The United Republic of Tanzania

National Examinations Council

Advanced Certificate of Secondary Education Examination

141 Basic Applied Mathematics – 2024

Instructions:

• This paper consists of ten (10) questions.
• Answer all the questions. Each question carries ten (10) marks.
• All work done in answering each question must be shown clearly.
• Non-programmable calculators and NECTA mathematical tables may be used.
• All writing must be in blue or black ink, except drawings which must be in pencil.
• Communication devices and unauthorized materials are not allowed in the examination room.
• Write your Examination Number on every page of your answer booklet(s).

Question 1

1. Compute the value of 1000 × (1 + 0.12/8760)^4×8760 correct to 2 decimal places.
2. Compute the value of sin(x)/x correct to five significant figures.
3. Evaluate the integral of x dx from 0 to 1 correct to two decimal places.

Question 2

1. Given the function g(x) = 1 / (x - 3) + 2, find the vertical and horizontal asymptotes.
2. Given f(x) =
   • 1 if x > 0
   • 0 if x = 0
   1. Sketch the graph of f(x).
   2. State the domain and range of f(x).

Question 3

1. Solve the system: x + 2y = 4 and x² + 3xy = 10.
2. Evaluate √16.
3. The time t to complete a project varies inversely with the number of employees e. If 3 people complete the project in 10 days, how many days will 5 people take?

Question 4

1. Use first principles to find the derivative of f(x) = 3x² - 2.
2. Given y³ + x³ - 3xy = 4, find dy/dx.
3. Find the slope of the curve y³ = 64x at x = -1.

Question 5

1. Evaluate ∫(2x - 1)(4x - 4x) dx.
2. Find the area enclosed between y = 4 - x² and y = x² - 2x.

Question 6

Given:

• Marks: 1–5, 6–10, 11–15, 16–20, 21–25
• Frequency: 9, 8, 13, 12, 8

1. Represent the data using a histogram.
2. Calculate the variance correct to four significant figures.

Question 7

1. If outcomes a, b, c, and d have probabilities 0.1, 0.3, 0.5, and 0.1 respectively, and A = {a,b,d}, B = {b,c,d}, find P(A ∩ B).
2. A four-digit number is formed from 1, 2, 3, 5 with no repetition. Find the probability it is divisible by 5.
3. A machine breaks down with probability 0.1 in a month. If two machines are installed, find the probability that only one breaks down.

Question 8

1. If cos A = -1/2, evaluate cos(A/2) in surd form.
2. Given sin θ - 2sin θ cos θ = 0 and 0° ≤ θ ≤ 180°, find the values of θ.
3. Calculate side EF of triangle EFG with angle E = 75°, angle F = 40°, and side EG = 14 cm.

Question 9

1. Evaluate ∫ dx / (7x + 2).
2. The population N(t) of bacteria over time is graphed. Formulate the equation for N(t).

Question 10

1. Given matrices A and B, evaluate 3A - 2B where A = [[3,4],[2,5]] and B = [[1,2],[2,2]].
2. A manufacturer produces nuts and bolts with different machine time requirements. Maximize profit given constraints and profits for nuts (250,000 Tsh) and bolts (100,000 Tsh).

NECTA 2024 - Basic Applied Mathematics

Full Solutions for Questions 1 to 5

NECTA 2024: Basic Mathematics - Questions 1 to 10 with Smart Solutions