.('q-10,1' 'ro. 1. (a) Find the binomial expansion of 1,, 9. 3+x 9. ('q - tox)' (b) Hence, or otherwise, find the expansion of.

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1 m 1. (a) Find the binomial expansion of 1,, 9 ('q - tox)' I"l ' lo in ascending powers of x up to and including the term in x2. Give each coeffrcient as a simplified fraction. (s) (b) Hence, or otherwise, find the expansion of 3+x 9.('q-10,1' l,'l 'ro in ascending powers of x, up to and including the term in x2. Give each coefficient as a simplified fraction. (3)

2 o) Q- ror,) -\' ' q- L ( r-te*)'l { * (r- '*4-t 't t, *?+,1(-,i*) *t-*ilf'er)--] iirt*?*" h) (.**)L b * t + ty- *#nn' 2*t + 9,t? TB* 51,-- F;*---*' 1 + \*"*t'

3 2. Figure 1 Figure 1 shows a sketch of part of the curve with equation y:(2-r)e2', xelr The finite region.r, shown shaded in Figure 1, is bounded by the curve, the x-axis and the y-axis. The table below shows corresponding values of r and y for y: 12 - x)e2',^\ h x v (a) Use the trapezium rule with all the values ofy in the table, to obtain an approximation for the area of R, giving your answer to 2 decimal places. (3) (b) Explain how the trapezium rule can be used to give a more accurate approximation for the area of R. (1) (c) Use calculus, showing each step in your working, to obtain an exact value for the area of R. Give your answer in its simplest form. (s) t (a.zfu*qlt-.j*-g)! 5,8tr )

4 3. x2+)p+10x+2y-4xy:la (a) pind 9 in terms of x and y, fully simpli$ing your answer. dx (s) (b) Find the values ofy for which * : O dx (s) & q+t(l-- \x--'?-q-l b') da Eo 3,.tzt :fu1+l?$ i

5 4. (a) Express -Artin partial fractions. Figure 2 ' xrl(2x + 1) Figure2 showsasketchof partofthe curve Cwithequation, = --l-, x) 0 The finite region R is bounded by the curve C, the x-axis, the line with equation n : 1 and the line with equationx: 4 This region is shown shaded in Figure 2 The region fi is rotated through 360' about the x-axis. (b) Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form a + bln c, where a, b and c are constants. (6) z,=t 1-s i 3^+38 fq LcD,ry 2"5 = 3A +.?s{-too i:. 3_4"_-I5o-i-ff'5>-q -E ry -SOlt)t- {-SO tn(z:l+t -\,t2' +Son

6 At time / seconds the radius of a sphere is r cm, its volume is Zcm3 and its surface area is S crn2. {7bu orn given that V:!or3 and that S: 4ttrzf The volume of the sphere is increasing uniformly at a constant rate of 3 cm3 s-1. (a) find * when the radius of the sphere is 4 cm, giving your answer to 3 significant d.t figures. (4) (b) Find the rate at which the surface area of the sphere is increasing when the radius is 4 cm. (2) ffi s, [- 5 Z

7 With respect to a fixed origin, the point A with position vector i + 2i + 3k lies on the line l, with equation r = [i]-..,.[j]' where 't is a scarar parameter' and the point B with position vector 4i + pi t 3k, where p ts a constant, lies on the line /, with equation ' : [i]. '[ ;] where p is ascarar parameter (a) (b) Find the value of the constantp. Show that l, and lrintersect and find the position vector of their point of intersection, C. (1) $) (c) Find the size of the angle ACB, giving your answer in degrees to 3 significant at*".,5 (d) Find the area of the triangle ABC, giving your answer to 3 significant figures. 1+31,t= { *) }Pr.= - J :-i;-\--- 3-r\ 2 -t +3ylx*-6- z-

8 , fabs,nc t sf{(qe) Srn e3'?- Are{' ' lq'tr/\- "/ re\-,.ffi'=tti3 i;lr=sqct" s.l-r- ;R Cd = or4o+\5 'sl 0'(or-\(;f,ha s' ar '(q.(s5 - ' 0'rt-+.

9 1 The rate of increase of the number, N, of fish in a lake is modelled by the differential equation ditr = (kl-1xs000-n) dtt t>0,0<n<5000 In the given equation, the time / is measured in years from the start of January 2000 and ft is a positive constant. (a) By solving the differential equation, show that 1/: Arc-kl where A is a positive constant. (s) After one yeal atthe start of January 200i, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake. (b) Find the exact value of the constant A andthe exact value of the constant k (4) (c) Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish. V*a -lnt + c ^ smo^; *t4 -'* \l z ;-\;-ffi k:t- --frl, SfrD:aAe3: &b (1) ;ft =9C2.I^ ll= '''-'"rc,l --) tu

10 8. Figure 3 The curve shown in Figu:e 3 has parametric equations x:t-4sin/,!:l-2cos/, -*t J _2n J The point A, wrth coordinates (k, 7),lies on the curve. Given thalk> 0 (a) find the exact value of ft, (b) find the gradient of the curve at the point A. (2) (4) There is one point on the curve where the gradienl is equal * -* (c) Find the value of t atthis point, showing.uln,,"0 in your working and giving your answer to 4 decimal places. fsolutions based entirely on graphical or numerical methods are not acceptable.] (6)

11 L) tsr^a l*kb.lk a -l ') -?S (bs LE a -- 3 *?.mtf ( ba Zo?k.: t = 0'60+t' Z 4S,^t s4cot-l l6s,n"k ' 16Cotk -8(st + \ lb -t6co#t ' l6(rl)'k -8cot t \ :) ;) 34trn't-t&ht' -l( = O dctv' 6k (ot, 0' (o'? = -0'5?09? l+rrrt Z r-j3r 7!,r 0'60+?" ; 2-l+8s"