Topic 1 Part 3 [483 marks]

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1 Topic Part 3 [483 marks] The complex numbers z = i and z = 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 marks] a b 3, where a, b Z + b Calculate [3 marks] AOB in terms of π An arithmetic sequence has first term a and common difference d, d 0 The 3 rd 4 th 7 th, and terms of the arithmetic sequence are the first three terms of a geometric sequence a a = 3 d [3 marks] b the 4 th 6 th term of the geometric sequence is the term of the arithmetic sequence 3a Factorize z 3 + into a linear and quadratic factor [ marks] 3b Let γ = +i 3 (i) γ is one of the cube roots of γ (iii) = γ ( γ) 6 Hence find the value of [9 marks]

2 4 In the arithmetic series with n th u n u 4 u 9 term, it is given that = 7 and = Find the minimum value of n so that u + u + u 3 ++ u n > a Prove by mathematical induction that, for [8 marks] n Z +, 3 n + ( ) + 3 ( ) + 4 ( ) ++n ( n + ) = 4 n 5b Using integration by parts, show that [7 marks] e x sin xdx = ( sin x cosx) + C 5 ex Solve the differential equation = y e x sin x, given that y = 0 when x = 0, dy dx writing your answer in the form y = f(x) (i) Sketch the graph of y = f(x), found in part, for 0 x 5 Determine the coordinates of the point P, the first positive intercept on the x-axis, and mark it on your sketch The region bounded by the graph of y = f(x) and the x-axis, between the origin and P, is rotated 360 about the x-axis to form a solid of revolution Calculate the volume of this solid The integral I n I n is defined by = e x sin x dx, for n N (n+)π nπ 6a I 0 = ( + e π ) 6b By letting [4 marks] y = x nπ, show that I n = e nπ I 0

3 6c Hence determine the exact value of e 0 x sin x dx 7 Solve the equation 4 x = x [0 marks] sin nx = sin((n + )x)cosx cos((n + )x)sin x Hence prove, by induction, that sin nx cosx + cos3x + cos5x + + cos((n )x) =, sin x for all n Z +, sin x 0 Solve the equation cosx + cos3x =, 0 < x < π 9 The system of equations x y + 3z = 3x + y + z = x + y + az = b is known to have more than one solution Find the value of a and the value of b 0 Solve the equation z 3 = + i, giving your answers in modulus-argument form Hence show that one of the solutions is + i when written in Cartesian form [7 marks] Find the sum of all three-digit natural numbers that are not exactly divisible by 3

4 Consider the following sequence of equations [0 marks] = ( 3), = ( 3 4), = (3 4 5), 3 (i) Formulate a conjecture for the n th equation in the sequence Verify your conjecture for n = 4 A sequence of numbers has the n th term given by u n = n + 3, n Z + Bill conjectures that all members of the sequence are prime numbers Bill s conjecture is false Use mathematical induction to prove that 5 7 n + n Z + is divisible by 6 for all 3 Consider π 3 π 3 ω = cos( ) + isin( ) (i) ω 3 = ; + ω + ω = 0 (i) Deduce that e iθ + e i(θ+ π ) 3 + e i(θ+ 4π ) 3 = 0 Illustrate this result for θ = π on an Argand diagram (i) Expand and simplify F(z) = (z )(z ω)(z ω ) where z is a complex number Solve F(z) = 7, giving your answers in terms of ω

5 4 Expand and simplify ( x ) x 4 [4 marks] 5 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 6 Find the value of the 5 th term of the sequence 6 The sum, [8 marks] S n n th u n, of the first n terms of a geometric sequence, whose term is, is given by S n 7 = n a n, where a > 0 7 n Find an expression for u n (i) Find the first term and common ratio of the sequence Consider the sum to infinity of the sequence Determine the values of a such that the sum to infinity exists Find the sum to infinity when it exists 7 Consider the complex number [9 marks] ω = z+i, where z+ z = x + iy and i = If ω = i, determine z in the form z = rcis θ ω = Prove that Hence show that when Re(ω) = the points (x, y) lie on a straight line, l ( x +x+ y +y)+i(x+y+) (x+) + y, and write down its gradient (d) Given arg(z) = arg(ω) = π, find 4 z

