Hints offered by N Hopley.

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Paper 1

Question 1

Hint 1: recognise that with mixed fractions you can deal with the whole numbers separately from the fractions

Hint 2: process 3 subtract 1, and then focus on the fractions 2/3 and 1/4

Hint 3: to subtract fractions they need to have a common denominator

Hint 4: the common denominator of 3 and 4 is 12

Hint 5: find equivalent fractions to 2/3 and 1/4 that have a denominator of 12

Hint 6: process 8/12 subtract 3/12

Hint 7: present the final answer that has a whole number and a single fraction

Question 2

Hint 1: evaluating f(7) requires us to substitute 7 into the given expression

Hint 2: know that for (7 + 3)² we have to simplify the expression in the brackets first

Question 3

Hint 1: know that each term in the first bracket will multiply each term in the second bracket

Hint 2: write out the resulting 6 terms from perfoming both sets of multiplications

Hint 3: gather together similar terms, taking care when combining similar terms with positive and negative coefficients

Question 4

Hint 1: know that 3a + b means 3 times the first vector plus the second vector

Hint 2: in vector a, multiply each component by 3

Hint 3: add the corresponding components together from 3a and b to give one vector with 3 components

Question 5

5a) Hint 1: know that the median of a list of six numbers is half way between the third and fourth numbers

5a) Hint 2: know that the lower quartile will be the second number

5a) Hint 3: know that the upper quartile will be the fifth number

5a) Hint 4: know that the interquartile range is the upper quartile subtract the lower quartile

5b) Hint 5: write one sentence comparing the shop median of £200 with the website median of £195

5b) Hint 6: write one sentence comparing the shop interquartile range of £70 with the website interquartile range of £73

Question 6

Hint 1: know that √75 can be written as √(25 × 3)

Hint 2: √(25 × 3) = √25 × √3

Hint 3: √25 × √3 = 5 × √3

Hint 4: now combine 5√3 with -√3 in the same way you might simplify 5y - y

Question 7

Hint 1: notice that one equation has a negative coefficient of r, and the other has a positive coefficient of r

Hint 2: we need to multiply each equation by a number so that the coefficients of r in each are the same magnitude

Hint 3: multiply the first equation through by 2

Hint 4: multiply the second equation through by 7

Hint 5: add the two resulting equations to eliminate the terms in r

Hint 6: you should have the equation 25p = 50

Hint 7: solve for p

Hint 8: substitute the value of p = 2 back into either of the original equations

Hint 9: solve for r

Hint 10: clearly present your final solution of p = 2 and r = -1

Question 8

8a) Hint 1: notice that the amplitude of the graph is 7, when the graph of y = cos(x) would normally have an amplitude of 1

8a) Hint 2: this tells you the value of 'a'

8b) Hint 3: notice that the period of the graph is 180° when the graph of y = cos(x) would normally have a period of 360°

8b) Hint 4: this tells you the value of 'b'

Question 9

9a) Hint 1: realise that you have the coordinates of two points (3, 26) and (10, 12)

9a) Hint 2: calculate the gradient between these two points

9a) Hint 3: assemble the information obtained so far to give D = -2T + c

9a) Hint 4: substitute in T = 3 and D = 26 into this equation

9a) Hint 5: solve for c

9a) Hint 6: write the final equation in terms of D and T, not y and x

9b) Hint 7: realise that we have T = 7

9b) Hint 8: substitute T = 7 into the equation from part (a) to obtain a value for D

9b) Hint 9: clearly state the final answer, including the units of distance from the question

Question 10

Hint 1: mark on the diagram all of the right angles for where the tangents AC and CE meet the circle

Hint 2: now 'work backwards' from angle BCD to determine which angle might be helpful to know

Hint 3: identify that angle BOD would be helpful to know

Hint 4: think how to work from the angle 125° towards the angle BOD

Hint 5: work out angle OFD, knowing that OFE is a straight line

Hint 6: look at the triangle OFD. What sort of triangle is it?

Hint 7: knowing that triangle OFD is isoceles, you can now work out angle FDO then angle FOD

Hint 8: work out angle BOD from angle OFD, given that BOF is a straight line

Hint 9: now focus on the kite OBCD for which you have 3 of its 4 angles

Hint 10: either know that the angles in a kite add up to 360°, or imagine the kite split vertically into two symmetrical halves

Hint 11: either way, you have enough knowledge to now work out angle BCD, as required.

