Question 1 (Unit 1 LO3 Differentiation)
Find the stationary points of the function
and determine their nature.
marks (7)
412)( 3 xxxf
Solution 1
Know to differentiate
Differentiate
Derivative equal to zero
Factorise and solve
Find y coordinates
Nature Table
State nature of TPs
............)( xf1
2
3
ans:
4
0123 2 x
123 2 x
2,20)2)(2(3 xxx
Max TP at (-2,20) Min TP at (2,-12)
5 )12,2()20,2(
6
7
x 2
)(xf
shape
2 00
Max TP at (-2,20) Min TP at (2,-12)
Solution 2
Prepare to integrate
Integrate
Substitute
Answer
2
1
247 dxxx1
2
3
ans:
4
2
1
12
472
xx
x
)47()142( 21
24
213
213
2
122
2
2
3 47dx
xx
x
x
x
Question 3 (Unit 2 LO3 Trigonometry)
If x is an acute angle such that 3
1tan x
find exact values for
marks (4,2)
(a)
(b)
x2sin
x2cos
Solution 3a
Construct a right angled triangle and use Pythagoras
Find sin x and cos x
Formula for sin2x
Substitute and answer
1
2
3
ans:
4
10
3cos
10
1sin xx
xxcossin2
10
3
10
12
5
3
X
1
3
10
6
10
Solution 3b
Formula for cos2x
Substitute and answer
1
2
ans:
xxx 22 sincos2cos
22
10
1
10
3
5
4
10
8
10
1
10
9
Note
usedbemayxorxx 22 sin211cos22cos
Solution 4
ans: 23
32
23
,,,x
1 Use double angle formula
Factorise and form equations
Solve
Solve
xxx cossinsin 22
032 )sin(cos xx2
230 xorx sincos
23
2 orx3
23xsin
32
3 orx4
0xcos
Question 5 (Unit 1 LO1 Straight Line)
The line with equation meets the
x and the y axes at the points A and B respectively
123 yx
(a) Determine the coordinates of A and B
marks (2,4)
(b) Find the equation of the perpendicular bisector of AB.
Solution 5b
ans: 163 xy
1 Rearrange to y =…… and find gradient
Perpendicular gradient
Midpoint of AB
Find equation of line
431 xy
3m2
31m
)(, 2
402
012 3
)6(32 xy4
),( 26
Question 6 (Unit 1 LO2 Functions and graphs)
The diagram below shows part of the graph of .
The function has stationary points at and as shown
)(xgy)3,0(
marks (3)
)0,2(y
x
-3
2O
)(3 xgy Sketch the graph of the related function
Question 7 (Unit 1 Recurrence Relations)
A doctor administers 40ml of a drug to Mr Sick each week. Over the same period 80% of the drug in the bloodstream is removed.If the level in the bloodstream rises above 55ml the drug becomes toxic
(a) Write down a recurrence relation to model this situation.
(b) Find a limit and explain what it means in the context of the question.
marks (2,4)
Solution 7b
a
bL
1
1
3
ans:
201
40
.L
toxicnotso5550
justify limit
use limit formula
calculate limit
explanation
50
4 toxicnotso5550
12.01 asexistslimit
2
Question 8 (Unit 2 LO1 Polynomials)
(a) For what value of k is
when k takes this value
marks (3,2)
2x
8223 kxxx
a factor of
?
(b) Hence fully factorise the expression 8223 kxxx
Solution 8a
Use synthetic division
Complete division
Calculate k
1
2
3
ans:
0204 k
5k
-2 1 -1 2k -8
-2 1 -1 2k -8
-2 6 -4k-12
1 -3 2k+6 -4k-20
5k
Solution 8b
Find quotient
Factorise fully
1
2
ans:
)43( 2 xx
-2 1 -1 -10 -8
-2 6 8
1 -3 -4 0
))(( 432 2 xxx
)4)(1)(2( xxx
)4)(1)(2( xxx
Question 9 (Unit 2 LO1 Polynomials)
For what value of p does the equation
have equal roots?
01)1( 2 pxxp
marks (4)
Solution 9
Use the discriminant
Find values of a, b and c
Substitute and simplify
Calculate value of p
042 acb1
3
ans:
0142 )( pp
4 0)2)(2( pp
2p
2p
11 cpbpa2
0442 pp
Question 10 (Unit 2 LO4
Circle)
A block of wood of thickness t has to pass between two rollers
046422 yxyx
The equations of the two circles are
marks (4)
and
0246302222 yxyx
x
t
y
Find the maximum possible value of t
Solution 10
Find centre and radius of small circle
Find centre and radius of large circle
Calculate distance between centres
Calculate distance, t
3)3,2( rcentre1
2
3
ans:
4
10)15,11( rcentre
22 )315()211( d
15d
)310(15 t 2t
2t
Question 11 (Unit 1 LO3 Differentiation)
Part of the graph of
marks (5)
is shown in the diagram
The tangent to the curve at the point where x = 1is also shown
1 x
y
o
Find the equation of the tangent at the point where x = 1
)65( 2 xxxy
Solution 11
ans:
1 knowing to differentiate differentiate
gradient at x = 1
y coordinate
equation
.......dxdy
6103 2 xxdxdy2
16103 m
3
3 xy
26511 yxat ,4
)1(12 xy5
1 xatm dxdy
Question 12 (Unit 1 LO2 Functions and
graphs)
The functions 9)( 2 xxf
are defined on the set of real numbers.
marks (1,2,3)
and xxh 23)(
(a) Evaluate ))3(( fh
(b) Find an expression for ))(( xhf
(c) For what value(s) of x does )())(( xfxhf
Solution 12a
Evaluate h(f (3)) 3)0())3(( hfh1
1
2
ans:
)23( xf
9)23( 2 x
3
Solution 12b
Apply h
Apply f
ans: 9)23( 2 x