ACJC EJC NJC RVHS FM 2024 Prelim P1

2024 H2 FM 9649 Prelim Paper 1 (ACJC_EJC_NJC_RVHS) P1 Q1 The transformation T is a linear mapping from 4 to 3 defined by the matrix A. (a) Explain why the null space of A is not the zero vector space. [2] It is given that the matrix A is defined as follows. 2 3 4 8 1 1 1 3 3 6 9 15 − =− − A (b) The kernel of T, ( )ker T , is defined as the set of vectors x y z w       in 4 that maps to the zero vector in 3 . Showing all necessary working, use row operations on A to find ( )ker T . [4] P1 Q2 Let 3 2 1 e sin dxI x x= . (a) Use a calculator to find the value of I, giving your answer to 2 decimal places. [1] (b) Use Simpson’s Rule with 3 ordinates to approximate the value of I, correct to 2 decimal places. Hence find the approximate percentage error when using this method. [3] (c) Explain with the aid of a suitable diagram, why Simpson’s Rule does not give a good approximation of the value of I in (b) and suggest how the method can be improved to give a better approximation. [3] P1 Q3 The sequence  nM is given by the recurrence relation 2 1 2 1 2 2 nn n nn MMM MM + + + =+  for 1n , and the set of positive initial values 1M and 2M . (a) For the case when 1 1M = and 2 2M = , by considering the subsequences  2kM and  21kM − where k + , of the sequence  nM , determine the behaviour of the sequence  nM as n→ . (A subsequence of  nM is a sequence obtained by removing some terms without changing the order of the remaining terms.) [3] (b) For the case when 1 2M = and 2 2M = , explain how the behaviour of the sequence will differ from part (a). [2] (c) For the case when 12MM == , the sequence  nM becomes a constant sequence. Find the exact value of  . [2]

2024 H2 FM 9649 Prelim Paper 1 (ACJC_EJC_NJC_RVHS) P1 Q4 The function f is given by f ( ) 8sin 2 5 2 xxx = − + . It is known that the equation f ( ) 0x = has a single root x = . (a) Show that 4.5 5 . [2] (b) Show that the iterative process defined by 0 4.5x = , 1 54sin 22 n n xx + =+  is not suitable to find a good approximation to  . [2] (c) Using the Newton-Raphson method, write down an iterative formula of the form ( )1 Fnnxx+ = that can be used to approximate the value of  . Hence calculate the value of  , correct to 3 decimal places. [3] (d) By considering the stationary points of the curve f ( )yx= , explain whether 0 2x = is a suitable starting value for using the Newton-Raphson method to find an approximation to  . [2] P1 Q5 A curve C is given by the equation 4 4 2 2 (1 )x y x y y+ = − . (a) Show that C has the polar equation sin 2 cosr = for 0 π . [3] (b) Sketch C, stating the equation of the line of symmetry (if any). [2] (c) Find the length of curve C. [4] P1 Q6 A Bernoulli Equation is a differential equation of the form d p( ) q( ) ,d ny x y x yx += where p( )x and q( )x are functions of x an