1. Given that 2 + z z = 2 i, z , find z in the form a + ib.(Total 4 marks)2. The diagram below shows a circle with centre O. The points A, B, C lie on the circumference of the circle and [AC] is a diameter.Let b a OB and OA.(a) Write down expressions for CB and AB in terms of the vectors a and b.(2)(b) Hence prove that angle C BA is a right angle.(3)(Total 5 marks)3. The points A(1, 2, 1), B(3, 1, 4), C(5, 1, 2) and D(5, 3, 7) are the vertices of a tetrahedron.(a) Find the vectors AC and AB.(2)(b) Find the Cartesian equation of the plane that contains the face ABC. (4)(c) Find the vector equation of the line that passes through D and is perpendicular to. Hence, or otherwise, calculate the shortest distance to D from. (5)(d) (i) Calculate the area of the triangle ABC.(ii) Calculate the volume of the tetrahedron ABCD.(4)(e) Determine which of the vertices B or D is closer to its opposite face.(4)(Total 19 marks)4. In the diagram below, [AB] is a diameter of the circle with centre O. Point C is on the circumference of the circle. Let c b OC and OB.IB Questionbank Mathematics Higher Level 3rd edition 1 (a) Find an expression for CB and for AC in terms of b and c.(2)(b) Hence prove that B CA is a right angle.(3)(Total 5 marks)5. Port A is defined to be the origin of a set of coordinate axes and port B is located at the point (70, 30), where distances are measured in kilometres. A ship S1 sails from port A at 10:00 in a straight line such that its position t hours after 10:00 is given by r =

,_

2010t.A speedboat S2 is capable of three times the speed of S1 and is to meet S1 by travelling the shortest possible distance. What is the latest time that S2 can leave port B?(Total 7 marks)6. Consider the functions f(x) = x3 + 1 and g(x) = 113+ x. The graphs of y = f(x) and y = g(x) meet at the point (0, 1) and one other point, P.(a) Find the coordinates of P.(1)(b) Calculate the size of the acute angle between the tangents to the two graphs at the point P.(4)(Total 5 marks)7. The points P(1, 2, 3), Q(2, 1, 0), R(0, 5, 1) and S form a parallelogram, where S is diagonally opposite Q.(a) Find the coordinates of S.(2)(b) The vector product PS PQ

,_

m713. Find the value of m.(2)(c) Hence calculate the area of parallelogram PQRS.(2) (d) Find the Cartesian equation of the plane, 1, containing the parallelogram PQRS.(3)(e) Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to the plane 1.(1)(f) Hence find the point on the plane that is closest to the origin.(3)(g) A second plane, 2, has equation x 2y + z = 3. Calculate the angle between the two planes.(4)(Total 17 marks)8. (a) Show that the two planes1 : x + 2y z = 12 : x + z = 2are perpendicular.(3)(b) Find the equation of the plane 3 that passes through the origin and is perpendicular to both 1 and 2.(4)(Total 7 marks)9. Consider the vectors b a b a + OC and OB , OA. Show that if a=b then(a + b)(a b) = 0. Comment on what this tells us about the parallelogram OACB.(Total 4 marks)10. The three vectors a, b and c are given by

,_

,_

,_

674,34,232c b axyxxxy where x, y .(a) If a + 2b c = 0, find the value of x and of y.(3)(b) Find the exact value of a + 2b.(2)(Total 5 marks)11. (a) Consider the vectors a = 6i + 3j + 2k, b = 3j + 4k.(i) Find the cosine of the angle between vectors a and b.(ii) Find a b.(iii) Hence find the Cartesian equation of the plane containing the vectors a and b and passing through the point (1, 1, 1).(iv) The plane intersects the x-y plane in the line l. Find the area of the finite triangular region enclosed by l, the x-axis and the y-axis.(11)IB Questionbank Mathematics Higher Level 3rd edition 3(b) Given two vectors p and q,(i) show that p p = p2;(ii) hence, or otherwise, show that p + q2 = p2 + 2p q + q2;(iii) deduce that p + qp + q.(8)(Total 19 marks)12. A plane has vector equation r = (2i + 3j 2k) + (2i + 3j + 2k) + (6i 3j + 2k).(a) Show that the Cartesian equation of the plane is 3x + 2y 6z = 12.(6)(b) The plane meets the x, y and z axes at A, B and C respectively. Find the coordinates of A, B and C.(3)(c) Find the volume of the pyramid OABC.(3)(d) Find the angle between the plane and the x-axis.(4)(e) Hence, or otherwise, find the distance from the origin to the plane .(2)(f) Using your answers from (c) and (e), find the area of the triangle ABC.(2)(Total 20 marks)13. The three planes2x 2y z = 34x + 5y 2z = 33x + 4y 3z = 7intersect at the point with coordinates (a, b, c).(a) Find the value of each of a, b and c.(2)(b) The equations of three other planes are2x 4y 3z = 4x + 3y + 5z = 23x 5y z = 6.Find a vector equation of the line of intersection of these three planes.(4)(Total 6 marks)14. Consider the plane with equation 4x 2y z = 1 and the line given by the parametric equationsx = 3 2y = (2k 1) + z = 1 + k.Given that the line is perpendicular to the plane, find(a) the value of k;(4)(b) the coordinates of the point of intersection of the line and the plane.(4)(Total 8 marks)15. Consider the vectors a = sin(2)i cos(2)j + k and b = cos i sin j k, where 0 < < 2.Let be the angle between the vectors a and b.(a) Express cos in terms of .(2)(b) Find the acute angle for which the two vectors are perpendicular.(2)(c) For = 6 7, determine the vector product of a and b and comment on the geometrical significance of this result.(4)(Total 8 marks)16. The diagram shows a cube OABCDEFG.Let O be the origin, (OA) the x-axis, (OC) the y-axis and (OD) the z-axis.Let M, N and P be the midpoints of [FG], [DG] and [CG], respectively.The coordinates of F are (2, 2, 2).(a) Find the position vectors OP and ON , OM in component form.(3)(b) Find MN MP.(4)(c) Hence,IB Questionbank Mathematics Higher Level 3rd edition 5(i) calculate the area of the triangle MNP;(ii) show that the line (AG) is perpendicular to the plane MNP;(iii) find the equation of the plane MNP.(7)(d) Determine the coordinates of the point where the line (AG) meets the plane MNP.(6)(Total 20 marks)17. A triangle has vertices A(1, 1, 1), B(1, 1, 0) and C(1, 1, 1).Show that the area of the triangle is 6.(Total 6 marks)18. (a) Show that a Cartesian equation of the line, l1, containing points A(1, 1, 2) and B(3, 0, 3) has the form 121121 + z y x.(2)(b) An equation of a second line, l2, has the form 132211 z y x. Show that the lines l1 and l2 intersect, and find the coordinates of their point of intersection.(5)(c) Given that direction vectors of l1 and l2 are d1 and d2 respectively, determine d1 d2.(3)(d) Show that a Cartesian equation of the plane,, that contains l1 and l2 is x y + 3z = 6.(3)(e) Find a vector equation of the line l3 which is perpendicular to the plane and passes through the point T(3, 1, 4).(2)(f) (i) Find the point of intersection of the line l3 and the plane. (ii) Find the coordinates of T, the reflection of the point T in the plane. (iii) Hence find the magnitude of the vector T T .(7)(Total 22 marks)19. Find the angle between the lines 21 x = 1 y = 2z and x = y = 3z.(Total 6 marks)20. Consider the planes defined by the equations x + y + 2z = 2, 2x y + 3z = 2 and5x y + az = 5 where a is a real number.(a) If a = 4 find the coordinates of the point of intersection of the three planes.(2)(b) (i) Find the value of a for which the planes do not meet at a unique point.(ii) For this value of a show that the three planes do not have any common point.(6)(Total 8 marks)21. The position vector at time t of a point P is given byOP = (1 + t)i + (2 2t)j + (3t 1)k, t 0.IB Questionbank Mathematics Higher Level 3rd edition 7(a) Find the coordinates of P when t = 0.(2)(b) Show that P moves along the line L with Cartesian equationsx 1 = 3122 + z y.(2)(c) (i) Find the value of t when P lies on the plane with equation 2x + y + z = 6.(ii) State the coordinates of P at this time.(iii) Hence find the total distance travelled by P before it meets the plane.(6)The position vector at time t of another point, Q, is given by

,_

2211 OQttt, t 0.(d) (i) Find the value of t for which the distance from Q to the origin is minimum.(ii) Find the coordinates of Q at this time.(6)(e) Let a, b and c be the position vectors of Q at times t = 0, t = 1, and t = 2 respectively.(i) Show that the equation a b = k(b c) has no solution for k.(ii) Hence show that the path of Q is not a straight line.(7)(Total 23 marks)22. Given that a = 2 sin i + (1 sin )j, find the value of the acute angle , so that a is perpendicular to the line x + y = 1.(Total 5 marks)23. The vector equation of line l is given as