6 8 Consider the polynomial [7 marks] p(x) = x 4 + a x 3 + b x + cx + d, where a, b, c, d R Given that + i and i are zeros of p(x), find the values of a, b, c and d A geometric sequence u, u, u 3, has u = 7 and a sum to infinity of 8 Find the common ratio of the geometric sequence 9a [ marks] An arithmetic sequence 9b v, v, v 3, is such that v = u and v 4 = u 4 Find the greatest value of N such that N vn n= > 0 0a Write down the expansion of (cosθ + isin θ) 3 in the form a + ib, where a and b are in terms of sin θ and cosθ [ marks] Hence show that 0b cos3θ = 4cos 3 θ 3 cosθ [3 marks] Similarly show that 0c cos5θ = 6cos 5 θ 0cos 3 θ + 5 cosθ [3 marks] 0d Hence solve the equation cos5θ + cos3θ + cosθ = 0, where π π θ [, ] By considering the solutions of the equation 0e cos5θ = 0, show that π 5+ 5 cos = and state the value of 0 8 cos 7π 0 [8 marks] Write down the quadratic expression a x + x 3 as the product of two linear factors [ mark]

7 Hence, or otherwise, find the coefficient of b x in the expansion of ( x 8 + x 3) [4 marks] Solve the following system of equations log x+ y = log y+ x = 4 The equations of three planes, are given by ax + y + z = 3 x + (a + )y + 3z = x + y + (a + )z = k where a R Given that 3a a = 0, show that the three planes intersect at a point [3 marks] Find the value of a such that the three planes do not meet at a point 3b 3c 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 x x 0 l y = y 0 + λ m z n z 0 4 Express in the form where a, b Z a ( i 3) 3 b 5 The common ratio of the terms in a geometric series is x State the set of values of x for which the sum to infinity of the series exists If the first term of the series is 35, find the value of x for which the sum to infinity is 40 6a Find the sum of the infinite geometric sequence 7, 9, 3,, [3 marks] 6b Use mathematical induction to prove that for n Z +, [7 marks] a + ar + a r ++a r n a( r n ) = r

8 7 Let w = cos z 5 w is a root of the equation = 0 w 4 + w 3 + w + w + = 0 π 5 + isin π 5 (w )( w 4 + w 3 + w + w + ) = w 5 π 5 Hence show that 4π 5 cos + cos = and deduce that [ marks] [4 marks] 8 An 8 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 5 metres and 75 metres respectively, find the values of n and d 9a Use de Moivre s theorem to find the roots of the equation z 4 = i z z z z Draw these roots on an Argand diagram If is the root in the first quadrant and is the root in the second quadrant, find in the form a + ib [ marks] 9b Expand and simplify [3 marks] (x )( x 4 + x 3 + x + x + ) Given that b is a root of the equation z 5 = 0 which does not lie on the real axis in the Argand diagram, show that + b + b + b 3 + b 4 = 0 If u = b + b 4 and v = b + b 3 show that (i) u + v = uv = ; u v = 5 u v > 0, given that 30 A geometric sequence has a first term of and a common ratio of 05 Find the value of the smallest term which is greater than 500

9 3 Find the set of values of k for which the following system of equations has no solution x + y 3z = k 3x + y + z = 4 5x + 7z = 5 Describe the geometrical relationship of the three planes represented by this system of equations 3 Consider the complex numbers z = + i and w = + ai, where a R Find a when w = z ; ; Re(zw) = Im(zw) 33a If z is a non-zero complex number, we define [9 marks] L(z) by the equation L(z) = ln z + iarg(z), 0 arg(z) < π when z is a positive real number, L(z) = lnz Use the equation to calculate (i) L( ) ; L( i) ; (iii) L( + i) Hence show that the property L( z z ) = L( z ) + L( z ) z z and does not hold for all values of

