UNIVERSITI TEKNOLOGI MALAYSIA FACULTY OF SCIENCE... FINAL EXAMINATION SEMESTER I SESSION 2015/2016 COURSE NAME : ENGINEERING MATHEMATICS I

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1 UNIVERSITI TEKNOLOGI MALAYSIA FACULTY OF SCIENCE... FINAL EXAMINATION SEMESTER I SESSION 05/06 COURSE CODE : SSCE 693 COURSE NAME : ENGINEERING MATHEMATICS I PROGRAMME : SKAW/SKEE/SKEL/SKEM/SKMM/SKMV/ SKMT/SKMO/ SKMI/SKMP/SKMB/SKKK/ SKKB/SKPG/SKPN/SKPP/SMBE LECTURER : PN WAN RUKAIDA BINTI WAN ABDULLAH (C) ASSOC. PROF. HAZIMAH BINTI ABD HAMID DR AMIDORA BINTI IDRIS DR MASLAN BIN OSMAN DR MOHD ARIFF BIN ADMON DR NUR ARINA BAZILAH BINTI AZIZ DR ZAITON BINTI MAT ISA EN IBRAHIM BIN MOHD JAIS EN MAHAD BIN AYEM EN NIKI ANIS BIN ABD KARIM EN TAUFIQ KHAIRI BIN AHMAD KHAIRUDDIN PN NORASLINDA BINTI MOHD ISMAIL DATE : 8 DECEMBER 05 DURATION : 3 HOURS INSTRUCTIONS : ANSWER ALL QUESTIONS IN PART A AND ANY THREE (3) QUESTIONS IN PART B. (THIS EXAMINATION BOOKLET CONSISTS OF 9 PRINTED PAGES)

2 SSCE 693 Part A (55%) QUESTION (6 MARKS) Using the efinition of hyperbolic function, solve the following equation sinh x 4 cosh x +=0. 5 QUESTION (6 MARKS) Fin y x for y3 sinh (xy) =0. QUESTION 3 (5 MARKS) Evaluate 7+6x x x. QUESTION 4 (5 MARKS) Determine whether the integral converges or iverges 9 3 x 9 x. QUESTION 5 (7 MARKS) If ln y =sinh x,showthat(+x ) ( ) y x = y an (+x ) y Hence, using the Maclaurin series, show that e sinh x =+x + x +. x +x y x y =0.

3 SSCE 693 QUESTION 6 (5 MARKS) Fin the equation of a line that passes through (,, ) an parallel to the line of intersection between the planes x+3y z =anx 3y+z =3. QUESTION 7 (7 MARKS) Fin the eigenvalues an eigenvectors for the matrix 0 A =. 0 3 QUESTION 8 (7 MARKS) Given r =6sinθ is an equation in polar coorinates. (i) Obtain the cartesian equation in the form of (x a) +(y b) = c, where a, b an c are constants. (ii) Fin the intersection point between r =6sinθ an r =inthefirst quarant. (3 marks) QUESTION 9 (6 MARKS) z Given z = i anz =+4i. Express in the form of a+ib, where z z a an b are real numbers. Hence etermine the moulus an the argument z of. z z 3

4 Part B (45%) SSCE 693 QUESTION 0 (5 MARKS) Given the equation of two lines, l an l. l : x =+3t, y = 4+t, an z = t, l : x = +s, y = 4+s, an z = s. (i) Fin the point of intersection between the lines l an l. (ii) Fin an acute angle between l an l at the point of intersection. (iii) Obtain the shortest istance from the point of intersection between l an l to the plane 3x 7y + z =5. (3 marks) (iv) Fin an angle between the plane an the line l. QUESTION (5 MARKS) 7 (a) Given A = ajoint metho. (b) Given a system of linear equations. Fin an inverse matrix of A using the x +y +4z =5 x +y + z =6 x +y +3z =9, solve the above system using the following methos (5 marks) (i) Gauss elimination metho, (ii) Cramer s rule. (5 marks) (5 marks) 4

