Download Exercise 2 - Elgin Academy

Elgin Academy
Mathematics Department
Higher
Unit 2 Revision
Please do not write on these sheets.
Return this booklet to your teacher when you are finished with it.
Good Luck.
Outcome 1
Use the Factor/Remainder Theorem and apply quadratic theory.
PC(a): Apply the Factor/Remainder theorem to a polynomial function.
Example
Show that x  2 is a factor of f x   x 3  4 x 2  x  6 and express f (x) in
fully factorised form.
x 2  2x  3
x  2 x 3  4x 2  x  6
 (x 3  2x 2 )
2x 2  x
 (2 x 2  4 x)
 3x  6
2 1
OR
4
1 6
2 4 6
1 2 3 0
 ( 3 x  6)
0
Hence ( x  2) is a factor of f(x) and
f ( x)  ( x  2)( x 2  2 x  3)
 ( x  2)( x  1)( x  3)
Exercise 1
1. Show that x  1 is a factor of f x   x 3  7 x  6 . Express f x  in fully factorised form.
2. Show that x  2 is a factor of f x   3x 3  2 x 2  19 x  6 . Express f x  in fully factorised
form.
3. Show that x  3 is a factor of f x   x 3  3x 2  16 x  48 . Express f x  in fully factorised
form.
PC(b) Determine the nature of the roots of a quadratic equation using the discriminant.
Example
Use the discriminant to determine the nature of the roots of x 2  3x  5  0 .
b 2  4ac  (3) 2  4.1.5
 11  0
so there are no real roots.
Exercise 2
Use the discriminant to determine the nature of the roots of the following equations:
1. x 2  4 x  1  0
2. x 2  3x  4  0
3. 2 x 2  5  0
4. 3x 2  x  5  0
5. 2 x 2  2 x  3  0
6. 5 x 2  10 x  5
Outcome 2
Use basic integration.
PC(a): Integrate functions reducible to sums of powers of x.
Example
Find
dx
 5x
2
dx
 5x

2
1
dx   15 x  2 dx   15 x 1  C
2
5x
Exercise 3
6
x
1. Find
2
2. Find
dx
4. Find  x  3x  4 dx
3
x
4
3. Find
dx
5. Find  6 x dx
6. Find
2
x

3
dx
x2  3
x
dx
PC(b): Find the area between a curve and the x-axis using integration.
Example
Calculate the area between the curve and the x-axis in the diagram:
4  3x  x 2  0
y
y  4  3x  x 2
4  x 1  x   0
x  4 or 1
1
 4  3x  x
4
-4
1
x
2

 4.1 
 4 x  32 x 2  13 x 3
 20 56
3
2

1
4
 
.12  13 .13  4. 4  32 . 4  13 . 4
2
3

Exercise 4
Calculate the area between the following curves and the x-axis: 1.
y  12 x  1 between x = 1 and x = 3.
2.
y  (3  x)(1  x)
3.
y  (2  x)( 2  x)
4.
y  (6  x)( 2  x)
PC(c): Find the area between two curves using integration.
Example
Find the area enclosed by the graphs of the functions y = 2x and y = x2:
First find the x coordinates of the points of contact:
2x  x 2
x 2  2x  0
x( x  2)  0
x  0,2
y
Use a sketch to decide which graph is uppermost:

2
0
4
y = 2x
(2 x  x 2 ) dx ( this is all that is needed for the unit test)


 2  2   0
 4    0  1
 x 2  13 x 3
1
3
2
2
0
3
8
3
2
 13 0 3

0
y = x2
2
1
3
Exercise 5
Find the area enclosed by the graphs of each pair of the following functions:
1. y = 2x2 and y = 4x
2. y = x2 and y = x + 2
3. y = 2 x – x2 and y = 5x – 4
x
Outcome 3
Solve trigonometric equations and apply trigonometric formulae.
PC(a): Solve a trigonometric equation in a given interval.
Example
Solve 2sin2x = 1, 0  x  2
sin 2 x 
2x 
x