10 33b Let f be a function with domain [4 marks] R that satisfies the conditions, f(x + y) = f(x)f(y), for all x and y and f(0) 0 f(0) = Prove that f(x) 0, for all x R Assuming that f (x) exists for all x R, use the definition of derivative to show that f(x) satisfies the differential equation f (x) = k f(x) k = f (0), where (d) Solve the differential equation to find an expression for f(x) 34 Given that [8 marks] z = and z = + 3 i z 3 + b z + cz + d = 0 where b, c, d R, are roots of the cubic equation write down the third root, z 3, of the equation; find the values of b, c and d ; write z z 3 re iθ and in the form 35 Prove by mathematical induction n r(r!) = (n + )!, r= n Z + [8 marks]

11 36 The complex number z is defined as z = cosθ + isin θ z n (d) (e) θ = π 4 (f) State de Moivre s theorem = isin(nθ) Use the binomial theorem to expand giving your answer in simplified form Hence show that Check that your result in part (d) is true for Find 0 sin 5 θdθ (g) z n (z ) z 5 6sin 5 θ = sin 5θ 5 sin 3θ + 0 sin θ π π Hence, with reference to graphs of circular functions, find 0 cos 5 θdθ, explaining your reasoning [ marks] 37 Find the values of n such that ( + 3i) n is a real number [4 marks] 38a The sum of the first six terms of an arithmetic series is 8 The sum of its first eleven terms is 3 Find the first term and the common difference The sum of the first two terms of a geometric series is 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 The r th r th r th r th term of a new series is defined as the product of the term of the arithmetic series and the term of the geometric series above the term of this new series is (r + ) r 38b Using mathematical induction, prove that [7 marks] n (r + ) r = n n, n Z + r=

12 39a Let [8 marks] z = x + iy be any non-zero complex number (i) Express z in the form u + iv If z + z = k, k R, show that either y = 0 or x + y = (iii) if x + y = k then 39b Let [4 marks] w = cosθ + isin θ (i) w n + w n = cosnθ, n Z Solve the equation 3w w + w + 3 w = 0, giving the roots in the form x + iy [7 marks] 40 Three Mathematics books, five English books, four Science books and a dictionary are to be placed on a student s shelf so that the books of each subject remain together In how many different ways can the books be arranged? In how many of these will the dictionary be next to the Mathematics books? 4 Consider the arithmetic sequence 8, 6, 44, Find an expression for the n th term Write down the sum of the first n terms using sigma notation Calculate the sum of the first 5 terms [4 marks]

13 The three planes x y z = 3 4x + 5y z = 3 3x + 4y 3z = 7 intersect at the point with coordinates (a, b, c) 4a Find the value of each of a, b and c [ marks] 4b The equations of three planes are [4 marks] x 4y 3z = 4 x + 3y + 5z = 3x 5y z = 6 Find a vector equation of the line of intersection of these three planes 43 Given that [7 marks] z = cosθ + isin θ show that Im( z n + ) = 0, n Re( z z+ z n Z + ) = 0, z ; 44 [ marks] The interior of a circle of radius 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 θ radians θ = π( k) The perimeter of the third sector is half the perimeter of the first sector Find the value of k and of θ

14 When x ( + ), n N, is expanded in ascending powers of x, the coefficient of x 3 is n x Find the value of Hence, find the coefficient of Consider the equation z 3 + a z + bz + c = 0, where a, b, c R 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 + 3i, find the other two roots; 46 a, b and c [7 marks] 47 the complex number i is a root of the equation Find the other roots of this equation x 4 5 x 3 + 7x 5x + 6 = 0 48 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 International Baccalaureate Organization 07 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for Sierra High School

Find the common ratio of the geometric sequence. (2) 1 + 2

. Given that z z 2 = 2 i, z, find z in the form a + ib. (Total 4 marks) 2. A geometric sequence u, u 2, u 3,... has u = 27 and a sum to infinity of 8. 2 Find the common ratio of the geometric sequence.