Hints offered by N Hopley, with video solutions by DLB Maths

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Paper 1

Question 1

Hint 1: decide if you wish to add the fractions in the bracket first, or whether to multiply out the brackets first (both methods will work)

Hint 2: know that adding fractions requires them to have a common denominator

Hint 3: know that multiplying fractions involves multiplying numerator by numerator, and denominator by denominator

Hint 4: once you have completed the additions/multiplications, decide if your final answer can be simplified any further

Hint 5: and here is a video of the solution:

Question 2

Hint 1: know that f(-3) means that you will replace the letter x with the number -3

Hint 2: remember to write (-3)³ and not just -3³ so that you are cubing the number (-3) and not just the number 3

Hint 3: know that a negative number subtracting another number will still be negative, and be further away from zero

Hint 4: and here is a video of the solution:

Question 3

Hint 1: check the formula for the volume of a cone from the formula sheet

Hint 2: know that you need the radius, r, but the diagram gives us the diameter

Hint 3: faced with multiplying ¹/_{3} and 3.14 and 100 and 60
together,
decide which two numbers would be easiest to combine together first.

Hint 4: repeat dealing with pairs of numbers, and avoid trying to multiply everything together all at once

Hint 5: remember to include the units with your final answer

Hint 6: and here is a video of the solution:

Question 4

Hint 1: notice that there are lots of radii that you can mark as equal

Hint 2: as radius OE equals radius OD, this means that triangle EOD is isosceles

Hint 3: this means that angle OED = angle ODE = (180 - 68) ÷ 2

Hint 4: notice that CD is a diameter

Hint 5: so triangle CED is a 'triangle in a semi-circle' which means that we know angle CED

Hint 6: consider triangle CED, and we know the size of angles E and D

Hint 7: after calculating angle C, we need to know angle ACD

Hint 8: know that AB is a tangent to the circle, so it meets the radius OC at a right angle

Hint 9: angle ACE will be angle ACD + angle DCE

Hint 10: and here is a video of the solution:

Question 5

5a) Hint 1: use your preferred standard technique to complete the square

5a) Hint 2: if done correctly, the value of a is positive, and the value of b is negative

5b) Hint 3: know that you can read off the required coordinates from the answer in part (a), being careful with the signs.

Hint 4: and here is a video of the solution:

Question 6

Hint 1: consider drawing a diagram, plotting the two points on a set of axes and draw a line through them

Hint 2: recognise that the gradient is going to be a negative value

Hint 3: after calculating the gradient, know that we are looking for an equation of the form y = mx + c

Hint 4: use your preferred standard technique to obtain the value of c

Hint 5: and here is a video of the solution:

Question 7

Hint 1: consider what operations are happening to the letter B, and in what order

Hint 2: notice that B first has 4 added to it, and then divided by C²

Hint 3: perform the inverse of dividing by C², which is multiplying both sides of the equation by C²

Hint 4: perform the inverse of adding 4, which is subtracting 4 from both sides of the equation

Hint 5: and here is a video of the solution:

Question 8

8a) Hint 1: know that a indicates the amplitude of the trigonometric function

8b) Hint 2: know that b indicates the frequency of the function, the number of times a complete wave happens within 360°

8b) Hint 3: notice that one complete wave happens in 45°, so how many will there be in 360°?

Hint 4: and here is a video of the solution:

Question 9

Hint 1: recognise that the cosine rule for working out an angle is required

Hint 2: refer to the formula on the formula sheet, but notice that the letters are not quite correctly placed for the diagram in the question

Hint 3: decide you wish to re-label the diagram to match the formula, or re-write the formula to match the diagram

Hint 4: substitute the correct values into the correct letters in your formula

Hint 5: once you have calculated the numerator and denominator values, decide if the fraction can be simplified any further

Hint 6: and here is a video of the solution:

Question 10

Hint 1: recognise that this is a 'backwards' percentage question, where the £16.10 is the cost AFTER a downwards percentage change

Hint 2: know that £16.10 is after 30% has been deducted, so it must represent 100% - 30% = 70%

Hint 3: write it as a ratio £16.10 : 70%

Hint 4: divide both sides of the ratio by 7, to calculate the cost representing 10%

Hint 5: multiply both sides of the ratio by 10, to calculate the cost representing 100%, and thus the original price

Hint 6: and here is a video of the solution:

Question 11

Hint 1: know the index law of (m^{a})^{b} = m^{ab}

Hint 2: use this rule, using a = -2 and b = 4

Hint 3: know the index law of m^{a} × m^{b} =
m^{a+b}

Hint 4: use this rule, using a = -8 and b = -5

Hint 5: know that m^{-c} = ^{1}/m^{c}

Hint 6: write your final answer as a fraction with a numerator of 1 and a denominator of m to a positive power.

