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MATH1131 Mathematics 1A and MATH1141 Higher Mathematics 1A PAST EXAM PAPERS AND SOLUTIONS Copyright 2020 School of Mathematics and Statistics, UNSW Contents PAST EXAM PAPERS 4 NOVEMBER 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 JUNE 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 JUNE 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 JUNE 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 JUNE 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 JUNE 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 PAST HIGHER EXAM PAPERS 33 JUNE 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 JUNE 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 JUNE 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 JUNE 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 JUNE 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 PAST EXAM SOLUTIONS 49 NOVEMBER 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 JUNE 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 JUNE 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 JUNE 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 JUNE 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 JUNE 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 PAST HIGHER EXAM SOLUTIONS 89 JUNE 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 JUNE 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 JUNE 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 JUNE 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 JUNE 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Table of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 The 2017 to 2019 exam papers will be provided on Moodle. 3 4PAST EXAM PAPERS Please note that the emphasis of the exam may change from year to year and that it is not possible to test all aspects of the course in a 2-hour examination. Hence, these papers are only a guide as to the style and level of difficulty of our first year examinations. The exam format may be different in 2020 — check Moodle for details. The solutions to the examination papers contained here have been written by many members of staff of the School of Mathematics and Statistics. While every care is taken to excluded errors, we cannot guarantee that the solutions are error-free. Please report any serious errors to the Director of First Year Mathematics. Exam papers from 2019 will be provided on Moodle for practice before the exam. Students are encouraged to produce their own solutions to the 2019 exam papers and discuss these on Moodle. c 2020 School of Mathematics and Statistics, UNSW Sydney 5MATH1131 November 2010 1. i) Let z = 1 + i. a) Find |z|. b) Find Arg(z). c) Use the polar form of z to evaluate z28 and then express your answer in Cartesian form. ii) Suppose that (x+ iy)(3 + 2i) = (4 + 7i) where x, y ∈ R. Find the value of x and the value of y. iii) a) Find all solutions to the equation z6 = 1, where z ∈ C. b) Hence, or otherwise, express z6 1 as a product of three quadratic polynomials with real coefficients. iv) Let the region S be defined as S = {z ∈ C : |z + i| = 1}. Sketch the region S on a carefully labelled Argand diagram. v) Consider the following MAPLE session. > with(LinearAlgebra): > A:=<<1,0,-sqrt(3)>|<0,2,0>|>; A := 1 0 √ 3 0 2 0 √3 0 1 > A^2; 2 0 2√3 0 4 0 2√3 0 2 > A^4; 8 0 8√3 0 16 0 8 √ 3 0 8 > A^6; 64 0 0 0 64 0 0 0 64 Use the above Maple session to find the inverse of the matrix A2. c 2020 School of Mathematics and Statistics, UNSW Sydney 6vi) A car dealer sells three brands of cars: Audis, BMWs and Chevrolets. Her costs consist of registration, GST and the manufacturer’s price of the car. She has to pay registration costs of $200 per Audi, $600 per BMW and $400 per Chevrolet. The GST charges are $1, 800 per Audi, $2, 400 per BMW and $2, 000 per Chevrolet. To the manufacturer she pays $20, 000 per Audi, $30, 000 per BMW and $26, 000 per Chevrolet. Last year her total registration bill was $12, 000, the total GST tax bill was $65, 600 and she paid a total of $784,000 to manufacturers. Let a, b and c denote the number of Audis, BMWs and Chevrolets she sold last year. a) Write down a system of linear equations in a, b and c determined by the above information. b) Convert the system of equations to an augmented matrix form and hence find a, b and c by performing Gaussian elimination on the augmented matrix. 