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Higher Mathematics
Practice Unit Tests
Contents & Information

9 Practice Assessments....(3 for each unit)

Answers
Detailed marking schemes
Pegasys Educational Publishing
Pegasys 2010
FORMULAE LIST
( UNIT 2 )
The equation x 2  y 2  2 gx  2 fy  c  0 represents a circle centre ( g ,  f ) and radius
g
2

 f 2c .
The equation (x  a )2  ( y  b)2  r 2 represents a circle centre ( a , b ) and radius r.
Trigonometric formulae:
sin  A  B  sin A cos B  cos A sin B
cos  A  B  cos A cos B  sin A sin B
cos 2 A  cos2 A  sin2 A
 2 cos2 A  1
 1  2 sin2 A
sin 2 A  2 sin A cos A
Pegasys 2010
FORMULAE LIST ( UNIT 3 )
Scalar Product:
a
.b

a b cos , where  is the angle between a and b .
or
a
.b

Trigonometric formulae:
a1 b1  a2 b2  a3 b3
 a1 
 b1 
 
 
where a   a2  and b   b2 
 
 
 a3 
 b3 
sin  A  B  sin A cos B  cos A sin B
cos  A  B  cos A cos B  sin A sin B
cos 2 A  cos2 A  sin2 A
 2 cos2 A  1
 1  2 sin2 A
sin 2 A  2 sin A cos A
Table of standard derivatives:
Table of standard integrals:
f (x )
f  (x )
sin ax
a cos ax
cos ax
 a sin ax
sin ax
cos ax
Pegasys 2010
 f ( x) dx
f ( x)