5 SSCE 693 QUESTION (5 MARKS) Given the polar equation r = cos θ. (i) Test the symmetries of the above polar equation. (3 marks) (ii) Construct a table for (r, θ)withappropriatevaluesansketchthegraph of r = cos θ. (Use the polar gri provie) (5 marks) (iii) Sketch the graph tan θ =onthesameiagram. (3 marks) (iv) Fin the intersection points between the curves r = cosθ an tan θ =. QUESTION 3 (5 MARKS) (a) Given a complex number u =+3i. (i) Determine z = u +3 4i in the form a + ib. ( marks) (ii) Express z in polar form. ( marks) (iii) Solve w 3 = z an sketch the roots on a single Argan Diagram. (b) Use e Moivre s theorem to show that sin 3θ =3sinθ 4sin 3 θ. Hence, obtain all solutions of x for the following equation: 4x 3 3x +=0. (7 marks) 5

6 FORMULA SSCE 693 Trigonometric cos x +sin x = + tan x =sec x cot x + = cosec x sin(x ± y) =sinx cos y ± cos x sin y cos(x ± y) = cos x cos y sin x sin y tan x ± tan y tan(x ± y) = tan x tan y sin x =sinxcos x cos x = cos x sin x = cos x = sin x tan x = tan x tan x sinxcos y =sin(x + y)+sin(x y) sinxsin y = cos(x + y) + cos(x y) cos x cos y = cos(x + y) + cos(x y) Logarithm a x = e x ln a log a x = log b x log b a Hyperbolic sinh x = ex e x cosh x = ex + e x cosh x sinh x = tanh x =sech x coth x = cosech x sinh(x ± y) =sinhx cosh y ± cosh x sinh y cosh(x ± y) = cosh x cosh y ± sinh x sinh y tanh x ± tanh y tanh(x ± y) = ± tanh x tanh y sinh x =sinhxcosh x cosh x = cosh x +sinh x = cosh x =+sinh x tanh x = tanh x + tanh x Inverse Hyperbolic sinh x =ln(x + x + ), <x< cosh x =ln(x + x ), x tanh x = ( ) +x ln, <x< x 6

7 FORMULA SSCE 693 Differentiations [k] =0, x k constant. x [xn ]=nx n x [ex ]=e x. x [ln x ] = x. [cos x] = sin x. x [sin x] = cos x. x x [tan x] =sec x. Integrations kx= kx + C, k constant. x n x = xn+ n + + C, n. e x x = e x + C. x x x [cot x] = cosec x. =ln x + C. sin xx= cos x + C. cos xx=sinx + C. sec xx= tan x + C. cosec xx= cot x+c. Differentiations [sec x] =secx tan x. x [cosec x] x = cosec x cot x. [cosh x] =sinhx. x [sinh x] = cosh x. x x [tanh x] =sech x. [coth x] x = cosech x. [sech x] x = sech x tanh x. [cosech x] x = cosech x coth x. Integrations sec x tan xx=secx + C. cosec x cot xx = cosec x + C. sinh xx= cosh x + C. cosh xx=sinhx + C. sech xx= tanh x + C. cosech xx= coth x+c. sech x tanh xx = sech x + C. cosech x coth xx = cosech x + C. 7

8 FORMULA SSCE 693 Differentiations of Inverse Functions x [sin x]=, x <. x x [cos x]=, x <. x x [tan x]= +x. x [cot x]= +x. x [sec x]= x [cosec x]= x, x >. x x, x >. x x [sinh x]= x +. x [cosh x]=, x >. x x [tanh x]=, x <. x x [coth x]=, x >. x x [sech x]= x x, 0 <x<. x [cosech x]= x +x,x 0. Integrations Resulting in Inverse Functions x x =sin (x)+c. x +x = tan (x)+c. x x x =sec (x)+c. x x + =sinh (x)+c. x x = cosh (x)+c, x > 0. x x = ( ) +x ln + C, x <. x x x = ( ) x ln + C, x >. x + x x x = sech (x)+c, x <. x x +x = cosech x + C, x 0. 8