,_

+

,_

,_

121631zyx.Find the Cartesian equation of the plane containing the line l and the point A(4, 2, 5).(Total 6 marks)24. Two lines are defined by l1 : r = 4734: and2236432+

,_

+

,_

y xl = (z + 3).(a) Find the coordinates of the point A on l1 and the point B on l2 such that AB is perpendicular to both l1 and l2.(13)(b) Find AB.(3)(c) Find the Cartesian equation of the plane that contains l1 and does not intersect l2.(3)(Total 19 marks)25. The points A, B, C have position vectors i + j + 2k, i + 2 j + 3k, 3i + k respectively and lie in the plane .(a) Find(i) the area of the triangle ABC;(ii) the shortest distance from C to the line AB;(iii) the cartesian equation of the plane .(14)The line L passes through the origin and is normal to the plane , it intersects at the point D.(b) Find(i) the coordinates of the point D;(ii) the distance of from the origin.(6)(Total 20 marks)26. Given any two non-zero vectors a and b, show that a b2 = a2b2 (a b)2.(Total 6 marks)27. Consider the points A(1, 1, 4), B(2, 2, 5) and O(0, 0, 0).(a) Calculate the cosine of the angle between OA and AB.(5)(b) Find a vector equation of the line L1 which passes through A and B.(2)The line L2 has equation r = 2i + 4j + 7k + t(2i + j + 3k), where t .(c) Show that the lines L1 and L2 intersect and find the coordinates of their point of intersection.(7)(d) Find the Cartesian equation of the plane that contains both the line L2 and the point A.(6)(Total 20 marks)28. A ray of light coming from the point (1, 3, 2) is travelling in the direction of vector

,_

214 and meets the plane : x + 3y + 2z 24 = 0.Find the angle that the ray of light makes with the plane.(Total 6 marks)29. Find the vector equation of the line of intersection of the three planes represented by the following system of equations.IB Questionbank Mathematics Higher Level 3rd edition 92x 7y + 5z =16x + 3y z = 1 14x 23y +13z = 5(Total 6 marks)30. Three distinct non-zero vectors are given by c b a OC and , OB , OA.If OA is perpendicular to BC and OB is perpendicular to CA, show that OC is perpendicular to AB.(Total 6 marks)31. The angle between the vector a = i 2j + 3k and the vector b = 3i 2j + mk is 30.Find the values of m.(Total 6 marks)32. (a) Find the set of values of k for which the following system of equations has no solution.x + 2y 3z = k3x + y + 2z = 45x + 7z = 5(4)(b) Describe the geometrical relationship of the three planes represented by this system of equations.(1)(Total 5 marks)33. (a) Write the vector equations of the following lines in parametric form.r1 =

,_

+

,_

212723mr2 =

,_

+

,_

114241n(2)(b) Hence show that these two lines intersect and find the point of intersection, A.(5)(c) Find the Cartesian equation of the plane that contains these two lines. (4)(d) Let B be the point of intersection of the plane and the line r =

,_

+

,_

283038.Find the coordinates of B.(4)(e) If C is the mid-point of AB, find the vector equation of the line perpendicular to the plane and passing through C.(3)(Total 18 marks)34. The line L is given by the parametric equations x = 1 , y = 2 3, z = 2.Find the coordinates of the point on L that is nearest to the origin.(Total 6 marks)35. (a) Show that the following system of equations will have a unique solution when a 1.x + 3y z = 03x + 5y z = 0x 5y + (2 a)z = 9 a2(5)(b) State the solution in terms of a.(6)(c) Hence, solvex + 3y z = 03x + 5y z = 0x 5y + z = 8(2)(Total 13 marks)36. Consider the points A(1, 2, 1), B(0, 1, 2), C(1, 0, 2) and D(2, 1, 6).(a) Find the vectors AB and BC.(2)(b) Calculate BC AB.(2)(c) Hence, or otherwise find the area of triangle ABC.(3)(d) Find the Cartesian equation of the plane P containing the points A, B and C.(3)(e) Find a set of parametric equations for the line L through the point D and perpendicular to the plane P.(3)(f) Find the point of intersection E, of the line L and the plane P.(4)(g) Find the distance from the point D to the plane P.(2)(h) Find a unit vector that is perpendicular to the plane P.(2)(i) The point F is a reflection of D in the plane P. Find the coordinates of F.(4)(Total 25 marks)37. (a) Show that lines 133212 z y x and 244312 z y x intersect and find the coordinates of P, the point of intersection.(8)(b) Find the Cartesian equation of the plane that contains the two lines.(6)(c) The point Q(3, 4, 3) lies on . The line L passes through the midpoint of [PQ]. Point S is on L such that QS PS = 3, and the triangle PQS is normal to the plane . Given that there are two IB Questionbank Mathematics Higher Level 3rd edition 11possible positions for S, find their coordinates.(15)(Total 29 marks)38. A triangle has its vertices at A(1, 3, 2), B(3, 6, 1) and C(4, 4, 3).(a) Show that AC AB = 10.(3)(b) Find C A B.(5)(Total 8 marks)39. Consider the matrix A =

,_

12 0a.(a) Find the matrix A2.(2)(b) If det A2 = 16, determine the possible values of a.(3)(Total 5 marks)40. The equations of three planes, are given byax + 2y + z = 3x + (a + 1)y + 3z = 12x + y + (a + 2)z = kwhere a .(a) Given that a = 0, show that the three planes intersect at a point.(3)(b) Find the value of a such that the three planes do not meet at a point.(5)(c) Given a such that the three planes do not meet at a point, find the value of k such that the planes meet in one line and find an equation of this line in the form

,_

+

,_

,_

nmlzyxzyx000.(6)(Total 14 marks)41. Consider the matrix A =

,_

cos 2 sinsin 2 cos, for 0 < < 2.(a) Show that det A = cos .(3)(b) Find the values of for which det A2 = sin .(3)(Total 6 marks)42. Consider the matrices

,_

,_

2 23 1,4 52 3B A.(a) Find BA.(2)(b) Calculate det (BA).(2)(c) Find A(A1B + 2A1)A.(3)(Total 7 marks)43. The system of equations2x y + 3z = 23x + y + 2z = 2x + 2y + az = bis known to have more than one solution. Find the value of a and the value of b.(Total 5 marks)44. Let A, B and C be non-singular 22 matrices, I the 22 identity matrix and k a scalar. The following statements are incorrect. For each statement, write down the correct version of the right hand side.(a) (A + B)2 = A2 + 2AB + B2(2)(b) (A kI)3 = A3 3kA2 + 3k2A k3(2)(c) CA = B C = AB(2)(Total 6 marks)45. Consider the matrix

,_

2 01 2 01 1k kkkFind all possible values of k for which the matrix is singular.(Total 5 marks)46. Consider the matrix A =

,_

+1 e 2e exx x, where x .Find the value of x for which A is singular.(Total 6 marks)47. Let M =

,_

a bb a where a and b are non-zero real numbers.(a) Show that M is non-singular.(2)(b) Calculate M2.IB Questionbank Mathematics Higher Level 3rd edition 13(2)(c) Show that det(M2) is positive.(2)(Total 6 marks)48. Matrices A, B and C are defined as

,_

,_

,_

408,1 3 00 1 31 2 1,7 3 93 1 31 5 1C B A.(a) Given that AB =

,_

aaa0 00 00 0, find a.(1)(b) Hence, or otherwise, find A1.(2)(c) Find the matrix X, such that AX = C.(2)(Total 5 marks)49. Let M be the matrix .1 11 00 2

,_

Find all the values of for which M is singular.(Total 6 marks)50. Find the determinant of A, where A =

,_

6 4 78 5 92 1 3.(Total 5 marks)51. If A =

,_

12 1k and A2 is a matrix whose entries are all 0, find k.(Total 5 marks)52. Given that M =

,_

4 31 2 and that M2 6M + kI = 0 find k.(Total 5 marks)53. The square matrix X is such that X3 = 0. Show that the inverse of the matrix (I X) isI + X + X2.(Total 6 marks)54. (a) Write down the inverse of the matrixA =

