Tiêu đề Topic 1 Part 1 T.pdf ✅

9a. [3 marks] Consider the complex numbers and . (i) Write down in Cartesian form. (ii) Hence determine in Cartesian form. z1 = 2 3cis – √ 3π 2 z = −1 + i 2 3 – √ z1 (z1 + z2 ) ∗ 9b. (i) Write [6 marks] in modulus-argument form. (ii) Hence solve the equation . z2 z 3 = z2 9c. Let , where and . Find all possible values of r and , [6 marks] (i) if ; (ii) if . z = r cisθ r ∈ R + 0 ⩽ θ < 2π θ z 2 = (1 + z2 ) 2 z = − 1 z2 9d. Find the smallest positive value of n for which [4 marks] ( ) ∈ . z1 z2 n R + 10a. [3 marks] Consider a function f , defined by . Find an expression for . f(x) = for 0 ⩽ x ⩽ 1 x 2−x (f ∘ f)(x) 10b. Let [8 marks] , where . Use mathematical induction to show that for any . Fn(x) = x 2 −( −1)x n 2 n 0 ⩽ x ⩽ 1 n ∈ Z + (f ∘ f∘...∘f) (x) = (x)  n times Fn 10c. Show that [6 marks] is an expression for the inverse of . F−n(x) Fn 10d. (i) State [6 marks] . (ii) Show that , given 0 < x < 1, . (iii) For , let be the area of the region enclosed by the graph of , the x-axis and the line x = 1. Find the area of the region enclosed by and in terms of . Fn(0) and Fn(1) Fn(x) < x n ∈ Z + n ∈ Z + An F −1 n Bn Fn F −1 n An
[4 marks] 11. Find the sum of all the multiples of 3 between 100 and 500. 12. A metal rod 1 metre long is cut into 10 pieces, the lengths of which form a geometric sequence. The length of the longest piece [6 marks] is 8 times the length of the shortest piece. Find, to the nearest millimetre, the length of the shortest piece. 13. Let [7 marks] . Find, in terms of , the modulus and argument of . ω = cosθ + isin θ θ (1 − ω 2) ∗ 14. [5 marks] Find the value of k such that the following system of equations does not have a unique solution. kx + y + 2z = 4 −y + 4z = 5 3x + 4y + 2z = 1 [3 marks] 15a. Three boys and three girls are to sit on a bench for a photograph. Find the number of ways this can be done if the three girls must sit together. [4 marks] 15b. Find the number of ways this can be done if the three girls must all sit apart. 16a. [9 marks] (i) Express each of the complex numbers and in modulus-argument form. (ii) Hence show that the points in the complex plane representing , and form the vertices of an equilateral triangle. (iii) Show that where . z = + i, = − + i 1 3 – √ z2 3 – √ z = −2i 3 z1 z2 z3 z + = 2 3n 1 z 3n 2 z 3n 3 n ∈ N
16b. [9 marks] (i) State the solutions of the equation for , giving them in modulus-argument form. (ii) If w is the solution to with least positive argument, determine the argument of 1 + w. Express your answer in terms of . (iii) Show that is a factor of the polynomial . State the two other quadratic factors with real coefficients. z = 1 7 z ∈ C z = 1 7 π z − 2zcos( ) + 1 2 2π 7 z − 1 7 17a. [2 marks] Consider the system of equations Express the system of equations in matrix form. 0.1x − 1.7y + 0.9z = −4.4 −2.4x + 0.3y + 3.2z = 1.2 2.5x + 0.6y − 3.7z = 0.8. 17b. [3 marks] Find the solution to the system of equations. 18a. [2 marks] The arithmetic sequence has first term and common difference d = 1.5. The geometric sequence has first term and common ratio r = 1.2. Find an expression for in terms of n. {un : n ∈ Z } + u1 = 1.6 {vn : n ∈ Z } + v1 = 3 un − vn 18b. [3 marks] Determine the set of values of n for which un > vn. 18c. [1 mark] Determine the greatest value of un − vn. Give your answer correct to four significant figures. 19. [7 marks] Use the method of mathematical induction to prove that is divisible by 576 for . 5 − 24n − 1 2n n ∈ Z +