2. i) Let P = ( 1 0 1 2 3 0 ) . a) Evaluate the matrix product PP T . b) State the size of the matrix P TP . ii) Let A,B and C be points in R3 with position vectors a = 12 3 , b = 10 1 and c = 34 7 respectively. a) Find → AB and → AC. b) Find a parametric vector equation of the plane that passes through the points A,B and C. iii) Determine the coordinates of the point of intersection of the line xy z = 12 5 + t 1 1 1 for t ∈ R and the plane x 3y + z = 15. iv) Let v = 1 3 1 5 and w = 2 6 β 10 be two vectors in R4. Find the value of β so that the vectors v and w are perpendicular. v) Let u = 21 7 be a vector in R3. a) Find the magnitude of u. b) Write down a vector with a magnitude of 10 which is parallel to u. c 2020 School of Mathematics and Statistics, UNSW Sydney 7vi) Find the determinant of the matrix C = 3 1 04 1 7 1 2 0 . vii) Suppose that u, v and w are distinct non-zero vectors in Rn with the property that projw(u) = projw(v). Prove that (u v) is perpendicular to w. 3. i) Evaluate the limit lim x→0 x2ex 1 cos(πx) . ii) Determine all real values of a and b such that the function f given by f(x) = { x2 for x ≤ 1, x2 + ax+ b for x > 1, is differentiable at x = 1. iii) Let L = lim x→∞ x2 2 x2 + 1 . a) Evaluate L. b) Given any ε > 0, find a constant M (which may depend on ε) such that we have∣∣∣∣x2 2×2 + 1 L ∣∣∣∣ < ε whenever x > M . iv) Let f(x) = x3 + √ 3 x 5 for all real x. a) Use the Intermediate Value Theorem to prove that f has at least one positive real root. b) By considering f ′, or otherwise, show that f has only one real root. v) Use logarithmic differentiation to calculate dy dx for y = (sinx)x. vi) A curve in R2 is given in polar coordinates as r = 6 sin θ, where 0 ≤ θ ≤ π/2. a) Express the equation of the curve using Cartesian coordinates x and y and state the range of x and the range of y. b) Hence, or otherwise, sketch the curve in the xy-plane. 4. i) Evaluate the following integrals: c 2020 School of Mathematics and Statistics, UNSW Sydney 8a) I1 = ∫ 1 x (1 + x)3 dx; b) I2 = ∫ π 0 x cos 2x dx. ii) Determine whether the improper integral K = ∫ ∞ 1 1 + sinx 3×2 dx converges or diverges. iii) a) State the definitions of sinhx and coshx in terms of the exponential function. b) Hence find an expression for d dx cosh(ax) in terms of sinh(ax), where a is a constant. c) Simplify the expression cosh 1 ( cosh( 4726)). iv) A continuous function f satisfies the equation ∫ x 0 f(t) dt = ∫ 1 x t2f(t) dt+ x16 8 + x18 9 1 9 . By differentiating this equation with respect to x, and using the First Fundamental Theorem of Calculus where needed, find f(x). v) Let g be a function defined by g(x) = tan 1(x) + tan 1(1/x). a) What is the (maximal) domain of g Give reasons for your answer. b) By examining g′(x), show that g is piecewise constant on its domain. c) Hence, or otherwise, determine the exact value of tan 1( eα) + tan 1( e α), for all α ∈ R. c 2020 School of Mathematics and Statistics, UNSW Sydney 9MATH1131 JUNE 2011 1. i) Let z = 1 i. a) Find |z|. b) Find Arg(z). c) Use the polar form of z to evaluate z102 and then express your answer in Cartesian form. ii) a) Simplify (2 + 4i)2. b) Hence, or otherwise, solve the quadratic equation z2 4z + (7 4i) = 0. iii) Sketch the following region on the Argand diagram S = {z ∈ C : 0 ≤ Arg(z i) ≤ π 4 }. iv) Evaluate the limit lim x→∞ 1 x √x2 6x 4 . v) Evaluate the improper integral ∫ ∞ 1 x 5/4 dx . vi) A curve in the plane is defined implicitly by the equation x2 3xy2 + 11 = 0 . a) Show that the curve has slope at the point (x, y) given by dy dx = 2x 3y2 6xy . b) Find the equation of the tangent to the curve at the point (1, 2). c) Write a Maple command to plot the curve in the region 1 ≤ x ≤ 4 and 5 ≤ y ≤ 5. 