,_

3 5 11 2 21 3 1(2)(b) Hence, find the point of intersection of the three planes.x 3y + z = 12x + 2y z = 2x 5y + 3z = 3(3)(c) A fourth plane with equation x + y + z = d passes through the point of intersection. Find the value of d.(1)(Total 6 marks)55. A geometric sequence u1, u2, u3, ... has u1 = 27 and a sum to infinity of 281.(a) Find the common ratio of the geometric sequence.(2)An arithmetic sequence v1, v2, v3, ... is such that v2 = u2 and v4 = u4.(b) Find the greatest value of N such that >Nnnv10.(5)(Total 7 marks)56. (a) Show that 2 cos 12 sin+ = tan.(2)(b) Hence find the value of cot8 in the form a + 2 b, where a, b .(3)(Total 5 marks)57. The diagram shows a tangent, (TP), to the circle with centre O and radius r. The size ofA OP is radians. (a) Find the area of triangle AOP in terms of r and .(1)IB Questionbank Mathematics Higher Level 3rd edition 15(b) Find the area of triangle POT in terms of r and .(2)(c) Using your results from part (a) and part (b), show that sin < < tan .(2)(Total 5 marks)58. (a) (i) Sketch the graphs of y = sin x and y = sin 2x, on the same set of axes,for 0 x 2.(ii) Find the x-coordinates of the points of intersection of the graphs in thedomain 0 x 2.(iii) Find the area enclosed by the graphs.(9)(b) Find the value of xxxd410 using the substitution x = 4 sin2 .(8)(c) The increasing function f satisfies f(0) = 0 and f(a) = b, where a > 0 and b > 0.(i) By reference to a sketch, show that b ax x f ab x x f010d ) ( d ) (.(ii) Hence find the value of xxd4arcsin20

,_

.(8)(Total 25 marks)59. The cumulative frequency graph below represents the weight in grams of 80 apples picked from a particular tree.(a) Estimate the(i) median weight of the apples;(ii) 30th percentile of the weight of the apples.(2)(b) Estimate the number of apples that weigh more than 110 grams.(2)(Total 4 marks)60. GivenABC, with lengths shown in the diagram below, find the length of the line segment [CD].diagram not to scale(Total 5 marks)61. The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm. If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two arcs AB. (Total 7 marks)62. The points P and Q lie on a circle, with centre O and radius 8 cm, such that Q OP = 59.IB Questionbank Mathematics Higher Level 3rd edition 17Find the area of the shaded segment of the circle contained between the arc PQ andthe chord [PQ].(Total 5 marks)63. The vertices of an equilateral triangle, with perimeter P and area A, lie on a circle with radius r. Find an expression for AP in the form rk, where k +.(Total 6 marks)64. (a) Given that > 1, use the substitution u = x1 to show thatuuxxd11d1111212 ++.(5)(b) Hence show that arctan + arctan2 1.(2)(Total 7 marks)65. If x satisfies the equation

,_

,_

+3sin sin 23sin x x, show that 11 tan x = a + b3,where a, b +.(Total 6 marks)66. Throughout this question x satisfies 0 x < 2.(a) Solve the differential equation sec2 x 2ddyxy , where y = 1 when x = 0.Give your answer in the form y = f(x).(7)(b) (i) Prove that 1 sec x 1 + tan x.(ii) Deduce that 2 ln214d sec440+ x x.(8)(Total 15 marks)67. The graph below shows y = a cos (bx) + c.Find the value of a, the value of b and the value of c.(Total 4 marks)68. In the right circular cone below, O is the centre of the base which has radius 6 cm.The points B and C are on the circumference of the base of the cone. The height AO of the cone is 8 cm and the angle C OB is 60.diagram not to scaleCalculate the size of the angle C AB.(Total 6 marks)69. Points A, B and C are on the circumference of a circle, centre O and radius r.A trapezium OABC is formed such that AB is parallel to OC, and the angle C OAIB Questionbank Mathematics Higher Level 3rd edition 19is , 2 < .diagram not to scale(a) Show that angle C OB is .(3)(b) Show that the area, T, of the trapezium can be expressed asT = 2 sin21sin212 2r r .(3)(c) (i) Show that when the area is maximum, the value of satisfiescos = 2 cos 2.(ii) Hence determine the maximum area of the trapezium when r = 1.(Note: It is not required to prove that it is a maximum.)(5)(Total 11 marks)70. Consider the triangle ABC where C AB = 70, AB = 8 cm and AC = 7 cm. The point D on the side BC is such that DCBD = 2.Determine the length of AD.(Total 6 marks)71. (a) Consider the set A = {1, 3, 5, 7} under the binary operation *, where * denotes multiplication modulo 8.(i) Write down the Cayley table for {A, *}.(ii) Show that {A, *} is a group.(iii) Find all solutions to the equation 3 * x * 7 = y. Give your answers in theform (x, y).(9)(b) Now consider the set B = {1, 3, 5, 7, 9} under the binary operation , where denotes multiplication modulo 10. Show that {B, } is not a group.(2)(c) Another set C can be formed by removing an element from B so that {C, } is a group.(i) State which element has to be removed.(ii) Determine whether or not {A, *} and {C, } are isomorphic.(3)(Total 14 marks)72. Consider the function f : x x arccos4.(a) Find the largest possible domain of f.(4)(b) Determine an expression for the inverse function, f1, and write down its domain.(4)(Total 8 marks)73. Let be the angle between the unit vectors a and b, where 0 .(a) Express a b and a + b in terms of .(3)(b) Hence determine the value of cos for which a + b = 3a b.(2)(Total 5 marks)74. Triangle ABC has AB = 5cm, BC = 6 cm and area 10 cm2.(a) Find Bsin.(2)(b) Hence, find the two possible values of AC, giving your answers correct to two decimal places.(4)(Total 6 marks)75. The diagram below shows a curve with equation y = 1 + k sin x, defined for 0 x 3.IB Questionbank Mathematics Higher Level 3rd edition 21The point A

,_

2 ,6 lies on the curve and B(a, b) is the maximum point.(a) Show that k = 6.(2)(b) Hence, find the values of a and b.(3)(Total 5 marks)76. (a) Show that 431arctan21arctan ,_

+ ,_

.(2)(b) Hence, or otherwise, find the value of arctan (2) + arctan (3).(3)(Total 5 marks)77. The diagram below shows two straight lines intersecting at O and two circles, each with centre O. The outer circle has radius R and the inner circle has radius r.diagram not to scaleConsider the shaded regions with areas A and B. Given that A : B = 2 : 1, find the exact value of the ratio R : r.(Total 5 marks)78. A triangle has sides of length (n2 + n + 1), (2n + 1) and (n2 1) where n > 1.(a) Explain why the side (n2 + n + 1) must be the longest side of the triangle.(3)(b) Show that the largest angle, , of the triangle is 120.(5)(Total 8 marks)79. Two non-intersecting circles C1, containing points M and S, and C2, containing points N and R, have centres P and Q where PQ = 50. The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is , the size of the obtuse angle NQR is , and the size of the angle MPQ is . The arc length MS is l1 and the arc length NR is l2. This information is represented in the diagram below.The radius of C1 is x, where x 10 and the radius of C2 is 10.(a) Explain why x < 40.(1)(b) Show that cos = 5010 x.(2)(c) (i) Find an expression for MN in terms of x.(ii) Find the value of x that maximises MN.(2)(d) Find an expression in terms of x for(i) ;(ii) .(4)(e) The length of the perimeter is given by l1 + l2 + MN + SR.(i) Find an expression, b(x), for the length of the perimeter in terms of x.(ii) Find the maximum value of the length of the perimeter.(iii) Find the value of x that gives a perimeter of length 200.(9)(Total 18 marks)IB Questionbank Mathematics Higher Level 3rd edition 2380. The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.The points P and Q lie on the larger circle and Q OP = x, where 0 < x < 2. (a) Show that the area of the shaded region is 8 sin x 2x.(3)(b) Find the maximum area of the shaded region.(4)(Total 7 marks)81. In triangle ABC, AB = 9 cm, AC =12 cm, and B is twice the size of C.Find the cosine of C.(Total 5 marks)82. In the diagram below, AD is perpendicular to BC.CD = 4, BD = 2 and AD = 3. D AC = and D AB = .Find the exact value of cos ( ).(Total 6 marks)83. The diagram below shows the boundary of the cross-section of a water channel.The equation that represents this boundary is y = 16 sec

