A1: 99 Problems Remainder & Factor Theorems Easy Question

Remainder & Factor Theorems

Find p & q given: (x + 1) is a factor of f(x)When f(x) is divided by (2x + 1), the remainder is 7Sketch f(x1)

So so Question:

A1: 99 Problems

. Find p & r

Medium Question:

#01:

#02:

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

The remainder when f(x) isdivided by (x+1) is 36Find p, q & r

Hard Question:#03:

Remainder & Factor Theoremsa) 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:

In the expansion of:

The coefficient of the coefficient of . Find C

#04:

#05:

Binomial Expansions

Medium Question:

Find 1st 3terms in expansion:Tricky Question:#06:

In the expansion of:

The coefficient of is 7

Find n (given n is positive)

The 2nd and 3rd terms of

are and

Find A and B

#07:

#08:

Hard Question:

Hard Question:

Binomial Expansions Discriminant

The curve: y = kx² (k+1)x + (k+1) has no real rootsa) Find the range of possible values of kAt x = 3, the curve is parallel to the line 2y 18x = 3b) Find kc) 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

#09:Easy Question:

The curve y = x3 2kx2 + (k+1)x has only one rootFind the range of possible values of k

#10:Hard Question:

Solve:

Solve:

Solve:

#12:

#13:

#14:

Easy Question:

Easy Question:

Easy Question:

Solving EquationsSolve:#11:

Halfwit Question:

Solve:

Solve:

Solve:

#16:

#17:

#18:

Medium Question:

Medium Question:

Solving Equations Easy Question:#15: Solve:

So so Question:

Solve:

Solve simultaneously

#19:

#21:

#22:

Medium Question:

Medium Question:

Easy Question:Solving Equations

Tricky Question:Solve:#20:

Solve these simultaneously

Solve:

#23:

#24:

#25:

Medium Question:

Medium Question:

Medium Question:

Solving Equations

Hard Question:#26:

a) Solve:

Hence, solve:

b)

c)

d)

#28:

Medium Question:Solving Equations

Medium Question:

#27: 12sin2x + 7cosx − 13 = 0, 360° ≤ x < 540° Find the equation of line , through P(1, 9) andperpendicular to the line connecting A(2,4) & B(6,8)b) Find the area of triangle ABP

Q=(x,y) is a point along line such that

c) Show gradient AQ is: and find gradient of QB

d) Hence find the coordinates of Q

e) A circle passes through A, Q and B. Find its equation

#29:

Plane GeometryTricky Question:

ℓ

ℓ

Sketch the cubic: y = (x+2)²(x5)showing clearly the intercepts with the axesFind the line y = 8(x+4) is a tangent to thecurve at the point P.Find the coordinates of PThe tangent at Q on the curve is parallelto the tangent at P. Find the xcoord of Q

#30:

Finding an EquationMedium Question:

y = x³ 9x² + px

Find p & q has a turning point at (2, 32)Find the other turning pointand sketch the curve

#31: #32:

Medium Question: Medium Question:Finding an Equation

71

f(x) = x³ + nx² + 15x + 9

g(x) = mx 12

Find the roots of f(x) and the points where f(x) = g(x)#33:

Medium Question:Finding an Equation

Given that f(x) passes through, find the two possible values of k

#35:

Easy Question: 2

46

The graph shown is y = f(x)How many roots does i) f(x)=3x ii) f(x)=⅔x have?

Find the roots of: y = f(x2) 12

#34:

Easy Question: Finding an Equation

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

5

1P

Q

Hence find P and Qand find the first of the 4 'roots' shown on the diagram

#36:Medium Question:Finding an Equation

Finding an Equation

Find A, B and C#37:

Jo can only move her boat when tide level > 9.5m How long is that each afternoon?

#38:Hard Question:Superhard Question:

The graph of y = log x is translated byThe same effect could be achieved by stretch of kparallel to the xaxis. State k

Transformations#39:

Easy Question:

#40:Hard Question:

A sketch of is shown. Sketch

b) State the eqn of f(x)

The curve shown is: . Sketch

0

x

y

Transformations Medium Question:

In each case, state the transformation(s) andshow clearly the turning point and the asymptote

#41:

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 2

Line 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

#42:

Medium Question:Equation of a Circle

a) Identify the centre and radius of the circleb) Is the origin inside the circle?

