2023 SAJC Promo (Qn)

2023 SAJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. 1 The sum of the first n terms of a sequence, nS , is given by 32 n bcadSn nn  , where a, b, c and d are constants . It is given that 12 3 45, 20, 57 and 128SS S S   . Find nS .             [4] 2 (a) Differentiate  1 2 sin 2 1, for 1 12 24 xx x  , with respect to x, simplifying your answer as much as possible as a single fraction. [3] (b) It is given that 24 3e 4xy  . By repeated differentiation, show that 22 2 2 dd 8dd yyy kxx y   , where k is a constant to be determined. [4] 3 (a) Given that    1 2 1 12 16 n r rn n n   , find  1 61 n r rr   , in terms of n. [2] (b) (i) Express   1 12rr r in partial fractions.  [1] (ii) Hence find NS in terms of N, where  1 1 12 N N r S rr r   , and deduce that 1 4 NS  . [6] 4 An arithmetic sequence 123,,,,, nuuu u  is known to have a common difference of ln 3. The rth term, rw , of another sequence 123,,,www , is given by e ru rw  . (i) Show that the sequence 123,,,,, nwww w  , is a geometric progression with common ratio 1 3 . [2] (ii) Explain why 1 r r w    converges. [1] (iii) Given that 1 ln 3u  , find the smallest possible value of n such that the sum of the first n terms of the geometric progression given by 123,,,www  is within 0.5% of the sum to infinity of the geometric series. [3]

5 The position vectors of the points A and B relative to the origin O are a and b respectively. (i) Point M on AB is such that :1 : 2AM MB  . Find OM  in terms of a and b. If the area of triangle OBM is 4 units2, find || ab . [5] (ii) Another point P has position vector p and p0 . Given () pa ba 0 , what can you deduce about the relationship between the vectors AP  and AB  ? Hence find the vector equation of line l that passes through points A and P in terms of b and a. [2] (iii) Given instead that 4aa b b , describe in words, the relation between a and b and find |b|. [3] 6 (a)                     O y x x x

The diagram above shows the graph of . The curve crosses the x-axis at  3.5, 0 and has a minimum point at  2, 0 . The lines 3x , 0x and 3y are asymptotes of the curve. On separate diagrams, sketch the graphs in (i) and (ii), stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes and of any turning points in exact form, if possible. (i)  fyx , and [2] (ii) f '( )yx . [4] (b) The transformations A, B and C are given as follows: A: Reflection about the x-axis; B: Translation of 4 units in the positive x-direction; C: Scaling parallel to the x-axis by a factor of 2. A curve undergoes in succession, the transformations A, B and C, and the equation of the resulting curve is  21 13yx . Determine the original equation of the curve. [3