,_

36x 32 where x and y are both measured in cm.The top of the channel is level with the ground and has a width of 24 cm. The maximum depth of the channel is 16 cm.Find the width of the water surface in the channel when the water depth is 10 cm. Give your answer in the form a arccos b where a, b .(Total 6 marks)84. In triangle ABC, BC = a, AC = b, AB = c and [BD] is perpendicular to [AC].(a) Show that CD = b c cos A.(1)(b) Hence, by using Pythagoras Theorem in the triangle BCD, prove the cosine rule for the triangle ABC.(4)(c) If C BA = 60, use the cosine rule to show that c = 2 24321a b a t.(7)(Total 12 marks)85.The above three dimensional diagram shows the points P and Q which are respectively west and south-IB Questionbank Mathematics Higher Level 3rd edition 25west of the base R of a vertical flagpole RS on horizontal ground. The angles of elevation of the top S of the flagpole from P and Q are respectively 35 and 40, and PQ = 20 m.Determine the height of the flagpole.(Total 8 marks)86. The depth, h (t) metres, of water at the entrance to a harbour at t hours after midnight on a particular day is given byh (t) = 8 + 4 sin . 24 0 ,6 ,_

tt(a) Find the maximum depth and the minimum depth of the water.(3)(b) Find the values of t for which h (t) 8.(3)(Total 6 marks)87. Consider triangle ABC with C AB = 37.8, AB = 8.75 and BC = 6.Find AC.(Total 7 marks)88. (a) Sketch the curve f(x) = sin 2x, 0 x .(2)(b) Hence sketch on a separate diagram the graph of g(x) = csc 2x, 0 x , clearly stating the coordinates of any local maximum or minimum points and the equations of any asymptotes.(5)(c) Show that tan x + cot x 2 csc 2x.(3)(d) Hence or otherwise, find the coordinates of the local maximum and local minimum points on the graph of y = tan 2x + cot 2x, 0 x 2.(5)(e) Find the solution of the equation csc 2x = 1.5 tan x 0.5, 0 x 2.(6)(Total 21 marks)89. In a triangle ABC, A = 35, BC = 4 cm and AC = 6.5 cm. Find the possible values of B and the corresponding values of AB.(Total 7 marks)90. Solve sin 2x = 2cos x, 0 x .(Total 6 marks)91. The obtuse angle B is such that tan B = 125. Find the values of(a) sin B;(1)(b) cos B;(1)(c) sin 2B;(2)(d0 cos 2B.(2)(Total 6 marks)92. Given that tan 2 = 43, find the possible values of tan .(Total 5 marks)93. Let sin x = s.(a) Show that the equation 4 cos 2x + 3 sin x cosec3 x + 6 = 0 can be expressed as8s4 10s2 + 3 = 0.(3)(b) Hence solve the equation for x, in the interval [0, ].(6)(Total 9 marks)94. (a) If sin (x ) = k sin (x + ) express tan x in terms of k and. (3)(b) Hence find the values of x between 0 and 360 when k = 21 and = 210.(6)(Total 9 marks)95. The angle satisfies the equation 2 tan2 5 sec 10 = 0, where is in the second quadrant. Find the value of sec .(Total 6 marks)96. The lengths of the sides of a triangle ABC are x 2, x and x + 2. The largest angle is 120.(a) Find the value of x.(6)(b) Show that the area of the triangle is 43 15.(3)(c) Find sin A + sin B + sin C giving your answer in the form r q p where p, q, r .(4)(Total 13 marks)97. A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the angle between these two sides is 60.(a) Calculate the length of the third side of the field.(3)(b) Find the area of the field in the form 3 p, where p is an integer.(3)IB Questionbank Mathematics Higher Level 3rd edition 27Let D be a point on [BC] such that [AD] bisects the 60 angle. The farmer divides the field into two parts by constructing a straight fence [AD] of length x metres.(c) (i) Show that the area o the smaller part is given by 465x and find an expression for the area of the larger part.(ii) Hence, find the value of x in the form 3 q, where q is an integer.(8)(d) Prove that 85DCBD.(6)(Total 20 marks)98. Consider the functions given below.f(x) = 2x + 3g(x) = x1, x 0(a) (i) Find (g f)(x) and write down the domain of the function.(ii) Find (f g)(x) and write down the domain of the function.(2)(b) Find the coordinates of the point where the graph of y = f(x) and the graph ofy = (g1 f g)(x) intersect.(4)(Total 6 marks)99. The diagram below shows the graph of the function y = f(x), defined for all x ,where b > a > 0.Consider the function g(x) = b a x f ) (1.(a) Find the largest possible domain of the function g.(2)(b) On the axes below, sketch the graph of y = g(x). On the graph, indicate any asymptotes and local maxima or minima, and write down their equations and coordinates.(6)(Total 8 marks)100. The quadratic function f(x) = p + qx x2 has a maximum value of 5 when x = 3.(a) Find the value of p and the value of q.(4)(b) The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.(2)(Total 6 marks)101. The diagram shows the graph of y = f(x). The graph has a horizontal asymptote at y = 2.IB Questionbank Mathematics Higher Level 3rd edition 29(a) Sketch the graph of y = ) (1x f.(3)(b) Sketch the graph of y = x f(x).(3)(Total 6 marks)102. A function is defined by h(x) = 2ex xx,e1. Find an expression for h1(x).(Total 6 marks)103. The function f(x) = 4x3 + 2ax 7a, a leaves a remainder of10 when dividedby (x a).(a) Find the value of a.(3)(b) Show that for this value of a there is a unique real solution to the equation f(x) = 0.(2)(Total 5 marks)104. Sketch the graph of f(x) = x + 982 xx. Clearly mark the coordinates of the two maximum points and the two minimum points. Clearly mark and state the equations of the vertical asymptotes and the oblique asymptote.(Total 7 marks)105. Given that Ax3 + Bx2 + x + 6 is exactly divisible by (x + 1)(x 2), find the value of Aand the value of B.(Total 5 marks)106. Shown below are the graphs of y = f(x) and y = g(x).If (f g)(x) = 3, find all possible values of x.(Total 4 marks)107. Solve the equation 4x1 = 2x + 8.(Total 5 marks)108. The graph of y = cx bx a++ is drawn below.IB Questionbank Mathematics Higher Level 3rd edition 31(a) Find the value of a, the value of b and the value of c.(4)(b) Using the values of a, b and c found in part (a), sketch the graph of y = x acx b++on the axes below, showing clearly all intercepts and asymptotes.(4)(Total 8 marks)109. (a) Express the quadratic 3x2 6x + 5 in the form a(x + b)2 + c, where a, b, c .(3)(b) Describe a sequence of transformations that transforms the graph of y = x2 to the graph of y = 3x2 6x + 5.(3)(Total 6 marks)110. A function f is defined by f(x) = 13 2xx, x 1.(a) Find an expression for f1(x).(3)(b) Solve the equation f1(x) = 1 + f1(x).(3)(Total 6 marks)111. Let f(x) = xx442.(a) State the largest possible domain for f.(2)(b) Solve the inequality f(x) 1.(4)IB Questionbank Mathematics Higher Level 3rd edition 33(Total 6 marks)112. (a) Find the solution of the equationln 24x1 = ln 8x+5 + log21612x,expressing your answer in terms of ln 2.(4)(b) Using this value of x, find the value of a for which logax = 2, giving your answer to three decimal places.(2)(Total 6 marks)113. The function f is defined byf(x) = 212 3) 10 3 6 ( + + x x x, for x D,where D is the greatest possible domain of f.(a) Find the roots of f(x) = 0.(2)(b) Hence specify the set D.(2)(c) Find the coordinates of the local maximum on the graph y = f(x).(2)(d) Solve the equation f(x) = 3.(2)(e) Sketch the graph of y= f(x), for x D.(3)(f) Find the area of the region completely enclosed by the graph of y = f(x).(3)(Total 14 marks)114. The functions f, g and h are defined byf(x) = 1 + ex, for x ,g(x) = x1, for x / {0},h(x) = sec x, for x /'+nn,21 2;.(a) Determine the range of the composite function g f.(3)(b) Determine the inverse of the function g f, clearly stating the domain.(4)(c) (i) Show that the function y = (f g h)(x) satisfies the differential equationxydd = (1 y) sin x.(ii) Hence, or otherwise, find x x y d sin, as a function of x.(iii) You are given that the domain of y = (f g h)(x) can be extended to the whole real axis. That part of the graph of y = (f g h)(x), between its maximum at x = 0 and its first minimum for positive x, is rotated by 2 about the y-axis. Calculate the volume of the solid generated.(14)(Total 21 marks)115. Find the set of values of x for which x 1>2x 1.(Total 4 marks)116. The diagram shows the graphs of a linear function f and a quadratic function g.On the same axes sketch the graph of gf. Indicate clearly where the x-intercept and the asymptotes occur.(Total 5 marks)117. Let g(x) = log52log3x. Find the product of the zeros of g.(Total 5 marks)118. (a) Let a > 0. Draw the graph of y = 2ax for a x a on the grid below.(2)IB Questionbank Mathematics Higher Level 3rd edition 35(b) Find k such that xax k xaxaad2d200 .(5)(Total 7 marks)119. Let f be a function defined by f(x) = x arctan x, x .(a) Find f(1) and f(3 ).(2)(b) Show that f(x) = f(x), for x .(2)(c) Show that x 2) (2+ < < x x f, for x .(2)(d) Find expressions for f(x) and f(x). Hence describe the behaviour of the graph of f at the origin and justify your answer.(8)(e) Sketch a graph of f, showing clearly the asymptotes.(3)(f) Justify that the inverse of f is defined for all x and sketch its graph.(3)(Total 20 marks)120. When the function q(x) = x3 + kx2 7x + 3 is divided by (x + 1) the remainder is seven times the remainder that is found when the function is divided by (x + 2).Find the value of k.(Total 5 marks)121. A function is defined as f(x) = x k, with k > 0 and x 0.(a) Sketch the graph of y = f(x).(1)(b) Show that f is a one-to-one function.(1)(c) Find the inverse function, f1(x) and state its domain.(3)(d) If the graphs of y = f(x) and y = f1(x) intersect at the point (4, 4) find the value of k.(2)(e) Consider the graphs of y = f(x) and y = f1(x) using the value of k found in part (d).(i) Find the area enclosed by the two graphs.(ii) The line x = c cuts the graphs of y = f(x) and y = f1(x) at the points P and Q respectively. Given that the tangent to y = f(x) at point P is parallel to the tangent to y = f1(x) at point Q find the value of c.(9)(Total 16 marks)122. Let f(x) = xx+11 and g(x) = 1 + x, x > 1.Find the set of values of x for which f (x) f(x) g(x).(Total 7 marks)123. Let f be a function defined by f(x) = x + 2 cos x, x [0, 2]. The diagram below shows a region S bound by the graph of f and the line y = x.A and C are the points of intersection of the line y = x and the graph of f, and B is the minimum point of f.(a) If A, B and C have x-coordinates 2and6,2c b a, where a, b, c , find the values of a, b and c.(4)(b) Find the range of f.(3)IB Questionbank Mathematics Higher Level 3rd edition 37(c) Find the equation of the normal to the graph of f at the point C, giving your answer in the form y = px + q.(5)(d) The region S is rotated through 2 about the x-axis to generate a solid.(i) Write down an integral that represents the volume V of this solid.(ii) Show that V = 62.(7)(Total 19 marks)IB Questionbank Mathematics Higher Level 3rd edition 39124. (a) The graph of y = ln(x) is transformed into the graph of y = ln(2x + 1).Describe two transformations that are required to do this.(2)(b) Solve ln(2x + 1) > 3 cos (x), x [0, 10].(4)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 41125. The cubic curve y = 8x3 + bx2 + cx + d has two distinct points P and Q, where the gradient is zero.(a) Show that b2 > 24c.(4)(b) Given that the coordinates of P and Q are

