2019 A Level H2 FM 9649 P2 (Qns & Solutions))(30 Sep 2024)

2019 GCE A Level H2 Further Maths 9649 Paper 2 Solutions Section A: Pure Mathematics [50 marks] Question 1 For a given non-zero column vector x y = u , let M be the set of all 22 matrices X for which there exists a real scalar constant  such that =Xu u . (i) Show that M contains the zero 22 matrix and also that it is closed under the usual operations of matrix addition and multiplication by a scalar. [4] (ii) Write down a 22 matrix that is not in M for any non-zero u . Justify your answer. [2] [Solution] (i) Given M is the set of 22 matrices X for which there exists a real scalar constant  such that =Xu u , that is u is the eigenvector of X. 00 000 xx yy      =         , thus M contains the zero 22 matrix. Let X1 and X2  M, that is there exists real scalar constants 1 and 2 such that 11 =X u u and 22 =X u u Then 1 2 1 2 1 2 1 2( ) ( )    + = + = + = +X X u X u X u u u u , 12+ is a real scalar constant. 12+XX  M . Thus M is closed under matrix addition. Consider   , (X1)u = ( 1Xu ) =  1 u , 1 is a real scalar. Thus X1  M . Thus M is closed under matrix multiplication by a scalar. (ii) Consider X = 01 10  − then =Xu u  01 10 x x y x y y x y           =  =        −−         No solution for  for any non-zero u = x y   . Thus 01 10  −  M Or any other 2x2 matrix such that =Xu u has no real solution for  .

Question 2 A conic section has polar equation 1 , 0 2 .2 sinr =  − (i) By finding the equation of this curve in a standard Cartesian form, show that it is an ellipse [5] (ii) Determine the eccentricity of this curve and the coordinates of the two foci [4] (i) 1 2 sinr = − 2 sin 1rr −= 2221 x y y+ = + ( ) ( ) 22241x y y+ = + ( ) 2 2 24 1 2x y y y+ = + + 223 2 4 1y y x− + = 2 2113 4 1 33yx − − + = 2 21434 33yx − + = 2 2 19 3 314 y x  − += 2 2 22 1 3 1 21 33 yx  −+=       (ii) ( )1 ; 0,02e= and 20, 3  

Question 3 (i) On an Argand diagram, shade the region of the complex plane which represents the set of all complex numbers z for which 1 i 2z− +  . [3] (ii) For this set of complex numbers, determine the maximum value of ( )arg 2z+ , where ( )arg 2 ,z−  +  giving your answer in the form 1tan p− for some rational number p. [7] [Solution] (i) 1 i 2z− +   (1 i) 2z− −  The shaded region is inside the circle with centre at 1 – i and radius = 2 units (ii) Let max arg(z + 2) = arg(z – (−2)) =  AB is tangent to the circle at B, thus ABC = 90 0 AC = 1 ( 2) 10i− − − = and BC = 2 Thus AB = 10 2 2 2−= tanCAB = 21 222 = Let  be the acute angle AC made with the Real axis, thus tan  1 3= . Thus