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

041 BASIC MATHEMATICS

Time: 3hours  Year: 2022

INSTRUCTIONS

1. This paper consists of section A and B with a total of fourteen (14) questions
2. Answer all questions
3. Each question in section A carried six (06) marks while each question in section B carried ten(10) marks
4. All necessary working and answers for each question must be shown clearly
5. NECTA mathematical tables and non-programmable calculators may be used.
6. All communication devices and any unauthorized materials 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)Find the percentage of numbers which are multiples of 5 from the set [1, 2, 3, 4, . . . . . . up to 52] . Write the answer correct to one decimal place

(b)(i)Arrange the following fractions in ascending order:

(ii) Simplify the expression

, hence write the answer in standard form.

2. (a)Find the value of x if 8^x-1 = 16

(b)(i)Simplify the expression

(ii) Rationalize the denominator of the expression

3. (a)(i)If P ={ all multiples of 5 less than 35} and Q={all numbers between 14 and 30} find P

(ii)In a village of 50 farmers, 25 grow cashew nut and 16 grow both cashew nut and maize. If 10 farmers grow neither cashew nut nor maize, find the number of famers who grow maize only. Do not use Venn diagram.

(b)A farmer was given three seeds to germinate in a nursery. The probability that a seed will germinate is . Using a tree diagram, find the probability that at least two seeds will germinate.

4.(a) If  a = (4, 3), b=(-4, 1) and c=(2, 5), determine which of the vectors a + 2b and 3a + c is longer then the other.

(b)If a line passing through the point (4, 2) is perpendicular to another line whose equation is 2x + 3y + 14 = 0, find the equation of the line.

5. (a)The sides of a triangle are 4cm, 5cm and 6cm. If the longest side of a similar triangle is 18cm, find the lengths of the other sides.

(b)(i)The perimeter of a regular hexagon inscribed in a circle is 72cm. Find radius of the circle

(ii)The area of a triangle ABC is 70cm². If AB= 14cm and AC=20cm,   find the angle BAC

6. (a)Anna walks 24km every day. Find, in metres, the distance she walks in 2 days.

(b)(i)A dealer sells mattresses whose cost price (C) is directly proportional to the selling price(s). If the selling price and the cost price of one mattress are Tsh. 20,000 and Tsh. 18,000, respectively, find the constant of proportionality

(ii) By using the answer obtained in part (b)(i), determine the equation that relates the cost price and the selling price.

7. (a)A damaged table that costs Tshs. 20,000 was sold at a loss of 15%. Find the loss made and the selling price.

(b)Extract a Trial Balance from the following Mabala’s cash account.

8. (a)The fifth and eleventh terms of an arithmetic progression are 8 and -34 respectively. Find the sum of the first ten terms

(b)A school wishes to invest Tshs. 100,000,000 in a bank which pays an interest rate of 2% compounded annually

1. Find the total amount of money that will be accumulated after two years.
2. Calculate the interest after two years

9. (a)(i)Find the value of  without using mathematical table

(ii)A rectangular garden ABCD is 400m long and 300m wide. The seedlings are to be planted along the diagonal BD at equal intervals of 1.25m as shown in the following figure.

Find the number of seedlings that were planted

(b)From the top of a tower which is 50m high, the angle of depression of a car parked on the ground is 30. How far is the car from the base of the tower? Leave the answer in surd form.

10. (a)Express the equation 2t^-10 - 3t^-5 + 1 = 0 in term of x where

(b)From the equation you obtained in part (a), find the value(s) of x that satisfy it by using the quadratic formula.

SECTION B (40 Marks)

Answer all questions in this section.

11. (a)Find the central angle (in degrees) made by an arc of length 22cm in a circle whose radius is 63cm. Use .

(b) in the following figure, prove that angles x and y are supplementary given that a and b are the angles at the centre of the circle.

(c)In the following figure, AC=8cm, BE=3cm and CE=4cm. find length of DE 

12. (a)A bus leaves town A (3°S, 39°E) at a constant speed of 40km/h. How many hours will the bus take to reach town B (12°S, 39°E)? Use π=3.14 and radius of the Earth, R= 6,400km.

(b)A box has a rectangular base UVXY with plane PQRS being vertically above UVXY as shown in the following figure.

If

1. The length of  and , Correct to one decimal place
2. The angle between the diagonal  and the base UVXY

13. (a) Find the values of x, y, z and w in the following matrix equation

.

(b)By using the matrix method, find the image of the point (3, -2) after a reflection in the line y= -x followed by another reflection in the line x=0

(c)A translation takes point (5, 5) to the point (-7, -7). If it takes point (x, y) to (-4, -4), find the values of x and y

14. (a)Given the function

 , find;

1. The domain and range
2. 

(b)Antony wishes to buy black shirts and white shirts. He intends to buy at most black shirts. A black shirt costs Tsh. 24,000 while a white shirt costs Tsh. 30,000 and he is planning to spend up to Tsh. 180,000 for buying shirts.

1. How many shirts of each kind should be bought so as to have maximum number of shirts?
2. Find the greatest number of shirts that should be bought