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Session 1Paper 2 Questions and
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Question 1 (Unit 1 LO1 Straight Line)
Triangle ABC has as its vertices A(-18,6) , B(2,4) and C(10,-8) .
(a) Find the equation of the median from A to BC
(b) Find the equation of the perpendicular bisector of side AC.
(c) Find the coordinates of T, the point of intersection of these
lines.
A
B
C
marks 3, 4, 3
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Solution 1(a)
mid-point of BC
gradient of median
equation of line
1
),( 26
2
3
32
)18(3
16 xy
3
1
24
8
618
26
AMm
(a) ans: xy3
1
284
2102 )(
,
M
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Solution 1(b)
mid-point of AC
gradient of AC
perpendicular gradient
equation of line
1
2
3
2
1
28
14
1018
86
ACm
2m
),( 14
72 xy(b) ans:
4 )4(21 xy
286
21018 )(
,
N
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Solution 1(c)
solving a system of
equations
x-coordinate
y-coordinate
1
2
3
723
1 xx
3x
1
7)3(2
y
y
ans: T(-3,1)
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Question 2(Unit 1 LO3 Differentiation)
The diagram below shows the parabola with equation
and a line which is a tangent to the curve at the point T(1,5). Find the size of the angle marked θ, to the nearest degree.
238 xxy
marks (4)
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Solution 2
Know to differentiate
Find gradient of tangent at x = 1
Use m = tanθ
Complete calculations
1
2
3
xdx
dy68
4
2tan
2)1(68 tangentm
21tan
ans: 63
63
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Question 3(Unit 2 LO4 Circle )
058422 yxyx
yx 7
The circle in the diagram has equation
.The line AB is a chord of the circle and has equation
.
(a) Show that the coordinates of A and B are (-1, -8) and (6, -1) respectively.
(b) Establish the equation of the circle which
has AB as its diameter.
marks (4,3)
x
A
B
y
O
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Solution to question 3a
ans:
substituting into circle equation
multiplying brackets and tidying up
factorising and values of y
corresponding values of x
1
2
3
4
058)7(47 22 yyyy
016182 2 yy
8or1
0812
y
yy
A(-1, -8) and B(6, -1)
A(-1, -8) and B(6, -1)
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Solution to question 3b
ans:
knowing to find midpoint of AB
finding radius
substituting into equation
1
2
3
(25, -45)
524)541()526( 22 r
52454y52x 22
52454y52x 22
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The graph of the cubic function y = f (x) is shown in the diagram. There are turning points at (1,1) and (3,5).
Sketch the graph of y = f '(x)
Question 4(Unit 1 LO3 Differentiation)
x
y (3,5)
(1,1)
)(xfy
marks (3)
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Solution 4
ans: sketch
Interpret stationary points
Parabola
Maximum TP
1
2
3
31
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Session 2Paper 2 Questions and
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Question 5 (Unit 2 LO3 Trigonometry)
022cos3sin oxx
3600 x
Solve algebraically the equation
where
marks 5
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Solution 5
double angle formula
re-arrange to zero and factorise
find roots
answers from
answers from
1
2
4
3
5
02sin213sin 2 xx
0)1sin2)(1sin3(
01sinsin6 2
xx
xx
21
31 xx sinsin or
5.160,5.19
330,210
ans: 1950, 16050 , 2100, 3300
31sin x
21sin x
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An open box is designed in the shape of a cuboid with a square base.
The total surface area of the base andfour sides is 1200cm2 x
x
Question 6(Unit 1 LO3 Differentiation)
(a) If the length of the base is x centimetres, show that the volume V (x) is given by
3
4
1300)( xxxV
(b) Find the value of x that maximises the volume of the box.
h
marks (3,5)
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Solution 6 (a)
ans: proof
Equation for surface area
Rearrange with h = …….
Find V
1
2
3
xhx 41200 2
3
22
4
1300
4
1200
xx
x
xxV
x
xh
4
1200 2
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knowing to differentiate
differentiate
set derivative to zero
solve for x
nature table
Solution 6 (b)
ans: 20x
1
2
3
4
............)( xV
2
4
3030)( xxV
04
3300 2 x
5r 20
)(xV
shape
0
20x
max TP at x = 20
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Question 7 (Unit 1 LO3 Differentiation Unit 2 LO1 Polynomials)
Also shown is the tangent to the curve at the point P
where
Part of the curve is shown in the diagram
xxxy 2410 23
1x
(a) Find the equation of the tangent.
