EXAMINATION 1

PREPARATION FOR THE CERTIFICATE OF SECONDARY EDUCATION EXAMINATION (CSEE) 2025

BASIC MATHEMATICS – FORM FOUR

Time: 3 Hours

INSTRUCTIONS:

1. This paper consists of fourteen (14) questions divided into sections A and B.
2. Answer all questions.
3. Show clearly all necessary working and answers.
4. Use NECTA mathematical tables and non-programmable calculators as needed.
5. Write in blue or black ink; diagrams must be drawn in pencil.

SECTION A: (60 Marks)

Answer all ten (10) questions. Each question carries 6 marks.

1. (a) Simplify: \( \frac{5}{6} + \frac{3}{4} - \frac{1}{3} \)
   (b) Write 0.000394 in standard form.
   (c) Solve: \( 3x - 5 = 2x + 7 \)
2. (a) Rationalize: \( \frac{5}{\sqrt{3} + 1} \)
   (b) Evaluate: \( \log_{10} 1000 \)
   (c) Simplify: \( 2^3 \times 4^2 \div 8 \)
3. (a) List the elements of \( A = \{x : x \text{ is a prime number less than 10} \} \)
   (b) Given \( P(A) = 0.4, P(B) = 0.5, P(A \cup B) = 0.7 \), find \( P(A \cap B) \)
   (c) Find \( A \cup B \), where \( A = \{1,3,5\}, B = \{2,3,4\} \)
4. (a) Find the gradient of the line joining points A(2, 3) and B(5, 9)
   (b) Write the equation of a line with gradient 2 passing through (0, -1)
   (c) Find the distance between (1, 2) and (4, 6)
5. (a) Find the area of a triangle with base 12 cm and height 5 cm
   (b) Calculate the perimeter of a rectangle with length 8 cm and width 3 cm
   (c) Two triangles are similar. One has sides 3 cm, 4 cm, 5 cm. The shortest side of the second is 6 cm. Find the longest side.
6. (a) Convert 4 hours 30 minutes to seconds
   (b) If 5 men complete a job in 12 days, how many men are needed to finish it in 4 days?
   (c) A car travels 120 km in 2 hours. What is its speed in m/s?
7. (a) A radio is bought for TZS 60,000 and sold at 15% profit. Find selling price.
   (b) Express the ratio 3 kg to 500 g
   (c) Profit is TZS 250,000 from a revenue of TZS 1,000,000. Find the profit percentage.
8. (a) Write the first four terms of the sequence defined by \( T_n = 3n - 1 \)
   (b) Find the sum of the first 10 natural numbers
   (c) Determine the 8th term of the arithmetic sequence: 2, 5, 8, ...
9. (a) Find \( \sin 30^\circ \)
   (b) Use Pythagoras' Theorem to find the hypotenuse of a right triangle with legs 6 cm and 8 cm
   (c) In triangle ABC, angle A = 90°, angle B = 60°. Find angle C
10. (a) Solve: \( x^2 - 5x + 6 = 0 \)
    (b) Factorize: \( x^2 - 9 \)
    (c) Expand: \( (x + 2)(x - 3) \)

SECTION B: (40 Marks)

Answer all four (4) questions. Each question carries 10 marks.

11. The marks scored by 40 students are grouped as follows:

    Marks	0–9	10–19	20–29	30–39	40–49	50–59
    Frequency	2	5	8	10	9	6

    (a) Draw a histogram and frequency polygon
    (b) Calculate the mean
12. A circle has radius 7 cm:
    (a) Find its circumference
    (b) Find its area
    (c) A sector of the circle has central angle 60°. Find the arc length and area of the sector
13. (a) A box has 3 red, 2 green, and 5 blue balls. One ball is picked at random. Find the probability that it is:
      (i) Red
      (ii) Not green
      (iii) Blue or red
    
    (b) If matrix \( A = \begin{bmatrix} 2 & 3 \\\\ 1 & 4 \end{bmatrix} \), find:
      (i) The determinant of A
      (ii) The inverse of A, if it exists
14. A farmer has 80 m of fencing to build a rectangular enclosure against a wall. The wall forms one side:
    (a) Let width be \( x \) m. Express the length in terms of \( x \)
    (b) Write a formula for the area \( A \) in terms of \( x \)
    (c) Find the value of \( x \) that gives maximum area
    (d) What is the maximum area?

END OF EXAMINATION