2022 H2 Maths TMJC P1.pdf

Tampines Meridian Junior College 2022 JC2 Preliminary Examination H2 Mathematics: The exam consists of 10 questions, with a total of 100 marks, and candidates have 3 hours to complete it.

1. (i) Express 3/2 + 5/12 + 1/2 - 1/4 - 1/3 - 1/6 as a single fraction. (1 mark)
(ii) Find the sum of the series 1/4 + 3/4 + 5/4 + ... + (2n-1)/4. (3 marks)
(iii) Use the result from (ii) to find the sum of the series 3/4 - 5/4 - 7/4 - ... - (2n+1)/4. (2 marks)

2. (a) Find d(1 - sin(x))/dx. (2 marks)
(b) Find d(2x - 4)/(x^2 - 2x + 1). (3 marks)

3. The curve C is defined by parametric equations x = ae^(-2t) and y = e^(-t) for t ≥ 0, where a is a positive constant. Find the exact area enclosed by C, the axes, and the line x = 8. (5 marks)

4. Sketch the curves y = x^2 - 5x + 3 and y = -x^3, labeling the axial intercepts. (2 marks)
Hence, solve the inequality x^2 - 5x + 3 < -x^3 without using a calculator. (4 marks)

5. Given ln(sin(y)) = kx, where k is a non-zero constant, show that d^2y/dx^2 + (k^2y)/2 = 0. (5 marks)
Hence, obtain the expansion of y in ascending powers of x up to the term in 2x. (5 marks)
Using the standard series, verify the result and determine the coefficient of 3x. (4 marks)

6. (a) A cone with radius 4 cm and height 8 cm is being filled with water at a constant rate of 1.5 cm^3/s. Find the exact rate at which the water level is decreasing when the depth of the water is 2 cm. (4 marks)

(b) A trapezium ABCD has height 5 cm, with DA + AB + BC = 20 cm. Points M and N are the feet of the perpendiculars from A and B to line DC, respectively, with ∠DAM = ∠CBN = θ, where 0 < θ < π/2.
(i) Show that the area A of trapezium ABCD is 100 - 50sec(θ) + 25tan(θ). (3 marks)
(ii) Hence, find the maximum area of trapezium ABCD by differentiation. (4 marks)

7. (a) The function f is defined as f(x) = (ax - 1)/(x - 1) for x ≠ 1, where a > 1.
(i) Sketch the graph of f, indicating the coordinates of the turning points and the equations of the asymptotes. (3 marks)
(ii) Explain why f does not have an inverse. (1 mark)
The function g is defined as g(x) = (x - 1)/(x + 1) for x ≠ -1, x ≠ 1.
(iii) Determine whether gf exists, with a reason. (2 marks)

(b) A curve C has the equation y = 12e^(-2x) for x > 0. The curve undergoes transformations: A (stretch by factor 1/2 parallel to the x-axis), B (translate by 1 unit in the negative y-direction), and C (reflection in the line y = x). Find the equation of the new curve in the form q(y) = x and state the domain of q. (5 marks)

8. The plane P1 has the cartesian equation x + 3z = 1. The plane P2 is perpendicular to P1 and contains the line L1 with equation x + 2y + 3z = 5.
(i) Show that the cartesian equation of P2 is 4x - y + z = 1. (2 marks)
(ii) Find a vector equation of the line L2, given that P1 and P2 intersect at L2. (2 marks)
(iii) The point B(0, 4, 3) is on P1, and the perpendicular distance from B to P2 is k. Find the position vector of the foot of the perpendicular from B to P2 and deduce the value of k. (4 marks)
(iv) Hence, find the vector equations of the lines in P1 such that the perpendicular distance from each line to P2 is k. (3 marks)

9. (a) The complex number z is given by z = x + iy, where x and y are non-zero real numbers. Given z^(-1) = 1, find the possible values of z for which z^2 is real. (6 marks)

(b) Without using a calculator, find the roots of the equation z^2 + 33z + 56i = 0, expressing the answer in cartesian form x + iy. (4 marks)
Hence, find the roots of the equation w^2 + 33w + 56i = 0 in cartesian form. (2 marks)

10. A squirrel falls vertically from a tall tree, with the distance x meters fallen from the tree after t seconds given by the differential equation d^2x/dt^2 + 0.1dx/dt + 10 = 0.
(i) By substituting dy/dt = dx/dt, show that the differential equation can be written as dy/dt = 10 - 0.1y. (1 mark)
(ii) Find y in terms of t and hence find x in terms of t. (8 marks)
(iii) How far has the squirrel fallen after 2 seconds? (1 mark)
(iv) Find the terminal velocity of the falling squirrel. (2 marks)