Question 11

Hint 1: know that you must try to rearrange the equation into the form y = mx + c

Hint 2: first, make 4y the subject

Hint 3: then divide everything through by 4

Hint 4: read off the coefficient of x, which is the required gradient

Question 12

12a) Hint 1: use your standard practiced method to include some extra terms, so that you can factorise the first part

12a) Hint 2: the first part should be (x - 3)² and then simplify the constant terms

12a) Hint 3: the final result should be (x - 3)² - 1

12b) Hint 4: know that the values from the answer to part (a) will help provide the coordinates of the turning point

12b) Hint 5: think what value of x would make the expression (x - 3)² - 1 as numercially small as possible

12c) Hint 6: know that the turning point is on the axis of symmetry of the parabolic graph

12c) Hint 7: know that half way between P and Q, the x coordinate is 3

12c) Hint 8: this means that the x coordinate of Q is 6

12c) Hint 9: the y-coordinate of Q is the same as the y-coordinate of P

Question 13

Hint 1: know that the 'x' outside the bracket will multiply each term inside the bracket

Hint 2: know that x is really x1

Hint 3: know that multiplying terms will involve adding their indices

Hint 4: so x1.x1/2 is equal to x3/2

Hint 5: and x1.x-1 is equal to x0

Hint 6: know that x0 is equal to 1 (but not when x is itself zero)

Hint 7: present a final answer of two terms

Question 14

Hint 1: consider drawing two separate diagrams: one of triangle ABC and one of triangle ADE

Hint 2: mark on your diagrams of similar triangles all of the lengths that you were given

Hint 3: realise that we need to work out length AB, in order to then work out length BD

Hint 4: identify the scale factor of reduction to go from triangle ADE to triangle ABC

Hint 5: to obtain length AB, multiply length AD by the scale factor of reduction

Hint 6: obtain length BD = length AD - length AB

Paper 2

Question 1

Hint 1: notice the word depreciate

Hint 2: know that depreciating by 26% means using a decimal multiplier of (100 - 26) / 100 = 0.74

Hint 3: know that for each year, the starting amount needs to be multipled by 0.74

Hint 4: think of a short way to write 460 × 0.74 × 0.74 × 0.74

Question 2

Hint 1: the number of ants will be the quantity in a set unit of area, multiplied by the number of area units

Hint 2: a hectare is a unit of area

Hint 3: we have 250 hectares and each hectare has 1,220,000 ants in it

Hint 4: the total number of ants will involve multiplying these two numbers together

Hint 5: you then need to write the answer of 305,000,000 back into scientific notation

Question 3

Hint 1: notice that we have 3 side lengths and we want to calculate one angle, but there are no right angles in the triangle

Hint 2: decide whether it requires the sine rule or the cosine rule

Hint 3: use the formula sheet to write down the cosine rule version that starts cos(A) = ..

Hint 4: on the diagram label the side lengths with 'a', 'b' and 'c'

Hint 5: substitute the values of 'a', 'b' and 'c' into the formula to evaluate for cos(A)

Hint 6: use inverse cosine on your scientific calculator to give an answer that should be larger than 90°

Question 4

Hint 1: expand the brackets on the left side

Hint 2: simplify the -10 and +4 terms, that are now on the left side

Hint 3: decide which side of the inequality you want to collect the variables on, and which side for the constants

Hint 4: add and subtract terms to both sides, in order to obtain your objective

Hint 5: perform a division to get 'x' on its own

Hint 6: the convention is to write the final answer with x > .. rather than .. < x

Question 5

Hint 1: know that an increase of 16% will involve a decimal multiplier of 1.16

Hint 2: set up an equation that looks like: 'last year' × 1.16 = 'this year'

Hint 3: substitute in the value of 278.40 for 'this year'

Hint 4: divide both sides of your equation by 1.16 to obtain 'last year'

Hint 5: present your final answer with a £ sign, and use 2 decimal places if required

Question 6

6a) Hint 1: look for a common factor in both terms

6a) Hint 2: recognise that 'y' is a common factor

6a) Hint 3: start writing the answer as: y( .. - .. ) where you can fill the gaps in with either letters or numbers

6a) Hint 4: to check, multiply out your brackets to see if you get back to the question's expression

6b) Hint 5: knowing the answer from part (a) to be y(y - 6) will help with simplifying the algebraic fraction in part (b)

6b) Hint 6: we expect some terms to simplify, so see whether y² -3y -18 can be factorised into two brackets, where one of the brackets is (y - 6)

6b) Hint 7: you ought to find that (y - 6) is now in both numerator and denominator

6b) Hint 8: this means that (y - 6)/(y - 6) is equivalent to the number 1, so long as y ≠ 6

6b) Hint 9: simplifying the whole expresion should leave you with y / (y + 3)

Question 7

Hint 1: recognise that the volume of clear glass = volume of the cuboid - volume of hemisphere

Hint 2: know that the volume of a cuboid = length × breadth & times; height

Hint 3: know that the volume of a hemisphere = half of the volume of a sphere

Hint 4: look at the formulae sheet for the formula to work out the volume of a sphere

Hint 5: notice that the formula for the volume of a sphere requires the radius

Hint 6: decide whether the radius of the hemisphere is 6cm, or something else

Hint 7: calculate the volume of the hemisphere using all the pieces of information that you have so far

Hint 8: calculate the volume of clear glass by performing the subtraction

Hint 9: remember to write down an answer with full accuracy, and then an answer rounded to 2 significant figures

Hint 10: it is a good habit to also include the units of volume

Question 8

Hint 1: recognise this as a standard situation in which to use the quadratic formula, as it asks for the answers correct to 2 decimal places

Hint 2: identify the values of 'a', 'b' and 'c' from the quadratic equation

Hint 3: write out the formula that you will use, using the formula sheet to help if required