,_

,_

20 ,23and 12 ,21, respectively, find the values of b, c and d.(4)(Total 8 marks)126. When 3x5 ax + b is divided by x 1 and x + 1 the remainders are equal. Given that a, b , find(a) the value of a;(4)(b) the set of values of b.(1)(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 43127. Consider the function f, where f(x) = arcsin (ln x).(a) Find the domain of f.(3)(b) Find f1(x).(3)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 45128. The real root of the equation x3 x + 4 = 0 is 1.796 to three decimal places.Determine the real root for each of the following.(a) (x 1)3 (x 1) + 4 = 0(2)(b) 8x3 2x + 4 = 0(3)(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 47129. A tangent to the graph of y = ln x passes through the origin.(a) Sketch the graphs of y = ln x and the tangent on the same set of axes, and hence find the equation of the tangent.(11)(b) Use your sketch to explain why ln x ex for x > 0.(1)IB Questionbank Mathematics Higher Level 3rd edition 49(c) Show that xe ex for x > 0.(3)(d) Determine which is larger, e or e.(2)(Total 17 marks)IB Questionbank Mathematics Higher Level 3rd edition 51130. Find the values of k such that the equation x3 + x2 x + 2 = k has three distinct real solutions.(Total 5 marks)131. Consider the function g, where g(x) = 253xx+.(a) Given that the domain of g is x a, find the least value of a such that g has an inverse function.(1)IB Questionbank Mathematics Higher Level 3rd edition 53(b) On the same set of axes, sketch(i) the graph of g for this value of a;(ii) the corresponding inverse, g1.(4)(c) Find an expression for g1(x).(3)(Total 8 marks)IB Questionbank Mathematics Higher Level 3rd edition 55132. The functions f and g are defined as:f (x) = , e2x x 0g (x) = . 3 ,31 +xx(a) Find h (x) where h (x) = g f (x).(2)IB Questionbank Mathematics Higher Level 3rd edition 57(b) State the domain of h1 (x).(2)(c) Find h1 (x).(4)(Total 8 marks)IB Questionbank Mathematics Higher Level 3rd edition 59133. The polynomial P(x) = x3 + ax2 + bx + 2 is divisible by (x +1) and by (x 2).Find the value of a and of b, where a, b .(Total 6 marks)134. Let f (x) = 2 ,24 +xx and g (x) = x 1.If h = g f, find(a) h (x);(2)IB Questionbank Mathematics Higher Level 3rd edition 61(b) h1 (x), where h1 is the inverse of h.(4)(Total 6 marks)135. (a) Sketch the curve f (x) = 1 + 3 sin (2x) , for 0 x . Write down on the graph the values of the x and y intercepts.(4)IB Questionbank Mathematics Higher Level 3rd edition 63(b) By adding one suitable line to your sketch, find the number of solutions to the equation f (x) = 4( x).(2)(Total 6 marks)136. A system of equations is given bycos x + cos y = 1.2sin x + sin y = 1.4.IB Questionbank Mathematics Higher Level 3rd edition 65(a) For each equation express y in terms of x.(2)(b) Hence solve the system for 0 < x < , 0 < y < .(4)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 67137. The graph of y = f (x) for 2 x 8 is shown.On the set of axes provided, sketch the graph of y = ( ),1x f clearly showing any asymptotes and indicating the coordinates of any local maxima or minima.(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 69138. Find the set of values of x for which . log 3 2 1 . 0102x x x < + (Total 6 marks)139. When f(x) = x4 + 3x3 + px2 2x + q is divided by (x 2) the remainder is 15,and (x + 3) is a factor of f(x).Find the values of p and q.(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 71140. (a) Sketch the curve y = ln x cos x 0.1, 0 < x < 4 showing clearly the coordinates of the points of intersection with the x-axis and the coordinates of any local maxima and minima.(5)(b) Find the values of x for which ln x > cos x + 0.1, 0 < x < 4.(2)(Total 7 marks)IB Questionbank Mathematics Higher Level 3rd edition 73141. The function f is defined by f(x) =

,_

+ 3arcsin 2 92xx x.(a) Write down the largest possible domain, for each of the two terms of the function, f, and hence state the largest possible domain, D, for f.(2)(b) Find the volume generated when the region bounded by the curve y = f(x), the x-axis, the y-axis and the line x = 2.8 is rotated through 2 radians about the x-axis.(3)IB Questionbank Mathematics Higher Level 3rd edition 75(c) Find f(x) in simplified form.(5)(d) Hence show that

,_

+ 3arcsin 4 9 2 d92 11222pp p xxxpp, where p D.(2)IB Questionbank Mathematics Higher Level 3rd edition 77(e) Find the value of p that maximises the value of the integral in (d).(2)(f) (i) Show that f(x) = 2322) 9 () 25 2 (xx x.IB Questionbank Mathematics Higher Level 3rd edition 79(ii) Hence justify that f(x) has a point of inflexion at x = 0, but not at x = 225t.(7)(Total 21 marks)142. Find all values of x that satisfy the inequality 112< xx.(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 81143. The polynomial f(x) = x3 + 3x2 + ax + b leaves the same remainder when divided by (x 2) as when divided by (x +1). Find the value of a.(Total 6 marks)144. The functions f and g are defined by f : x ex, g : x x + 2.Calculate(a) f1(3) g1(3);(3)(b) (f g)1(3).(3)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 83145. Let f(x) = 14++xx, x 1 and g(x) = 42xx, x 4. Find the set of values of x such that f (x) g(x).(Total 6 marks)146. (a) Write down the expansion of (cos + i sin )3 in the form a + ib, where a and b are in terms of sin and cos .(2)IB Questionbank Mathematics Higher Level 3rd edition 85(b) Hence show that cos 3 = 4 cos3 3 cos .(3)(c) Similarly show that cos 5 = 16 cos5 20 cos3 + 5 cos .(3)IB Questionbank Mathematics Higher Level 3rd edition 87(d) Hence solve the equation cos 5 + cos 3 + cos = 0, where 1]1