6
1
2
(30 O ),
5
(150 O )
6
 5
,
12 12
2

2



 5  
 13  5
 17
So x  , ,     
,
  
 12  12
 12
12 12  12



the period of sin2 x 
Exercise 6
Solve the following equations for 0  x  2:
1. 2cos2x = 1
2. tan2x = 3
3. 2sin2x = 3
4. 3tan2x = -1
PC(b): Apply trigonometric formulae in the solution of a geometric problem.
D
A
Example
The diagrams show two
right angled triangles,
ABC and DEF.
E
12
C
Write down the values of sin xo and cos yo.
AC  12 2  5 2  13, sin x o 
ED  4 2  3 2  5, cos y o 
b)
y
x
B
a)
3
5
5
13
4
5
Show that the exact value of sin(x + y) o =
sin( x  y )  sin x cos y  cos x sin y
12
3
cos x o 
and sin y o 
13
5
5 4 12 3 56
sin( x  y ) o     
13 5 13 5 65
56
.
65
4
F
Exercise 7
1.
The diagrams show two right angled triangles, PQR and STV.
S
P
4
8
x
Q
2.
y
T
R
15
V
3
a)
Write down the values of sin xo and cos yo.
b)
By expanding cos(x – y), show that cos(x – y) =
77
85
The diagrams show two right angled triangles, JKL and MNP.
M
J
4
7
x
K
y
N
24
L
P
3
a) Write down the values of sin xo and cos yo.
b) By expanding cos(x + y), show that cos(x + y) =
44
125
PC(c): Solve a trigonometric equation involving an addition formula in a given interval.
Example
Express cos xocos 60o + sin xosin 60o in the form cos(A-B)o.
Use this result to solve the equation
cos xocos 60o + sin xosin 60o =
2
, 0  x  360.
5
cos x o cos 60 o  sin x o sin 60 o  cos( x  60) o
2
cos( x  60) o 
5
x  60  66.4, 293.6
x  126.4, 353.6.
Exercise 8
1. Express sin xocos 20o + cos xosin 20o in the form sin(A+B)o.
Hence solve the equation sin xocos 20o + cos xosin 20o = 0.6, 0  x  360.
2. Express cos xocos 30o - sin xosin 30o in the form cos(A+B)o.
Hence solve the equation cos xocos 30o + sin xosin 30o = 0.2, 0  x  360.
Outcome 4
Use the equation of the circle.
PC(a): Given the centre (a,b) and radius r, find the equation of the circle in the form
(x - a)2 + (y - b)2 = r2.
Example
Write down the equation of the circle with centre (2, 7) and radius 4.
x  22   y  72  16
Exercise 9
Write down the equations of the circles with these centres and radii:
1. radius 7 and centre (3,-1)
3. radius 1 and centre (-2, 5)
2.
radius 2.5 and centre (-5,-7)
PC(b): Find the radius and centre of a circle given the equation in the form
x2 + y2 +2g x + 2f y + c = 0.
Example
Write down the centre and radius of the circle
x 2  y 2  4 x  8 y  17  0
Centre   g , f   2,4
Radius  g 2  f 2  c 
 22   42  17 
3
Exercise 10
Write down the centre and radius of each of the following circles:
1. x 2  y 2  6 x  10 y  9  0
3. x 2  y 2  2 x  2 y  5  0
2. x 2  y 2  8x  2 y  13  0
PC(c): Determine whether or not a given line is a tangent to a given circle.
Show that the straight line y  2 x  3 is a tangent to the circle
x 2  y 2  4x  6 y  7  0 .
Example
x 2  2 x  3  4 x  62 x  3  7  0
2
x 2  4 x 2  12 x  9  4 x  12 x  18  7  0
5 x 2  20 x  20  0
x 2  4x  4  0
 x  2 2  0
OR b 2  4ac   4  4.1.4  0
2
Equal roots, so only one point of contact proving the line is a tangent to
the circle.
Exercise 11
1. Show that the straight line y  4 x  7 is a tangent to the circle
x 2  y 2  6x  4 y  4  0 .
2. Show that the straight line y  2 x  4 is a tangent to the circle
x 2  y 2  2x  6 y  5  0 .
3. Show that the straight line y  3x  20 is a tangent to the circle
x 2  y 2  14 x  22 y  160  0 .
PC(d): Determine the equation of the tangent to a given circle given the point of
contact.
Example
The point A(13, 9) lies on the circle with equation x 2  y 2  2 x  4 y  152  0 .
Find the equation of the tangent at A.
Coordinates of centre = (1, 4) so m r 
94 5

13  1 12
Since, at the point of contact, the tangent is perpendicular to the radius
mT  mr  1 so mT  
12
 2.4 .
5
Equation of tangent is therefore y  9  
12
x  13  12 x  5 y  201  0
5
Exercise 12
1. Find the equation of the tangent to the circle x 2  y 2  8x  6 y  40  0 at the point
B(3,-1).
2. Find the equation of the tangent to the circle x 2  y 2  2 x  4 y  1  0 at the point
C(-1, 2).
3. Find the equation of the tangent to the circle x 2  y 2  4 x  2 y  3  0 at the point
D(3,-2).
Answers
Exercise 1
1. (x - 1)(x - 2)(x + 3)
2. (x + 2)(x - 3)(3x + 1)
3. (x - 3)(x - 4)(x + 4)
Exercise 2
b 2  4ac  0 , distinct real roots; Q1,4,5.
b 2  4ac  0 , equal real roots; Q6.
b 2  4ac  0 , no real roots; Q2,3.
Exercise 3
1.
6
c
x
1
1
3. 2  c
c
3
x
x
2 5
x 6 x c
5
2.
5. 4 x 3  c
6.
Exercise 4
1. 4
2.
Exercise 5
2
1. 0 4 x  2 x 2  dx
10 2/3
2.
1.
 7 9 15
8
,
8
,
8
,
8
3. 10 2/3
 x  2  x  dx
2
2
3.
1
Exercise 6
2.
,
3
,
6
,
1 3 1 2
x  x  12 x  c
3
2
4. 85 1/3
 2x  x
2
1
 2 7 4
6
4.
3.
3
2



 5x  4 dx    x 2  3x  4 dx
2
1
5 11 17 23
,
,
,
12 12 12 12
Exercise 7
1. a) sin x  = 8/17, cos y  = 3/5
2. a) sin x  = 7/25, cos y  = 3/5
b) Proof.
b) Proof.
Exercise 8
1. sin(x + 20), x = 16.9, 123.1
2. cos(x + 30), x = 48.5, 251.5
Exercise 9
1. (x – 3)2 + (y +1)2 = 49 2. (x + 5)2 + (y +7)2 = 6.25 3. (x + 2)2 + (y +5)2 = 1
Exercise 10
1. C(-3, 5), R = 5. 2. C(-4, -1), R = 2. 3. C(1, -1), R = 7.
Exercise 11
Proofs
Exercise 12
1. y + 1 = 7/4(x – 3) or 4y = 7x – 25
3. y + 2 = 1(x – 3) or y = x – 5
2. y - 2 = 1/2(x + 1) or 2y = x + 5