CZ MAY 131. The first three terms ofa geometric series are

18,12 and p

respectively, where p is a constant.

Find

(a) the value of the common ratio of the series,

(b) the value ofp,

(1)

(1)

(c) the sum of the first 15 terms of the series, giving your answer to 3 decimal places.(2)

ILLd\) c.Grz

tLrk ? I =6

c) s,s = K(l:la.:) = *a++-12-

€

2. (a) Use the binomial theorem to find all the tems of the expansion of

(2 + 3;r)a

Give each term in its simplest form.

(b) Write down the expansion of

(2 - 3x)a

in ascending powers ofx, giving each term in its simplest fom.

15.-kl' o b+ q6r--+Ll6*+ A6r1+glc*

b) t6-%c+ Ll6zf -Lt6*+8lx\

J. f(x):2r3 5x2 + ar + 18

whereaisaconstant.

Given that (r - 3) is a factor of f(:r),

(a) show that a: 9

(b) factorise f(x) completely.

Given that

g(r,) - 2(3rr) - 5(34'1 9(3.) + 18

(c) find the values of ;r that satisfy g(v) : 0, giving your answers to 2 decimal places

where appropriat". ,r,

a) +(3) = Z(g)3 - S(S)'+3a+ \8 =O D 3a " -n -'- a.=-1

(2)

(4)

b) Za: +>L -bx

ec-

-3

c) !Ci3 1 = l:5rOL=-L

3s= l-5 -r Soto3sls = o.3+

35 ='Z =) \= lOqskD no so\gr,hovr,\./ U

-

5, -

1*, + l.y

(a) Complete the table betow, giving the missing value of y to 3 decimal places'

Figure I

Figure 1 shows the region -R which is bounded by the curve with equation y =:;'the x-axis and the lines x = 0 and x: 3 (x' + 1)

(b) Use the trapezium rule, with all the values ofy from your table' to find an approximate

value for the area ofR. (4)

(c) Use your answer to part (b) to find an approximate value for

J,[-..h)*giving your answer to 2 decimal places'

(2)

5-

Figure 2

Figure 2 shows a plan view of a garden'

The plan of the garden ,4BCuf'aionsists of a lrt'anglre ABEjoined to a sectot BCDE of a

circle uith radius l2m and centre B'

ifr" p"iti, A,, B andC lie on a straight line with AB :23m and BC : 12m'

Given that the size of angle IBE is exactly 0 64 radians' find

(a) the area of the garden, giving yout answer in m2' to 1 decimal place' (4)

(b) the perimeter of the garden, giving your answer in metres' to 1 decimal place' (5)

q,) A NEE 7 t (,zxzs)S,c.,o'6\ -- 32'+tr,r+- - -

Grj* =\23 +LzL- z(rz)(zg)CosO6tk

-'> hg = lS.lt--.

rz(Z:sorsil

0.64 rad / '.

? , z3+ 12+ tS:l?-- t jo'ol1- -

6.

Figure 3

Figure 3 shows a sketch of part of the curve C with equation

y=x(x+$(x-2)

The curve C crosses the x-axis at the origin O and at the points I and B'

(a) Write down the x-coordinates of the points ,4 and B.(1)

The finite region, shown shaded in Figure 3, is bounded by the curve C and the r-axis.

(b) Use integration to find the total area of the finite region shown shaded in Figure 3.

= tzrL+

.'. fotw( or"eo.. c S+? = l€

7. (i) Find the exact value ofx for which

logr(2x): logr(5x + 4) - 3

(ii) Given that

log,y + 31o9"2: 5

express Jr' in terms of a.

Give your answer in its simplest fotm.

(4\

(3)

.-) 5 a+.t+ o 2-7 = 8 r) 5l-+ 1= J6rLI

jlL,l$*$. 1qS*2:-a-

8. (i) Solve, for -1 80" ( x < 180",

tan(,r - 40') = 1.5

giving your answers to 1 decimal place'

(ii) (a) Show that the equation

sitq tanq = 3cose + 2

can be written in the fotm

4cos2 0 * 2 cos0 - 1 :0

O) Hence solve, for 0 < d<360',

sin9 tanq : 3cose + 2

showing each stage of your working'

(3)

(3)

(s)

-129.+==-Y:Y,:ftg

-a- 1-CoszB =gCn

zgr2(psS

9. The curve with equation

!:x2-32"'l(i +20, .r>o

has a stationary point P.

Use calculus

(a) to find the coordinates ofP,

(b) to determine the nature of the stationary point P.

(6)

(3)

10.

Figure 4

The circle c has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4.

(a) Write down an equation for the circle C that is shown in Figure 4'(3)

A line through the point P(8, - 7) is a tangent to the circle C at the point 7'

(b) Find the length of PZ.(3)