Higher Certificate in Engineering Exam: Mathematics Questions and Answers, Exams of Mathematics

Download Higher Certificate in Engineering Exam: Mathematics Questions and Answers and more Exams Mathematics in PDF only on Docsity! Cork Institute of Technology Higher Certificate in Engineering in Maintenance Technology – Award (NFQ – Level 6) Summer 2007 Mathematics (Time: 3 Hours) Answer FIVE questions. Attempt Question 1 and FOUR others. Examiners: Mr. M. Walsh Mr. J. Connolly Dr. P. Delassus Q1. (a) Find dx dy for 15 32 + + x x (4 marks) (b) Evaluate dxx∫ +4 2 2 2 )1( (4 marks) (c)           −= 513 402 132 A           −= 435 215 034 B Find A·B (3 marks) (d) Solve i i 43 832 + + (3 marks) (e) jiu ρρρ 43 += jiv ρρρ −= 6 Find vu ρρ 23 − in terms of i ρ and j ρ . (3 marks) (f) A footballer scores a penalty 4 times out of 5. In a match he takes 3 penalties. What is the probability that he misses all 3? (3 marks) 2 Q2. (a) If 102 += ty and 33 −= tx , find dx dy (5 marks) (b) If 11563 23 +−−= xxxy (i) Find local max, min and point of inflection. (5 marks) (ii) Find equation of tangent to the curve at (2,1). (5 marks) (iii) Find the equation of another tangent with the same slope. Include a sketch. (5 marks) Q3. (a) Solve ∫ + dxxSin )73( (6 marks) (b) Solve ∫ ⋅ dxxSinx 24 (6 marks) (c) Solve ∫ −− − )2)(3( 2436 xxx x (8 marks) Q4. (a)           − − − = 511 240 432 A Prove that A·A-1 = I (12 marks) (b) Solve for X, Y and Z, if 1142 932 4 =++ =++ =++ ZYX ZYX ZYX (8 marks) Q5. (a) Prove that to multiply 2 complex numbers together in polar form, you must multiply their moduli and add their arguments. Verify this by multiplying 21 ZZ ⋅ , where iZ iZ 34 43 2 1 −= += (14 marks) (b) If iZ 23+= , find 5Z using De Moivres Theorem. (6 marks)