HCI 2023 JC1 Promos 9649

1 1. With respect to the origin as the pole, a curve C has polar equation of the form sin 2ra b  for   , where a and b are constants, and b is non-zero. (a) Find the range of values of a b so that there are tangents to C at the pole. [2] (b) For the case where a = 1 and b = 2, sketch C, stating clearly the equations of the tangents to C at the pole, and the equations of the line of symmetry of C. [4] 2. Consider the equation f( ) 0x  , where f( ) c o s 1xx x    and  is very close to zero. (a) By sketching the curve cos 1y xx  , show that f( ) 0x  has a single root , and that  is very close to 0. [3] (b) Use two iterations of the Newton-Raphs on method, with initial approximation 0 0x  , to show that 21 21     . [You may assume for  small, sin  , 2 cos 1 . 2   ] [5] 3. Points F and G lie on the x-axis at the points (, 0 )c and (, 0 )c respectively, with c > 0. The curve L is defined as the locus of points P for which the total distance 2FP PG a , where ac . It is given that FP r and GFP  . (a) Show that 1c o s lr k   , giving the constants l and k in terms of a and c. [4] A, B, C, D are points on L such that the chords AB and CD both pass through F, and AB is perpendicular to CD. (b) Find 11 FA FB in terms of a and c. [2] (c) Find 11 ABC D in terms of a and c. [3]

2 4. A sequence 123, , , uuu  is such that 1 2 3u  and 21 1 4 1 nn n uu   , for all 2n . (a) Find the values of 23 4, and uu u , giving each answer as a fraction in its lowest term. Make a conjecture for nu in terms of n. [2] (b) Use Mathematical Induction to prove your conjecture in (i) for all positive integers n. [ 4 ] (c) Deduce the value of 2 2 1 41n n    . [ 3 ] 5. A function f is defined as 31f( )x xx  , x  , 1x . (a) Using the trapezium rule with 3 ordinates, the value for 2 f( )d a x x , where a  , 2a , is approximately 5.56. Find the value of a correct to 1 decimal place. [3] An approximate value for 2.5 2 f( )dx x is to be found using Simpson’s rule. (b) Explain why it is not possible to use Simpson’s rule with 3 strips to find an approximate value for 2.5 2 f( )dx x . [ 2 ] (c) Estimate 2.5 2 f( )dx x using Simpson’s rule with 4 strips. Leave your answer correct to 1 decimal place. [2] (d) A student made the following claim: ‘Since Simpson’s rule will give an exact value for 2.5 2 f( )dx x if f( )x is of degree 3 or less, the value for 2.5 2 f( )dx x calculated in part (c) should be taken instead of the value for 2.5 2 f( )dx x given in part (a).’ Comment on the student’s claim. [2]

3 6. Let 33:T  be a function given by 25 2 36 3 xx y z k Ty x y zk zy z k            , where k is a constant. (a) State the value of k for which T is a linear transformation, justifying your a n s w e r .