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1 DBM0: ENGINEERING MATHEMATICS SECTION A : 7 MARKS BAHAGIAN A : 7 MARKAH INSTRUCTION: This section consists of THREE () structured questions. Answer ALL questions. ARAHAN : Bahagian ini mengandungi TIGA () soalan berstruktur. Jawab SEMUA soalan. QUESTION SOALAN CLO C a) Simplify the following expressions to the lowest term. Permudahkan ungkapan berikut kepada sebutan terendah. i. xyz 8xy a 0ab [ marks] [ markah] c d e ii. e c d [ marks] [ markah] iii. a a a a [ marks] [ markah]

2 DBM0: ENGINEERING MATHEMATICS CLO C b) Solve the following quadratic equations. Selesaikan persamaan kuadratik berikut. i. x 8x (By using the quadratic formula.) [6 marks] [6 markah] ii. x x 0 (By using the completing the square.) [9 marks] [9 markah]

3 DBM0: ENGINEERING MATHEMATICS QUESTION SOALAN a) State the order of matrix and form the transposition of matrix P and Q: 0 9, Q P Nyatakan peringkat bagi matrik dan dapatkan tansposisi bagi matrik P dan Q: 0 9, Q P [ marks] [ markah] CLO C

4 DBM0: ENGINEERING MATHEMATICS b) Given matrix, Di beri matrik, find: cari: i) A [ marks] [markah] ii) (A + B) [ marks] [ markah] ii) B T A [ marks] [ markah] iii) CD [ marks] [ markah] 0 8 7, 0, D and C B A CLO C and, 0, D C B A

5 DBM0: ENGINEERING MATHEMATICS CLO C c) Solve the following equation by using the inverse matrix method. Selesaikan persamaan berikut dengan menggunakan kaedah matrix songsang. x y z 9 x y z x 6y z 0 [ marks] [ markah] 6

6 DBM0: ENGINEERING MATHEMATICS QUESTION SOALAN CLO C (a) Given that ~ p i 9 j and q~ i j, find each of the following vector. Diberi ~ p i 9 j dan q~ i j, dapatkan vektor bagi setiap yang berikut. i. ~ p q ~ [ marks] [markah] ii. q ~ ~ p [ marks] [markah] CLO C CLO C (b) Given that vector OP is and vector OQ is. Find: 6 Diberikan bahawa vektor OP ialah dan vector OQ ialah. Dapatkan: 6 i) OPOQ [ marks] [ markah] ii) OP OQ [ marks] [ markah] (c) Calculate the angle between the vectors i + j k and i j + k. Kirakan sudut antara vektor i + j k dan i j + k. [ marks] [ markah] 7

7 DBM0: ENGINEERING MATHEMATICS (d) Given, vectors OC = i j -k, OD = i j k and OE = i j + k. Calculate: Diberi vector OC = i j -k, OD = i j k dan OE = i j + k. Kirakan: CD DE [7 marks] [7 markah] ( e ) Given vectors OM = (,, ) and ON = (0,, 7). Find unit vector in the direction of MN. : Diberi vector OM = (,, ) dan ON = (0,, 7). Dapatkan unit vector bagi arah MN. : [ marks] [ markah] 8

8 DBM0: ENGINEERING MATHEMATICS SECTION B: MARKS BAHAGIAN B: MARKAH INSTRUCTION: This section consists of THREE () structured questions. Answer ONE () question only. ARAHAN: Bahagian ini mengandungi TIGA () soalan berstruktur. Jawab SATU () soalan sahaja. QUESTION SOALAN a) Express each of the following in partial fractions. Nyatakan pecahan yang berikut kepada pecahan separa. CLO C x 7 x x [ marks] [ markah] b) Solve the following partial fractions: Selesaikan pecahan separa berikut: CLO C i. 8x 0 x [6 marks] [6 markah] ii. x x x [7 marks] [7 markah] iii. x x 7x x 0 [8 marks] [8 markah] 9

9 DBM0: ENGINEERING MATHEMATICS QUESTION SOALAN CLO C a) Find all the angle for the trigonometric equation below for the range 0 x 60. Cari semua sudut untuk persamaan trigonometri di bawah bagi julat 0 x 60. i. cos x sin x 0 ii. cos x cot x [ marks] [ markah] [ marks] [ markah] CLO C b) Prove each of the following trigonometric identities. (Use the equation on the right to prove the identity of the equation on the left). Buktikan setiap identiti trigonometri berikut. (Gunakan persamaan di sebelah kanan untuk membuktikan identity persamaan di sebelah kiri ). i. cot A tan A cot A [ marks] [ markah] ii. sin sin A B cos A cos B sin A B sin B [ marks] [ markah] iii. cosec A cot A tan Acos A [ marks] [ markah] 0

10 DBM0: ENGINEERING MATHEMATICS QUESTION 6 SOALAN 6 CLO C a) Given x = + i, w = + i and z = 6i. Express each of the following in the form of a + bi Diberi persamaan x = + i, w = + i dan z = 6i. Ungkapkan persamaan berikut dalam bentuk a + bi i. x + w [ marks] [ markah] ii. iii. x w x z [ marks] [ markah] [ marks] [ markah] CLO C b) Given J = i, K = + i. Find the modulus, the argument and sketch the Argand s diagram for : Diberi J = i, K = + i. Dapatkan modulus, hujah dan lakarkan gambarajah Argand s bagi: i. J ii. J K [6 marks] [6 markah] [9 marks] [9 markah] SOALAN TAMAT

11 DBM0: ENGINEERING MATHEMATICS FORMULA QUADRATIC EQUATION X = b ± b ac a (x + b ) ( b ) + c = 0 FORMULA OF TRIANGLE a Sine Rules ; sina = b sinb = c sinc Cosine Rules; a = b + c bc cosa Area of Triangle = ab sinc MATRIX Cofactor, C = ( )(i + j)m ij Adjoin, Adj(A) = C T Inverse of Matrix, A = A Adj(A) COMPLEX NUMBER Modulus of z = a + b Argument of z = tan ( b a ) Cartesian Form ; z = a + bi Polar Form ; z = r θ Exponential Form ; z = re iθ TRIGONOMETRY Pythagoras Theorem c a ɵ b c = a + b Trigonometry Identities tanθ = sinθ cosθ cos θ + sin θ = + tan θ = sec θ + cot θ = cosec θ VECTOR & SCALAR Unit Vector, u = u u A B = a a + b b + c c i j k A B = a b c a b c Area of parallellogram ABC = AB bc COMPOUND-ANGLE DOUBLE-ANGLE sin(a ± B) = sinacosb ± cosasinb cos(a ± B) = cosacosb sinasinb tan(a ± B) = tana ± tanb tanatanb sina = sinacosa cosa = cos A sin A = sin A = cos A tana = tana tan A

12 DBM0: ENGINEERING MATHEMATICS

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