PURE MATHEMATICS 1
(9709/1)
CHAPTER SUMMARIES
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Chapter 1: QUADRATICS
Quadratic equations can be solved by:
●
factorisation
●
completing the square
●
using the quadratic formula x =
−b ± b 2 − 4ac
.
2a
Solving simultaneous equations where one is linear and one is quadratic
●
Rearrange the linear equation to make either x or y the subject.
●
Substitute this for x or y in the quadratic equation and then solve.
Maximum and minimum points and lines of symmetry
For a quadratic function f( x ) = ax 2 + bx + c that is written in the form f( x ) = a( x − h )2 + k :
b
● the line of symmetry is x = h = −
2a
●
if a . 0, there is a minimum point at ( h, k )
●
if a , 0, there is a maximum point at ( h, k ).
Quadratic equation ax 2 + bx + c = 0 and corresponding curve y = ax 2 + bx + c
●
Discriminant = b 2 − 4ac.
●
If b 2 − 4ac . 0, then the equation ax 2 + bx + c = 0 has two distinct real roots.
●
If b 2 − 4ac = 0, then the equation ax 2 + bx + c = 0 has two equal real roots.
●
If b 2 − 4ac , 0, then the equation ax 2 + bx + c = 0 has no real roots.
●
The condition for a quadratic equation to have real roots is b 2 − 4ac ù 0.
Intersection of a line and a general quadratic curve
●
If a line and a general quadratic curve intersect at one point, then the line is a tangent to the curve at that point.
●
Solving simultaneously the equations for the line and the curve gives an equation of the form ax 2 + bx + c = 0.
●
b 2 − 4ac gives information about the intersection of the line and the curve.
b 2 − 4 ac
Nature of roots
Line and parabola
.0
two distinct real roots
two distinct points of intersection
=0
two equal real roots
one point of intersection (line is a tangent)
,0
no real roots
no points of intersection
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1
Chapter 2: FUNCTIONS
Functions
●
A function is a rule that maps each x value to just one y value for a defined set of input values.
●
A function can be either one-one or many-one.
●
The set of input values for a function is called the domain of the function.
●
The set of output values for a function is called the range (or image set) of the function.
Composite functions
●
fg( x ) means the function g acts on x first, then f acts on the result.
●
fg only exists if the range of g is contained within the domain of f.
●
In general, fg( x ) ≠ gf( x ).
Inverse functions
●
The inverse of a function f( x ) is the function that undoes what f( x ) has done.
f f −1( x ) = f −1 f( x ) = x or if y = f( x ) then x = f −1( y )
●
The inverse of the function f( x ) is written as f −1 ( x ).
●
The steps for finding the inverse function are:
Step 1: Write the function as y =
Step 2: Interchange the x and y variables.
Step 3: Rearrange to make y the subject.
●
The domain of f −1 ( x ) is the range of f( x ).
●
The range of f −1 ( x ) is the domain of f( x ).
●
An inverse function f −1 ( x ) can exist if, and only if, the function f( x ) is one-one.
●
The graphs of f and f −1 are reflections of each other in the line y = x.
●
If f( x ) = f −1 ( x ), then the function f is called a self-inverse function.
●
If f is self-inverse then ff( x ) = x.
●
The graph of a self-inverse function has y = x as a line of symmetry.
Transformations of functions
●
 0
The graph of y = f( x ) + a is a translation of y = f( x ) by the vector   .
 a
●
The graph of y = f( x + a ) is a translation of y = f( x ) by the vector
●
The graph of y = − f( x ) is a reflection of the graph y = f( x ) in the x-axis.
●
The graph of y = f( − x ) is a reflection of the graph y = f( x ) in the y-axis.
 −a 
.
 0 


The graph of y = a f( x ) is a stretch of y = f( x ), stretch factor a, parallel to the y-axis.
1
● The graph of y = f( ax ) is a stretch of y = f( x ), stretch factor , parallel to the x-axis.
a
●
Combining transformations
●
When two vertical transformations or two horizontal transformations are combined, the order
in which they are applied may affect the outcome.