1
cos ax  C
a
1
sin ax  C
a
Practice Assessment (1)
Unit 1 - Mathematics 1 (H)
Outcome 1
1.
Marks
A line passes through the points (4, –5) and (7, 4).
Find the equation of this line.
2.
A line makes an angle of 42o with the
positive direction of the x-axis, as shown
in Diagram 1.
The scales on the axes are equal.
Find the gradient of the line giving your answer
correct to 3 significant figures.
3.
(2)
y
42o
o
x
Diagram 1
(2)
A line L has equation y  34 x  2 .
Write down the gradient of a line which is:
(a)
parallel to L
(1)
(b)
perpendicular to L
(1)
Pegasys 2010
Outcome 2
4.
y
The graph of a cubic function y  f (x)
is shown in Diagram 2.
(4, 6)
On separate diagrams sketch the graphs of :
(a)
y   f (x) .
(b)
y  f ( x  1) .
(2)
o
1
The graphs with equations y  2 x
And y  a x are shown in Diagram 3.
(2)
x
Diagram 2
–3
5.
8
y
(2, 16)
14
12
If the graph with equation y  a x
passes through the point (2, 16), find
the value of a.
10
8
(1)
6
y2
x
4
(2,4)
2
-1
o
1
2
3
x
Diagram 3
y
6.
x
The graphs of y = 4 and its
inverse function are shown in Diagram 4.
y  4x
Write down the equation of the
inverse function.
(1)
(0, 1)
o
x
(1, 0)
Diagram 4
7.
Functions f and g are defined on suitable domains by f ( x)  x 2  5 and g ( x)  2 x  1 .
Obtain an expression for f ( g ( x)) .
Pegasys 2010
(2)
Outcome 3
8.
Given y 
dy
x5  3
.
, x ≠ 0, find
3
dx
x
(2)
y  x 2  5x  6
9.
A sketch of the curve with equation
y  x 2  5x  6 is shown in Diagram 5.
A tangent has been drawn at the point P(4, 2).
Find the gradient of the tangent at P.
10.
y
P(4, 2)
o
x
Diagram 5
(3)
A curve has equation y  31 x 3  x 2  8 x  1 .
Using differentiation, find the coordinates of the stationary points on this curve and determine
their nature.
(6)
Outcome 4
11.
In a small colony 25% of the existing insects are eaten by predators each day but during
the night 500 insects are hatched.
There are Un insects at the start of a particular day.
(a)
(b)
Write down a recurrence relation for Un+1, the number of insects at the start
of the next day.
(1)
It is known that the colony cannot survive if there are more than 2100 insects.
(i)
Find the limit of the sequence generated by this recurrence relation
as n
∞.
(ii)
In the long term, can the colony survive?
(3)
Pegasys 2010
Practice Assessment (2)
Unit 1 - Mathematics 1 (H)
Outcome 1
1.
Marks
A line passes through the points (2, –5) and (–3, 5).
Find the equation of this line.
2.
A line makes an angle of 65o with the
positive direction of the x-axis, as shown
in Diagram 1.
(2)
y
65o
The scales on the axes are equal.
Find the gradient of the line giving your answer
correct to 3 significant figures.
o
x
(2)
Diagram 1
3.
A line L has equation y  5 x  7 .
Write down the gradient of a line which is:
(a)
parallel to L
(1)
(b)
perpendicular to L
(1)
Pegasys 2010
Outcome 2
4.
The graph of a cubic function y  f (x)
is shown in Diagram 2.
y
(–1,
5)
On separate diagrams sketch the graphs of :
(a)
y   f (x) .
(b)
y  f ( x  3) .
(0, 3)
(2)
–3
(2)
o
3
x
Diagram 2
5.
y
The graphs with equations y  3 x
and y  a x is shown in Diagram 3.
.
10
.
8
If the graph with equation y  a x
passes through the point (1, 7), find
the value of a.
(2, 9)
(1, 7)
6
y  3x
(1)
4
2
-1
.
o
1
2
3
x
Diagram 3
y
y  5x
6.
The graphs of y  5 x and its
inverse function are shown in Diagram 4.
Write down the equation of the inverse function.
(0, 1)
o
(1)
x
(1, 0)
Diagram 4
7.
Functions f and g are defined on suitable domains by f ( x)  4 x 2 and g ( x)  x  3 .
Obtain an expression for f ( g ( x) .
Pegasys 2010
(2)
Outcome 3
8.
Given y 
dy
x6  4
.
, x ≠ 0, find
2
dx
x
(2)
y
9.
A sketch of the curve with equation
y  x 2  10 x  16 is shown in Diagram 5.
A tangent has been drawn at the point
P(6, –8).
Find the gradient of the tangent at P.
10.
y  x 2  10 x  16
o
P(6, –8)
x
Diagram 5
(3)
A curve has equation y  13 x 3  x 2  15x  3 .
Using differentiation, find the coordinates of the stationary points on this curve
and determine their nature.
(6)
Outcome 4
11.
In a small rabbit colony one sixth of the existing rabbits are eaten by predators each
summer, however during the winter 300 rabbits are born.
There are Un rabbits at the start of a particular summer.
(a)
(b)
Write down a recurrence relation for Un+1, the number of rabbits at the start
of the next summer.
(1)
It is known that the colony cannot sustain more than 2000 rabbits at any one time.
(i)
Find the limit of the sequence generated by this recurrence relation
as n
∞.
(ii)
In the long term, can the colony sustain the number of rabbits?
Pegasys 2010
(3)
Practice Assessment (3)
Unit 1 - Mathematics 1 (H)
Outcome 1
1.
Marks
A line passes through the points (–1, 3) and (3, –5).
Find the equation of this line.
2.
A line makes an angle of 35o with the
positive direction of the x-axis, as shown
in Diagram 1.
The scales on the axes are equal.
Find the gradient of the line giving your answer
correct to 3 significant figures.
3.
(2)
y
35o
o
x
Diagram 1
(2)
A line L has equation y   2 x  9 .
3
Write down the gradient of a line which is:
(a)
parallel to L
(1)
(b)
perpendicular to L
(1)
Pegasys 2010
Outcome 2
y
4.
The graph of a function y  f (x)
is shown in Diagram 2.
(3, 4)
On separate diagrams sketch the graphs of :
(a)
y   f (x) .
(b)
(2)
o
-5
5
x
y  f ( x)  1 .
(2)
(–3, –4)
Diagram 2
y
5.
The graphs with equations y  5 x
and y  a x is shown in Diagram 3.
10
y  5x
8
.
6
If the graph with equation y  a x
passes through the point (2, 4), find
the value of a.
.
(1, 5)
4
2
-1
(2, 4)
(1)
.
o
1
2
3
x
Diagram 3
6.
The graphs of y  2 x and its
inverse function are shown in Diagram 4.
y
y  2x
Write down the equation of the inverse function.
(1)
(0, 1)
o
x
(1, 0)
Diagram 4
7.
Functions f and g are defined on suitable domains by f ( x)  2 x 2  x and g ( x)  2 x  5 .
Obtain an expression for f ( g ( x)) .
Pegasys 2010
(2)
Outcome 3
8.
9.
dy
5  x6
Given y 
.
, x ≠ 0 , find
2
dx
x
(2)
y
A sketch of the curve with equation
y  x 2  2 x  3 is shown in Diagram 5.
A tangent has been drawn at the point
P(–1, 0).
y  x 2  2x  3
.
P(– 1, 0) o
x
Find the gradient of the tangent at P.
(3)
Diagram 5
10.
A curve has equation y  x 3  8x 2  12 x  1.
Using differentiation, find the coordinates of the stationary points on this curve
and determine their nature.
(6)
Outcome 4
11.
For an established ant hill 8% of the worker ants fail to return at the end of each day.
However, during the night 460 worker ants are hatched.
There are Un worker ants at the start of a particular day.
(a)
(b)
Write down a recurrence relation for Un+1, the number of worker ants at the start
of the next day.
(1)
It is known that the colony cannot survive if there are more than 6000 ants.
(i)
Find the limit of the sequence generated by this recurrence relation
as n
∞.
(ii)
In the long term, will the colony survive?
Pegasys 2010
(3)
Answers
Unit 1 - Practice Assessments
Practice Assessment 1
1.
m3
2.
m  0 900
4.
(a) diagram (reflection in x-axis)
5.
a4
7.
(a)
Outcome 3 :
8.
dy
 2 x  9 x 4
dx
Outcome 4 :
12.
(a)
Outcome 1 :
Outcome 2 :
y  3x  17
,
( y  4  3( x  7) or y  5  3( x  4) )
3.
6.
m
(a)
3
4
m   43
(b)
(b) diagram (translated 1 unit left)
y  log 4 x
f ( g ( x))  (2 x  1) 2  5  4 x 2  4 x  4
9.
m3
U n1  0  75U n  500
(b)
10.
(4,  773 ) , max ; (2, 313 ) , min
(i)
L  2000 , (ii) will survive
since 2000 < 2100.
Practice Assessment 2
1.
m  2 ,
2.
m  214
4.
(a) diagram (reflection in x-axis)
5.
a7
7.
(a)
Outcome 3 :
8.
dy
 4 x 3  8 x 3
dx
Outcome 4 :
11.
(a)
Outcome 1 :
Outcome 2 :
y  2 x  1
3.
6.
( y  5  2( x  2) or y  5  2( x  3) )
m5
(a)
m   15
(b)
(b) diagram (translated 3 units right)
y  log 5 x
f ( g ( x))  4( x  3) 2  4 x 2  24 x  36
9.
10. (5, 184
3 ) , max
m2
U n1  56 U n  300
(b)
(i)
L  1800
,
(ii)
(3,24 ) , min
will sustain
since 1800 < 2000.
Practice Assessment 3
Outcome 1 :
Outcome 2 :
1.
m  2
,
y  2 x  1
2.
m  0 700
3.
4.
(a) diagram (reflection in x-axis)
5.
a  2 6.
m   23
(a)
(b)
m
3
2
(b) diagram (translated 1 unit down)
y  log 2 x
f ( g ( x))  2(2 x  5) 2  (2 x  5)  8x 2  42 x  55
7.
(a)
Outcome 3 :
8.
dy
 10 x 3  4 x 3
dx
Outcome 4 :
11.
(a)
Pegasys 2010
( y  5  2( x  3) or y  3  2( x  1) )
9.
U n1  0  92U n  460
m  4
10. (6, 145) , max
(b) (i) L  5750
,
( 23 ,  85
27 ) , min
(ii) will survive
since 5750 < 6000.
Practice Assessment (1)
Unit 2 - Mathematics 2 (H)
Outcome 1
1.
2.
Marks
(i)
Show that ( x  3) is a factor of g ( x)  x 3  6 x 2  5 x  12 .
(ii)
Hence factorise g(x) fully.
(5)
Determine the nature of the roots of the equation 2 x 2  7 x  1  0 using the discriminant.
(3)
Outcome 2
3.
4.
Find

