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A1 Final RevisionTopics we've learnt so far: Remainder & Factor Theorems Binomial Expansions Solving Equations (Exponential, Logarithmic & Trigonometric) Geometric Progressions Finding the Equation Connecting Two Variables Equation of a Circle Radians: Arcs, Sectors & Segments Optimisation Exponential Growth and Decay Calculus: Differentiation (1st Principles) and Integration Trapezium Rule
Remainder & Factor Theorems
eg
Show (x + 1) is a factor of f(x)
Hence solve f(x) = 0
Easy Question:
Remainder & Factor Theorems
eg
(x + 1) is a factor of f(x)
When f(x) is divided by (2x + 1), the remainder is 7Find p & q
Standard Question:
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Remainder & Factor Theorems
eg
Find p & r
Hard Question:
x
y
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Remainder & Factor Theorems
A graph of y = f(x) is shown:
The remainder when f(x) isdivided by (x+1) is 36Find p, q & r
Hard Question:
Binomial Expansions
eg a) Expand up to the term in
b) Hence estimate (without a calculator):
c) Using the calculator, what is the percentage error in the estimate made in part (b)
Easy Question:
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Binomial Expansions
eg In the expansion of:
The coefficient of the coefficient of
Find C
Medium Question:
Binomial Expansions
eg In the expansion of:
The coefficient of is 7
Find n (given n is positive)
Hard Question:
Binomial Expansions
eg The 2nd and 3rd terms of
are and
Find A and B
Hard Question:
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eg Solve:
Solving EquationsEasy Question:
Solving Equations
eg Solve:
Easy Question:Solving Equations
eg Solve:
Medium Question:
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Solving Equations
eg Solve:
Medium Question:Solving Equations
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Easy Question:Solving EquationsMedium Question:eg Solve these simultaneous equations
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Solving Equations
eg Solve:
Medium Question:Solving Equations
eg Solve:
Medium Question:Solving Equations
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Medium Question:
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Solving Equations
eg Solve:Medium Question:
Solving Equations
eg Solve:
Medium Question:Solving Equations
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Medium Question:
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Solving Equations
eg Solve:
Hard Question:Solving Equations
eg Solve:
Hard Question:
Solving Equations
eg Solve:
Hard Question:
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Solving Equations
eg Solve:
Easy Question:Solving Equations
eg Solve:
Hard Question:Solving Equations
eg
(x + 1) is a factor of f(x)
When f(x) is divided by (2x + 1), the remainder is 7
a) Find p & q
b) Solve:
c) Solve:
Hard Question:
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Solving Equations
eg
a) Show (x + 1) is a factor of f(x) and solve f(x) = 0
b) Solve:
c) Solve:
d) Solve:
Hard Question: Easy Question:eg The equation: y = kx² (k+1)x + (k+1) has no real rootsa) Find the range of possible values of k
At x = 3, the curve is parallel to the line 2y 18x = 3b) Find kc) Hence state the equation in the form: y = A(x p)² + q and explain (using a sketch) why this confirms the equation has no real roots
Finding an Equation
egThe graph of y = x3 2kx2 + (k+1)x is shownIt has one root, at x = 0Find the range of possible values of k
Easy Question:Finding an Equation
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Finding an Equation
eg
Given that f(x) passes through, find the two possible values of k
Easy Question:Finding an Equation
eg has a turning point at (2, 32)
Find the other turning point
Hard Question:
x
y y = x³ 9x² + px
1 4
y + 2x = q
Finding an Equation
eg Find p and qMedium Question:
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f(x) = x³ + nx² + 15x + 9
g(x) = mx 12
eg Find the roots of f(x) and the points where f(x) = g(x)
Finding an EquationMedium Question: eg The graph shown is y = A + B sin(2x)
By first thinking about the AMPLITUDE, state A and B
5
1P
Q
Hence find P and Qand find the first of the 4 'roots' shown on the diagram
Finding an EquationHard Question:
x
y
12
4
360
Finding an Equation
Find A , B and C
Hard Question:
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eg A sketch of is shown
Sketch
Curve SketchingEasy Question:
egShown is:Sketch
0
x
yCurve SketchingMedium Question:
eg Triangle ABC has vertices
Find the area of Triangle ABC
Plane GeometryEasy Question:
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Plane GeometryEasy Question:eg Triangle ABC has vertices
Find the area of triangle ABC
Plane GeometryEasy Question:eg Find the equation of the line through P(1, 9) that is perpendicular to the line connecting A(2,4) and B(6,8)
Find the area of triangle ABP
eg Line 1 has equation: Line 2 has the same root as Line 1, but Line 2 is perpendicular to Line 1
a) Find the equation of Line 2Line 1 has yintercept at R,The lines meet at PQ is a point on Line 2; area PQR is 26 units2
b) Find the possible coordinates of Q
Medium Question:Plane Geometry
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Medium Question:Plane Geometry
eg Line ℓ1 has equation
Line ℓ2 is perpendicular to line ℓ1Lines ℓ1 and ℓ2 cross the line x=5 at A and B respectively
|AB|= 7.5 units
The lines meet at C
Find the area of triangle ABC
Equation of a CircleEasy Question:
egA circle has eqn:
a) Identify the centre and radius of the circle
b) Is the origin inside the circle?