2. i) Use De Moivre’s Theorem to prove that cos(4θ) = 8 cos4 θ 8 cos2 θ + 1. ii) Consider the line with parametric vector equation xy z = 12 4 + 32 5 t, for t ∈ R. a) Give two points on the line. b) Give a vector parallel to the line. c) Explain why the line is perpendicular to the plane P with Cartesian equation 9x+ 6y + 15z = 24. c 2020 School of Mathematics and Statistics, UNSW Sydney 10 d) Find a point on the line whose y-coordinate is 0. iii) Consider the following Maple session: > with(LinearAlgebra): > A:=<<1,1,0>|<-1,1,0>|<0,0,sqrt(2)>>; A := 1 1 0 1 1 0 0 0 √ 2 > A^2; 0 2 0 2 0 0 0 0 2 > A^4; 4 0 0 0 4 0 0 0 4 > A^8; 16 0 0 0 16 0 0 0 16 Use the above MAPLE session to find the inverse of A7. iv) Evaluate the limit lim x→1 (x 1)2 1 + cos(πx) . v) Evaluate the indefinite integral ∫ x sin(2x) dx. vi) The function f has domain [0, 1] and is defined by f(x) = ex + ax, where a is a positive constant. a) Prove that 2 is in the range of f . b) Prove that f has an inverse function f 1. c) Find the domain of f 1. c 2020 School of Mathematics and Statistics, UNSW Sydney 11 3. i) Let A = ( 1 4 7 2 1 3 ) and B = 1 30 5 1 0 . a) Find AB. b) What is the size of the matrix BA ii) A pet shop has x hamsters, y rabbits and z guinea pigs. Each hamster eats 50g of dry food and 40g of fresh vegetables, and needs 1m2 of space. Each rabbit eats 300g dry food and 320g of fresh vegetables, and needs 5m2 of space. Each guinea pig eats 100g of dry food and 200g of fresh vegetables, and needs 3m2 of space. Altogether they eat 2900g of dry food and 3920g of fresh vegetables, and need 63m2 of space. a) Explain why 5x+ 30y + 10z = 290. b) Write down a system of linear equations that determine x, y and z. c) Reduce your system to echelon form and solve to find the number of hamsters, rabbits and guinea pigs. iii) The points A, B and C in R3 have position vectors a = 10 1 , b = 23 4 and c = 41 5 respectively. Write down a parametric vector equation of the plane passing through A, B and C. iv) Consider the following system of linear equations. x+ y z = 2 2x+ 3y + z = 6 a) Using Gaussian Elimination find the general solution to the system of equations. b) Hence, or otherwise, find a solution to the system with the property that the sum of the x, y and z coordinates of the solution is 0. c 2020 School of Mathematics and Statistics, UNSW Sydney 12 v) Suppose A,B are two points in R3 with position vectors 12 4 and 31 5 respectively. We let O denote the origin. a) Find | → OB |. b) Find the area of triangle AOB. c) Hence, or otherwise, find the perpendicular distance from A to the line through O and B. vi) Suppose that u and v are non-zero, non-parallel vectors in R3 of the same magnitude. Prove that u v is perpendicular to u+ v. 4. i) a) Give the definition of cosh x. b) Use the definition to prove that 4 cosh3 x = cosh 3x+ 3cosh x . ii) Find a) d dx ∫ x 0 cos t√ 1 + t2 dt ; b) d dx ∫ sinhx 0 cos t√ 1 + t2 dt . Give your answer in simplest form. iii) a) Sketch the curve whose equation in polar coordinates is r = 6 sin 2θ. b) Find the gradient, dydx , of this curve at the point where θ = 1 6π. c 2020 School of Mathematics and Statistics, UNSW Sydney 13 iv) A statue 2 metres high stands on a pillar 2.5 metres high. A person, whose eye is 1.5m above the ground, stands at a distance x metres from the base of the pillar. θ 2 1 1.5 A B C φ x The diagram shows the above information, with the person’s eye being at