Hint 7: and here is a video of the solution:

Question 12

Hint 1: know that to divide by a fraction, you multiply by the reciprocal of the dividing fraction

Hint 2: in other words (^{a}/_{b}) ÷
(^{c}/_{d}) =
(^{a}/_{b}) × (^{d}/_{c})

Hint 3: look to see if a term in the numerator can be simplified with a term in the denominator

Hint 4: and here is a video of the solution:

Question 13

Hint 1: expand the first set of brackets

Hint 2: know that √10 × √10 = √(10 × 10) = √100 = 10

Hint 3: know that √10 × √2 = √(10 × 2) = √20 = √(4 × 5) = √4 × √5 = 2√5

Hint 4: gather together similar terms and simplify

Hint 5: and here is a video of the solution:

Question 14

Hint 1: know that the function cuts the x-axis when y = 0

Hint 2: solve (x + 1)(x - 3) = 0 to obtain the x-coordinates of x-axis intercepts

Hint 3: know that the function cuts the y-axis when x = 0

Hint 4: evaluate the function y when x = 0 to obtain the y-coordinate of the y-axis intercept

Hint 5: know that the turning point will have an x-value that is the mid-point between the x-axis intercepts

Hint 6: evaluate the function y when x = midpoint value, to obtain the y-coordinate of the turning point

Hint 7: plot all the four known points on the blank set of axes

Hint 8: draw a symmetric function curve through them all, making it clear that the turning point is NOT on the y-axis

Hint 9: and here is a video of the solution:

Question 15

15a) Hint 1: know that the formula for the area of a triangle =
^{1}/_{2}
× base × height

15a) Hint 2: use a base of '3' and a height of '(x + 12)'

15b) Hint 3: know that the formula for the area of a rectangle = base × height

15b) Hint 4: use a base of '(8 - x)' and a height of '6'

15b) Hint 5: make the two area expressions equal to each other, to form an equation

15b) Hint 6: deal with the fraction by multiplying both sides of the equation by 2

15b) Hint 7: expand out the brackets on each side

15b) Hint 8: gather together terms and solve for x

Hint 9: and here is a video of the solution:

Paper 2

Question 1

Hint 1: use your standard method for multiplying out brackets

Hint 2: know to expect 6 different terms, before simplifying

Hint 3: check that x terms multiplied by x² terms gives x³ terms

Hint 4: check that negative terms multiplied by negative terms give positive terms

Hint 5: know that if you have a negative term and you subtract a similar term, you'll end up with a term that is 'more negative'

Hint 6: and here is a video of the solution:

Question 2

Hint 1: an increase of 3% is 100% + 3% = 103% = 1.03

Hint 2: multiply by this decimal for each year of increase

Hint 3: be sure to round your answer to the accuracy stated in the question

Hint 4: and here is a video of the solution:

Question 3

Hint 1: appreciate that the total volume = volume of sphere + volume of cuboid

Hint 2: use the formula sheet to provide the volume of a sphere

Hint 3: note that the volume of a sphere formula requires knowing the radius, not the diameter

Hint 4: using the diagram, deduce the height of the cuboid

Hint 5: read the question carefully to see what shape the base of the cuboid is

Hint 6: volume of a cuboid = area of base × height

Hint 7: and here is a video of the solution:

Question 4

4a) Hint 1: write down the two letters you will use to represent mangoes and apples and define clearly what they start for (e.g. is it the cost of them? the weight of them? the size of them?)

4a) Hint 2: check that your equation does not include any units of money - it is just numbers and the two letters you chose, and nothing else.

4c) Hint 3: write your two simultaneous equations from parts (a) and (b), one above the other and decide which letter you will try to eliminate first

4c) Hint 4: after solving for one letter, substitute its value back in to either of the original equations to work out the value of the other letter

4c) Hint 5: be sure to write a sentence at the end, clearly stating what you have found out, including units.

Hint 6: and here is a video of the solution:

Question 5

5a) Hint 1: Add up all the values

5a) Hint 2: Divide this number by 7, to calculate the mean

5a) Hint 3: Create a table with 2 columns: (each value subtract the mean) and (this answer squared)

5a) Hint 4: The first column should have a mix of positive and negative answers

5a) Hint 5: The second column should all be positive after squaring

5a) Hint 6: Total this second column

5a) Hint 7: Write out the standard deviation formula carefully from the formula sheet

5a) Hint 8: Substitute your total into the numerator and n = 7 into the denominator

5a) Hint 9: Type into your calculator and write your answer to a few decimal places

5a) Hint 10: Round to 2 decimal places (or more)

5b) Hint 11: Compare the means by saying which sports team members did more press-ups, on average

5b) Hint 12: Compare the standard deviation by saying which sports team members had a more consistent number of press-ups

Hint 13: and here is a video of the solution:

Question 6

Hint 1: recognise that the triangle is not right angled, so you have to use the triangle area formula given on the formula sheet

Hint 2: decide if you wish to re-label the diagram with A, B and C, or whether you want to re-write the formula in terms of F, G and H

Hint 3: be sure to use capital letters to represent angles and lower-case letters to represent lengths of sides

Hint 4: substitute the correct values into the correct letters in your formula

Hint 5: write your final answer to at least 1 decimal place, and include the units of area.

