A1 Final Revision - Math'scool

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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:


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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eg

Find p & r

Hard Question:

x

y

2


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

eg Solve:

Easy Question:Solving EquationsMedium Question:eg Solve these simultaneous equations

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Solving Equations

eg Solve:


eg Solve:


eg Solve:

Medium Question:

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Solving Equations

eg Solve:Medium Question:

Solving Equations

eg Solve:


eg Solve:

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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71

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:


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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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:


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:


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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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:


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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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:


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:

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A1 Final Revision - Math'scool