Remainder & Factor Theorems

1)

Show (x + 1) is a factor of f(x). Solve f(x) = 0

Easy Question:

2)

(x + 1) is a factor of f(x)

When f(x) is ÷ by (2x + 1), the remainder is 7Find p & q

Medium Question:

C2 Final Revision

Remainder & Factor Theorems

4) A graph of y = f(x) is shown:

The remainder whenf(x) is ÷ by (x+1) is 36Find p, q & r

3) . Find p & r

HARD Question:

HARD Question:

Binomial Expansions

1 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)

2)In the expansion of:

The coefficient of the coefficient of

Find C

Medium Question:

Easy Question:

Binomial Expansions

3) In the expansion of:

The coefficient of is 7

Find n (given n is positive)

HARD Question:

4) The 2nd and 3rd terms of

are and

Find A and B

HARD Question:

1) Solve:

Solving Equations

Easy Question:

2) Solve:

Easy Question:

3) Solve:

Medium Question:

4) Solve:

Medium Question:

Solving Equations

Medium Question:6) Solve these simultaneous equations

7) Solve:

Medium Question:

8) Solve:

5) Solve:

Easy Question:

Easy Question:

Solving Equations

8) Solve:

0) Solve:

Medium Question:

9) Solve:

Medium Question:

1) Solve:

Medium Question:

Easy Question:

Solving Equations

2) Solve:

HARD Question:

3) Solve:

4) Solve:

5) Solve:

Medium Question:

Medium Question:

Easy Question:

Medium Question:

6) Solve:

Solving Equations

13)

(x + 1) is a factor of f(x)

When f(x) is ÷ by (2x + 1), the remainder is 7

a) Find p & q

b) Solve:

c) Solve:

HARD Question:

Solving Equations

14)

Show (x + 1) is a factor of f(x) and hence solve

a) f(x) = 0

b)

c)

d)

HARD Question:

Geometric Progressions

1) Find

Hence find:

And then find:

Easy Question:

Easy Question:2) Riki sells secondhand i­phones. He says a 3­yr old i­phone will sell for £216 ...and a 5­yr old i­phone will sell for £121.50 Assuming that the i­phone's depreciation (from new) is at a constant rate, work out how much a new phone would cost.

Geometric Progressions

3) An investor buys a bond for £1000It pays £300 at the end of the first year, butpayments at the end of each subsequent yearare 20% less than the previous payment...

a) Show payment at end of the 5th yr is just > £120

b) Find the sum of the first 5 payments

c) After how many payments will the total received 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 paymentsalso decrease by 20% year on year. However, this policypromises the total payout is double the initial investment

e) Find the value of x

Medium Question:

Geometric Progressions

4) A ball is dropped from 10 m After each bounce, it rises to 4/5 of the height it 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

5) A polygon has n sides and there 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?

6) Starting on 1st Jan 2010, I put £500 into an account atthe start of each year and at the end of each year, theyadd 10% interest to the balance of my account...

Once the savings in this account have reached £50,000I'm gonna tell my boss to stuff his job and I'll start myown business (doing what? Mind your own business!)Tell me; in what year will that be?

HARD Question:

HARD Question:

Geometric Progressions

HARD Question:7) An insurance policy salesman has managed to snagan amazing contact with his employer...

He'll be selling insurance policies and he'll earncommission payments on each policy he sellsFor each policy that he sells, the employer agrees to payhim £75 that month, and then £60 the following month,and so on... (each new payment being 80% of the previous payment)

He started work in Jan 2016 and he sold one policythat month......but, being quite lazy, he's decided he's only going tosell one policy each month, and then play Playstationfor the rest of that month!

a) Work out how much commission he earns in total in Jan 2017

Actually, in August 2017, he's going on holiday for thewhole month!b) Re­calculate his commission for Jan 2018

Geometric Progressions

HARD Question:8) A teacher decided to quit and invest in property instead

In 2008 he purchased an acre of land, which heplanted with Camelia Senensis (Darjeeling ­ I meant,proper­tea, not property).In the summer of 2009, this produced a crop of45 kg of top grade tea. He knows that the yield fromthis planation will increase by 15% year on year...

But, in 2009, he purchased a second acre and plantedit in the same way ­ and he did so again in 2010 etc

How much tea will all of his plantations produce inthe summer of 2020(Top quality Darjeeling sells for around £500/kg)

Finding an Equation

1)

Given that f(x) passes throughfind the two possible values of k

Easy Question:

2)

has roots at

and at

Find its maximumpoint

Medium Question:

3) has a turning point (2, 32).