The line y = ¾x + 3 meets the circle:x² 6x + y² + 2y 15 = 0 at (0, 3)Show that this line is a tangent to the circle

#43:

#44:

Easy Question:

Easy Question:

Equation of a Circle

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

#45:


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

#46:


is a diameter of a circle of r = 10 units

which passes through the origin. Find eqn of the circle

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

#47:

#48:

Medium Question:

Hard Question:



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!

#49:Hard Question:

A company started trading in 1980, makinga £20K profit that year. Since then, its profitshave increased by £10K, year on year...a) How much did they make in 2010b) In what year did their profit reach £160K

#50:

Find, to 3 s.f:

b) The sum 25 terms of this series is 126. Find x

#51:

Medium Question:

Easy Question:A-ri-thm-etic Series

An oilrig, with a 50m well costs £225millionIf you need to go deeper, the next 5m will cost £15 million more, and then, each additional 5mcosts £15million more than the previous 5ma) Show cost of a rig & 60m well is £270millionb) Find the cost of rig + 95m wellAn investor has £1.8billion. How deep can he go?

#52:A-ri-thm-etic Series

Balls are stacked into tetrahedrons. The 5th tetrahedron (T5) is shownIts bottom layer is also shownHow many balls are in the bottom layer of T25?Express (in sigma notation) the total of balls in T25

Hard Question:

Hard Question:#53:

The figure shows a dark cross surroundedby 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 white 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?

A-ri-thm-etic Series

#54:Hard Question:

Find Hence find:

Now find:

are in geometric progression

a) Find 't'b) Find the 6th term (to 3 s.f.)c) Find the sum to infinity of the series

#55:

#56:

Easy Question:

Easy Question:

G-eo-metr-ic series An investor buys a bond for £1000. It pays £300 at

the end of the first year, but payments at the end of each subsequent year are 20% less than the previous paymenta) Show the payment at the end of the 5th year is £120b) Find the sum of the first 5 paymentsc) After how many payments will the total exceed £1300d) 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 also decrease by 20% year on year. However, this policy promises the total payout is double the initial investment

e) Find the value of x

#57:Medium Question:

≈

G-eo-metr-ic series

A ball is dropped from 10mEach bounce, it rises to of the height it just fell froma) Calculate the height reached after the 10th bounceb) After how many bounces will the 'rise' be < 10cmc) Find the total distance travelled up to the 10th bounce

A polygon has n sides and the lengths form a GPThe shortest side is 1cm, the longest side is 1024cmand the perimeter is 2047cmHow many sides does the polygon have?

#58:

#59:

Medium Question:

Hard Question:

G-eo-metr-ic series

When Rikin reached 25 years old, he decided to start saving for his pension...On his 35th birthday, he put £2000 into a pension fundAt the end of that year they added 10% interest to his fund, and then, on his 36th birthday, he added another £2000 of his 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?

G-eo-metr-ic series

#60:Hard Question:

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?

#61:Hard Question:

Superhard Question: A GP has terms 4,a,b. A 2nd GP has terms 4,p,qWhen the terms of the two series are added is generates a new series, with terms: 8, 3, 1.25Find the sum to infinity of the new series

#62:

G-eo-metr-ic series

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 then sold (as scrap) at its current value and taken off the booksa) If a vehicle costs £24,000 new, show that its scrap value is approximately £2,577

Starting in 2002, when they purchased their 1st vehicle for £24,000; they've brought a new vehicle every yearb) Show that, in 2006, after they purchased their 5th vehicle, the total book value of the vehicles in their fleet is just over £80,000c) What's the book value of all vehicles in their current fleet?

#63:Hard Question:G-eo-metr-ic series

Expand: . If

show eqn for is:

#66:Hard Question:

Recurrrrsive Series

A sequence is defined by:

a) Starting with find U2 and U3

b) . Find A and B

#65:

5

A sequence is defined by:

a) If and U3 = K + U1. Find K#64:

Medium Question:

Easy Question: Radians: Arcs, Sectors & Segments

Starfleet have a new sy mm etrical logo:The logo is on an equilateraltriangle of side cmThe shaded area is paintedwith Klingon bloodFind the bloody area!

#67:Hard Question:

Radians: Arcs, Sectors & Segments

Calculatethe entire shadedarea

not too scale0

#68:Hard Question:

6 cm 6 cm

6 cm

The new logo forHAGnDAG icecream looks like this:Find the total areaof the logo


#69:Hard Question: Radians: Arcs, Sectors & Segments

The crosssection of the jellybean is made of two semicircles attached to the sides of a sector of a circle of r=6cm

Find the area of the crosssection

#70:Hard Question:


A circle of radius4cm fits perfectly insidea sector of a circle of radius 12 cm

Find theshadedarea

#71:Hard Question:

The gradient on the curve

is minimum at R. The normal at R meets the curve

again at P and Q. Find length |PQ|

Differentiation

#72:

#73:

Easy Question:

Easy Question:

Given that find when:

The tangent to at T is

. The normal at N is parallel to the tangent at T

Find the possible xcoords of N.