●
When one horizontal and one vertical transformation are combined, the order in which they are
applied does not affect the outcome.
●
Vertical transformations follow the ‘normal’ order of operations, as used in arithmetic
●
Horizontal transformations follow the opposite order to the ‘normal’ order of operations, as
used in arithmetic.
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Chapter 3: COORDINATE GEOMETRY
Midpoint, gradient and length of line segment
Q (x2, y2)
M
P (x1, y1)
●
x + x2 y1 + y2 
,
Midpoint, M, of PQ is  1
 .

2
2
●
Gradient of PQ is
●
Length of segment PQ is
y2 − y1
.
x2 − x1
( x2 − x1 )2 + ( y2 − y1 )2
Parallel and perpendicular lines
●
If the gradients of two parallel lines are m1 and m2 , then m1 = m2.
●
If the gradients of two perpendicular lines are m1 and m2 , then m1 × m2 = −1.
The equation of a straight line is:
●
y − y1 = m( x − x1 ), where m is the gradient and ( x1, y1 ) is a point on the line.
The equation of a circle is:
●
( x − a )2 + ( y − b )2 = r 2 , where ( a, b ) is the centre and r is the radius.
●
x 2 + y 2 + 2 gx + 2 fy + c = 0 , where ( − g, − f ) is the centre and
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g 2 + f 2 − c is the radius.
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Chapter 4: CIRCULAR MEASURE
Radians and degrees
r
r
1 rad
O
r
●
One radian is the size of the angle subtended at the centre of a circle, radius r, by an arc of length r.
●
π radians = 180°
●
To change from degrees to radians, multiply by
π
.
180
●
To change from radians to degrees, multiply by
180
.
π
Arc length and area of a sector
B
r
θ
O
r
A
When θ is measured in radians, the length of arc AB is rθ .
1 2
● When θ is measured in radians, the area of sector AOB is
r θ.
2
●
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Chapter 5: TRIGONOMETRY
Exact values of trigonometric functions
sin θ
cos θ
tan θ
θ =
= 30° =
π
6
1
2
3
2
1
3
θ = 45° =
π
4
1
2
1
2
1
θ = 60° =
π
3
3
2
1
2
3
Positive and negative angles
●
Angles measured anticlockwise from the positive x-direction are positive.
●
Angles measured clockwise from the positive x -direction are negative.
Diagram showing where sin, cos and tan are positive
90°
Sin
180°
All
0°, 360°
O
Tan
Cos
151
270°
●
Useful mnemonic: ‘A ll Students Trust Cambridge’.
Graphs of trigonometric functions
y
1
y = sin x
–360 –270 –180
–90 O
90
180
270
360 x
270
360 x
–1
y
1
y = cos x
–360 –270 –180 –90 O
90
180
–1
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1
y
y = tan x
–360
–270
–180
–90
O
90
180
270
360 x
●
The graph of y = a sin x is a stretch of y = sin x , stretch factor a, parallel to the y -axis.
●
The graph of y = sin( ax ) is a stretch of y = sin x , stretch factor
●
●
1
, parallel to the x -axis.
a
0
The graph of y = a + sin x is a translation of y = sin x by the vector   .
a
 −a 
.
 0 
The graph of y = sin( x + a ) is a translation of y = sin x by the vector 
Inverse trigonometric functions
y
π
–
2
–1
y=
y
π
sin –1x
O
1
π
––
2
y = sin −1 x
domain: −1 < x < 1
π
π
range: − < sin −1 x <
2
2
π
–
2
x
–1
O
y
π
–
2
y = cos –1 x
1
y = tan –1 x
O
x
y = cos −1 x
domain: −1 < x < 1
range: 0 < cos −1 x < π
x
π
––
2
y = tan −1 x
domain: x ∈ R
π
π
range: − < tan −1 x <
2
2
Trigonometric identities
sin x
● tan x ≡
cos x
●
sin2 x + cos2 x ≡ 1
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1
Chapter 6: SERIES
Binomial expansions
 n
Binomial coefficients, denoted by n C r or   , can be found using:
 r 
● Pascal’s triangle
●
 n
 n
n!