C. a) Prove that d dt ( cot 1 t ) = 1 1 + t2 . b) Show that θ = cot 1 (x 3 ) cot 1 x c) Hence find the distance x that maximises the angle θ. c 2020 School of Mathematics and Statistics, UNSW Sydney 14 MATH1131 June 2012 1. i) Let u = 3 + 2i and w = 1 5i. a) Find u 2w in Cartesian form. b) Find u/w in Cartesian form. ii) Let z = √ 3 i. a) Calculate |z| and Arg(z). b) Express z in polar form. c) Hence, or otherwise, express z10 + (z)10 in Cartesian form. iii) Evaluate the determinant ∣∣∣∣∣∣ 1 1 4 0 2 7 0 3 1 ∣∣∣∣∣∣ . iv) a) Evaluate lim x→∞ 3×2 + sin(2×2) x2 . b) Evaluate lim x→0 3×2 + sin(2×2) x2 . v) Consider the curve in the plane defined by x2 5x sin y + y2 = 4. Find the equation of the tangent line to this curve at the point (2, 0). vi) Let p(x) = x5 + 5x+ 7. a) Explain why p has at least one real root. b) Prove that p has exactly one real root. 2. i) Use De Moivre’s Theorem to show that cos(3θ) = 4 cos3 θ 3 cos θ. ii) Find the intersection of the line x = 10 2 + λ 11 2 , λ ∈ R, with the plane 5x 2y + z = 17. iii) Consider the following MAPLE session: > with(LinearAlgebra): > A:=<<0,0,1>|<0,1,0>|<-1,0,0>>; A := 0 0 1 0 1 0 1 0 0 > A^2; 1 0 0 0 1 0 0 0 1 c 2020 School of Mathematics and Statistics, UNSW Sydney 15 > A^3; 0 0 1 0 1 0 1 0 0 > A^4; 1 0 0 0 1 0 0 0 1 Using the Maple session above, find the inverse of A2001. iv) The points C and D have position vectors c = 12 3 and d = 41 5 . a) Find the cross product c× d. b) Hence, or otherwise, find the area of the parallelogram with adjacent sides OC and OD, where O is the origin. v) Evaluate the indefinite integral ∫ x4 lnx dx. vi) Consider the three functions: f : R→ R, f(x) = x 2 1 + x2 , g : (0, 3) → R, g(x) = (x 1)2, h : [1, 5]→ R, h(x) = √ 1 + lnx+ sinx cos x. Only one of these functions has a maximum value (on its given domain). Which one is it Give reasons for your answer. vii) Sketch the polar curve r = 2 2 cos θ. You should show any lines of symmetry, and clearly identify where the curve intersects the x and y axes. viii) Suppose that y = xsinx. Find dy dx . 3. i) The points A and B in R3 have position vectors a = 10 1 and b = 12 4 . a) Find a parametric vector equation of the line l passing through A and B. c 2020 School of Mathematics and Statistics, UNSW Sydney 16 b) By evaluating an appropriate dot product, show that the line l from part (a) is perpendicular to the line xy z = 14 6 + 1 3 2 t; t ∈ R. ii) Let P = ( 1 2 1 3 1 4 ) and Q = ( 1 1 1 2 5 0 ) . a) Evaluate PQT . b) What is the size of P TQ c) Does the matrix product PQ exist Explain your answer. iii) A system of three equations in three unknowns x, y and z has been reduced to the following echelon form 1 2 3 50 2 4 8 0 0 α2 9 α 3 . a) For which value of α will the system have no solution b) For which value of α will the system have infinitely many solutions c) For the value of α determined in part (b), find the general solution. iv) The number of $10, $20, and $50 notes in the cash register at Bill’s Burger Barn is x, y and z respectively. The total value of all the notes in the register is $1020. There are 44 notes in total. Also, the number of $10 notes is equal to the sum of the number of $20 notes and the number of $50 notes. a) Explain why x+ 2y + 5z = 102. b) By setting up two further equations and solving the system of three equations in three unknowns x, y and z, determine how many of each type of note is in the cash register. v) The (non-zero) point Q has position vector ab c . The vector →OQ makes angles α, β and γ respectively with the X,Y and Z axes. a) By considering the vector e1

