WXY Z and base ABCD has

AB = 9cm, BC = 12cm and WA = 3cm.

Calculate:

(i)  The length of AC ,

(ii)  The angle between WC and AC .

(c)  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.

14.  The following trial balance was extracted from the businessman books’ of Chericho Ramaji, at

31^st December 2006.

Prepare Trading, Profit and Loss account for the year ended 31^st December 2006.

 15.(a) Given matrices

And

Such that

Find elements of matrix P

(b) Determine the matix A from the equation

(c)  Given 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),

(ii)  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.

16.  (a) The function f is defined as follows:

(i)  Sketch the graph of f(x),

(ii)  Determine the domain and range of f(x) .

(b)  Jeremia has two shirts, a white one and a blue one. He also has 3 trousers, a black, green and a yellow one. What is the probability of Jeremia putting on a white shirt and a black trouser?

(c)  If a number is to be chosen at random from the integers 1, 2, 3, ..., 11, 12. find the probability that:

(i)  It is an even number,

(ii)  It is divisible by 3.

(d)  If in part 16(c) above, E₁ is the set of even numbers and E₂ the set of numbers that are divisible by 3, show whether E₁ and E₂ are mutually exclusive events.