OCR 2023 GCE FURTHER MATHEMATICS B MEI Y414/01: NUMERICAL METHODS AS LEVEL QUESTION PAPER & MARK SCHEME (MERGED)

The table shows some spreadsheet output. The values displayed in cells K21 and K22 are exact, but the values displayed in rows 21 and 22 in columns L, M and N have been found using appropriate cell formulae. ◢ K L M N 20 x e x 10x e x - 10x 21 0....30258E-09 22 1E-10 1 1 1.30258E-10 (a) (i) Write the value displayed in cell N21 in standard mathematical notation. [1] The formula in cell N22 is . (ii) Explain why the value displayed in cell N22 is not zero. [2] (b) (i) Determine the absolute error when e 0.1 is chopped to 3 decimal places. [2] (ii) Without doing any further calculation, explain whether the absolute error found in part (b)(i) will be the same as the absolute error when e 0.1 is chopped to 3 decimal places. [1] 2 The table gives three values of x and the associated values of y. x 3 5 8 y 9.26 19.3 37.96 Use Lagrange’s method to construct the interpolating polynomial of degree 2 for the values in the table, giving your answer in the form y = ax2 + bx+ c, where a, b and c are constants to be determined. [4] 3 © OCR 2023 Y414/01 Jun23 Turn over x-y n n n n 3 You are given that 43.2 x+ y and Q = 43.2 and that x = 2.17 and y = 2.14. The values of x and y have been rounded to 2 decimal places. (a) Calculate the range of possible values of P. [2] (b) Calculate the range of possible values of Q. [2] (c) Explain why the answer to part (b) is much larger than the answer to part (a). [1] 4 It is given that the function g(x) is continuous and differentiable and that gl(1.8) = 0.576, correct to 3 decimal places. (a) Use this result to determine the error when g(1.8) is used to approximate g(1.802), giving your answer correct to 3 significant figures. [2] The largest root, a, of the equation x 3 - x 2 - 2x + 1 = 0 is approximately 1.8. The iterative formula xn = g(x ) where g(x ) = 3 x 2 +2x - 1 is to be used to find this root. (b) Use the iterative formula with x0 = 1.8 to find a correct to 5 decimal places. [3] The other positive root of the equation, b, is approximately 0.445. It is given that gl(0.445) = 4.87, correct to 2 decimal places. (c) Explain why the iterative formula may not be used to find b. [1] (d) Use the relaxed iteration xn +1 = (1 - m) xn + mg(xn ), with m =-0.258 and x0 = 0.445, to determine b correct to 8 decimal places.