A-Level H2 Math
NJC Paper 1
๐KEY FORMULAE (MF27)
Paul is helping his friends to convert their foreign currencies back to Singapore Dollars. The amounts of foreign currencies converted, and the total amount received in Singapore Dollars are shown in the following table.
| Alex | Nicholas | Palmer | Maybelline | |
|---|---|---|---|---|
| USD | 150 | 250 | 425 | a |
| JPY | 5500 | 9500 | 1000 | 0 |
| CNY | 1000 | 2200 | 2000 | 1200 |
| Total SGD | 419.30 | 797.20 | 913.10 | 568.40 |
However, he has forgotten the amount of US Dollar that Maybelline has passed to him. Assuming that, for each foreign currency, the exchange rate quoted for each of the friends is the same, calculate the value of a.
Do not use a calculator in answering this question.
(i) Find the values of z and w that satisfy the equations
expressing your answers in the form \(c + d\mathrm{i}\), where \(c, d \in \mathbb{R}\). [4]
(ii) Points W and Z represent w and z found in part (i). Find \(\frac{w}{z}\) in the form \(p + \mathrm{i}q\), where \(p, q \in \mathbb{R}\). Hence, state the transformation that maps line segment OZ onto line segment OW. [2]
(i) On the same axes, sketch the graphs of \(y = -\dfrac{b}{x - a}\) and \(y = \left|x - a\right|\), where a and b are positive constants and \(a > b > 1\).
State, in terms of a and b, the coordinates of the points where the curves cross the x- and y-axes. [3]
(ii) Hence or otherwise, solve the inequality \(-\dfrac{b}{x - a} < \left|x - a\right|\). [4]
A curve has equation \(y = f(x)\), where \(f(x) = 1 - \sqrt{q^2 - x^2}\) for \(q > 1\). State the shape of \(y = f(x)\). [1]
(i) Sketch the curve \(y = \dfrac{1}{f(x)}\), giving the equations of any asymptotes and the coordinates of the end-points. [3]
(ii) Describe the transformations that map the graph of \(y = f(x)\) to \(y = -\sqrt{1 - x^2}\). [3]
The curve C has equation
where k and p are constants.
It is given that C has a vertical asymptote \(x = 2\) and a stationary point at \(x = -4\).
(i) Find the equation of the oblique asymptote of C. [5]
(ii) Sketch C, clearly labelling the equations of asymptotes and the coordinates of stationary points. [3]
(a) An infinite geometric series S has first term 1 and non-zero common ratio r. It is given that the sum to infinity of S is equal to the square of the sum of the first three terms of S.
(i) Show that r satisfies the equation \(r^4 + ar^3 + br^2 + cr + d = 0\), where a, b, c and d are constants to be determined. [3]
(ii) Find the possible values of r. [1]
(b) An arithmetic progression with 4n terms has first term 7 and common difference d. Every 4th term is removed. Find the sum of the remaining terms in terms of n and d. [4]
The curve C is defined parametrically by \(x = e^{4t}\), \(y = t^2\), where \(t \geq 0\).
(i) Find the Cartesian equation of C. [2]
(ii) The tangent at point P has the steepest gradient. Find the exact coordinates of P.
[You do not need to show that the gradient at P is the steepest.] [3]
(iii) Sketch C, indicating the coordinates of P and the point where C crosses the axes clearly. [2]
In this question, you may use expansions from the List of Formulae and Results (MF27).
It is given that \(a > 0\).
(i) Find, in terms of a, the series expansion of \(\dfrac{a}{a - x} - 1\), in ascending powers of x, up to and including the term in \(x^2\). State, in terms of a, the range of x for which the expansion is valid. [4]
(ii) Hence, find the Maclaurin expansion of \(e^{\frac{a}{a-x}-1}\) in ascending powers of x, up to and including the term in \(x^2\). [2]
(iii) Use the expansion in part (ii) to approximate \(\displaystyle\int_0^{\frac{a}{2}} e^{\frac{a}{a-x}-1} \, dx\). Explain why this approximation is an under-estimation. [4]
(a) Find \(\displaystyle\int \cos(3\ln x)\, dx\). [4]
(b) Let I be the indefinite integral \(\displaystyle\int \frac{P(x)}{1 - \sqrt{x}}\, dx\), where 0 < x < 1 and P(x) is a polynomial in x.
(i) Find I when P(x) = 1 โ x. [2]
(ii) By using the substitution \(u = 1 - \sqrt{x}\), find I when P(x) = 1. [3]
Hence find I when P(x) = x. [2]
A sequence of numbers \(u_1, u_2, u_3, \ldots\) has a sum \(S_n = \sum_{r=1}^{n} u_r\). It is given that \(S_n = A - \dfrac{2}{(n+1)!}\), where A is a non-zero constant.
(i) Find the value of A if \(u_1 = 1\). [1]
(ii) Show that \(u_n = \dfrac{2}{(n+1)\left[(n-1)!\right]}\) for \(n \geq 1\). [3]
(iii) Find a recurrence relation in the form \(u_{n+1} = \left[f(n)\right] u_n\). [2]
(iv) Explain why \(S_n\) converges as \(n \to \infty\). [1]
(v) Hence, find the least value of m such that the sum of the infinite series \(u_m + u_{m+1} + u_{m+2} + \cdots\) does not exceed \(10^{-10}\). [3]
The function f is defined by
where a is a positive constant.
(i) Find the range of f. [3]
(ii) Find \(f^{-1}(x)\) and state its domain. [3]
The function g is defined by \(g(x) = 3 + e^x\), for \(x \in \mathbb{R}\).
(iii) Show that fg exists. [1]
(iv) Find the exact value of k for which \(fg(k) = \dfrac{a}{7}\). [3]
A model of a triangular canopy that provides shade outdoors is shown in Figure 1 below.
With point O taken as the origin, the canopy ABC is held taut using three vertical columns given by OA, DB and EC. The unit vectors i and k are defined with i along OD, k along OA, and unit vector j is perpendicular to both. The bases of the vertical columns are anchored to the horizontal ground ODE, which is perpendicular to OA.
(i) State the cartesian equation of plane OAB. [1]
Points A and B have position vectors given by \(\vec{OA} = 3\mathbf{k}\) and \(\vec{OB} = 10\mathbf{i} + 4\mathbf{k}\). A marking, given by point M, is to be placed on the line segment AB.
(ii) Find \(\vec{OM}\) in terms of a parameter ฮป, stating the range of ฮป. [2]
Point C has position vector given by \(\vec{OC} = 6\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}\). The plan is to lay cables to connect M to C and then C to E. All cables are laid in straight lines and have negligible thickness.
(iii) Explain why it is not possible for angle MCE to be 90ยฐ. [1]
With reference to Figure 2 below, points F and G lie on lines AB and OD respectively. Party streamers, with negligible thickness, are laid in straight lines to connect E, C, F and G. The quadrilateral formed lies on a plane with cartesian equation \(x - y = 2\).
(iv) Show that quadrilateral ECFG is a trapezium. [3]
(v) Find the shortest distance between F and the line CE. Hence or otherwise, find the area enclosed by the streamers. [5]
Figure 1
Figure 2
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Syllabus alignment based on the SEAB H2 Mathematics 9758 syllabus (2026).
Exam paper: NJC Senior High 2 Preliminary Examination, 15 September 2025.