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

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Paper 1

Question 1

Hint 1: recognise that there is a mixed fraction and a division sign, both of which need to be dealt with

Hint 2: change the mixed fraction into a top heavy, or vulgar fraction

Hint 3: change the division sign into a multiplication sign, and think what that does to the fraction 8/9

Hint 4: so you now have 13/6 × 9/8

Hint 5: you can simplify the denominator of 6 with the numerator of 9, as 3 is a factor of each of them

Hint 6: so you now have 13/2 × 3/8

Hint 7: numerator times numerator, denominator times denominator

Hint 8: turn the top heavy (or vulgar fraction) back into a mixed fraction, to match how the question was posed

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

Question 2

Hint 1: write (x + 7)² as (x + 7)(x + 7)

Hint 2: expand (x + 7)(x + 7) to get four terms, two of which can be combined

Hint 3: expand 6(x² - 10) to get two terms

Hint 4: with the five terms, look to see which can be gathered together

Hint 5: your final answer should have threee terms

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

Question 3

Hint 1: decide whether you will try to eliminate the variable x, or y, first

Hint 2: look at the coefficients of the variable you're trying to eliminate

Hint 3: find the lowest common multiple of those coefficients, as that will tell you what you need to multiply each equation by

Hint 4: after multiplying each equation, you should find the variable you're trying to eliminate in each equation has the same coefficient

Hint 5: subtract one equation from the other: choose the order of subtraction to avoid introducing negatives (if you can)

Hint 6: this should eliminate one variable, leaving a linear equation in the other variable

Hint 7: solve the linear equation in one variable

Hint 8: substitute the solution for one variable back into either one of the original equations

Hint 9: solve the linear equation in the second variable

Hint 10: clearly present your final answers for the values of x and y

Hint 11: and here is a video of the solution:

Question 4

4a) Hint 1: notice that this is the graph of y = x² moved to the right, and moved up

4a) Hint 2: given that it has moved 3 to the right, x² becomes (x - 3)²

4a) Hint 3: given that it has moved up by 2, then y = ... Becomes y = ... +2

4a) Hint 4: combining these gives y = (x - 3)² + 2

4a) Hint 5: now compare with y = (x + a)² + b

4a) Hint 6: notice that variable a will take on a negative value

4b) Hint 7: recognise that P(0, c) means that the x-value is zero....

4b) Hint 8: ... and that c is the y-value

4b) Hint 9: using your answer from part (a), calculate the the y-value when x = 0

4b) Hint 10: be clear to state the value of c in your final line of working

4b) Hint 11: and here is a video of the solution:

Question 5

Hint 1: the phrase 'nature of roots' means that the discriminant will need to be used

Hint 2: know that the discriminant of ax² + bx + c is b² - 4ac

Hint 3: identify the values of a, b and c from the function f(x)

Hint 4: evaluate the discriminant using your values of a, b and c

Hint 5: notice that the discriminant is greater than zero

Hint 6: know that this means that the roots are 'real and distinct'

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

Question 6

Hint 1: notice that we are given the value of cos(C), so the cosine rule is the likely tool to use

Hint 2: length AB would be labelled 'c' so we need to use the cosine rule that starts with c² = ....

Hint 3: substitute the values of BC = a, and AC = b, into the cosine rule

Hint 4: evaluate the expression to work out c²

Hint 5: know that the value of c will be either +7 or -7, but that we are only seeking the positive value

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

Question 7

7a) Hint 1: know that for an equation, we need a gradient and an intercept

7a) Hint 2: to work out the gradient, we need two coordinate points

7a) Hint 3: use (5, 20000) and (25, 50000) as two points

7a) Hint 4: calculate the value of the gradient, m, from these points

7a) Hint 5: this gives P = 1500 × T + c

7a) Hint 6: use one of the coordinates to substitute into the equation to then work out c

7a) Hint 7: present the final equation in terms of variables P and T

7b) Hint 8: recognise that '8 years' means that T = 8

7b) Hint 9: substitute the value of T = 8 into your equation from part (a)

7b) Hint 10: present your final answer as a salary value, with £ symbol

7b) Hint 11: and here is a video of the solution:

Question 8

Hint 1: know that 'rationalise the denominator' means that we have to remove the square root from the denominator

Hint 2: know that we shall multiply the given fraction by another fraction that is essentially equal to the number 1

Hint 3: multiply the given fraction by √15 / √15

Hint 4: simplify the denominator to give just 15

Hint 5: simplify part of the numerator with the denominator, by looking for a common factor