Find the other turning point

HARD Question:

Finding an Equation

4) Find p and qHARD Question:

5)The graph shown is y = A + B sin(2x)By thinking about the AMPLITUDE, state A and B

5

­1P

Q

Hence find P and Q and find the firstof the 4 'roots' shown on the diagram

Finding an Equation

HARD Question:

Finding an Equation

6) The graph of is shown:

Find A , B and C

HARD Question:

Equation of a Circle

1)A circle centre (4, 2) passes through A(7, 5)

a) Find the equation of the circle

b) Find the tangent at A

Easy Question:

3) is a diameter of a circle of radius 10 units

The circle passes through the origin

Find the equation of the circle

HARD Question:

Medium Question:2)Find the equation of the circle that passes

through: , and

Equation of a Circle

4) The y­axis is a chord of the circle:

a) Find the length of the chordThe chord splits the circle into two segmentsb) Find the area of the minor segment

5) In order to find the eqnof the circle C (centre Z),you are given:

1) X=(0, 3)2) Y=(6, 11)3) Z has y­coordinate = 10

Off you go!

HARD Question:

Medium Question:

Radians: Arcs, Sectors and Segments

1)The cross section of the jelly­bean is made of two semi­circles attached to the sides ofa sector of a circle of radius 6 cm.

Find the area of the cross­section

Easy Question:

Radians: Arcs, Sectors and Segments

2)Find the entire shaded area

not too scale0

HARD Question:

Radians: Arcs, Sectors and Segments

3)Star­fleet have a new symmetrical logo:

The whole logo is on an equilateral triangle

of side cm

The shaded areais painted withKlingon blood

Find thebloodyarea!

HARD Question:

Radians: Arcs, Sectors and Segments

4)A circle of radius 4 cm fits perfectly inside

a sector of a circle of radius 12 cm

Find the shaded area

HARD Question:

Optimisation

1) . Find when f(x) is increasing

2) A straight line passes through the point (2, 3) an crosses the positive x­axis at A and the positive y­axis at B Find the gradient of the line 'm', that will producethe smallest area OAB

Medium Question:

3) Mala is taking part in a triathalon500 m run, 1500 m swim and 4 km cycleShe knows that, if she runs at x km/h, then shewill be able to cycle at 2x km/hBut, she also knows that if she runs too fast,the time taken for the swim will increase; In fact,the time taken for the swim is always 10x hoursWhat speed should she run at to win?

Medium Question:

Easy Question:

Optimisation

4) The open­topped cylinder is made from 48л cm3 of blue semi­transparent plastic

Nice, isn't it?

a) Find its volume in terms of r b) Find the value of r that produces the largest possible container volume

Medium Question:

Optimisation

5) The roof is the curved surface of a hemisphereThe wall is the curved surface of acircular cylinder The floor is a circular discThe volume is 800л

a) Show the surface area, A , is:

b) Find the value of R when A is min

c) Find the values of H and A which corresponds to this value for R

Medium Question:

Optimisation

6)The diagram shows a semi­circleattached to a triangle

Find the value of 'x' tominimises the areaof the shape

HARD Question:

Optimisation

7)This bread bin is a prism with the endsthe shape of a sector of a circle

The volume of the bread bin must be 9000 cm3

Find the values ofx and θthat enablethe bin tobe madewith the minimum amount of metal

HARD Question:

Optimisation

8) A box of matches is designed according to these rules:

The length (l) must be 1.5 times the width (w) The volume must be 108 cm3

a) Find an equation for the area of card used in terms of w

b) What box dimensions will allow these rules to be obeyed and keep the production cost as low as possible...

HARD Question:

Integration

1) Find the area between these curves

( and )

Medium Question:

IntegrationMedium Question:2)The diagram shows the curve

Find the shaded area

The tangent at x=2 divides the area into 2 parts.What is the ratio of these areas?

Integration

3)Find the area between the curve

and the lines

Medium Question:

Integration

4) The curve and the line meet at

Find this shaded area

HARD Question:

Integration

5) The curve and the line meet at

Find this shaded area

HARD Question:

Integration

6)The curve and the line meet at

Find the shaded area

HARD Question:

Integration

7) Find the area between the curve

and its tangent at x=1

HARD Question:

Integration

8) Find the area between the curve

and its tangent at x=16 (PC) and the line connecting itsminimum point to the root of this tangent (MC)

Serious Question:

Trapezium Rule

1) Using 4 ordinates, estimate the shaded area:

a) Giving your answer in the form:

b) Giving your answer in the form:

HARD Question:

Trapezium Rule

2)a) Using 4 strips, estimate the shaded area:

b) Use integration to find the exact area

HARD Question:

Trapezium Rule

3)Using 3 ordinates, estimate the shaded area,giving your ans exactly

Integration gives the area as

b) Find the percentage error in your estimate

HARD Question:

Trapezium Rule

4)The shape of a horn is given by:

Its volume can be found using:

Estimate the volume, using 3 strips

y = 2x

HARD Question:

­1 2