= 10 0 show that a = √ a2 + b2 + c2 cosα. b) Deduce that cos2 α+ cos2 β + cos2 γ = 1. c 2020 School of Mathematics and Statistics, UNSW Sydney 17 c) If the angles α and β are complementary, that is, their sum is 90 , what can be said about the vector → OQ 4. i) Find a quadratic function q(x) = x2 + bx+ c such that the function h(x) = { e3x, if x ≤ 0 q(x), if x > 0 is differentiable at x = 0. ii) Each of the following calculations is expressed in MAPLE. Write each in normal math- ematical notation and evaluate. a) arcsin(sin(7*Pi/3)); b) diff(int(exp(t^2),t=0..x^2),x); iii) Prove that lim x→∞ x2 2 x2 + 3 = 1 as follows: Given any real number > 0, find a real number M (expressed in terms of ), such that if x > M then ∣∣∣∣x2 2×2 + 3 1 ∣∣∣∣ < . iv) Let f : R→ R be defined by f(x) = x3 + sinhx+ 1. a) Explain why f has a differentiable inverse g. b) What is the domain of g c) Evaluate g′(1). v) A chemical process produces Factor X, which flows into a 50 litre tank which is initially empty. At time t ≥ 0, Factor X flows into the tank at the rate of 100 10 + t2 litres per hour. Will the tank eventually overflow Explain your answer. c 2020 School of Mathematics and Statistics, UNSW Sydney 18 MATH1131 June 2013 1. i) Evaluate the following limits: a) lim x→∞ 10x2 + 3x+ sinx 5x2 + 3x 2 , b) lim x→0 e3x 1 sin (7x) . ii) A function f : [0, 5]→ R has the following properties: lim x→2+ f(x) = 3, lim x→2 f(x) = 1, f(2) = 4. Draw a possible sketch of the graph of f . (You do not need to give a formula for your function.) iii) a) State the definitions of coshx and sinhx in terms of the exponential function. b) Prove that cosh2 x sinh2 x = 1. iv) Let z = 5 + 5i and w = 2 + i. a) Find 2z + 3w. b) Find z(w 1). c) Find z/w. v) Suppose that (x+ iy)(3 + 4i) = 13 + 9i, where x, y ∈ R. Find the value of x and the value of y. vi) Let the set S in the complex plane be defined by S = { z ∈ C : |z| ≤ 3 and 0 ≤ Im(z) ≤ 3 } . a) Sketch the set S on a labelled Argand diagram. b) By considering your sketch, or otherwise, find the area of the region defined by S. c 2020 School of Mathematics and Statistics, UNSW Sydney 19 vii) Consider the following MAPLE session. > with(LinearAlgebra): > A:=<<0,1,-1>|<1,0,1>|<-1,1,0>>; A := 0 1 1 1 0 1 1 1 0 > B:=A^2; B := 2 1 1 1 2 1 1 1 2 > C:=A^3; C := 2 3 3 3 2 3 3 3 2 > F:=3*A-C; F := 2 0 0 0 2 0 0 0 2 Without carrying out any row reduction, use the above Maple session to find the inverse of the matrix A. 2. i) A function h is defined by h(x) = { ax2 + 3x, if x ≥ 1 2x+ d, if x < 1. Given that h is differentiable at x = 1, find the values of a and d. ii) Evaluate ∫ ln 2 0 9xe3x dx. iii) Find the equation of the tangent at the origin to the curve implicitly defined by ex + sin y = xy + 1. iv) Sketch the polar curve whose equation in polar coordinates is given by r = 1 + cos 2θ. v) Let z = √ 2 √2i. a) Find |z|. b) Find Arg(z). c 2020 School of Mathematics and Statistics, UNSW Sydney 20 c) Use the polar form of z to evaluate z6. Express your answer in Cartesian form. vi) Let A = ( 2 0 1 1 ) and B = ( 3 0 1 1 2 0 ) . a) Evaluate AB or explain why this product does not exist. b) Evaluate ABT or explain why this product does not exist. vii) a) Find, in polar form, all solutions to the equation z5 = 1, where z ∈ C. b) Hence, or otherwise, express z5 + 1 as a product of real linear and real quadratic factors. 3. i) Determine the point of intersection of the line xy z = 1 1 1 + t 12 1 for t ∈ R . and the plane 4x 5y + 3z = 0. ii) Let M = 1 2 02 5 1 0 2 α . a) Evaluate the determinant of M . b) Determine the value(s) of α for which M does not have an inverse. c) Find the inverse of M when α = 1. iii) Let u = 21 0 and v = 31 1 . a) Find the cross product u× v. b) Hence find the Cartesian equation of the plane parallel to u and v and passing through the point 14 2 . iv) Let u = 2 1 4 3 and v = 0 3 3 β be two vectors in R4. a) Find the value of β so that the vectors u and v are orthogonal. b) For the value β = 1, find the projection, proju(v), of v onto u. v) Let A, B and D be three points on some circle with centre C in R2 with position vectors a = ( 2 1 ) , b = ( 2 2 ) , and d = ( 4 2 ) . a) Let M be the midpoint of the line joining A and B. Find the position vector m of the point M . c 2020 School of Mathematics and Statistics, UNSW Sydney 21 b) Find a non-zero vector u that is perpendicular to → AB. c) Hence or otherwise, find the parametric vector equation for the line whose points are equidistant from A and B. d) Given that the parametric vector equation for the line whose points are equidistant from B and D is x = ( 3 2 ) + μ ( 0 1 ) , μ ∈ R, find the centre C of the circle. 