Topic 3 - Quadratics (revision)
● E.g.: y = x2 - 4x - 5
Roots: x2 - 4x - 5 = 0, (x - 5)(x + 1) = 0; x = 5, -1
Y-intercept (when x = 0)
Lowest point of curve - complete the square: y = (x - 2)2 - 9 {x = 2}
● Discriminant: b2 - 4ac
If discriminant......
○ >0: 2 distinct real roots
○ =0: 1 repeated root
○ <0: no real roots (graph line does NOT touch x-axis)
●
Topic 6 - Remainder theorem
b
● “Similarly, when f(x) is divided by (ax - b) the remainder is f ( ).”
a
Topic 7 - Factor theorem
b
● “If f ( ) = 0, then (ax - b) is a factor of f(x).”
a
Topic 8 - Counting (Permutations)
1
, ● n distinct objects, number of different arrangements = n!
● If identical objects present (will have counted too many), need to divide by number
of equivalent arrangements
○ E.g.: How many different ways are there to rearrange the letters in the word
REARRANGE?
A:
R E A N G
3 2 2 1 1
9!
∴ = 15120//
3! × 2! × 2!
n!
● nPr =
(n−r )!
n!
OR ***( ❑r )
n
● nCr =
(n−r )! × r !
nPr
● ***↓↓ ∴ nCr=
r!
Topic 9 - Binomial expansion
● Pascal’s triangle:
● ∴ (a + b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7
○ ∵ a7 is 0th term, equation power is 7, ∴ coefficient of a 7 = 7C0 = 1
(∅)
n
b
● In general, (a+ b)n=a n (1+ )
a
● (up to n terms) ...... + O(xn)
Topic 10 & 11 - Probability distributions & Binomial distribution
● X ~ B (n, p)
P (X = x) ＝ nCx * px * (1 - p)n - x
● ***E.g.:
∵ P (X = 0) = 0.430, P (X = 1) = 0.383,
∴ P (X ≥ 2) = 1 - P (X ≤ 1)
2
, = 1 - 0.430 - 0.383
= 0.187//
Topic 12 - Coordinate geometry
● ***y - y1 = m(x - x1)
Topic 13 - Circles
● By Pythagoras’ theorem, x2 + y2 = r2
○ E.g.: (x - a)2 + (y - b)2 = r2
Centre: (a, b) Radius: r {Point of intersection with circumference: (x, y)}
● Exam-styled questions ———
○ Q: (a) Show that the point (1, 5) lies on the circle x2 + y2 - 6x - 4y = 0. (b) Find
the equation of the tangent at that point.
A (answer to part b):
(x - 3)2 - 9 + (y - 2)2 - 4 = 0
(x - 3)2 + (y - 2)2 = 13, centre (3, 2), radius √13
Gradient of radius to (1, 5)
5−2 3
↳ midpoint = =-
1−3 2
−3 2
Gradient of TANGENT = −1 ÷( )= /¿
2 3
Using y - y1 = m(x - x1):
2 2
y−5= x−
3 3
3y - 15 = 2x - 2
2 13
3y = 2x + 13 OR y= x + (equivalent form)//
3 3
○ Q: A tangent is drawn from the point (8, 2) to touch the circle x2 + y2 - 4x - 8y -
5 = 0. Find the length of the tangent.
A:
(x - 2)2 + (y - 4)2 = 25 ≡ 52, ∴ radius = 5
Length OB = √ ❑
Using Pythagoras’:
AB2 = OB2 - r2 = 40 - 25 = 15, ∴ AB = √❑
○ Q: Show that the line y = 7x + 10 is a tangent to the circle x2 + y2 = 2.
