National Examinations Council

NECO Further Mathematics · Paper II — Essay

Written answers, marked against a rubric.

8 exam-style questions, written to the National Examinations Council format and free to read. The answers are here too — what they do not tell you is how many you would get right with a clock running.

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1. A particle moves so that its displacement is s = 2t³ − 3t² + 4t metres after t seconds. (a) Find expressions for its velocity and acceleration. [8 marks] (b) Find its velocity and acceleration when t = 2 s. [8 marks] (c) State the time at which the acceleration is zero. [4 marks]
2. Given matrices A = (2 1; 3 4) and B = (1 0; 2 5): (a) Find AB. [7 marks] (b) Find the determinant of A and hence its inverse. [10 marks] (c) State the condition for a matrix to have no inverse. [3 marks]
3. The data 4, 7, 9, 11, 14, 16, 18 are given. (a) Calculate the mean. [4 marks] (b) Calculate the variance and the standard deviation, correct to two decimal places. [12 marks] (c) State the effect on the standard deviation if every value is increased by 3. [4 marks]
4. (a) The third term of an arithmetic progression is 11 and the seventh is 27. Find the first term and the common difference. [10 marks] (b) Find the sum of the first 20 terms. [6 marks] (c) State the condition for an arithmetic progression to be increasing. [4 marks]
5. Given p = 5i − 2j and q = i + 3j: (a) Find 2p − q. [6 marks] (b) Find |p| and |q|, correct to two decimal places. [8 marks] (c) Find the angle between p and q, correct to the nearest degree. [6 marks]
6. (a) Given f(x) = x³ + 2x² − 5x − 6, show that (x + 1) is a factor. [6 marks] (b) Factorise f(x) completely. [8 marks] (c) State the three roots of f(x) = 0. [6 marks]

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NECO Further Mathematics · Paper II — Essay

1. A particle moves so that its displacement is s = 2t³ − 3t² + 4t metres after t seconds. (a) Find expressions for its velocity and acceleration. [8 marks] (b) Find its velocity and acceleration when t = 2 s. [8 marks] (c) State the time at which the acceleration is zero. [4 marks]

2. Given matrices A = (2 1; 3 4) and B = (1 0; 2 5): (a) Find AB. [7 marks] (b) Find the determinant of A and hence its inverse. [10 marks] (c) State the condition for a matrix to have no inverse. [3 marks]

3. The data 4, 7, 9, 11, 14, 16, 18 are given. (a) Calculate the mean. [4 marks] (b) Calculate the variance and the standard deviation, correct to two decimal places. [12 marks] (c) State the effect on the standard deviation if every value is increased by 3. [4 marks]

4. (a) The third term of an arithmetic progression is 11 and the seventh is 27. Find the first term and the common difference. [10 marks] (b) Find the sum of the first 20 terms. [6 marks] (c) State the condition for an arithmetic progression to be increasing. [4 marks]

5. Given p = 5i − 2j and q = i + 3j: (a) Find 2p − q. [6 marks] (b) Find |p| and |q|, correct to two decimal places. [8 marks] (c) Find the angle between p and q, correct to the nearest degree. [6 marks]

6. (a) Given f(x) = x³ + 2x² − 5x − 6, show that (x + 1) is a factor. [6 marks] (b) Factorise f(x) completely. [8 marks] (c) State the three roots of f(x) = 0. [6 marks]

Reading these is not the same as sitting them.

7. (a) Integrate (4x³ − 6x + 2) with respect to x. [6 marks] (b) Evaluate the definite integral of the same expression between x = 0 and x = 2. [8 marks] (c) State why a constant of integration is not needed for a definite integral. [6 marks]

8. (a) Differentiate y = (2x + 1)⁴ with respect to x. [6 marks] (b) Find the equation of the tangent to y = x² − 4x + 3 at the point where x = 3. [10 marks] (c) State what the gradient of the tangent represents. [4 marks]