1
dx , x  0
x5
The curve with equation y  x( x  4) 2 is shown in
Diagrams 1.
(3)
y
y  x( x  4) 2
Calculate the shaded area shown in Diagram 1.
o
x
4
Diagram 1
5.
(5)
The line with equation y  x and the curve with the equation y  4 x  x 2 are shown
in Diagram 2.
The line and the curve meet at the points where
x = 0 and x = 3.
y
yx
Calculate the shaded area shown in Diagram 2.
y  4x  x 2
o
3
4
x
Diagram 2
(6)
Pegasys 2010
Practice Assessment (2)
Unit 2 - Mathematics 2 (H)
Outcome 3
1
for 0  x   .
6.
Solve the equation sin 2 x 
7.
Diagram 3 shows two right-angled triangles.
2
(2)
2
1
yo
x
o
3
2
Diagram 3
(a)
Write down the values of cos x  and sin y  .
(b)
Show that the exact value of cos ( x  y )  is
(2)
4
(2)
.
80
8.
(a)
Express sin x  cos10   cos x  sin 10  in the form sin ( x   )  .
(b)
Using the results from (a),
(1)
solve sin x  cos10   cos x  sin 10   13 for 0  x  180 .
(4)
A circle has radius 7 units and centre C(2, –5).
Write down the equation of the circle.
(2)
A circle has equation x 2  y 2  6 x  4 y  3  0 . Write down the coordinates
of its centre and the length of its radius.
(2)
Outcome 4
9.
(a)
(b)
10.
Show that the line with equation y  3x  4 is a tangent to the circle with the
equation x 2  y 2  8x  4 y  10  0 .
11.
(5)
y
The point P(4, 2) lies on the circle with centre C(2 ,–1) ,
as shown in Diagram 4.
P(4, 2)
Find the equation of the tangent at P.
(3)
x
C(2, –1)
Diagram 4
Pegasys 2010
Practice Assessment (2)
Unit 2 - Mathematics 2 (H)
Outcome 1
1.
2.
Marks
(i)
 10 .
Show that ( x  2) is a factor of f ( x)  x 3  2 x 2  13x•
(ii)
Hence factorise f (x) fully.
(5)
Determine the nature of the roots of the equation 3x 2  4 x  3  0 using the discriminant.
(3)
Outcome 2
3.
Find

