factor-theorem-11-1.pdf

Questions Q1. Figure 2 shows a solid right pentagonal prism ABCDEFGHJK which is made by fixing a solid right triangular prism ABCDEF onto a solid cuboid ACDFGHJK. The triangle ABC is isosceles with BA = BC and the height of the triangle is x cm. AH = FG = CJ = DK = 3x cm AC = HJ = FD = GK = 2x c...

Questions Q1. Figure 2 shows a solid right pentagonal prism ABCDEFGHJK which is made by fixing a solid right triangular prism ABCDEF onto a solid cuboid ACDFGHJK. The triangle ABC is isosceles with BA = BC and the height of the triangle is x cm. AH = FG = CJ = DK = 3x cm AC = HJ = FD = GK = 2x cm HG = JK = AF = CD = (x − 2) cm The volume of the pentagonal prism is 1008 cm3 (a) Show that x3 − 2x2 − 144 = 0 (4) Given that f(x) = x3 − 2x2 − 144 (b) use the factor theorem to show that (x − 6) is a factor of f(x) (2) (c) (i) Find the value of p, the value of q and the value of r so that f(x) = (x − 6)(px2 + qx + r) (ii) Hence explain why the equation f(x) = 0 has only one solution. (4)  (Total for question = 10 marks) (Q08 4MB1/02R, Jan 2021) Q2. (a) Use the factor theorem to show that (x + 1) is a factor of 18x3 – 9x2 – 17x + 10 (2) (b) Factorise fully 18x3 – 9x2 – 17x + 10 Show clear algebraic working............................................................ (4) (Total for question = 6 marks) (Q24 4MB1/01, Jan 2023) Q3. Given that, for all values of x, 4x3 − 8x2 − 12x + 11 = (2x + k)Q(x) + 11 where Q(x) is a quadratic expression in x, find the positive value of k. k =...........................................................  (Total for question = 3 marks) (Q12 4MB1/01, Jan 2020) Q4. The three functions, f, g and h, are defined as (a) Write down the value of x that must be excluded from any domain of g (1) (b) Find g(2) (1) (c) Express the inverse function g−1 in the form g−1(x) =... (3) (d) Solve the equation g(x) = h(x) (4) (e) (i) Use the factor theorem to show that (2x + 3) is a factor of f(x) (2) (ii) Hence solve the equation f (x) = 0 Show clear algebraic working. (4) (Total for question = 15 marks) (Q11 4MB1/02, Nov 2021) Q5. (a) Use the factor theorem to show that (x - 5) is a factor of x3 - 6x2 - 7x + 60 (2) (b) Hence factorise completely x3 - 6x2 - 7x + 60........................................................... (3) (Total for question = 5 marks) (Q29 4MB1/01, Nov 2020) Q6. f(x) = x3 + ax − 3 where a is an integer. (x − 3) is a factor of f(x) (a) Use the factor theorem to show that a = −8 (2) Given that f(x) = (x − 3)Q(x) where Q(x) is a quadratic expression in x (b) find Q(x) Q(x) =........................................................... (2) (Total for question = 4 marks) (Q19 4MB1/01, Jan 2021) Q7. One solution of the equation Find the other 2 solutions of the equation. Show clear algebraic working....................................................................................................................... (Total for question = 5 marks) (Q24 4MB1/01, June 2021) Q8. (x + 2) is a factor of 6x3 + 31x2 + kx + 30 (a) Use the factor theorem to show that k = 53 (2) (b) Factorise fully 6x3 + 31x2 + 53x + 30........................................................... (3) (Total for question = 5 marks) (Q21 4MB1/01R, Jan 2023) Q9. f(x) = 12x3 − 4x2 25x + 14 (a) Use the factor theorem to show that (3x − 4) is not a factor of f(x) (2) g(x) = f(x) − 2 (b) Factorise fully g(x) Show clear algebraic working............................................................ (4) (Total for question = 6 marks) (Q28 4MB1/01, Nov 2023) Q10. f(x) = 3x3 + ax2 – 20x + b where a and b are integers. (x + 4) is a factor of f(x) (x – 2) is a factor of f(x) (a) Use the factor theorem to find the value of a and the value of b (3) One solution of the equation (b) Without using a calculator and showing all your working, find the other 2 solutions of the equation. Give your answers in exact form. (5) (Total for question = 8 marks)  (QU09 4MB1/02, June 2023) Q11. Given that (x − 5) is a factor of 3x3 − 20x2 + kx + 10 where k is a constant, (a) use the factor theorem to show that k = 23 (2) (b) Solve the equation 3x3 − 20x2 + 23x + 10 = 0 Show clear algebraic working............................................................ (4) (Total for question = 6 marks) (Q29 4MB1/01R, Jan 2022) Powered by TCPDF (www.tcpdf.org)