A:
Intersection of line & circle
x2 + (7x + 10)2 = 2
x2 + 49x2 + 140x + 100 = 2
50x2 + 140x + 98 = 0
Discriminant: b2 - 4ac = 1402 - 4 x 50 x 98 = 0/
∴ The line intersects the circle at exactly 1 point, so it must be a
tangent.//
3
● E.g.: y = x2 - 4x - 5
Roots: x2 - 4x - 5 = 0, (x - 5)(x + 1) = 0; x = 5, -1
Y-intercept (when x = 0)
Lowest point of curve - complete the square: y = (x - 2)2 - 9 {x = 2}
● Discriminant: b2 - 4ac
If discriminant......
○ >0: 2 distinct real roots
○ =0: 1 repeated root
○ <0: no real roots (graph line does NOT touch x-axis)
●
Topic 6 - Remainder theorem
b
● “Similarly, when f(x) is divided by (ax - b) the remainder is f ( ).”
a
Topic 7 - Factor theorem
b
● “If f ( ) = 0, then (ax - b) is a factor of f(x).”
a
Topic 8 - Counting (Permutations)
1
, ● n distinct objects, number of different arrangements = n!
● If identical objects present (will have counted too many), need to divide by number
of equivalent arrangements
○ E.g.: How many different ways are there to rearrange the letters in the word
REARRANGE?
A:
R E A N G
3 2 2 1 1
9!
∴ = 15120//
3! × 2! × 2!
n!
● nPr =
(n−r )!
n!
OR ***( ❑r )
n
● nCr =
(n−r )! × r !
nPr
● ***↓↓ ∴ nCr=
r!
Topic 9 - Binomial expansion
● Pascal’s triangle:
● ∴ (a + b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7
○ ∵ a7 is 0th term, equation power is 7, ∴ coefficient of a 7 = 7C0 = 1
(∅)
n
b
● In general, (a+ b)n=a n (1+ )
a
● (up to n terms) ...... + O(xn)
Topic 10 & 11 - Probability distributions & Binomial distribution
● X ~ B (n, p)
P (X = x) ＝ nCx * px * (1 - p)n - x
● ***E.g.:
∵ P (X = 0) = 0.430, P (X = 1) = 0.383,
∴ P (X ≥ 2) = 1 - P (X ≤ 1)
2
, = 1 - 0.430 - 0.383
= 0.187//
Topic 12 - Coordinate geometry
● ***y - y1 = m(x - x1)
Topic 13 - Circles
● By Pythagoras’ theorem, x2 + y2 = r2
○ E.g.: (x - a)2 + (y - b)2 = r2
Centre: (a, b) Radius: r {Point of intersection with circumference: (x, y)}
● Exam-styled questions ———
○ Q: (a) Show that the point (1, 5) lies on the circle x2 + y2 - 6x - 4y = 0. (b) Find
the equation of the tangent at that point.
A (answer to part b):
(x - 3)2 - 9 + (y - 2)2 - 4 = 0
(x - 3)2 + (y - 2)2 = 13, centre (3, 2), radius √13
Gradient of radius to (1, 5)
5−2 3
↳ midpoint = =-
1−3 2
−3 2
Gradient of TANGENT = −1 ÷( )= /¿
2 3
Using y - y1 = m(x - x1):
2 2
y−5= x−
3 3
3y - 15 = 2x - 2
2 13
3y = 2x + 13 OR y= x + (equivalent form)//
3 3
○ Q: A tangent is drawn from the point (8, 2) to touch the circle x2 + y2 - 4x - 8y -
5 = 0. Find the length of the tangent.
A:
(x - 2)2 + (y - 4)2 = 25 ≡ 52, ∴ radius = 5
Length OB = √ ❑
Using Pythagoras’:
AB2 = OB2 - r2 = 40 - 25 = 15, ∴ AB = √❑
○ Q: Show that the line y = 7x + 10 is a tangent to the circle x2 + y2 = 2.
A:
Intersection of line & circle
x2 + (7x + 10)2 = 2
x2 + 49x2 + 140x + 100 = 2
50x2 + 140x + 98 = 0
Discriminant: b2 - 4ac = 1402 - 4 x 50 x 98 = 0/
∴ The line intersects the circle at exactly 1 point, so it must be a
tangent.//
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