4
dx, x  0
x6
(3)
y
4.
The curve with equation y  12  4 x  x 2 is shown in
Diagram 1.
Calculate the shaded area shown in Diagram 1.
y  12  4 x  x 2
o
Diagram 1
x
6
(5)
5.
The curve with equation y  x 2  4 x  6 and the curve with the equation y  6  2 x  x 2 are shown
in Diagram 2.
The curves meet at the points where
x = 0 and x  3 .
y
y  x 2  4x  6
Calculate the shaded area shown in Diagram 2.
(6)
y  6  2x  x 2
o
3
Diagram 2
Pegasys 2010
x
Outcome 3
6.
Solve the equation tan 2 x  1 for 0  x  2 .
7.
Diagram 3 shows two right-angled triangles.
(2)
15
25
yo
xo
24
20
Diagram 3
8.
(a)
Write down the values of sin x  and cos y  .
(2)
(b)
Show that the exact value of sin ( x  y )  is 54 .
(2)
(a)
Express cos x  cos 30   sin x  sin 30  in the form cos ( x   )  .
(1)
(b)
Using the results from (a),
solve the equation cos x  cos 30   sin x  sin 30   92 for 0  x  360 .
(4)
A circle has radius 4 units and centre (–2, 3).
Write down the equation of the circle.
(2)
A circle has equation x 2  y 2  8x  14 y  1  0 .
Write down the coordinates of its centre and the length of its radius.
(2)
Outcome 4
9.
(a)
(b)
10.
Show that the line with equation y  17  2 x is a tangent to the circle with
equation x 2  y 2  6 x  2 y  10  0 .
11.
(5)
The point P(–3, 1) lies on the circle with C(–1, –2),
y
as shown in Diagram 4.
P(–3, 1)
Find the equation of the tangent at P.
x
C(–1, –2)
(3)
Diagram 4
Pegasys 2010
Practice Assessment (3)
Unit 2 - Mathematics 2 (H)
Outcome 1
1.
2.
Marks
(i)
Show that ( x  1) is a factor of f ( x)  x3  x 2  5x  3 .
(ii)
Hence factorise f (x) fully.
(5)
Determine the nature of the roots of the equation 4 x 2  12 x  9  0 using the discriminant.
(3)
Outcome 2
3.
4.
Find