Find the eqn of the tangent to at x = 16

Find when the tangent meets the curve again

Differentiation

#74:

#75:Easy Question:

So so Question:

Optimisation

. Find when f(x) is increasing#76:

The opentopped cylinder ismade from 48л cm2 of semitransparent plastic. Nice, isn't it?

a) Find its volume in terms of rb) What value of r willproduce the largest possiblecontainer volume

#77:

Easy Question:

Easy Question:

Optimisation The diagram is a cardboard model for a buildingThe roof is the curved surface of a hemisphereThe wall is the curved surface of a circular cylinder The floor is a circular discThe volume is 800л & r < 10cm

a) Show the surface area, A, is:

b) Find the value of R, to 3 s.f., for which A is a minimum.c) Find the value of H which corresponds to this value for R


A diesel particulate filter is the shape of a prism, with crosssection the shape of a sector of a circle:Regulations state that filters must have a min volume (cm3) of 10 times the engine size of the vehicle (litres)Serena is designing the filter for a car with engine size = 1.4litres. As the filters walls are made of (expensive) platinum, she wants to minimise the surface area (A) of the filtera) Find an eqn for A in terms of r onlyb) What height will minimise the volume of the filter

Optimisation#79:

Medium Question:

A section ofthe Monaco racetrackis modelled by the

equation:A photographer sat at (4,0) needs to aim hiscamera at the point onthe track that is closest to him (he can't afford a zoom lens)

What point should he aim for, and what is the distance from him, to that point on the track

Optimisation

#80:Medium Question: A mathematician is visiting an apple farm with

50 trees. Farmer Giles wants to plant more trees, but all his land is given up to his orchard"Couldn't you just plant a few more trees, in the gaps?" he asks"Well", explains Giles, "these trees currently give me a crop of 800 apples from each tree. If I plant any more trees the yieldfrom each tree will go down: A drop of 10 apples from each tree, for every tree added" he exclaimed! "So I don't think adding in more trees will maximise my crop Mr Maths". Tell him how many more apples he could be selling!

Optimisation#81:

Medium Question: Optimisation

The diagram shows a semicircle attached to a triangle

Find the value of 'x' that minimises the area of the shape

#82:Hard Question:

Optimisation

This bread binis a prism withthe ends shaped as sector of a circle

The volume of the bread bin must be 9000 cm3

Find the values of x and θ that enable the bin to be made with the minimum amount of metal

#83:Hard Question:

Optimisation

A matchbox isdesigned according to these rules:

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

a) Find an eqn for the area of card used in terms of wb) What box dimensions will allow these rules to be obeyed and keep the production cost as low as possible

#84:Hard Question:

Integration

Find the areabetween these curves &

Find the shaded area

#85:

#86:Medium Question:Medium Question:

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

IntegrationFind the area between

the curve and

the lines

#87:Medium Question: Integration

The curve meets the line at a) Find the green area b) Find the cyan area

#88:Medium Question: Integration

Find the area between the curve

and its tangent at x=1


Integration

Find the area between the curve

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

#90:Hard Question: Integration: vt graphs

#91:Hard Question:

A speeding motorist (Q), driving at 5m/s, passes a stationary copcar (P) waiting in a layby, eating donuts...By the time the policeman puts down his donut and starts to give chase, the reckless driver is already 101m ahead and getting further away...Once he gets moving, thecop's acceleration given by:

a) Find the times when P is catching Qb) Find when P overtakes Q

Using 4 ordinates, estimate the shaded area:

a) Give answer in the form:

b) Give answer in the form:

Trapezium Rule

#92:Hard Question:

Trapezium Rule

a) Using 4 strips, estimate the shaded area:b) Use integration to find the exact area

#93:Hard Question:

Trapezium Rule

Using 3 ordinates, estimate the shaded area, giving your ans exactly b) Find the percentage error in your estimate

#94:

Hard Question:

y = 2x

Trapezium RuleThe shape of a horn is given by

Its volume can be found using:

Find the % error in using the trapezium rule (3 strips) to estimate the area

#95:

Hard Question:

A1: 99 Problems Remainder & Factor Theorems Easy Question

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