EXAMINATION 2

CERTIFICATE OF SECONDARY EDUCATION EXAMINATION (CSEE)

ADVANCED BASIC MATHEMATICS – FORM FOUR

Time: 3 Hours

INSTRUCTIONS:

1. This paper consists of fourteen (14) questions in Sections A and B.
2. Answer all questions.
3. Show clearly all necessary working and answers.
4. Use NECTA tables and non-programmable calculators.
5. All writing must be in blue/black ink, diagrams in pencil.

SECTION A: (60 Marks)

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

1. (a) Simplify: \( \left(\frac{3}{4} - \frac{2}{5}\right) \div \frac{5}{8} \)
   (b) Express 0.0000825 in standard form.
   (c) Solve for \( x \): \( \frac{2x - 1}{3} = \frac{x + 2}{4} \)
2. (a) Simplify: \( \log_{10}(5x) - \log_{10}(\sqrt{x}) \)
   (b) Rationalize: \( \frac{7}{2 - \sqrt{3}} \)
   (c) Simplify: \( \frac{16^2 \cdot 2^{-3}}{8^3} \)
3. (a) Given sets \( A = \{x \in \mathbb{N}: x < 10, x \text{ even} \} \), \( B = \{2, 3, 4, 5\} \), find \( A \cap B \)
   (b) Find \( P(A^c) \) if \( P(A) = 0.36 \)
   (c) A card is drawn from a well-shuffled deck of 52. Find the probability of drawing a king or a red card.
4. (a) Find equation of the perpendicular bisector of line joining (1,2) and (5,4)
   (b) The midpoint between A(2,x) and B(6,8) is (4,6). Find x
   (c) Determine the distance between points (−1, 2) and (3, −2)
5. (a) A triangle has sides of 5 cm, 6 cm, and 7 cm. Use Heron’s formula to find its area
   (b) A sector has radius 10 cm and central angle 150°. Find area and arc length
   (c) Find the length of diagonal of a square with area 98 cm²
6. (a) A tank is filled by pipe A in 4 hours and by pipe B in 6 hours. How long to fill it together?
   (b) Convert 120 km/h to m/s
   (c) If y varies inversely as x and y = 4 when x = 3, find x when y = 6
7. (a) A trader sells an item for TZS 115,000 after 15% loss. Find cost price
   (b) Find ratio of 1.2 m to 80 cm in simplest form
   (c) If profit = TZS 120,000 and cost = TZS 480,000, find profit percentage
8. (a) Find sum of first 12 terms of A.P: 5, 8, 11, ...
   (b) The nth term of a sequence is \( T_n = 2n^2 - 3n \). Find 5th term
   (c) Find the sum: \( \sum_{n=1}^{6} (2n - 1) \)
9. (a) Solve for x: \( \cos x = 0.5 \), where \( 0^\circ \le x \le 360^\circ \)
   (b) A ladder 10 m leans against wall, foot 6 m from wall. Find angle with ground
   (c) Find \( \tan \theta \) if \( \sin \theta = 3/5 \) and \( \theta \) is acute
10. (a) Solve: \( 2x^2 - 7x + 3 = 0 \)
    (b) Factorize: \( x^2 - 2x - 35 \)
    (c) Expand: \( (2x - 5)^2 - (x + 3)^2 \)

SECTION B: (40 Marks)

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

11. The scores of students in a test are:

    Score	10–19	20–29	30–39	40–49	50–59	60–69
    Frequency	2	5	8	12	9	4

    (a) Draw a histogram and frequency polygon
    (b) Calculate mean and modal class
12. A cone has height 12 cm and radius 5 cm:
    (a) Find slant height and surface area
    (b) Find volume of cone
    (c) If cone is melted and recast into sphere, find radius of sphere
13. Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), \( B = \begin{bmatrix} 2 & 0 \\ 1 & -1 \end{bmatrix} \):
    (a) Find AB and BA
    (b) Find determinant and inverse of A
    (c) Solve for \( x \) and \( y \): \( A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 11 \end{bmatrix} \)
14. A company wants to maximize profit from two products A and B. Each unit of A requires 3 hours labor and 2 kg raw. B requires 4 hours and 3 kg. 120 hours and 100 kg available.
    (a) Let x = units of A, y = units of B. Form inequalities
    (b) Draw feasible region
    (c) If profit is TZS 3,000 per A and TZS 4,000 per B, find max profit

END OF EXAMINATION 2