2,2.(6)(e) By considering the solutions of the equation cos 5 = 0, show that85 510cos+ and state the value of 10 7cos.(8)(Total 22 marks)IB Questionbank Mathematics Higher Level 3rd edition 89147. The complex numbers z1 = 2 2i and z2 = 1 3 i are represented by the points A and B respectively on an Argand diagram. Given that O is the origin,(a) find AB, giving your answer in the form 3 b a, where a, b +;(3)(b) calculate B OA in terms of .(3)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 91148. An arithmetic sequence has first term a and common difference d, d 0.The 3rd, 4th and 7th terms of the arithmetic sequence are the first three terms of a geometric sequence.(a) Show that a = d23.(3)(b) Show that the 4th term of the geometric sequence is the 16th term of the arithmetic sequence.(5)(Total 8 marks)IB Questionbank Mathematics Higher Level 3rd edition 93149. (a) Factorize z3 + 1 into a linear and quadratic factor.(2)Let = 23 i 1+.(b) (i) Show that is one of the cube roots of 1.(ii) Show that 2 = 1.(iii) Hence find the value of (1 )6.(9)IB Questionbank Mathematics Higher Level 3rd edition 95The matrix A is defined by A =

,_

101.(c) Show that A2 A + I = 0, where 0 is the zero matrix.(4)(d) Deduce that(i) A3 = I;(ii) A1 = I A.(5)(Total 20 marks)IB Questionbank Mathematics Higher Level 3rd edition 97150. (a) Write down the quadratic expression 2x2 + x 3 as the product of two linear factors.(1)(b) Hence, or otherwise, find the coefficient of x in the expansion of (2x2 + x 3)8.(4)(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 99151. Solve the following system of equations.logx+1 y = 2logy+1 x = 41(Total 6 marks)152. (a) Given that A =

,_

cos sinsin cos, show that A2 =

,_

2 cos 2 sin2 sin 2 cos.(3)IB Questionbank Mathematics Higher Level 3rd edition 101(b) Prove by induction thatAn =

,_

n nn ncos sinsin cos, for all n +.(7)(c) Given that A1 is the inverse of matrix A, show that the result in part (b) is truewhere n = 1.(3)(Total 13 marks)IB Questionbank Mathematics Higher Level 3rd edition 103153. In the arithmetic series with nth term un, it is given that u4 = 7 and u9 = 22.Find the minimum value of n so that u1 + u2 + u3 + ... + un > 10 000.(Total 5 marks)154. Prove by mathematical induction that, for n +,1 + 11 3 222421...214213212+ ,_

+ + ,_

+ ,_

+ ,_

nnnn.(Total 8 marks)IB Questionbank Mathematics Higher Level 3rd edition 105155. Two players, A and B, alternately throw a fair sixsided dice, with A starting, until one of them obtains a six. Find the probability that A obtains the first six.(Total 7 marks)156. (a) Show that sin 2 nx = sin((2n + 1)x) cos x cos((2n + 1)x) sin x.(2)IB Questionbank Mathematics Higher Level 3rd edition 107(b) Hence prove, by induction, thatcos x + cos 3x + cos 5x + ... + cos((2n 1)x) = xnxsin 22 sin,for all n +, sin x 0.(12)(c) Solve the equation cos x + cos 3x = 21, 0 < x < .(6)(Total 20 marks)IB Questionbank Mathematics Higher Level 3rd edition 109157. (a) Consider the following sequence of equations.1 2 = 31(1 2 3),1 2 + 2 3 = 31(2 3 4),1 2 + 2 3 + 3 4 = 31(3 4 5),.... .(i) Formulate a conjecture for the nth equation in the sequence.IB Questionbank Mathematics Higher Level 3rd edition 111(ii) Verify your conjecture for n = 4.(2)(b) A sequence of numbers has the nth term given by un = 2n + 3, n +. Bill conjectures that all members of the sequence are prime numbers. Show that Bills conjecture is false.(2)IB Questionbank Mathematics Higher Level 3rd edition 113(c) Use mathematical induction to prove that 5 7n + 1 is divisible by 6 for all n +.(6)(Total 10 marks)158. Consider =

,_

+ ,_

32sin i3 2cos.(a) Show that(i) 3 = 1;(ii) 1 + + 2 = 0.(5)IB Questionbank Mathematics Higher Level 3rd edition 115(b) (i) Deduce that ei +

,_

+ ,_

++3 4i3 2ie e = 0.(ii) Illustrate this result for = 2 on an Argand diagram.(4)(c) (i) Expand and simplify F(z) = (z 1)(z )(z 2) where z is a complex number.IB Questionbank Mathematics Higher Level 3rd edition 117(ii) Solve F(z) = 7, giving your answers in terms of .(7)(Total 16 marks)159. (a) Solve the equation z3 = 2 + 2i, giving your answers in modulusargument form.(6)IB Questionbank Mathematics Higher Level 3rd edition 119(b) Hence show that one of the solutions is 1 + i when written in Cartesian form.(1)(Total 7 marks)160. Find the sum of all three-digit natural numbers that are not exactly divisible by 3.(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 121161. Three Mathematics books, five English books, four Science books and a dictionary are to be placed on a students shelf so that the books of each subject remain together.(a) In how many different ways can the books be arranged?(4)(b) In how many of these will the dictionary be next to the Mathematics books?(3)(Total 7 marks)162. Consider the arithmetic sequence 8, 26, 44, ....(a) Find an expression for the nth term.(1)IB Questionbank Mathematics Higher Level 3rd edition 123(b) Write down the sum of the first n terms using sigma notation.(1)(c) Calculate the sum of the first 15 terms.(2)(Total 4 marks)IB Questionbank Mathematics Higher Level 3rd edition 125163. (a) Simplify the difference of binomial coefficients

,_

,_

223n n, where n 3.(4)(b) Hence, solve the inequality

,_

,_

223n n > 32n, where n 3.(2)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 127164. Given that z = cos + i sin show that(a) Im ,_

+ nzznn, 01 +;(2)(b) Re

,_

+11zz = 0, z 1.(5)(Total 7 marks)IB Questionbank Mathematics Higher Level 3rd edition 129165. The interior of a circle of radius 2 cm is divided into an infinite number of sectors.The areas of these sectors form a geometric sequence with common ratio k. The angle of the first sector is radians.(a) Show that = 2(1 k).(5)(b) The perimeter of the third sector is half the perimeter of the first sector.Find the value of k and of .(7)(Total 12 marks)IB Questionbank Mathematics Higher Level 3rd edition 131166. Expand and simplify 422

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xx.(Total 4 marks)167. The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic sequence is 16. Find the value of the 15th term of the sequence.(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 133168. The sum, Sn, of the first n terms of a geometric sequence, whose nth term is un, is given bySn = nn na77 , where a > 0.(a) Find an expression for un.(2)(b) Find the first term and common ratio of the sequence.(4)IB Questionbank Mathematics Higher Level 3rd edition 135(c) Consider the sum to infinity of the sequence.(i) Determine the values of a such that the sum to infinity exists.(ii) Find the sum to infinity when it exists.(2)(Total 8 marks)169. Consider the complex number = 2i++zz, where z = x + iy and i = 1 .(a) If = i, determine z in the form z = r cis .(6)IB Questionbank Mathematics Higher Level 3rd edition 137(b) Prove that = 2 22 2) 2 () 2 2 i( ) 2 (y xy x y y x x+ ++ + + + + +.(3)(c) Hence show that when Re() = 1 the points (x, y) lie on a straight line, l1, and write down its gradient.(4)IB Questionbank Mathematics Higher Level 3rd edition 139(d) Given arg (z) = arg() = 4, find z.(6)(Total 19 marks)170. (a) A particle P moves in a straight line with displacement relative to origin given bys = 2 sin (t) + sin(2t), t 0,where t is the time in seconds and the displacement is measured in centimetres.IB Questionbank Mathematics Higher Level 3rd edition 141(i) Write down the period of the function s.(ii) Find expressions for the velocity, v, and the acceleration, a, of P.(iii) Determine all the solutions of the equation v = 0 for 0 t 4.(10)(b) Consider the functionf(x) = A sin (ax) + B sin (bx), A, a, B, b, x .Use mathematical induction to prove that the (2n)th derivative of f is given byf(2n)(x) = (1)n (Aa2n sin (ax) + Bb2n sin (bx)), for all n +.(8)(Total 18 marks)IB Questionbank Mathematics Higher Level 3rd edition 143171. Solve the equations1 ln yxln x3 + ln y2 = 5.(Total 5 marks)172. Consider the polynomial p(x) = x4 + ax3 + bx2 + cx + d, where a, b, c, d .Given that 1 + i and 1 2i are zeros of p(x), find the values of a, b, c and d.(Total 7 marks)IB Questionbank Mathematics Higher Level 3rd edition 145173. Consider the complex numbers z = 1 + 2i and w = 2 +ai, where a .Find a when(a) w = 2z;(3)(b) Re (zw) = 2 Im(zw).(3)(Total 6 marks)174. The diagram below shows a solid with volume V, obtained from a cube with edge a > 1 when a smaller cube with edge a1 is removed.diagram not to scaleIB Questionbank Mathematics Higher Level 3rd edition 147Let x = aa1.(a) Find V in terms of x.(4)(b) Hence or otherwise, show that the only value of a for which V = 4x is a = 25 1+.(4)(Total 8 marks)175. (a) Consider the set of numbers a, 2a, 3a, ..., na where a and n are positive integers.(i) Show that the expression for the mean of this set is 2) 1 ( + n a.(ii) Let a = 4. Find the minimum value of n for which the sum of these numbers exceeds its mean by more than 100.(6)IB Questionbank Mathematics Higher Level 3rd edition 149(b) Consider now the set of numbers x1, ... , xm, y1, ... , y1, ... , yn where xi = 0 for i = 1, ... , m and yi = 1 for i = 1, ... , n.(i) Show that the mean M of this set is given by n mn+ and the standard deviationS by n mmn+.(ii) Given that M = S, find the value of the median.(11)(Total 17 marks)176. If z is a non-zero complex number, we define L(z) by the equationL(z) = lnz + i arg (z), 0 arg (z) < 2.(a) Show that when z is a positive real number, L(z) = ln z.(2)IB Questionbank Mathematics Higher Level 3rd edition 151(b) Use the equation to calculate(i) L(1);(ii) L(1 i);(iii) L(1 +i).(5)(c) Hence show that the property L(z1z2) = L(z1) + L(z2) does not hold for all valuesof z1 and z2.(2)(Total 9 marks)IB Questionbank Mathematics Higher Level 3rd edition 153177. Given that z1 = 2 and z2 = 1 + 3 i are roots of the cubic equation z3 + bz2 + cz + d = 0where b, c, d ,(a) write down the third root, z3, of the equation;(1)(b) find the values of b, c and d;(4)IB Questionbank Mathematics Higher Level 3rd edition 155(c) write z2 and z3 in the form rei.(3)(Total 8 marks)178. Prove by mathematical induction + nrn n r r1, 1 )! 1 ( ) ! ( +.(Total 8 marks)IB Questionbank Mathematics Higher Level 3rd edition 157179. The complex number z is defined as z = cos + i sin .(a) State de Moivres theorem.(1)(b) Show that zn nz1 = 2i sin (n).(3)IB Questionbank Mathematics Higher Level 3rd edition 159(c) Use the binomial theorem to expand 51