7
x3
dx , x  0
The curve with equation y  x 2  9 x  18 is shown in
Diagram 1.
(3)
y
Calculate the shaded area shown in Diagram 1.
y  x 2  9 x  18
o
3
6
x
(5)
Diagram 1
5.
The line with equation y  13  x and the curve with equation y  x 2  7 x  13 are shown in
Diagram 2.
y
The line and the curve meet at the points where
y  x 2  7 x  13
x = 0 and x = 6.
y  13  x
o
Calculate the shaded area shown in Diagram 2.
Pegasys 2010
6
Diagram 2
x
(6)
Outcome 3
3
for 0  x   .
2
6.
Solve the equation cos 2 x  
7.
Diagram 3 shows two right-angled triangles.
(2)
.
15
Diagram 3
xo
yo
1
8.
3
(a)
Write down the values of sin x  and sin y  .
(b)
Show that the exact value of sin ( x  y )  is
(a)
Express cos x  cos 25  sin x  sin 25 in the form cos ( x   )  .
(b)
Using the results from (a),
(2)
 15 .
2 24
(2)
(1)
7 for 0  x  360 .
solve the equation cos x  cos 25  sin x  sin 25  11
(4)
A circle has a radius of 2 units and centre (–3, 1).
Write down the equation of the circle.
(2)
A circle has equation x 2  y 2  8 y  7  0 .
Write down the coordinates of its centre and the length of its radius.
(2)
Outcome 4
9.
(a)
(b)
10.
Show that the line with equation y  17  4 x is a tangent to the circle with
equation x 2  y 2  8x  2 y  51  0 .
11.
The point P(1, –1) lies on the circle with centre C(–2, 2)
(5)
y
C(– 2, 2)
as shown in Diagram 4.
Find the equation of the tangent at P.
P(1, –1)
Diagram 4
Pegasys 2010
x
(3)
Answers
Unit 2 - Practice Assessments
Practice Assessment 1
Outcome 1 :
1. proof , g ( x)  ( x  4)( x  3)( x  1) 2.
Outcome 2 :
3.
x 4
C
4
5.
4 12 units2
6.

Outcome 3 :
Outcome 4 :

8
3
8
,
b 2  4ac  41  real, distinct and irrational
 1

 4  C
 4x

4.

cos x 
7.
(a)
 29  5
8.
(a)
sin( x  10) 
9.
(a)
( x  2) 2  ( y  5) 2  49
10.
1 root, x  1,  a tangent
(b)
21 13 units2

3
, sin y 
10
, 170  5

2
proof
(b)
8
(b) C(3, –2) , r  4
y  2   23 ( x  4)
11.

3 y  2 x  14
Practice Assessment 2
Outcome 1 :
Outcome 2 :
Outcome 3 :
Outcome 4 :
1.
proof , f ( x)  ( x  1)( x  2)( x  5)
3.
4 x 5
C
5
5.
9 units2
6.

8.

