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1 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: 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.
2 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 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 ( ) ( ) (i) ( ) ( ) (ii) (i) From the following augmented matrix, state the solution to the linear system:
3 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 (ii) A = Find det(a). 5 2
4 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 C B Find (i) BC correct to 1 decimal place. (ii) ABC to the nearest degree (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 5. (a) Given z 1 = 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 (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
6 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 Q
7 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 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
8 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
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH
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