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zz giving your answer in simplified form.(3)(d) Hence show that 16 sin5 = sin 5 5 sin 3 + 10 sin .(4)IB Questionbank Mathematics Higher Level 3rd edition 161(e) Check that your result in part (d) is true for = 4.(4)(f) Find d sin205.(4)IB Questionbank Mathematics Higher Level 3rd edition 163(g) Hence, with reference to graphs of circular functions, find 205d cos , explaining your reasoning.(3)(Total 22 marks)180. (a) Show that the complex number i is a root of the equationx4 5x3 + 7x2 5x + 6 = 0.(2)(b) Find the other roots of this equation.(4)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 165181. Let A =

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1 0 01 1 01 1 1 and B =

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1 1 10 1 10 0 1.(a) Given that X = B A1 and Y = B1 A,(i) find X and Y;(ii) does X1 + Y1 have an inverse? Justify your conclusion.(5)(b) Prove by induction that An =

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+1 0 01 02) 1 (1nn nn, for n +.(7)IB Questionbank Mathematics Higher Level 3rd edition 167(c) Given that (An)1 =

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1 0 01 01xy x, for n +,(i) find x and y in terms of n,(ii) and hence find an expression for An + (An)1.(6)(Total 18 marks)IB Questionbank Mathematics Higher Level 3rd edition 169182. Six people are to sit at a circular table. Two of the people are not to sit immediately beside each other. Find the number of ways that the six people can be seated.(Total 5 marks)183. Consider the graphs y = ex and y = ex sin 4x, for 0 x 4 5.(a) On the same set of axes draw, on graph paper, the graphs, for 0 x 4 5.Use a scale of 1 cm to 8 on your x-axis and 5 cm to 1 unit on your y-axis.(3)IB Questionbank Mathematics Higher Level 3rd edition 171(b) Show that the x-intercepts of the graph y = ex sin 4x are 4 n, n = 0, 1, 2, 3, 4, 5.(3)(c) Find the x-coordinates of the points at which the graph of y = ex sin 4x meets the graph of y = ex. Give your answers in terms of .(3)IB Questionbank Mathematics Higher Level 3rd edition 173(d) (i) Show that when the graph of y = ex sin 4x meets the graph of y = ex, their gradients are equal.(ii) Hence explain why these three meeting points are not local maxima of thegraph y = ex sin 4x.(6)IB Questionbank Mathematics Higher Level 3rd edition 175(e) (i) Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima of y = ex sin 4x for 0 x 4 5. You do not need to show that they are maximum values, but the values should be simplified.(ii) Show that y1, y2 and y3 form a geometric sequence and determine the common ratio r.(7)(Total 22 marks)IB Questionbank Mathematics Higher Level 3rd edition 177184. Find the values of n such that (1 + 3i)n is a real number.(Total 5 marks)185. (a) The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven terms is 231. Find the first term and the common difference.(6)IB Questionbank Mathematics Higher Level 3rd edition 179(b) The sum of the first two terms of a geometric series is 1 and the sum of its first four terms is 5. If all of its terms are positive, find the first term and the common ratio.(5)(c) The rth term of a new series is defined as the product of the rth term of the arithmetic series and the rth term of the geometric series above. Show that the rth term of this new series is (r + 1)2r1.(3)IB Questionbank Mathematics Higher Level 3rd edition 181(d) Using mathematical induction, prove that +n n rnrn r, 2 2 ) 1 (11 +.(7)(Total 21 marks)186. (a) Let z = x + iy be any non-zero complex number.(i) Express z1 in the form u + iv.(ii) If + k kzz ,1, show that either y = 0 or x2 + y2 = 1.(iii) Show that if x2 + y2 = 1 then k 2.(8)IB Questionbank Mathematics Higher Level 3rd edition 183(b) Let w = cos + i sin .(i) Show that wn + wn = 2cos n, n .(ii) Solve the equation 3w2 w + 2 w1 + 3w2 = 0, giving the roots in theform x + iy.(14)(Total 22 marks)IB Questionbank Mathematics Higher Level 3rd edition 185187. When ,_

+ nxn,21, is expanded in ascending powers of x, the coefficient of x3 is 70.(a) Find the value of n.(5)(b) Hence, find the coefficient of x2.(1)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 187188. Consider the equation z3 + az2 + bz + c = 0, where a, b, c . The points in the Argand diagram representing the three roots of the equation form the vertices of a triangle whose area is 9. Given that one root is 1 + 3i, find(a) the other two roots;(4)(b) a, b and c.(3)(Total 7 marks)IB Questionbank Mathematics Higher Level 3rd edition 189189. Express ( )33 i 11 in the form ba where a, b .(Total 5 marks)190. A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence.The angle of the largest sector is twice the angle of the smallest sector.Find the size of the angle of the smallest sector.(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 191191. The common ratio of the terms in a geometric series is 2x.(a) State the set of values of x for which the sum to infinity of the series exists.(2)(b) If the first term of the series is 35, find the value of x for which the sum to infinity is 40.(4)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 193192. The function f is defined by f (x) = x e2x.It can be shown that f (n) (x) = (2n x + n 2n1) e2x for all n+, where f (n) (x) represents the nth derivative of f (x).(a) By considering f (n) (x) for n =1 and n = 2, show that there is one minimum point P on the graph of f, and find the coordinates of P.(7)(b) Show that f has a point of inflexion Q at x = 1.(5)IB Questionbank Mathematics Higher Level 3rd edition 195(c) Determine the intervals on the domain of f where f is(i) concave up;(ii) concave down.(2)(d) Sketch f, clearly showing any intercepts, asymptotes and the points P and Q.(4)IB Questionbank Mathematics Higher Level 3rd edition 197(e) Use mathematical induction to prove that f (n) (x) = (2nx + n2n1) e2x for all n+, where f (n) (x) represents the nth derivative of f (x).(9)(Total 27 marks)193. (a) Find the sum of the infinite geometric sequence 27, 9, 3, 1, ... .(3)IB Questionbank Mathematics Higher Level 3rd edition 199(b) Use mathematical induction to prove that for n+,a + ar + ar2 + ... + arn1 = ( ).11rr an(7)(Total 10 marks)194. Let w = cos .52sin i52 +(a) Show that w is a root of the equation z5 1 = 0.(3)IB Questionbank Mathematics Higher Level 3rd edition 201(b) Show that (w 1) (w4 + w3 + w2 + w + 1) = w5 1 and deduce thatw4 + w3 + w2 + w + 1 = 0.(3)(c) Hence show that cos .2154cos52 +(6)(Total 12 marks)IB Questionbank Mathematics Higher Level 3rd edition 203195. Determine the first three terms in the expansion of (1 2x)5 (1+ x)7 in ascending powers of x.(Total 5 marks)196. Find, in its simplest form, the argument of (sin + i (1 cos ))2 where is an acute angle.(Total 7 marks)IB Questionbank Mathematics Higher Level 3rd edition 205197. z1 = ( )m3 i 1+ and z2 = ( ) . i 1n(a) Find the modulus and argument of z1 and z2 in terms of m and n, respectively.(6)(b) Hence, find the smallest positive integers m and n such that z1 = z2.(8)(Total 14 marks)IB Questionbank Mathematics Higher Level 3rd edition 207198. Consider w = 12+ zz where z = x + iy, y 0 and z2 + 1 0.Given that Im w = 0, show that z = 1.(Total 7 marks)199. Let M2 = M where M = . 0 ,