Hint 4: substitute the values of 'a', 'b' and 'c' into the formula for the roots of a quadratic equation

Hint 5: carefully evaluate each part of the calculation

Hint 6: write down two decimal roots, to more than 2 decimal places

Hint 7: then write down the same roots, but this time rounded to 2 decimal places

Question 9

Hint 1: notice that the 'd' is firstly being multiplied by 2, then it has 3 added to it, and then it is divided by 'e'

Hint 2: know that we need to perform the inverse operation to each of these steps, and in reverse order

Hint 3: so multiply both sides of the equation by 'e'

Hint 4: then substract 3 from both sides of the equation

Hint 5: then divide both sides of the equation by 2

Hint 6: this should leave you with the equation d = (ef - 3) /2

Question 10

Hint 1: focus your attention on points A, B and C

Hint 2: the triangle ABC has side lengths 15cm, 10cm and 10cm

Hint 3: however, we don't know any of the angles in the triangle

Hint 4: we are being asked for the total width, so this will involve us working out some horizontal lengths

Hint 5: consider a horizontal line drawn through point C and where this line cuts AB, call it point D

Hint 6: consider triangle ADC and decide if it is a right-angled triangle

Hint 7: decide on how long length AD is, compared to length AB

Hint 8: decide how to use the right angled triangle ADC to work out length DC

Hint 9: think how far it is from point C to the far right side of the circle

Hint 10: think how the total width is calculated from all of the horizontal lengths that you now have

Question 11

Hint 1: know that we first want to get sin(x°) on its own

Hint 2: subtract a number from both sides to obtain 17sin(x°) = ..

Hint 3: divide both sides of your equation by 17, to obtain sin(x°) = ..

Hint 4: use inverse sine on your scientific calculator to obtain one possible value of x°

Hint 5: use your knowledge of the sine function and its graph to find a second value for x that also has sin(x°) = 8/17

Hint 6: present your two answers, ideally rounded to either 1 or 2 decimal places.

Question 12

Hint 1: know that to add two fractions, they need to have a common denominator

Hint 2: notice that the two fractions have denominators of (x + 5) and (x - 4)

Hint 3: multiply the first fraction by (x - 4)/(x - 4), and mutiply the second fraction by (x + 5)/(x + 5)

Hint 4: the numerators of the fractions ought to now be 2(x -4) and 3(x + 5)

Hint 5: now expand the two numerators and simplify the expression when they are added together

Hint 6: present a final answer of a single fraction, leaving the denominator in factorised form of (x + 5)(x - 4)

Question 13

Hint 1: notice that we have length AC, but not the angle ABC

Hint 2: use angle facts of a triangle to calculate angle ABC

Hint 3: decide whether to use sine rule or cosine rule to help work out either length AB or length BC (it doesn't matter which)

Hint 4: substitute in the correct values to the sine rule and obtain either length AB or length BC

Hint 5: now either draw out triangle ABD or triangle BDC, depending on which you now hold the most information about

Hint 6: notice that your triangle is now a right-angled triangle

Hint 7: decide on how to work out length BD using your triangle

Hint 8: use normal trigonometry of sin(angle) = opposite / hypotenuse

Hint 9: clearly present your final answer for length BD, including units.

Question 14

14a) Hint 1: look for a route that will take you from W to X

14a) Hint 2: vector WX = vector WZ + vector ZX

14a) Hint 3: notice that vector WZ is the same as -a

14a) Hint 4: notice that vector ZX is the same as b

14a) Hint 5: so vector WX = -a + b

14b) Hint 6: look for a route that will take you from W to M

14b) Hint 7: vector WM = vector WX + vector XM

14b) Hint 8: notice that we have vector WX from part (a)

14b) Hint 9: notice that vector XM = half of vector XY

14b) Hint 10: notice that vector XY = vector WZ

14b) Hint 11: so vector XM = -½a

14b) Hint 12: add the expressions for vector WX to vector XM, and simplify

Question 15

Hint 1: know that the fraction of (arc length)/(circumference) = (sector area)/(circle area)

Hint 2: know that the formula for a circumference = 2πr

Hint 3: know that the formula for a circle area = πr²

Hint 4: substitute all of the correct values into the fraction equation, from the start

Hint 5: multiply both sides of the equation by π × 12² to get 'sector area' on its own

Hint 6: it is possible to simplify this expression without using a calculator, as the π terms in the numerator and denominator can be simplified

Hint 7: you can also simplify the 12 that's also in the both numerator and denominator

Hint 8: write your final answer - which should be 90 - with the correct units for area.

Question 16

Hint 1: notice that the given expression has a cos²x° term in it, and the desired form has a sin²x° term in it

Hint 2: know that sin²x° + cos²x° = 1

Hint 3: rearrange this equation to give cos²x° = 1 - sin²x°

Hint 4: substitute the expression for cos²x° into the question's expression, remembering to use brackets

Hint 5: expand the brackets and simplify the constant terms

Hint 6: for completeness, it can be nice to then clearly state the values of 'a' and 'b'


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