CONFIDENTIAL 1 CS/FEB 2022/MAT438 © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION COURSE : FOUNDATION OF APPLIED MATHEMATICS COURSE CODE : MAT438 EXAMINATION : FEB 2022 TIME : 3 HOURS INSTRUCTIONS TO CANDIDATES 1. This question paper consists of five (5) questions. 2. Answer ALL questions in English. Start each answer on a new page. 3. Please check to make sure that this examination pack consists of: i) the Question Paper ii) A Appendix 1 DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO This examination paper consists of 4 printed pages
CONFIDENTIAL CS/FEB 2022/MAT438 © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL QUESTION 1 a) Solve the following function tan-1(2x) + tan-1(x) = tan-1(3) (Hint: tan (A+B) = tan A + tan B 1 - tan A tan B ) (6 marks) b) Use the triangle method to evaluate sin $2 cos-1(x)+ π 2 % (5 marks) c) Given y = tanh -1 (x) , x∈R, |x|<1. Use definition of hyperbolic functions in terms of exponentials, prove that y = 1 2 ln ( 1 + x 1 − x + (6 marks) QUESTION 2 a) Find the derivatives of the following functions: i. f(x) = x2 sin-1 (1-x) ii. f(x) = 3 sinh "x2# cosh-1(x) (8 marks) b) Let x = y2 sinh(4x) + cosh(y). Use Implicit differentiation to find the expression of . (5 marks) QUESTION 3 a) Use integration by parts to evaluate , (3x + 4) sin (3x) dx (6 marks) b) Using trigonometric substitution x = sin θ, evaluate , -1 − x2 dx (8 marks)
CONFIDENTIAL CS/FEB 2022/MAT438 © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL c) By partial fraction decomposition, show that 3x + 14 (x + 4)(x + 5) = 2 x + 4 + 1 x + 5 Hence, evaluate , 3x+14 (x+4)(x+5) dx (6 marks) QUESTION 4 a) Solve the following separable equation . (7 marks) b) Solve the homogeneous equation (8 marks) c) Consider the following linear differential equation . Show that the integrating factor is Hence, solve the differential equation at (8 marks) d) Solve the following second-order ordinary differential equation using an undetermined coefficient method . (9 marks) QUESTION 5 a) A sample of radioactive substance Uranium-238 has a mass of 300g. The half-life of the substance is 100 years. Determine the amount of the substance at any time, t. i) Predict the amount of the substance after 200 years. ii) How long does the substance take to reach 10g? (9 marks) ( ) ( 5) 2 1 1 2 2 = + + x y dx dy x 2 3 3 4 2 xy y x dx dy - = 3 1 3 3 - = - + x y dx x dy ( 3) . 3 x - y(4) = 8. y' '+y'-6y = 5sin x +10cos x
CONFIDENTIAL CS/FEB 2022/MAT438 © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL b) Samad plans to build a small metal table, so he heats a piece of metal to The metal is then placed in room with a temperature of to cool down. After 40 minutes, the temperature of the metal is . i) Estimate the temperature of the metal after 3 hours? ii) When will the temperature of the metal achieve to ? (9 marks) END OF QUESTION PAPER 500 C. o Co 25 Co 300 Co 100
CONFIDENTIAL CS/FEB 2022/MAT438 © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL APPENDIX 1 LIST OF INTEGRALS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. ï ï î ï ï í ì + + = - + ¹ - + + + = + ò ln | ax b | C; n 1 a 1 C; n 1 a(n 1) (ax b) (ax b) dx n 1 n dx ln | x | C x 1 = + ò cos(ax) C a 1 sin(ax)dx = - + ò sin(ax) C a 1 cos(ax) dx = + ò tan(ax) C a 1 sec (ax)dx 2 = + ò ln | sec (ax) tan(ax)| C a 1 sec (ax)dx = + + ò ax C a ax dx = - + ò cot( ) 1 csc ( ) 2 sec (ax) C a 1 sec (ax)tan(ax)dx = + ò cosh(ax) C a 1 sinh(ax) dx = + ò sinh(ax) C a 1 cosh(ax)dx = + ò tanh(ax) C a 1 sech (ax)dx 2 = + ò ò =- coth(ax)+C a 1 csc h (ax)dx 2 ò =- sec h(ax)+C a 1 sech(ax)tanh(ax)dx csch(ax) C a 1 csch(ax)coth(ax) dx = - + ò C a x dx sin a x 1 1 2 2 ÷ + ø ö ç è æ = - - ò
