C2 model answers may 2013

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CZ MAY 13
1. 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,
(1)
(b) the value ofp,
(1)
(c) the sum of the first 15 terms of the series, giving your answer to 3 decimal places.
(2)
ILL
d\) c.Grz
tLrk ? I =6
c) s,s = K(l:la.:) = *a++-
12-

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Page 2

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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.…read more

Page 3

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J. f(x):2r3 5x2 + ar + 18
whereaisaconstant.
Given that (r - 3) is a factor of f(:r),
(a) show that a: 9
(2)
(b) factorise f(x) completely.
(4)
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.…read more

Page 4

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Complete the table betow, giving the missing value of y to 3
Figure I
Figure 1 shows the region -R which is bounded by the curve with
equation y =:;'
(x' + 1)
the x-axis and the lines x = 0 and x: 3
ofy from your table' to find an approximate
(b) Use the trapezium rule, with all the values
value for the area ofR. (4)
(c) Use your answer to part (b) to find an approximate value
for
J,[-..…read more

Page 5

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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' :23m and BC : 12m'
ifr" p"iti, A,, B andC lie on a straight line with AB
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)
in metres' to 1 decimal…read more

Page 6

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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..…read more

Page 7

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Find the exact value ofx for which
logr(2x): logr(5x + 4) - 3
(4\
(ii) Given that
log,y + 31o9"2: 5
express in terms of a.
Jr'
Give your answer in its simplest fotm.
(3)
.-) 5 a+.t+ o 2-7 = 8 r) 5l-+ 1= J6r
LI
jlL,l$*$.…read more

Page 8

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Solve, for -1 80" ( x < 180",
tan(,r - 40') = 1.5
giving your answers to 1 decimal place'
(3)
(ii) (a) Show that the equation
sitq tanq = 3cose + 2
can be written in the fotm
4cos2 0 * 2 cos0 - 1 :0 (3)
O) Hence solve, for 0 < d<360',
sin9 tanq : 3cose + 2
showing each stage of your working' (s)
-129.…read more

Page 9

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The curve with equation
!:x2-32"'l(i +20, .r>o
has a stationary point P.
Use calculus
(a) to find the coordinates ofP,
(6)
(b) to determine the nature of the stationary point P.…read more

Page 10

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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.…read more

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T


what board is this OCR model answers ??

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