n × ( n − 1) × ( n − 2) × … × ( n − r + 1)
or   =
the formulae   =
.
r × ( r − 1) × ( r − 2) × … × 3 × 2 × 1
 r  r! ( n − r )!
 r 
If n is a positive integer, the Binomial theorem states that:
 n
 n
 n  n
 n
(1 + x ) n =   +   x +   x 2 + … +   x n , where the ( r + 1)th term =   x r.
 r 
 2
 n
 0  1
We can extend this rule to give:
 n  n−r r
 n
 n
 n
 n
b.
( a + b ) n =   a n +   a n − 1b1 +   a n − 2 b 2 + … +   b n , where the ( r + 1)th term =   a
 r 
 2
 1
 n
 0
We can also write the expansion of (1 + x ) n as:
(1 + x ) n = 1 + nx +
n( n − 1) 2 n( n − 1)( n − 2) 3
x +
x + … + xn
2!
3!
Arithmetic series
For an arithmetic progression with first term a, common difference d and n terms:
●
the kth term is a + ( k − 1)d
the last term is l = a + ( n − 1)d
n
n
( a + l ) = [2 a + ( n − 1)d ].
● the sum of the terms is Sn =
2
2
●
Geometric series
For a geometric progression with first term a, common ratio r and n terms:
●
the kth term is ar k − 1
●
the last term is ar n − 1
●
sum of the terms is Sn =
a (1 − r n ) a ( r n − 1)
=
.
1− r
r −1
The condition for an infinite geometric series to converge is −1 < r < 1.
When an infinite geometric series converges, S∞ =
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a
.
1− r
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1
Chapter 7: DIFFERENTIATION
Gradient of a curve
●
dy
represents the gradient of the curve y = f( x ).
dx
The four rules of differentiation
●
Power rule:
d
( x n ) = nx n − 1
dx
●
Scalar multiple rule:
d
d
[ kf( x )] = k
[f( x )]
dx
dx
●
Addition/subtraction rule:
d
d
d
[f( x ) ± g( x )] =
[g( x )]
[f( x )] ±
dx
dx
dx
●
Chain rule:
du
dy
dy
=
×
dx
dx
du
Tangents and normals
dy
at the point ( x1, y1 ) is m, then:
dx
● the equation of the tangent at that point is given by y − y1 = m( x − x1 )
1
( x − x1 ) .
● the equation of the normal at that point is given by y − y1 = −
m
If the value of
Second derivatives
●
d  dy 
d2 y
=
dx  dx 
dx 2
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1
Increasing and decreasing functions
•
•
•
y = f( x ) is increasing for a given interval of x if
dy
. 0 throughout the interval.
dx
y = f( x ) is decreasing for a given interval of x if
dy
, 0 throughout the interval.
dx
Stationary points
Stationary points (turning points) of a function y = f( x ) occur when
dy
= 0.
dx
First derivative test for maximum and minimum points
At a maximum point:
•
•
•
•
dy
= 0
dx
the gradient is positive to the left of the maximum and negative to the right.
At a minimum point:
dy
= 0
dx
the gradient is negative to the left of the minimum and positive to the right.
Second derivative test for maximum and minimum points
•
•
•
If
d2 y
dy
= 0 and 2 , 0, then the point is a maximum point.
dx
dx
If
d2 y
dy
= 0 and 2 . 0, then the point is a minimum point.
dx
dx
If
dy
d2 y
= 0 and 2 = 0, then the nature of the stationary point can be found using the
dx
dx
first derivative test.
Connected rates of change
•
When two variables, x and y, both vary with a third variable, t, the three variables can
be connected using the chain rule:
•
You may also need to use the rule:
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dy
dy
dx
.
=
×
dt
dx
dt
dx
1
.