Hint 6: present your final answer in its simplest form

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

Question 9

9a) Hint 1: know that to work out a median and quartiles, you need to first sort the values in increasing order

9a) Hint 2: identify the 'middle value' that happens to lie between two of the values in the list

9a) Hint 3: identify the lower quartile and the upper quartile

9a) Hint 4: calculate the interquartile range using the upper and lower quartiles

9b) Hint 5: write one comment comparing the medians, being sure to clearly use the context of the age of readers

9b) Hint 6: write one comment comparing the interquartile ranges, being sure to clearly use the context of the age of readers

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

Question 10

Hint 1: notice that the width is the sum of two lengths, one of which is the radius from C to the further right side of the circle

Hint 2: the other length is one side of a right-angled triangle that can be drawn to the left of point C

Hint 3: the right angled triangle has a hypotenuse of length 50cm and a shorter side of 30cm

Hint 4: use Pythagoras' theorem to work out the length of the third side.

Hint 5: calculate the required width of the slab using the recently calculated length, and the radius of 50cm

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

Question 11

Hint 1: draw a graph of y = sin(x)

Hint 2: add to the diagram the point (30, 0.5)

Hint 3: locate the point on the graph that has x co-ordinate of 330

Hint 4: notice from the symmetry of the graph that the point (30, 0.5) is symmetrically positioned to (330, -0.5)

Hint 5: conclude that sin(330) = -0.5

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

Question 12

Hint 1: notice that the denominator can be simplified

Hint 2: simplify the denominator to give c⁷

Hint 3: know that c^{-2} can be written as 1 / c²

Hint 4: simplify the denominators again

Hint 5: this gives a denominator of c⁹

Hint 6: present the final answer of 5 / c⁹

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

Question 13

13a) Hint 1: draw the graph of y = cos(x)

13a) Hint 2: notice that the given graph is the y = cos(x) graph moved to the right by 30° and moved up by 1.

13a) Hint 3: know that this gives y = cos(x - 30) + 1

13a) Hint 4: compare this to y = cos(x + a) + b

13a) Hint 5: identify the value of a, being careful with the sign

13b) Hint 6: identify the value of b

13b) Hint 7: and here is a video of the solution:

Question 14

Hint 1: notice that the best strategy will be first to remove the fractions

Hint 2: calculate the lowest common multiple of 3 and 5

Hint 3: multiple the whole equation through by 15

Hint 4: be sure that x + 1 is treated as (x + 1)

Hint 5: expand and simplify each term, as required

Hint 6: obtain the inequality 5x + 5 - 30 > 9x

Hint 7: simplify numerical terms

Hint 8: simplify algebriac terms

Hint 9: if you have -4x > 25, know what will happen when you divide both sides of the inequality by negative 4

Hint 10: present a final answer of the form x < ....

Hint 11: and here is a video of the solution:

Paper 2

Question 1

Hint 1: know that 11% depreciation involves a multiplier of 0.89

Hint 2: know that 6% depreciation involves a multiplier of 0.94

Hint 3: calculate the value after three years as 20000 × 0.89 × 0.94 × 0.94

Hint 4: present a final answer with a £ sign, and rounded to 2 decimal places

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

Question 2

Hint 1: know that the number of atoms will be the total mass divided by the mass of a single atom

Hint 2: enter the calculation of 300 / (6.64 x 10^{-24}) into your
calculator

Hint 3: use the EXP or EE button to type in the denominator, by typing 6.64 EXP -24 or typing 6.64 EE -24

Hint 4: be sure to round the final answer to 3 significant figures, and to write it in scientific notation

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

Question 3

Hint 1: write down: angle / 360 = arc length / circumference

Hint 2: know that we have an angle of 106° and the radius is 9.15

Hint 3: calculate the circumference using 2 × π × 9.15

Hint 4: substitute all the values into the original equation that you wrote down

Hint 5: multiply both sides of the equation by the circumference value, to make arc length the subject

Hint 6: be sure to write the final answer with units, and rounded appropriately

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

Question 4

Hint 1: notice that the angle needed is opposite the side length of 10m

Hint 2: realise that the sine rule is required to be used

Hint 3: write down the sine rule in terms of J, K and L

Hint 4: re-write the sine rule with the sine terms in the numerators as we require to work out an angle

Hint 5: manipulate the equation to make sin(K) the subject

Hint 6: use inverse sine to calculate the value of angle K

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

Question 5

Hint 1: knowing that the polygon has 10 sides, calculate the size of each of the 10 centre angles

Hint 2: recognise the existance of 10 isosceles triangles

Hint 3: calculate the two remaining corner angles in each isosceles triangle

Hint 4: know that the shaded angle = 360 - right angle - two corner angles from isosceles triangles

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

Question 6

Hint 1: know that an 8% increase means that you now have 108%

Hint 2: recognise that 94500 represents 108%

Hint 3: work back to the 100% by dividing 94500 by 1.08

Hint 4: state the final answer with a £ sign

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

Question 7

Hint 1: notice that the variable 'm' is multiplied by 'n' and one third, and then r subtracted from it, in that order

Hint 2: know that we need to process each of these steps in reverse order

Hint 3: add 'r' to both sides of the equation

Hint 4: multiply both sides of the equation by 3

Hint 5: divide both sides of the equation by 'n'

Hint 6: present the final answer as m = ...

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

Question 8

Hint 1: recognise that you have to use Pythagoras' theorem to check if a right angle is there

Hint 2: as one calculation, work out the square of the hypotenuse

Hint 3: as a separate calculation, work out the sum of the squares of the two shorter sides

Hint 4: compare the two answers from these calculations to determine if they are equal, or not

Hint 5: write a sentence explaining whether the wall is perpendicular to the ground, giving a reason for your answer

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

Question 9

Hint 1: recognise the block's volume = volume of large pyramid - volume of small pyramid

Hint 2: use the formula sheet to obtain the formula for the volume of a pyramid

Hint 3: calculate the total height of the large pyramid

Hint 4: note that the base of both pyramids are squares

Hint 5: calculate the two pyramid's volumes

Hint 6: subtract one volume from the other to obtain the volume of the block

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

Question 10

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

Hint 2: know that each fraction needs to be multiplied by something to achieve this aim

Hint 3: multiply the first fraction by x/x, and multiple the second fraction by (x -3)/(x - 3)

Hint 4: leave the denominators of the fractions as x(x - 3), and don't expand these brackets

Hint 5: simplify the numerators slowly and carefully as it involves multiplying (x - 3) by -2

Hint 6: write it as a single fraction with 5x + 6 as the numerator

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

Question 11

Hint 1: know that 150 is a value of h

Hint 2: in the given equation, replace h with 150

Hint 3: rearrange the equation to make cos(x) the subject

Hint 4: use inverse cosine to obtain one value of x

Hint 5: use your knowledge of the cosine function to obtain another value of x that also has a cosine value of 3/20 (= 0.15)

Hint 6: state the two values of x, rounded to an appropriate degree of accuracy

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

Question 12

Hint 1: know to expect that something in the numerator will simplify with something in the denominator

Hint 2: factorise the numerator into two brackets

Hint 3: factorise the denominator into two brackets, and one of them should match with part of the factorised numerator

Hint 4: simplify the parts of the numerator and denominator that effectively equal the value of 1 when they are divided

Hint 5: note that the simplification requires x to not be the value of 4, else dividing by x - 4 (which would equal zero) is very problematic!

Hint 6: present your final answer as a single fraction, with no brackets

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

Question 13

Hint 1: note that you can factorise the number 2 out of both terms

Hint 2: recognise that sin²(x) + cos²(x) is equal to 1

Hint 3: replace the trigonometric terms with the number 1

Hint 4: write the final answer of 2 × 1 = 2

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

Question 14

14a) Hint 1: know that the volume of a cuboid is length x breadth x height

14a) Hint 2: replace the dimensions of this formula with (x + 7), x and 2

14a) Hint 3: note that we are told that V = 45

14a) Hint 4: substitute V to be 45 to give 45 = 2x(x+7)

14a) Hint 5: expand the brackets and rearrange to give the stated quadratic equation

14b) Hint 6: notice that they instruct the answer to be given to 1 decimal place. This suggests that x has non-integer values.

14b) Hint 7: therefore decide to use the quadratic formula to solve, instead of trying to factorise

14b) Hint 8: follow your standard process for calculating the two values of x, using the quadratic formula

14b) Hint 9: be sure to write both answers rounded to 1 decimal place

14b) Hint 10: and here is a video of the solution:

Question 15

Hint 1: note that we want a missing length, but that we have areas and we can work out possible angles

Hint 2: so write down the formula for a triangle that uses lengths and the sine of an angle

Hint 3: from the diagram, work out the value of sin(A), using triangle ABC (note: you don't need the value of angle A, just what sin(A) equals)

Hint 4: now focus on triangle ADE, and work out the length of AD

Hint 5: so, using the formula: Area = 1/2 × AD × AE × sin(A), replace Area, AE and sin(A) with their values

Hint 6: after simplifying, rearrange this equation to make AE the subject

Hint 7: state the value of AE, remembering to include the units of length

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

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