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bcd cb a(a) (i) Show that a + d = 1.(ii) Find an expression for bc in terms of a.(5)IB Questionbank Mathematics Higher Level 3rd edition 209(b) Hence show that M is a singular matrix.(3)(c) If all of the elements of M are positive, find the range of possible values for a.(3)IB Questionbank Mathematics Higher Level 3rd edition 211(d) Show that (I M)2 = I M where I is the identity matrix.(3)(e) Prove by mathematical induction that (I M)n = I M for n+.(6)(Total 20 marks)IB Questionbank Mathematics Higher Level 3rd edition 213200. (a) The independent random variables X and Y have Poisson distributions and Z = X + Y. The means of X and Y are and respectively. By using the identity( ) ( ) ( ) nkk n Y k X n Z0P P Pshow that Z has a Poisson distribution with mean ( + ).(6)(b) Given that U1, U2, U3, are independent Poisson random variables each having mean m, use mathematical induction together with the result in (a) to show that nrrU1 has a Poisson distribution with mean nm.(6)(Total 12 marks)IB Questionbank Mathematics Higher Level 3rd edition 215201. Write ln (x2 1) 2 ln(x + 1) + ln(x2 + x) as a single logarithm, in its simplest form.(Total 5 marks)202. An 81 metre rope is cut into n pieces of increasing lengths that form an arithmetic sequence with a common difference of d metres. Given that the lengths of the shortest and longest pieces are 1.5 metres and 7.5 metres respectively, find the values of n and d.(Total 4 marks)IB Questionbank Mathematics Higher Level 3rd edition 217203. (a) Using mathematical induction, prove that

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nn nn nn,cos sinsin coscos sinsin cos +.(9)(b) Show that the result holds true for n = 1.(5)(Total 14 marks)IB Questionbank Mathematics Higher Level 3rd edition 219204. (a) Use de Moivres theorem to find the roots of the equation z4 = 1 i.(6)(b) Draw these roots on an Argand diagram.(2)IB Questionbank Mathematics Higher Level 3rd edition 221(c) If z1 is the root in the first quadrant and z2 is the root in the second quadrant, find 12zz in the form a + ib.(4)(Total 12 marks)205. (a) Expand and simplify (x 1)(x4 + x3 + x2 + x + 1).(2)IB Questionbank Mathematics Higher Level 3rd edition 223(b) Given that b is a root of the equation z5 1 = 0 which does not lie on the real axis in the Argand diagram, show that 1 + b + b2 + b3 + b4 = 0.(3)(c) If u = b + b4 and v = b2 + b3 show that(i) u + v = uv = 1;(ii) u v = 5, given that u v > 0.(8)(Total 13 marks)IB Questionbank Mathematics Higher Level 3rd edition 225206. A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the smallest term that is greater than 500.(Total 5 marks)207. There are six boys and five girls in a school tennis club. A team of two boys and two girls will be selected to represent the school in a tennis competition.(a) In how many different ways can the team be selected?(3)IB Questionbank Mathematics Higher Level 3rd edition 227(b) Tim is the youngest boy in the club and Anna is the youngest girl. In how many different ways can the team be selected if it must include both of them?(2)(c) What is the probability that the team includes both Tim and Anna?(1)IB Questionbank Mathematics Higher Level 3rd edition 229(d) Fred is the oldest boy in the club. Given that Fred is selected for the team, what is the probability that the team includes Tim or Anna, but not both?(4)(Total 10 marks)208. Given that 4 ln 2 3ln 4 = ln k, find the value of k.(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 231209. Solve the equation log3(x + 17) 2 = log3 2x.(Total 5 marks)210. Solve the equation 22x+2 10 2x + 4 = 0, x .(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 233211. Given that (a + bi)2 = 3 + 4i obtain a pair of simultaneous equations involving a and b. Hence find the two square roots of 3 + 4i.(Total 7 marks)212. Given that 2 + i is a root of the equation x3 6x2 + 13x 10 = 0 find the other two roots.(Total 5 marks)IB Questionbank Mathematics Higher Level 3rd edition 235213. Given that z = 10, solve the equation 5z + *10z = 6 18i, where z* is the conjugate of z.(Total 7 marks)214. Find the three cube roots of the complex number 8i. Give your answers in the form x + iy.(Total 8 marks)IB Questionbank Mathematics Higher Level 3rd edition 237215. Solve the simultaneous equationsiz1 + 2z2 = 3z1 + (1 i)z2 = 4giving z1 and z2 in the form x + iy, where x and y are real.(Total 9 marks)216. Find b where i109107i 1i 2+ +bb.(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 239217. Given that z = (b + i)2, where b is real and positive, find the value of b when arg z = 60.(Total 6 marks)218. (a) Show that p = 2 is a solution to the equation p3 + p2 5p 2 = 0.(2)IB Questionbank Mathematics Higher Level 3rd edition 241(b) Find the values of a and b such that p3 + p2 5p 2 = (p 2)(p2 + ap + b).(4)(c) Hence find the other two roots to the equation p3 + p2 5p 2 = 0.(3)IB Questionbank Mathematics Higher Level 3rd edition 243(d) An arithmetic sequence has p as its common difference. Also, a geometric sequence has p as its common ratio. Both sequences have 1 as their first term.(i) Write down, in terms of p, the first four terms of each sequence.(ii) If the sum of the third and fourth terms of the arithmetic sequence is equal to the sum of the third and fourth terms of the geometric sequence, find the three possible values of p.IB Questionbank Mathematics Higher Level 3rd edition 245(iii) For which value of p found in (d)(ii) does the sum to infinity of the terms of the geometric sequence exist?(iv) For the same value p, find the sum of the first 20 terms of the arithmetic sequence, writing your answer in the form a + c b, where a, b, c .(13)(Total 22 marks)IB Questionbank Mathematics Higher Level 3rd edition 247219. Use mathematical induction to prove that 5n + 9n + 2 is divisible by 4, for n +.(Total 9 marks)220. Consider the complex geometric series ei + 3i 2ie41e21+ + ....(a) Find an expression for z, the common ratio of this series.(2)IB Questionbank Mathematics Higher Level 3rd edition 249(b) Show that z < 1.(2)(c) Write down an expression for the sum to infinity of this series.(2)IB Questionbank Mathematics Higher Level 3rd edition 251(d) (i) Express your answer to part (c) in terms of sin and cos. (ii) Hence show thatcos + 21cos 2 + 41cos 3 + ... = cos 4 52 cos 4.(10)(Total 16 marks)IB Questionbank Mathematics Higher Level 3rd edition 253221. The roots of the equation z2 + 2z + 4 = 0 are denoted by and ?(a) Find and in the form rei.(6)(b) Given that lies in the second quadrant of the Argand diagram, mark and on an Argand diagram.(2)IB Questionbank Mathematics Higher Level 3rd edition 255(c) Use the principle of mathematical induction to prove De Moivres theorem, which states that cos n + i sin n = (cos + i sin )n for n +.(8)(d) Using De Moivres theorem find 23 in the form a + ib.(4)IB Questionbank Mathematics Higher Level 3rd edition 257(e) Using De Moivres theorem or otherwise, show that 3 = 3.(3)(f) Find the exact value of* +* where * is the conjugate of and* is the conjugate of. (5)IB Questionbank Mathematics Higher Level 3rd edition 259(g) Find the set of values of n for which n is real.(3)(Total 31 marks)222. A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum.(a) Write down an expression for the value of the investment after n full years.(1)IB Questionbank Mathematics Higher Level 3rd edition 261(b) What will be the value of the investment at the end of five years?(1)(c) The value of the investment will exceed $10 000 after n full years.(i) Write an inequality to represent this information.(ii) Calculate the minimum value of n.(4)(Total 6 marks)IB Questionbank Mathematics Higher Level 3rd edition 263