Hint 6: and here is a video of the solution:

Question 7

Hint 1: use the quadratic formula, as we're expecting answers that are decimals

Hint 2: correctly identify the values of a, b and c

Hint 3: when working out the expression under the sqaure root, be careful of the (-7) value

Hint 4: write the two unrounded decimals first, before writing the rounded decimals.

Hint 5: and here is a video of the solution:

Question 8

Hint 1: look to see where you can add a line to the diagram to create a right-angled triangle

Hint 2: draw a vertical line from point O down towards line AB, to create two right-angled triangles

Hint 3: pick one of the triangles to use, and know that you already have two of its side lengths

Hint 4: use Pythagoras' theorem to calculate the length of the vertical line you drew from point O to line AB

Hint 5: know that the height of the tunnel is the length you just calculated plus the distance from O to the top of the tunnel

Hint 6: draw a vertical line from O to the top of the tunnel, and recognise that it is a radius, so you know its length

Hint 7: and here is a video of the solution:

Question 9

Hint 1: rearrange the equation so that it says 3sin(x) = ...

Hint 2: rearrange the equation so that it says sin(x) = ...

Hint 3: use inverse sine to calculate one possible value for x

Hint 4: uses your knowledge of the sine function, or its graph, to calculate a second value for x

Hint 5: clearly present both your final answers for the values of x

Hint 6: and here is a video of the solution:

Question 10

Hint 1: note from the information provided that we have both the radius of the circle and the length of the major arc AB

Hint 2: know that the fraction of a full circle that we have is linked to the missing angle, divided by 360.

Hint 3: calculate the fraction of a full circle by dividing the length of major arc AB by the circumference of a full circle

Hint 4: multiply this fraction by 360 to obtain the missing angle

Hint 5: we are expecting the angle to be greater that 180°, as we were told it was a reflex angle

Hint 6: and here is a video of the solution:

Question 11

Hint 1: know that the space diagonal is the hypotenuse of right-angled triangle ECG

Hint 2: know that to work out length EG, you will first have to consider right-angled triangle EGH

Hint 3: draw a right-angled triangle EGH and mark in all the side lengths you know

Hint 4: use Pythagoras' theorem to calculate the length of the the side EG

Hint 5: draw a right-angled triangle ECG and mark in all the side lengths you now know

Hint 6: use Pythagoras' theorem to calculate the length of the the side EC

Hint 7: be sure to present your final answer for the space diagonal with the correct units

Hint 8: and here is a video of the solution:

Question 12

Hint 1: notice that both the numerator and the denominator will each be able to be factorised

Hint 2: fully factorise the numerator

Hint 3: now fully factorise the denominator, knowing that you'd expect one of the brackets to match the brackets in the numerator

Hint 4: simplify the expressions that are common to both the factorised numerator and the factorised denominator

Hint 5: know that no further simplification is possible, so stop there!

Hint 6: and here is a video of the solution:

Question 13

Hint 1: know that a fraction with two terms being added in the numerator can be split into two separate fractions being added together

Hint 2: write the expression as sin(x)/cos(x) + 2cos(x)/cos(x)

Hint 3: recognise that sin(x)/cos(x) can be re-written as something else

Hint 4: recognise that 2cos(x)/cos(x) can be simplifed, due to the common factor of cos(x)

Hint 5: present a final answer that will be a single trigonometric function plus a constant number.

Hint 6: and here is a video of the solution:

Question 14

Hint 1: use the angles already given to fill in as many other angles as possible

Hint 2: please do not (incorrectly) assume that length BC is the same as length CD !!

Hint 3: know that to work out length BC, you will need another length in triangle ABC

Hint 4: realise that it is length AC that you need to work out first

Hint 5: now focus on triangle ACD and decide on how to use all the existing information to help work out length AC

Hint 6: use the sine rule to work out length AC

Hint 7: now focus on right-angled triangle ABC for which you know length AC and angle C, and you need length BC

Hint 8: decide what trigonometric function you can use to work out length BC

Hint 9: clearly present your final answer, including the units of length

Hint 10: and here is a video of the solution:

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