CONFIDENTIAL CS/FEB 2022/MAT438 © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL 16. 17. 18. 19. 20. 21. 22. TRIGONOMETRIC IDENTITIES 1. 2. 3. 4. 5. HYPERBOLIC FUNCTIONS 1. 2. 3. C a x tan a 1 dx a x 1 1 2 2 ÷ + ø ö ç è æ = + - ò C a x sec a 1 dx x x a 1 1 2 2 ÷ + ø ö ç è æ = - - ò C ln | x a x | C a x dx sinh a x 1 1 2 2 2 2 ÷ + = + + + ø ö ç è æ = + - ò C ln | x x a | C, if x a a x dx cosh x a 1 1 2 2 2 2 ÷ + = + - + > ø ö ç è æ = - - ò ï ï î ï ï í ì ÷ + > ø ö ç è æ ÷ + < ø ö ç è æ + = - + = - - - ò C, if | x | a a x coth a 1 C, if | x | a a x tanh a 1 C x a x a ln 2a 1 dx a x 1 1 1 2 2 C, if 0 x a x a a x ln a 1 C a x sec h a 1 dx x a x 1 2 2 1 2 2 + < < + - = - + = - - - ò C, if x 0 x a a x ln a 1 C a x csc h a 1 dx x a x 1 2 2 1 2 2 + ¹ + + =- + =- + - ò sin x cos x 1 2 2 + = sin2x = 2sinxcos x x x x 2 2 cos 2 = cos - sin sin(A ± B) = sinAcosB ± sinBcos A cos(A ± B) = cos AcosB ! sin AsinB 2 e e sinh x x -x - = 2 e e cosh x x -x + = cosh x sinh x 1 2 2 - =
FINAL MAT438/ Solution FEB 2022
FINAL MAT438/ Solution FEB 2022 1 Q1.a) Solve the following function −1 (2) + −1 () = −1 (3) (6 marks) Solution : 6 valid invalid
FINAL MAT438/ Solution FEB 2022 2 Q1.a) Use the triangle method to evaluate (2 −1 () + 2 ) (5 marks) Solution : 5
FINAL MAT438/ Solution FEB 2022 3 Q1. c) Given = ℎ −1 (). Prove that = 1 2 ( 1 + 1 − ). (6 marks) Solution : 5 (Proven)
FINAL MAT438/ Solution FEB 2022 4 Q2.a.i) Differentiate () = 2 −1 (1 − ) with respect to x. (4 marks) Solution : Q2. a. ii) Differentiate () = 3 ℎ( 2 ) ℎ −1() with respect to x. (4 marks) Solution : 4
FINAL MAT438/ Solution FEB 2022 5 Q2. b) Let = 2 ℎ(4) + ℎ(). Use Implicit differentiation to find the expression of . (5 marks) Solution 5
FINAL MAT438/ Solution FEB 2022 6 Q3. a) Use integration by parts to evaluate ∫(3 + 4) (3) . (6 marks) Solution : 6
FINAL MAT438/ Solution FEB 2022 7 Q3. b) Evaluate ∫ √1 − 2 using the substitution = . (8 marks) Solution :: 8
FINAL MAT438/ Solution FEB 2022 8 Q3. c) By partial fraction decomposition, show that 3 + 14 ( + 4)( + 5) = 2 + 4 + 1 + 5 . Hence, evaluate ∫ 3 + 14 ( + 4)( + 5) . (6 marks) . Solution : 6
FINAL MAT438/ Solution FEB 2022 9 Q4. a) Solve the following separable equation 1 (2 2 + 1) = ( 2 + 5). (7 marks) . Solution : 7
FINAL MAT438/ Solution FEB 2022 10 Q4. b) Solve the homogeneous equation = 4 3 − 2 3 2 . (8 marks) Solution : 8
FINAL MAT438/ Solution FEB 2022 11 Q4. c) Consider the following linear differential equation + 3 − 3 = 1 − 3 . Show that the integrating factor is ( − 3) 3 . Hence solve the differential equation at (4) = 8. (8 marks) Solution : 8
FINAL MAT438/ Solution FEB 2022 12 Q4d) Solve the second order differential equation ′′ + ′ − 6 = 5 + 10 . (9 marks) Solution :
FINAL MAT438/ Solution FEB 2022 13 Q5.a) A sample radioactive substance Uranium-238 has a mass of 300g. The half-life of the substance is 100 years. Determine the amount of the substance at any time, t. i) Predict the amount of the substance after 200 years. ii) How long does the substance take to reach 10g? (9 marks) Solution : 9
FINAL MAT438/ Solution FEB 2022 14 Q5.b) Samad plans to build a small table, so he heats a piece of metal to 500oC. The metal is then placed in room with a temperature of 25 oC to cool down. After 40 minutes, the temperature of the metal is 300oC. i) Estimate the temperature of the metal after 3 hours. ii) When will the temperature of the metal achieve to 100oC (9 marks) Solution :