UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics & Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4701 SEMESTER: Autumn 2011/12 MODULE TITLE: Technological Mathematics 1 DURATION OF EXAMINATION: 2 1 2 hours LECTURER: J. O Shea PERCENTAGE OF TOTAL MARKS: 85 % EXTERNAL EXAMINER: Prof. T. Myers INSTRUCTIONS TO CANDIDATES: Questions One (Q1) is compulsory and carries 40 marks. Answer any other four questions worth 15 marks each. N.B. There are some useful formulae at the end of the paper.
1. (a) Find the equation of the line passing through the point (1, 3) and which is perpendicular to the line 2x y + 1 = 0. 4 (b) Solve the equation 1 + 2 ln (4x + 1) = 3. (ln = log e ) 4 (c) Find the first derivative of the following functions (i) y = 3x 4 + ln 2x (ii) f(x) = (x 2 4x + 1) 4 (iii) g(x) = e 4x sin x (iv) f(t) = 3t2 +2 2t+1. 8 (d) Find the equation of the tangent to the curve y = 3x 3 6x 2 5x + 11 at the point (2, 1). 4 (e) Use De Moivre s theorem to evaluate (1 + 2i) 8. 4 (f) (i) If z = 3 + 5i evaluate z z. (ii) Express e 3i in the form a + bi. 4 (g) ā = 3i + j 2k, find the unit vector in the direction of ā. 4 (h) Evaluate the matrix product ( ) ( ) 3 1 2 1 (i) 0 2 1 3 ( ) ( ) 2 1 1 (ii). 4 1 3 2 (i) From the following augmented matrix, state the solution to the linear system: 2 1 4 4 0 3 2 12. 4 0 0 1 3 1
2. (a) (i) Find f 1 (x) the inverse of the function f(x) = 2x + 1. (ii) f(x) = 3x 2 and g(x) = sin 4x, find the composite function f g(x). (iii) Prove that f(x) = (iv) Evaluate 3x is an odd function. x 2 +1 lim x 2x 5 x + 1. (v) Find the local minimum point of y = x 2 4x+7 and hence sketch the function. 10 (b) (i) Find the value of x R and y R given that ( ) ( ) ( ) x 4x 1 9 =. y 2y 2 20 1 3 4 (ii) A = 2 1 1 1 2 5 Find det(a). 5 2
3. (a) The function y = 4 e 0.002t denotes the process known as radioactive decay where 4 grams is the initial level of radium and t is the time in years. How long will it take for the radium to reduce to 2 grams? 5 (b) Determine the amplitude A, angular velocity ω, period T and frequency f of the function y = 4 sin 2t. Sketch the function. 5 (c) For ABC, AB = 3cm, AC = 4cm and BAC = 60. A 3 60 4 C B Find (i) BC correct to 1 decimal place. (ii) ABC to the nearest degree. 5 4. (a) Let f(x) = x 3 12x + 4. Determine the stationary points of f(x) and classify them as local maxima or minima. Find also the inflection point. Sketch the curve. 10 (b) A ball is thrown vertically upwards. The height h metres of the ball, t seconds after it is thrown is given by the formula h = 3t(10 t). Find (i) the speed of the ball after 1 second. (ii) the time when the ball reaches its maximum height. (iii) the maximum height reached. 5 3
5. (a) Given z 1 = 2 + 4 i and z 2 = 3 - i. Find (i) 4z 1 - z 2. (ii) 3 z 1 in the form a + bi. (iii) the real number k so that 5k = 1 + z 1. (iv) the polar form of z 2. 8 (b) Express -16 in general polar form. Find the four fourth roots of ( 16) 1/4. Plot the roots in the complex plane. 7 6. (a) Given a = 2i j + k b = 3i + 2j + 4k Find (i) 3a b. (ii) the scalar product a.b (iii) the vector product a b. 11 (b) An object is experiencing two perpendicular forces F 1 of 20N and F 2 of 15N as shown in the diagram. Calculate the magnitude of the resultant force F and determine the angle θ that F makes with the vertical. θ F 1 F 2 F 4 4
7. (a) Solve the following system of linear equations using Gaussian elimination, or otherwise x + 2y + 3z = 9 3x + y + 2z = 11 2x y + z = 8 8 (b) Verify by drawing a suitable graph that the values of Q and P given by the table below satisfy a law of the form Q = a + bp 2. Determine the best values for a and b using the following b = nσxy ΣxΣy and a = Σy nσx 2 (Σx) 2 n bσx n where a and b are the intercept and slope of the least squares line y = a+bx. P 1 2 3 4 5 Q 4 10 22 40 60 7 5
Formulae 1. m = slope PQ = y 2 y 1 x 2 x 1 equation PQ : y y 1 = m(x x 1 ) P (x 1,y 1 ) Q (x 2,y 2 ) 2. Logarithms a x = y log a y = x 3. De Moivre s theorem 4. Vector operations [r(cos θ + i sin θ)] n = r n (cos nθ + i sin nθ) = r n e inθ v 1 = x 1 i + y 1 j and v 2 = x 2 ī + y 2 j v 1 = x 2 1 + y1 2 v 1 v 2 = x 1 x 2 + y 1 y 2 = v 1 v 2 cos θ (norm) (scalar product) ( a b 5. Inverse of a matrix A = c d = ad bc 0 6. Calculus ) ( is 1 d b det (A) c a ) where det(a) f(x) x n f (x) nx n 1 ln x 1 e x e ax cos x sin x e x ae ax sin x cos x 6
Product Rule: Quotient Rule: y = uv dy dx = udv dx + v du dx 7. Trigonometry Tables Page 9 and π Rad. = 180 y = u v dy dx = v du u dv dx dx v 2 7