Equation of a Circle
eg A circle has centre (4, 2) and passes through A(7, 5)a) Find the equation of the circleb) Find the tangent at A
The tangent at B is parallel to the tangent at Ac) Find the equation of the tangent at B
The tangents at P and Q are perpendicular to thetangents are A & Bd) Find the coordinates of P and Q
Medium Question:
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Equation of a CircleMedium Question:eg P=(2, 5), Q=(9, 4) & R=(10, 11) are three points on the circumference of a circle
a):Show PQ and QR are perpendicular
b):Find the midpoint of PR
c):Hence find the equation of the circle
eg Find the equations of the two tangents tox² + y² 10y = 50that touch the circle at x=5The tangents touch the circle at A and BThe circle has centre CThe tangents meet at DDescribe the quadrilateral ACBD
Equation of a CircleMedium Question:
Equation of a Circle
eg is a diameter of a circle of radius 10 units
The circle passes through the origin
Find the equation of the circle
Hard Question:
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Equation of a Circle
eg The yaxis is a chord of:
a) Find the length of the chord
The chord splits the circle into two segments
b) Find the area of the minor segment
Hard Question:
Equation of a Circle
In order to find the equationof the circle C (centre Z),you are given:
1) X=(0, 3)2) Y=(6, 11)3) Z has ycoordinate = 10Off you go!
Hard Question:Arithmetic Series
egA company started trading in 1980, making a£20K profit that yearSince then, its profits have increased by £10K, year on year...
a) How much did they make in 2010
b) In what year did their profit reach £160K
Easy Question:
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Arithmetic SeriesEasy Question:eg A polygon has 10 sides The lengths form an APIts perimeter is 675 cm and the longest sideis twice as long as the shortest sideFind 'a' and 'd'
eg Find, to 3 s.f:
b) The sum 25 terms of this series is 126. Find x
Arithmetic SeriesMedium Question: eg The figure shows a dark cross surrounded
by white squares to create a bigger cross shapeHere, there are 13 dark squaresand there are 12 white squares,making 25 squares in total
a) Another (larger) cross has 32 whites squares; how many squares does it have in total?
A different cross has a total of 221 squaresb) How many whites squares does it contain?
Arithmetic SeriesMedium Question:
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Arithmetic SeriesMedium Question:egBradley builds skyscrapers.He can build a 50 m tall skyscraper for £225 millionAfter that, the next 5 m costs £15 million, and each additional 5 m costs £15 million more thanthe previous 5 m......so the total cost of a 60 m high skyscraper is £270 milliona) Show that a 95 m building costs £900 millionA developer has £1800 millionb) How high can he go?
Geometric Progressions
eg Find
Hence find:
Now find:
Easy Question:
Geometric Progressions
eg An investor buys a bond for £1000It pays £300 at the end of the first year, but payments at the end of each subsequent year are 20% less than the previous payment...
a) Show the payment at the end of the 5th year is just over £120
b) Find the sum of the first 5 payments
c) After how many payments will the total exceed £1300
d) Assuming the payments go on forever, find the total amount this policy will payout
Another policy also costs £1000. It pays £x at the end of the 1st year and the payments alsodecrease by 20% year on year. However, this policy promises the total payout is double the initial investment
e) Find the value of x
Medium Question:
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Geometric Progressions
eg A ball is dropped from 10 mAfter each bounce, it rises to 4/5 of the heightit just fell from
a) Calculate the height reached after the 10th bounce
b) After how many bounces will the 'rise' be less than 10 cm
c) Find the total distance the ball has travelled up to the 10th bounce
Hard Question:
Geometric Progressions
eg A polygon has n sides and the lengths form a G.P.The shortest side is 1 cm, the longest side is 1024 cmand the perimeter is 2047 cm
How many sides does the polygon have?
Hard Question:Geometric Progressions
eg I put £500 into an account at the start of each yearand at the end of each year, they add 10% interest to the balance of my account...
How much more will I have in 40 years time than if I stuck my money in an account that paid no interest?
Hard Question:
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egWhen Rikin reached 25 years old, he decided to start saving for his pension...
So, on his 35th birthday, he put £2000 into a pension fund.At the end of that year they added 10% interest to his fund,and then, on his 36th birthday, he added another £2000 ofhis money to the fund.a) Show that, before he puts in £2000 on his 38th birthday his fund is worth £7,282
Rikin says he will retire once the fund is worth more than£220,000 (coincidentally, the cost of a Ferrari 458 spider)b) How old will Rikin be, when he retires?
Geometric seriesHard Question:
eg A company has a fleet of vehicles...
In order to save tax, the book value of each vehicleis depreciated by 20% yearonyear.When a vehicle reaches 10years old, it is thensold (as scrap) at its current value and taken off the books
a) If a vehicle cost £24,000 new, show that its scrap value is approximately £2577
Starting in 2002, when they purchased their first vehiclefor £24,000; they have purchased a new vehicle every year.
b) Show that, in 2006, after they purchased their 5th vehicle, the total book value of the vehicles in their fleet is just over £80,000
c) What is the book value of the vehicles in their fleet currently?
Geometric ProgressionsHard Question:
Radians: Arcs, Sectors and Segments
Starfleet have a new symmetrical logo:
The whole logo is on an equilateral
triangle of side cm
The shaded area is paintedwith Klingon blood
Find the bloody area!
Hard Question:
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Radians: Arcs, Sectors and Segments
Find the entire shaded area
Hard Question:not too scale0
6 cm 6 cm
6 cm
eg The new logo for HAGnDAG icecream looks like this:
Find the total areaof the logo
Radians: Arcs, Sectors and SegmentsHard Question:
Radians: Arcs, Sectors and Segments
The cross section of thejellybean is made of two semicircles attached to the sides of a sector of a circle of radius 6 cm
Find the area of the crosssection
Hard Question: