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 (ma)b = mab
Hint 2: use this rule, using a = -2 and b = 4
Hint 3: know the index law of ma × mb = ma+b
Hint 4: use this rule, using a = -8 and b = -5
Hint 5: know that m-c = 1/mc
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: