Title Topic 1 Part 3 T.pdf ✅

6c. [5 marks] Hence determine the exact value of ∫ |sin x|dx . ∞ 0 e−x 7. [5 marks] Solve the equation 4 = + 8. x−1 2 x 8. [20 marks] (a) Show that . (b) Hence prove, by induction, that for all . (c) Solve the equation . sin 2nx = sin((2n + 1)x) cosx − cos((2n + 1)x)sin x cosx + cos3x + cos5x +... + cos((2n − 1)x) = , sin 2nx 2 sin x n ∈ Z , sin x ≠ 0 + cosx + cos3x = , 0 < x < π 1 2 9. [5 marks] The system of equations is known to have more than one solution. Find the value of a and the value of b. 2x − y + 3z = 2 3x + y + 2z = −2 −x + 2y + az = b 10. [7 marks] (a) Solve the equation , giving your answers in modulus-argument form. (b) Hence show that one of the solutions is 1 + i when written in Cartesian form. z = −2 + 2i 3 [5 marks] 11. Find the sum of all three-digit natural numbers that are not exactly divisible by 3.
12. [10 marks] (a) Consider the following sequence of equations. (i) Formulate a conjecture for the equation in the sequence. (ii) Verify your conjecture for n = 4 . (b) A sequence of numbers has the term given by . Bill conjectures that all members of the sequence are prime numbers. Show that Bill’s conjecture is false. (c) Use mathematical induction to prove that is divisible by 6 for all . 1 × 2 = (1 × 2 × 3), 1 3 1 × 2 + 2 × 3 = (2 × 3 × 4), 1 3 1 × 2 + 2 × 3 + 3 × 4 = (3 × 4 × 5), 1 3 ... . n th n th un = 2 + 3, n ∈ n Z + 5 × 7 + 1 n n ∈ Z + 13. [16 marks] Consider . (a) Show that (i) (ii) (b) (i) Deduce that . (ii) Illustrate this result for on an Argand diagram. (c) (i) Expand and simplify where z is a complex number. (ii) Solve , giving your answers in terms of . ω = cos( ) + isin( ) 2π 3 2π 3 ω = 1; 3 1 + ω + ω = 0 2 e + + = 0 iθ e i(θ+ ) 2π 3 e i(θ+ ) 4π 3 θ = π 2 F(z) = (z − 1)(z − ω)(z − ω ) 2 F(z) = 7 ω