MATH101 Exam Solutions for Year 1 Students in Math Sciences, Exams of Mathematics

Download MATH101 Exam Solutions for Year 1 Students in Math Sciences and more Exams Mathematics in PDF only on Docsity! PAPER CODE NO. MATH101 EXAMINER: Prof A.C. Irving DEPARTMENT: Math Sciences TEL.NO. 43782 JANUARY 2007 EXAMINATIONS Bachelor of Arts: Year 1 Bachelor of Science : Year 1 Master of Mathematics : Year 1 Master of Physics : Year 1 FOUNDATION MODULE I TIME ALLOWED : Two Hours and a Half INSTRUCTIONS TO CANDIDATES Answer all of Section A and THREE questions from Section B. The marks shown against questions, or parts of questions, indicate their relative weight. Section A carries 55% of the available marks. Paper Code MATH101 Page 1 of 5 CONTINUED/ S E C T I O N A 1. Determine the natural domain and range of each of the following func- tions and provide a rough sketch of its graph: (i) y = √ x + 1, (ii) y = |1 − x| + x . [5 marks] 2. Find the general solution of cos ( θ + π 4 ) = −1 2 . [3 marks] 3. Find the inverse function f−1(x) of the function f(x) = 1 − x 3x − 2 (x 6= 2 3 ) and verify that f(f−1(x)) = x . [5 marks] 4. Determine, giving reasons, whether the following functions are odd, even or neither: (i) p(x) = sin(x2) + |x| , (ii) q(t) = t 3 − t + 1 2t2 + 5 . [4 marks] 5. Find the following limits, where they exist: (i) lim x→−2 x2 − x − 6 x + 2 , (ii) lim x→1 x3 − 1 ln x , (iii) lim x→π 2 cos x 1 + cos(2x) . [6 marks] 6. Differentiate 3x2 − x with respect to x from first principles, i.e. by using an appropriate limiting process. [4 marks] Paper Code MATH101 Page 2 of 5 CONTINUED/