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2 MATH1015W1 SECTION A A1. (2 marks) Find an integrating factor for the differential equation t dx − x = t2 . dt (Do NOT solve the equation.) A2. (2 marks) Determine whether the equation ¡ ¢ dx 2xt3 − t sin x + 3x2 t2 + cos x = 0 dt is exact. (Do NOT solve the equation). A3. (2 marks) Find a particular integral of the differential equation dy d2 y + 2 + 5y = 20x + 23 . dx2 dx A4. (2 marks) Find a particular integral of the differential equation d2 x dx + 2 − x = −2e−3t . 2 dt dt A5. (2 marks) If a = −i + 2j + 3k and b = j − 2k, and c = −2i + 3j − k , obtain a · b × c. A6. (2 marks) Points A and B have position vectors 2i + j − 3k and i − j + k . Find the distance between the two points. A7. (2 marks) A particle P has position vector 3 sin t i − 3 cos t k at time t. Calculate the velocity and speed of the particle. 3 MATH1015W1 A8. (2 marks) Find the vector equation of the plane which is perpendicular to the vector 3i − 4j − 2k and passes through the point (−1, 2, −3). A9. (2 marks) Obtain the product AB of the matrices −2 1 A = 4 −1 −2 3 µ and B = 0 −1 2 −3 0 1 A10. (2 marks) Calculate the determinant of the matrix C = 1 −1 ¶ . 0 0 1 0 −2 −1 . 3 0 2 1 −4 2 A11. (2 marks) State a condition on the matrix A, for the system of n homogeneous equations in n unknowns given by AX = 0 to have a non-zero solution vector X . ¡ ¢ A12. (2 marks) If det D2 = 25, obtain det(D). ¡ 2¢ ∂ 2f A13. (2 marks) Given that f (x, y) = y x − x y + x sin y , find . ∂x∂y 3 3 A14. (2 marks) If f is a function of u and v where both u and v depend only on t, obtain an expression for df in terms of derivatives of f , u and v . dt ∞ X A15. (2 marks) Determine whether the series k=1 k is convergent. k+1 x2 A16. (2 marks) Using l’Hopital’s rule, or otherwise, evaluate lim . x→0 1 − cos 2x TURN OVER 4 MATH1015W1 A17. (2 marks) Write −2j in complex exponential form and hence, or otherwise, calculate the principal value of ln(−2j). ¡ ¡ ¢ ¡ ¢¢4 A18. (2 marks) Using de Moivre’s Theorem, or otherwise, calculate cos π6 + j sin π6 . A19. (2 marks) Find the Laplace transform of the function t2 e−t . 1 . A20. (2 marks) Find the inverse Laplace transform of the function 2 s + 4s + 5 5 MATH1015W1 SECTION B B1. (a) (7 marks) Find the general solution of differential equation ³ ´ dx 2 2 2 x2 2xte − t sin x + sec x + ex + 2t cos x + cos t = 0. dt (b) (8 marks) Obtain the general solution of the second order differential equation d2 x dx + 5 + 6x = 2e−t . 2 dt dt B2. A line L1 passes through the points A (1, 2, 3) and B (−2, 5, 0). A second line L2 passes through the origin and the point C (0, 6, 4). (a) (5 marks) Obtain the vector equations of L1 and L2 . If L1 and L2 intersect, find the point of intersection. (b) (5 marks) Determine the angle between the lines L1 and L2 . (c) (5 marks) Find a vector perpendicular to the lines L1 and L2 . Hence, or otherwise, obtain the cartesian equation of the plane containing the two lines. TURN OVER 6 MATH1015W1 B3. (a) (7 marks) Find the eigenvalues and eigenvectors of the matrix µ 1 −2 −2 4 ¶ . (b) (8 marks) Using the notation z = x + jy , find the locus of the point z in the Argand diagram satisfying the equation ¯ ¯ ¯z − j ¯ ¯ ¯ ¯ z − 1 ¯ = 2. Sketch the result on an Argand diagram and describe the locus geometrically. B4. (a) (4 marks) Evaluate the determinant of the matrix 1 1 1 A= 1 2 3. α 1 2 (b) (6 marks) Find A−1 when α = 1. (c) (5 marks) Hence, or otherwise, solve the system of linear equations x + y + z = 2 x + 2y + 3z = 2 x + y + 2z = 1 for the unknowns x, y and z . 7 MATH1015W1 B5. (a) (4 marks) Use the table of Laplace transforms to find ½ (i) L−1 (ii) L−1 (iii) L−1 ¾ 1 , s−1 ½ ¾ 1 , (s − 1)2 ¾ ½ 1 . s+3 (b) (6 marks) Use partial fractions and part (a) to obtain ½ L−1 8 (s + 3)(s − 1)2 ¾ . (c) (5 marks) Use Laplace transforms and part (b) to find the solution of the differential equation d2 x dx + 2 − 3 x = 8 et 2 dt dt dx = 0 when t = 0 . which satsifies the initial conditions x = 0 and dt 1 B6. (a) (8 marks) Use Maclaurin’s theorem to obtain the expansion of f (x) = (1 − x) 2 as 1 1 (1 − x) 2 = 1 − x + ax2 + bx3 + R3 . 2 Give the values of the coefficients a and b and find the form of the Lagrange remainder R3 , stating any associated inequality. What is the maximum error if √ 3 we use this expansion up to the x term to estimate the value of 0.95 ? (b) (7 marks) The constant density ρ of a sphere is related to its mass m and its radius r by the formula ρ= 3m . 4πr3 If the mass is measured with a maximal error of 1% and the radius with a maximal error of 0.5%, use partial derivatives to estimate the maximum percentage error in the density. END OF PAPER MATH1015W1 Full marks may be obtained for complete answers to ALL questions in Section A and FOUR questions in Section B. All questions in Section A are worth 2 marks and all questions in Section B are worth 15 marks. Only your best FOUR answers to Section B questions will be taken into account. The Engineering Data Book (Calvert and Farrar) will be provided. The Official University Calculator MAY be used c University of Southampton Copyright 2006 ° Number of Pages 7