5
8
,
 4

 5  C
 5x


sin x 
(a)
(a)
cos( x  30) 
(b)
9.
(a)
( x  2) 2  ( y  3) 2  16
10.
1 root, x  7 ,  a tangent
 47
b 2  4ac  20  not real
72 units2
4.
7.
8
2.

7
20
, cos y 
25
25
, 253
proof
(b)

(b) C(4, –7) , r  8
y  1  23 ( x  3)
11.

3 y  2x  9
Practice Assessment 3
Outcome 1 :
1.
proof , g ( x)  ( x  3)( x  1)( x  1)
Outcome 2 :
3.
7 x 2
C
2
5.
36 units2
6.

Outcome 3 :
Outcome 4 :
5
12
,
7
12
 7

 2  C
 2x


7.
sin x 
 75  5
8.
(a)
cos( x  25) 
9.
(a)
( x  3) 2  ( y  1) 2  4
10.
1 root, x  4,  a tangent
Pegasys 2010
b 2  4ac  0  equal roots
22 12 units2
4.
(a)
(b)
2.

15
24
, 334  5
, sin y 
15
4

(b) C(0, 4) , r  3
11.
y 1  x 1 
y  x2
(b)
proof
Practice Assessment (1)
Unit 3 - Mathematics 3 (H)
Outcome 1
1.
2.
Marks
The points A, B and C have coordinates ( 3 ,  2 , 4 ) , ( 0 ,1, 3) , (  6, 7 , 1) respectively.
(a)
Write down the components of AB.
(1)
(b)
Hence show that A, B and C are collinear.
(3)
S(15,–15, –2)
The point T divides RS in the ratio 3:1
as shown in Diagram 1.
T
Find the coordinates of T.
R(7, –3, 2)
3.
(3)
Diagram 1
Diagram 2 shows vectors AB and AC where
C
 7 
 
AC    2 
 1
 
 3 


AB   0 
  6


and
(a)
Find AB AC
.
(b)
Hence find the size of angle BAC.
(4)
(a)
Differentiate  3 cos x with respect to x.
(1)
(b)
Given y 
A
B
Diagram 2
(1)
Outcome 2
4.
dy
3
sin x , find
.
5
dx
(1)
5.
Find g (x) when g ( x)  ( 3x  1) 5 .
(2)
6.
(i)
Find
 3 cos x dx
(2)
(ii)
Integrate sin 2 x with respect to x.
(1)
2
7.
Evaluate
 ( x  3) dx
3
0
Pegasys 2010
(4)
Outcome 3
8.
(a)
Simplify log x 2  log x 4 .
(1)
(b)
Simplify 5 log 4 2  log 4 2
(4)
9.
Solve e x  2  4
(2)
10.
Solve log 2 ( x  3)  4
(2)
Outcome 4
11.
Express 2 cos x   3sin x  in the form k cos ( x  a ) where k  0 and 0  a  360 .
Pegasys 2010
(5)
Practice Assessment (2)
Unit 3 - Mathematics 3 (H)
Outcome 1
1.
2.
Marks
The points P, Q and R have coordinates ( 2, 6, –4 ) , (4, 4, –2) and (10, –2, 4) respectively.
(a)
Write down the components of PQ.
(1)
(b)
Hence show that P, Q and R are collinear.
(3)
Q(–4, 3, –1)
The point P divides QR in the ratio 2:3
as shown in Diagram 1.
Diagram 1
P
Find the coordinates of P.
(3)
R(6, –2, 19)
3.
Diagram 2 shows vectors ST and SU where
 2 