=
y
d
dy
dx
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Chapter 8: INTEGRATION
12 The diagram shows the curve y = 4 − x and the line x + 2 y = 4 that
intersect at the points (4, 0) and (0, 2).
y
2
a Find the volume obtained when the shaded region is rotated through
360° about the x-axis.
b Find the volume obtained when the shaded region is rotated through
360° about the y-axis.
13 A mathematical model for the inside of a bowl is obtained by rotating
the curve x 2 + y2 = 100 through 360° about the y-axis between y = −8
and y = 0. Each unit of x and y represents 1cm.
y =√ 4 – x
x + 2y = 4
4
O
x
y
x2 + y2 = 100
a Find the volume of the bowl.
The bowl is filled with water to a depth of 3 cm.
x
b Find the volume of water in the bowl.
–8
3
P
14 Use integration to prove that the volume, V cm , of a sphere with radius
4
r cm is given by the formula V = πr 3.
3
Integration as the reverse of differentiation
●
d
[ F( x ) ] = f( x ), then
dx
If
∫ f(x ) dx = F(x ) + c.
Integration formulae
1
●
∫ x dx = n + 1 x
●
∫ (ax + b) dx = a( n + 1) (ax + b)
n
n +1
+ c (where c is a constant and n ≠ −1)
1
n
n +1
+ c ( n ≠ −1 and a ≠ 0)
Rules for indefinite integration
●
●
∫ k f(x ) dx = k ∫ f(x ) dx, where k is a constant
∫ [ f(x ) ± g(x ) ] dx = ∫ f(x ) dx ± ∫ g(x ) dx
Rules for definite integration
●
●
∫
If
∫
b
a
b
f( x ) dx = F( x ) + c, then
k f( x ) d x = k
∫ f(x ) dx = [ F(x ) ] = F(b) − F(a ).
b
a
a
b
∫ f(x ) dx, where k is a constant.
a
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1
b
●
a
●
b
b
a
a
∫ [ f(x ) ± g(x ) ] dx = ∫ f(x ) dx ± ∫ g(x ) dx
b
a
a
b
∫ f(x ) dx = − ∫ f(x ) dx
Area under a curve
y
y = f(x)
A
a
O
●
x
b
The area, A, bounded by the curve y = f( x ), the x-axis and the lines x = a and x = b is given
by the formula:
b
∫ y dx when y ù 0
A=
(or A =
a
b
∫ f(x ) dx when f(x ) ù 0 ).
a
y
274
b
A
a
x = f(y)
O
●
x
The area, A, bounded by the curve x = f( y ), the y-axis and the lines y = a and y = b is given
by the formula:
A=
b
∫ x dy when x ù 0
(or A =
a
b
∫ f( y) dy when f( y) ù 0).
a
y
y = f(x)
A
O
a
b
x
y = g(x)
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●
The area, A, enclosed between y = f( x ) and y = g( x ) is given by the formula:
A=
b
∫ [ f(x ) − g(x ) ] dx
a
where a and b are the x-coordinates of the points of intersection of the
functions f and g.
Improper integrals
∞
●
Integrals of the form
∫ f(x ) dx can be evaluated by replacing the infinite limit with a finite
a
value, X , and then taking the limit as X → ∞, provided the limit exists.
b
●
Integrals of the form
∫ f(x ) dx can be evaluated by replacing the infinite limit with a finite
−∞
value, X , and then taking the limit as X → −∞, provided the limit exists.
b
●
Integrals of the form
∫ f(x ) dx where f(x ) is not defined when x = a can be evaluated by
a
replacing the limit a with an X and then taking the limit as X → a , provided the limit exists.
b
●
Integrals of the form
∫ f(x ) dx where f(x ) is not defined when x = b can be evaluated by
a
replacing the limit b with an X and then taking the limit as X → b, provided the limit exists.
Volume of revolution
●
The volume, V , obtained when the function y = f( x ) is rotated through 360° about the x-axis
between the boundary values x = a and x = b is given by the formula V =
b
∫ πy dx.
2
a
●
The volume, V , obtained when the function x = f( y ) is rotated through 360° about the y-axis
between the boundary values y = a and y = b is given by the formula V =
b
∫ πx d y.
2
a
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