UJIAN PRA-PENTAKSIRAN PRESTASI STPM 2014
954/1
MATHEMATICS T (MATEMATIK T) PAPER 1 (KERTAS 1)
One and a half hours (Satu
SMK (P) SULTAN IBRAHIM JOHOR BAHRU Instruction to candidates:
Answer all questions in Part A and answer any one question in Part B. Answers may be written in either English or Malay. All necessary working should be shown clearly. Non-exact numerical answers may be given correct to three significant figures, or one decimal place in the case of angles in degrees, unless a different d ifferent level of accuracy is specified in the question. Mathematical tables, a list of mathematical formulae and graph paper are provided. Arahan kepada calon: mana-mana a satu soalan pada Jawab semua soalan pada bahagian A dan jawab mana-man bahagian B.
Jawapan boleh ditulis dalam bahasa Inggeris atau bahasa Melayu. Semua kerja yang perlu hendaklah ditunjukkan dengan jelas. Jawapan berangka tak tepat boleh diberikan betul hingga tiga angka bererti, atau satu tempat perpuluhan dalam kes sudut dalam darjah, kecuali aras kejituan yang lain ditentukan dalam soalan. Sifir matematik, senarai rumus matematik, dan kertas graf dibekalkan. This question paper consists of 3 printed pages. Kertas soalan ini terdiri daripada 3 halaman bercetak.
Section A (45 marks) Answer all questions in this section.
1
(a) A function g is defined as g(x) = x3 – 3x2 + x + 1 Find g(1) and g(-2) and hence state a factor of g(x). Express in the form (x +a) (x 2 + bx + c) where a, b and c are numbers to be determined. [3 marks] (b) The functions f and g are defined by
1 f : x → , x ∈ ℜ \ {0}; (all real number except zero) x
g : x
→ 2 −1, x
x
∈ℜ .
Find composite f g and its domain
[4 marks]
1
2
1 + 2 Express in x3. as a series of ascending powers up to the term in x 1 + 2 x
[5 marks]
x
3
(a) Using elementary row operations, find the inverse of the matrix of A A=
[5 marks] (b) Hence, solve the system of linear equations x + y + z = 4, 2 x – y y + 3 z = 1, 3 x +2 y + z = 1. 4
(a) Given that ( x + iy)2 = –5 + 12i where x where x and y and y are real, find the possible values of x x and y. y. (b) Hence, solve the equation z2 + 4z = – 9 + 2i
5
6
[5 marks]
[4 marks] [4 marks]
An ellipse is given by the equation 16x2 + 4y2 – 64x – 40y + 100 = 0. (a) Rewrite Rewrite the equation equation in standard standard form of ellipse. ellipse.
[3 marks] marks]
(b) Find the centre, foci and vertices. Sketch the ellipse.
[6 marks]
Given that A(1, 1, 1) , B(1, 2, 0) and C(-1, 2, 1) relative to a fixed origin O, find the vector equation and Cartesian equation of the plane through the points. If P is the point (2, (2, -1, 3) find the equation equation of the line through P perpendicular to to the plane
through A, B and C.
[6 marks]
Section B (15 marks) Answer any one questions in this section.
7
Express 3 cos x cos x – 4 sin x sin x in the form cos ( x + α) where R > 0 and 0 < α < 90o. Hence find the maximum and minimum values of 3 cos x cos x – 4 sin x sin x and the corresponding values of x x 0 0 in the interval 0 ≤ x ≤ 360 [7 marks] (a) Sketch Sketch the the graph graph of y = 3 cos cos x x – 4 sin x sin x for 00 ≤ x ≤ 3600
[3 marks]
(b) By drawing appropriate appropriate lines on your graph, determine the number of roots in the 0 interval 0 ≤ x ≤ 1800 of each of the following equations: (i)
3 cos x cos x – 4 sin x sin x =
(ii)
3 cos x cos x – 4 sin x sin x =
[2 marks]
(c) Find the the set of values values of x in the the interval interval 00 ≤ x ≤ 3600 for which | 3 cos x cos x – 4 sin x sin x | > 2
8
[3 marks]
The position vectors p, q and r of three of three points P, points P, Q and R and R respectively, relative to a fixed origin, O are given by p = 2i + 3 j – k q = 5i – 2 j + 3k r = 4i + j – 2k
(a) Find Find the length length of of PQ, PQ, correct to 3 significant figures
[2 marks]
(b) Find Find the the angle angle QPR , correct to the nearest degree
[3 marks]
(c) Find the area area of triangle triangle PQR PQR,, correct to 3 significant figures
[3 marks]
Show that, for all values of the parameter t, the point S with S with position vector s = (2+3t)i + (3-5t) j + (-1+4t)k
lies on the line through P through P and and Q.
[3 marks]
Find s such that OS is OS is perpendicular to PQ to PQ..
[4 marks]