SU    2 
3 


 4 


ST    2 
5 


and
(a)
Find ST SU .
.
(b)
Hence find the size of angle TSU.
S
T
(1)
U
(4)
Outcome 2
4.
(a)
Differentiate 4 sin x with respect to x.
(b)
Given y  7 cos x , find
dy
.
dx
(1)
(1)
5.
Find h(x) when h( x)  (1  4 x ) 5 .
6.
(a)
Find
dx
(2)
(b)
Integrate cos 3x with respect to x.
(1)
3
 4 sin x
(2)
3
7.
Evaluate
 ( x  2)
1
Pegasys 2010
2
dx
(4)
Outcome 3
8.
(a)
Simplify log t 5  log t 4 .
(1)
(b)
Simplify 2 log 2 6  log 2 9
(4)
9.
Solve e x  3  1
(2)
10.
Solve log 3 ( x  4)  3
(2)
Outcome 4
11.
Express 5 sin x  3 cos x in the form k cos ( x  a ) where k  0 and 0  a  360 .
Pegasys 2010
(5)
Practice Assessment (3)
Unit 3 - Mathematics 3 (H)
Outcome 1
1.
2.
Marks
The points S, T and U have coordinates ( 2 , 7 ,  3 ) , ( 3, 2 ,  5 ) and ( 6,  13 ,  11) respectively.
(a)
Write down the components of ST.
(1)
(b)
Hence show that S, T and U are collinear.
(3)
B (3,13, –9)
The point P divides AB in the ratio 2 : 3
as shown in the Diagram 1.
P
Find the coordinates of P.
A (8, – 2, 1)
3.
(3)
Diagram 1
Diagram 2 shows vectors KL and KM where
 3 


KL   2 
 5 


K
2 
 
KM =   8 
  1
 
and
Diagram 2
L
.
M
(a)
Find the value of KL KM .
(1)
(b)
Hence find the size of angle LKM.
(4)
(a)
Differentiate  2 sin x with respect to x.
(1)
(b)
Given y   4 cos x , find
5.
Find
f (x ) when
6.
(a)
Find
(b)
Integrate  cos 2 x with respect to x.
Outcome 2
4.
dy
.
dx
(1)
f ( x)  ( 2  3x ) 4 .
(2)
  2 sin x dx
(2)
(1)
2
7.
Evaluate
 (3x  1)
1
Pegasys 2010
2
dx
(4)
Outcome 3
8.
(a)
Simplify log y 15  log y 3 .
(1)
(b)
Simplify 3 log 2 4  log 2 16
(4)
9.
Solve e x  1 8
(2)
10.
Solve log 6 ( x  5)  2
(2)
Outcome 4
11.
Express 3sin x  7 cos x in the form k sin ( x  a )  where k  0 and 0  a  360 .
(5)
Answers
Unit 3 - Practice Assessments
Practice Assessment 1
1.
  3


(a) AB   3 
1 


3.
(a)
AB . AC  15
4.
(a)
3 sin x
6.
(a)
3sin x  C
Outcome 3 :
8.
(a)
Outcome 4 :
11.

Outcome 1 :

Outcome 2 :
(b) proof

log x 8
(b)
(b)
(b)
T(13, –12, –1)
72  3
dy 3
 cos x
dx 5
(b)
2
2.
9.
g ( x)  15(3x  1) 6
5.
 12 cos 2 x  C
x  0 875
136
7.
x  13
10.
13 cos( x  56  3) 
Practice Assessment 2
2 


PQ    2 
2 



Outcome 1 :
1.
(a)
3.
(a)
ST .SU  11
4.
(a)
4 cos x
6.
(a)
3
 cos x  C
4
Outcome 3 :
8.
(a)
Outcome 4 :
11.

Outcome 2 :
(b) proof

(b)
(b)
log t 20
(b)
2
2.
P(0, 1, 7)
66  6 
dy
 7 sin x
dx
h( x)  20(1  4 x) 4
5.
(b)
1
sin 3 x  C
3
7.
9.
x  1131
10.
32
2
3
x  31
34 cos( x  59  0) 
Practice Assessment 3
1 


ST    5 
  2



Outcome 1 :
1.
(a)
3.
(a)
KL. KM  5
4.
(a)
 2 cos x
6.
(a)
2 cos x  C
Outcome 3 :
8.
(a)
Outcome 4 :
11.

Outcome 2 :
log y 5
(b) proof

(b)
58 sin( x  66  8) 
(b)
(b)
2
2.
P(6, 4 , – 3 )
95  6 
dy
 4 sin x
dx
f ( x)  12(2  3x) 3
5.
(b)
 12 sin 2 x  C
9.
x  0 588
7.
10.
x  41
13 units2