4. i) Find ∫ cos(ln(x)) x dx. ii) Use the Pinching theorem to evaluate lim x→∞ e x sin(x). iii) Show that the improper integral ∫ ∞ 0 dx x2 + ex converges. iv) The following calculation is expressed in MAPLE > F:=diff(int(sin(sqrt(t)),t=0..x^2),x); a) Write the calculation using standard mathematical notation. b) Evaluate F . v) Let p(x) = x3 + 4x 7. a) Use the Intermediate Value theorem to show that p has at least one real root in the interval [1, 2]. b) Show that p has exactly one real root in the interval [1, 2]. c) Let g be the inverse of p and α be the unique root of p, whose existence is guaranteed in part b). Express g′(0) in terms of α. vi) Use the Mean Value Theorem to prove that, for x > 0, ln(1 + x) > x 1 + x . c 2020 School of Mathematics and Statistics, UNSW Sydney 22 MATH1131 June 2014 1. i) Let z = 7 + i and w = 4 + 3i. a) Find 2z w in a+ ib form. b) Find 5(w i)/z in a+ ib form. c) Find |zw|. d) Find Arg(zw). e) Hence, or otherwise, show that Arg(z) + Arg(w) = π 4 . f) Use the polar form of zw to evaluate (zw)40. ii) Consider the following system of equations: x y z = 1 x 3y + z = 1 2x 3y z = 2. a) Write the system in augmented matrix form and reduce it to row echelon form. b) Solve the system. iii) Evaluate the limits a) lim x→∞ 6×2 + sinx 4×2 + cos x ; b) lim x→0 e2x 2x 1 4×2 . iv) A function g is defined by g(x) = |x2 16| x 4 if x 6= 4 α if x = 4. By considering the left and right hand limits at x = 4, show that no value of α can make g continuous at the point x = 4. v) Let f(x) = x5 + x3 + x 2. a) Prove that f has at least one real root in the interval [0, 2], naming any theorems you use. b) State, with reasons, the number of real roots of f . 2. i) Use a substitution to find the integral∫ dx x(1 + (log x)2) . ii) a) Give the definitions of sinhx and cosh x in terms of the exponential function. b) Use your definitions to prove that sinh(2x) = 2 sinhx coshx. iii) Evaluate the integral ∫ π/3 0 x sin(2x) dx. c 2020 School of Mathematics and Statistics, UNSW Sydney 23 iv) Simplify the matrix expression (ATA) 1(ATA)T , where A is an invertible matrix. v) Consider the three points A,B,C in R3 with position vectors 01 3 , 31 4 , and 32 4 respectively. a) Find a parametric vector form for the plane Π that passes through points A, B, and C. b) Calculate the cross product n = → AB × →AC, showing your working. c) Hence, or otherwise, find a Cartesian equation for the plane Π. d) Find the area of the triangle ABC. e) Find the minimal distance from the point P 53 0 to the plane Π. vi) Consider the following MAPLE session. > with(LinearAlgebra): > A := <<0,1,1>|<1,-1,-1>|<-2,1,0>>; A := 0 1 2 1 1 1 1 1 0 > B := A^2; B := 1 1 1 0 1 3 1 2 3 > C:=A^3; C := 2 3 3 2 2 1 1 0 4 > F := 2A + B + C; F := 1 0 0 0 1 0 0 0 1 Using the above MAPLE session, or otherwise, find the 3×3 matrix which is the inverse of the matrix A. 3. i) A block of wood is subject to 3 vector forces: F1 = 1, in the direction West, F2 = 1 in the direction South, F3 = 2 in the direction North-East, each measured in Newtons. Let F = F1 + F2 + F3 be the resultant force on the block. a) On a scale diagram draw the 3 forces F1, F2, F3 and the resultant force F. b) Find the exact value of |F| and the direction of F. c 2020 School of Mathematics and Statistics, UNSW Sydney 24 ii) Let M = 1 1 12 1 3 1 2 1 and N = (3 1 4 2 ) . a) Evaluate the determinant of M . b) Write down the inverse of N . iii) Let p(z) = z4 + 2z2 3. a) Show that p(1) = p( 1) = 0. b) Factor p(z) into two real quadratic polynomials q(z) and r(z). c) Find the roots of p(z). d) Factor p(z) into four complex linear polynomials. iv) Consider the line and the plane Π given by the following equations: : x = xy z = 32 1 + λ 12 3 , λ ∈ R, Π : 6x+ 8y 9z = 0. Determine the point of intersection of the line and the plane Π. v) a) Use De Moivre’s Theorem to prove that 4 cos3 θ = cos 3θ + 3cos θ. b) Deduce that 2 cos π 9 is a root of the polynomial q(z) = z3 3z 1. vi) Let u = 01 1 and v = 21 β be two vectors in R3. a) Find the value of β so that the vectors u and v are orthogonal. b) For the value β = 0, find the projection, projv u, of u onto v. c) Find the value of β so that the angle between u and v is π 4 . 4. i) Use the Fundamental Theorem of Calculus to find d dx ∫ x3 x2 cos ( 1 t ) dt. ii) Let f(x) = 175×2 350x + 10 x2 2x+ 2 . Consider the following MAPLE session. > f:=(175*x^2-350*x+10)/(x^2-2*x+2); 175×2 350x + 10 x2 2x+ 2 > subs(x=0.0,f); 5.0 > subs(x=4.0,f); 141.0 > solve(f=0,x); c 2020 School of Mathematics and Statistics, UNSW Sydney 25 1 + √ 1155/35, 1 √ 1155/35 > evalf(%); 1.971008312, 0.0289916875 > fdash:=diff(f,x); 350x 350 x2 2x+ 2 ( 175×2 350x + 10) (2x 2) (x2 2x+ 2)2 > solve(fdash=0,x); 1 > subs(x=1.0,f); 165.0 a) Use the information in the MAPLE output to give a rough sketch of f(x) = 175×2 350x + 10 x2 2x+ 2 , for 0 ≤ x ≤ 4. b) Hence, or otherwise, find the maximum and minimum values of f(x) = ∣∣∣∣175×2 350x+ 10×2 2x+ 2 ∣∣∣∣ over the closed interval [0, 4]. iii) Determine, with reasons, whether the improper integral K = ∫ ∞ 0 dx e2x + cos2 x converges or diverges. iv) Sketch in the xy-plane, the graph of the polar curve given by r = 1 cos θ. (You are NOT required to find the slope of the curve.) v) a) State carefully the Mean Value Theorem. b) Suppose 1 < x < y < 1. By applying the Mean Value Theorem to the function f(t) = sin 1 t on the interval [x, y], prove that sin 1 y sin 1 x ≥ y x. vi) Two poles A and B, of heights a metres and b metres respectively, are c metres apart on the horizontal ground. A single tight rope runs from the top of pole A to the point P on the ground between A and B and then to the top of pole B. Assume that the distance from A to P is x, and that the angles θ and φ are as shown in the diagram. c 2020 School of Mathematics and Statistics, UNSW Sydney 26 A BP a b q f x c E F a) Explain why the length L of the rope is given by L = √ a2 + x2 + √ b2 + (c x)2. b) Prove that cos θ = cosφ when dL dx = 0. c) Assuming that cos θ = cosφ minimizes L, using similar triangles, or otherwise, find the value of x that minimizes L. c 2020 School of Mathematics and Statistics, UNSW Sydney 27 MATH1131 June 2015 1. i) Find lim x→∞ x+ sinx 2x . (Give brief reasons for your answer.) ii) The function f is defined by f(x) = { 3 x, 0 ≤ x < 1, (x 2)2 + 1, 1 ≤ x ≤ 3, and f( x) = f(x) for all x. a) Find f(32). b) Sketch the graph of f(x) over the interval 3 ≤ x ≤ 3. c) Which, if any, of the following exists If it exists, state its value. (Give brief reasons for your answer.) lim x→1 f(x), f ′(0). iii) Find, for x > 0, I1 = ∫ dx 1 + √ x . iv) Evaluate lim x→0 1 cos (x2 ) x2 . v) Let z = 5 + i and w = 3 + 2i. a) Find z w in a+ ib form. b) Find 10w/(z 2) in a+ ib form. c) Find |(z/w)8|. d) Find Arg(zw). e) Use the polar form of zw to evaluate (zw)8. vi) Consider the three points A,B,C in R3 with position vectors 11 4 , 21 1 and 14 3 , respectively. Find a parametric vector form for the plane Π that passes through points A, B, and C. c 2020 School of Mathematics and Statistics, UNSW Sydney 28 vii) Consider the following MAPLE session. > with(LinearAlgebra): > A := <<1,1,3>|<1,0,1>|<4,1,5>>; A := 1 1 3 1 0 1 4 1 5 > B := A^2; B := 14 4 19 5 2 8 25 9 38 > C := A^3; C := 94 33 141 39 13 57 186 63 274 > F := C – 6B – 9A; F := 1 0 0 0 1 0 0 0 1 a) Using the above MAPLE session, or otherwise, find the 3 × 3 matrix which is the inverse of the matrix A. b) State (AT )2. 