(b) The tangent meets the curve again at Q. Find the coordinates of Q.
marks (4,4)
xxxy 2410 23
x
y
P
Q
O x=1
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Solution 7(a)
differentiate
gradient
y-coordinate
equation
1
2
3
32
(a) ans: 87 xy
24203 2 xxdx
dy
24)1(20)1(3,1 2 mxat
15)1(24)1(10)1( 23 y
4 )1(715 xy
724203 m
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Solution 7(b)
form equation
rearrange to zero
factorise
coordinates of Q
1
2
332
(a) ans: Q (8,64)
4
872410 23 xxxx
081710 23 xxx
)89)(1( 2 xxx
648)8(78 yx
1 1 -10 17 -8
1 -9 8
1 -9 8 0
)8)(1)(1( xxx
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Question 8(Unit 2 LO2 Integration )
marks (4, 4)
The diagram shows the parabolas and 244 xxy
422 xxy
(a) Find the coordinates of the point A
(b) Calculate the area enclosed between the two curves.
4
A
x
y
O
244 xxy
422 xxy
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Solution 8a
Form equation
Rearrange to = 0
Factorise and solve
Coordinates of A
(a) ans:
1
2
3
22 4442 xxxx
062 2 xx
30
0)3(2
xorx
xx
4 )7,3(A
)7,3(A
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Solution 8b
(b) ans:
Integrate
Substitute limits
Answer
1
2
3
dxxx )62(3
0
2
4
2332 3
3
0xx
0)3(3)3( 2332
unitssquare9
27)27(32
unitssquare9
dxbottomtop )(
4
A
x
y
O
244 xxy
422 xxy
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Session 3Paper 2 Questions and
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Question 9(Unit 2 LO4 Circle )
08420422 yxyx
A circle, centre C, has equation
.Show that the line with equation2y = x + 8 is a tangent to the circleand find the coordinates of the point of contact P
marks (5)
C
P
08420422 yxyx
82 xy
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Solution 9
ans:
substitute into circle equation
multiply out brackets and simplify
factorise and solve for y
complete proof
point of contact
1
2
3
4
08420)82(482 22 yyyy
0180605 2 yy
60665 yyy
One point of intersection so line is a tangent
P(4, 6)
5 48)6(26 xy
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Question 10(Unit 2 LO2 Integration)
The diagram shows a sketch of the graph of y = (x + 2)(x – 1)(x – 2) and the points P and Q
x
y
P Q
(0,4)
O(-2,0)
)2)(1)(2( xxxy
(a) Write down the coordinates of P and Q
(b) Find the total shaded area
marks (2,6)
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Solution 10a
ans:
Coordinates of P
Coordinates of Q
1
2
)0,1(
)0,2(
)0,2()0,1( QP
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Solution 10b
two integrals
multiply out brackets
integrate
integral from 0 to 1
integral from 1 to 2
total area
dxxxx )2)(1)(2(1
0
ans:
1
2
3
4
4423 xxxy
xxxx 42 23314
41
5
12111
127
6 127
12111
dxxxx )2)(1)(2(2
1
2212 units
2212 units
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Question 11(Unit 2 LO3 Trigonometry)
The diagram shows a sketch of part of the graph of a trigonometric function whose equation is of the form
Find the values of a, b and c
cbxay sin
x
5
-3
0
cbxay sin
π
y
marks (3)
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Solution 11
ans: 1,2,4 cba
Interpret amplitude
Interpret period
Interpret vertical displacement
1
2
3
4a
1c
2b
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(b) The second diagram shows lines OA and OB. The angle between these two lines is 300
Question 12(Unit 1 LO1 Straight Line)
marks 3,1
02 yxa
.(a) The diagram shows line OA with equations
.The angle between OA and the x-axis is Find the value of a.
x
y
O
A
a
B
30
.
Calculate the gradient of line OB correct to 1 decimal place
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Solution 12a
ans: 6.26
gradient of line
gradient = tan (angle) and apply
process
1
2
3
21gradient
gradient atan 0
6.262
1tan 1
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Solution 12b
ans: 5.1
angle = tan-1(angle)
1 5.1)6.2630tan( m