2. i) By writing z = a+ ib, or otherwise, solve z2 = 40+42i, giving your answers in Cartesian form. ii) Find condition(s) on b1, b2, b3 to ensure that the following system has a solution. x + 2y = b1 x + y z = b2 2x + y 3z = b3 iii) Sketch the following region on the Argand diagram: S = { z ∈ C : |z i 1| ≤ 1 or |Im(z)| ≥ 1 } . iv) Let u = 21 1 and v = 1 1 2 . a) Calculate the acute angle θ between the vectors u and v. b) Find u× v. v) Consider the function f(x) = { 1 + ax2, x ≤ 1, bx+ 2×3, x > 1, where a, b ∈ R. c 2020 School of Mathematics and Statistics, UNSW Sydney 29 a) Find all values of a and b such that f is continuous at x = 1. b) Find all values of a and b such that f is differentiable at x = 1. vi) Evaluate I2 = ∫ x3 lnx dx. vii) Consider the polar curve r = 1 + cos θ, for 0 ≤ θ ≤ 2π. a) Find the slope of the tangent to the curve at the point with Cartesian coordinates (0, 1). b) Find the polar coordinates of the points at which the tangent to the curve is hori- zontal. c) Sketch the polar curve. c 2020 School of Mathematics and Statistics, UNSW Sydney 30 3. i) Show that the set of points z in the complex plane that satisfy the equation 2|z 3i| = |z + 3i| lie on a circle. State the radius and centre of the circle. ii) Consider the complex polynomial p(z) = z4 3z3 + 6z2 12z + 8. a) Given that p(2i) = 0, factorise p into linear and quadratic factors with real coeffi- cients. b) Find all roots of p. iii) Consider the plane Π in R3 with Cartesian equation x 3y + 2z = 1 . a) Find a point-normal form for the plane Π. b) Show that the line with parametric vector form x = 31 4 + λ 20 1 , λ ∈ R is parallel to the plane Π. c) Find the shortest distance between the point P (4, 2, 2) and the plane Π. iv) a) Find the determinant of A = 1 1 33 a 2 2 1 1 . b) For what value of a does the matrix A not have an inverse. c) Determine the value of a and the value of b for which A 1 = B, where B = 1 4 51 b 7 1 3 4 . v) Let X be an n× n matrix with determinant |X| = 1. Prove that the inverse of X also has determinant equal to 1. c 2020 School of Mathematics and Statistics, UNSW Sydney 31 4. i) a) Define the functions sinhx and coshx in terms of the exponential function. b) From the definitions in (a), show that d dx (cosh 6x) = 6 sinh 6x. c) Show that, for x > 0, ln (sinhx) < x ln 2. ii) Prove carefully that the improper integral ∫ ∞ 1 lnx x3 dx converges. iii) Find d dx (∫ x3 0 cos ( t2 ) dt ) . iv) Consider the polynomial p(x) = x3 + 3x+ 1 defined on R. a) Use the Intermediate Value Theorem to show that the equation p(x) = 0 has at least one real root. b) Show that the function p has an inverse function g. c) Find the value of g′(1). v) a) State the Mean Value Theorem. b) Apply the Mean Value Theorem to f(t) = cos t, on a suitably chosen interval, to show that | cos y cos x| ≤ |y x|, for all x, y ∈ R. c 2020 School of Mathematics and Statistics, UNSW Sydney 32 c 2020 School of Mathematics and Statistics, UNSW Sydney 33 PAST HIGHER EXAM PAPERS c 2020 School of Mathematics and Statistics, UNSW Sydney 34 MATH1141 JUNE 2011 2. i) You may use the following Maple session to assist you in answering the question below. > with(LinearAlgebra): > A:=<<1,2,1>|<1,3,a>|<-1,a,3>>; A := 1 1 1 2 3 a 1 a 3 > t:=<1,3,2>; t := 1 3 2 > B:=; B := 1 1 1 1 2 3 a 3 1 a 3 2 > G:=GaussianElimination(B); G := 1 1 1 1 0 1 a+ 2 1 0 0 6 a2 a 2 a For which values of a will the system Ax = t have: a) no solutions, b) unique solution, c) infinitely many solutions ii) Evaluate the determinant ∣∣∣∣∣∣ 2 0 1 1 3 0 5 7 3 ∣∣∣∣∣∣ . iii) Find the point of intersection, if any, of the line x = λ 12 3 , λ ∈ R, with t