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Hints offered by N Hopley, with video solutions by 'DLBmaths'.

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Paper 1

Question 1

Hint 1: rearrange the equation into the form y = mx + c

Hint 2: read off the gradient

Hint 3: calculate the perpendicular gradient (by taking the negative reciprocal)

Hint 4: use a standard method to calculate the equation of a line through (-1, 6), with the gradient you just calculated

Hint 5: write your final answer in simplified, expanded form (either y = mx + c or ax + by = c)

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

Question 2

Hint 1: know that the logs cannot be combined whilst one of them has a coefficient of 2 in front of it

Hint 2: use laws of logarithms to move the 2 inside the log that will change the 6 to 6²

Hint 3: know that one log subtracting another log can be re-written as the log of a fraction (when have the same base, which they do here)

Hint 4: simplify the fraction that is 6²/4

Hint 5: know that you can simplify the final expression as the base of the log has a connection with the value inside the log(...) function

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

Question 3

Hint 1: start with the general definition of inverse functions that
h(h^{-1}(x)) =
x

Hint 2: use composition of function skills to re-write the left side of this equation

Hint 3: you should now have 4 + (1/3)h^{-1}(x) = x

Hint 4: rearrange to make h^{-1}(x) the subject

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

Question 4

Hint 1: know that the sqaure root term needs to be re-written before you can differentiate

Hint 2: know that √(x³) can be re-written as x^{3/2}

Hint 3: clearly communicate that you now have y'(x) when you differentiate both terms .... be careful with all the negatives and fractions!

Hint 4: consider re-writing the answer back in terms of square roots and fractions, given that the question had them in that form

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

Question 5

Hint 1: recognise that this question will make use of m = tan(θ)

Hint 2: know that θ is the angle measured anti-clockwise from the positive direction of the x-axis

Hint 3: recognise that the given angle of π/3 radians is not the angle you need, but rather you need to use its complementary angle

Hint 4: so, using θ = π/2 - π/3 which equals π/6, work out the gradient, m ...

Hint 5: ... using the appropriate exact value triangle

Hint 6: now use a standard method to obtain the equation of a line through (-2, 0) with the gradient you have just calculated.

Hint 7: write your final answer in simplified, expanded form (either y = mx + c or ax + by = c)

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

Question 6

Hint 1: use your chosen standard method for integrating a compound function, making sure you check your answer by differentiating back and using the chain rule

Hint 2: before substituting in the limits, re-write the expression in terms of roots, to remove the fractional powers

Hint 3: take care when substituting in the limits to the expression, especially with all the negative signs

Hint 4: take care with the arithmetic of simplifying each term ... there are no rewards for getting it wrong quickly!

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

Question 7

7a)i) Hint 1: focus on right-angled triangle ADE

7a)i) Hint 2: use Pythagoras' Theorem to calculate the length of side AD

7a)ii) Hint 3: focus on right-angled triangle ABC

7a)ii) Hint 4: use Pythagoras' Theorem to calculate the length of side BC

7b) Hint 5: know that sin(q - r) can be expanded to give four trigonometric terms

7b) Hint 6: use your triangles from part (a) to substitute in values for each of the four trigonometric terms

7b) Hint 7: simplify your answer into a single fraction

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

Question 8

Hint 1: know that the two log(...) expressions on the left side can be combined to give a single log(...) term

Hint 2: re-write the equation in terms of exponents

Hint 3: you should now have x(x + 5) = 6²

Hint 4: expand out the brackets and think what type of equation you have here to solve

Hint 5: rearrange the equation to give ... = 0

Hint 6: fully factorise the expression

Hint 7: write down two solutions for x from this equation

Hint 8: note that the question stated that x > 0, so one of these solutions needs to be rejected

Hint 9: clearly communicate your final answer, along with your reason for rejecting one solution

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

Question 9

Hint 1: know that cos(2x) can be re-written in one of three different ways

Hint 2: look at the rest of the equation to see that there is a cos(x) term

Hint 3: choose the cos(2x) expansion that also only has cos(x) terms

Hint 4: rearrange the equation to give ... = 0

Hint 5: recognise what type of equation you are having to solve here

Hint 6: fully factorise the expression

Hint 7: obtain two values for what cos(x) must equal

Hint 8: know that one of the values means that x cannot be calculated, so its solution is rejected

Hint 9: use the appropriate exact value triangle to calculate a possible value of x for the other, allowed value

Hint 10: use your knowledge of the cosine function, or its graph, to obtain a second possible value for x

Hint 11: clearly communicate your final answers, along with your reason for rejecting cos(x) = 2

Hint 12: and here is a video of the solution:

Question 10

10a) Hint 1: know that y = 2f(x) + 1 involves two different transformations of y = f(x)

10a) Hint 2: on one set of axes, sketch the unchanged y = f(x) recording the points (0, 3) and (4, 0)

10a) Hint 3: on a new set of axes, sketch the graph y = 2f(x) and record which one of the two points needs their coordinates changed.

10a) Hint 4: on a new set of axes, sketch the graph y = 2f(x) + 1 and record which one of the two points now needs their coordinates changed.

10b) Hint 5: know that y = f(^{1}/_{2}x) will affect the
horizontal
scaling of the function y = f(x)

10b) Hint 6: realise that the stationary point of (4, 0) on y = f(x) will
equate to
the stationary point of (8, 0) on y = f(^{1}/_{2}x)

10b) Hint 7: clearly write down the coordinates of both stationary points

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

Question 11

Hint 1: realise that the 2x² term means that it has to be dealt with before embarking upon completing the square

Hint 2: factorise 2 out from the first two terms of the expression

Hint 3: use a standard method to complete the square of the expression inside the brackets you just created

Hint 4: expand out the 'outer brackets' and simplify terms to obtain the format of the required answer

Hint 5: for completeness, you could also clearly state the values of p, q and r

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

Question 12

Hint 1: recognise that this differentiation will require the chain rule

Hint 2: after differentiating, simplify the expression ready for evaluating with π/6

Hint 3: after substituting in π/6, simplify the expression inside the cos(...) term

Hint 4: use the appropriate exact value triangle to calculate cos(π/6)

Hint 5: present a final answer, as an exact value

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

Question 13

13a)i) Hint 1: know that if (x + 2) is a factor then x = -2 will be a root

13a)i) Hint 2: proceed to show that f(-2) equals zero, thus confirming that x = -2 is a root

13a)ii) Hint 3: use synthetic division, or polynomial long division, to factorise the cubic using the knowledge that (x + 2) is a factor

13a)ii) Hint 4: once fully factorised, clearly list the values of x that are the roots of the original cubic function

13b) Hint 5: realise that we know one of the existing roots is x = -2

13b) Hint 6: notice that we now need this root to be located at x = 1

13b) Hint 7: realise that this is a total shift of 3 (from -2 to +1)

13b) Hint 8: think carefully whether this would be y = f(x - 3) or y = f(x +3)

13b) Hint 9: see if your thinking matches the constraint that k > 0 for y = f(x - k), as stated in the question

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

Question 14

14a)i) Hint 1: know that you can simply read off the circle's centre coordinates from the given, factorised equation

14a)i) Hint 2: know that the radius can also be read off, once you process the number 100 in the appropriate way.

14a)ii) Hint 3: sketch a diagram showing the centre point of the circle (7, -5) and the point P(-2, 7)

14a)ii) Hint 4: calculate the distance between these two points

14a)ii) Hint 5: determine whether this distance is greater than, or less than, the radius you stated in part (a)(i)

14b) Hint 6: sketch an accurate diagram showing the circle with centre (7, -5) and radius 5

14b) Hint 7: add to your diagram the point P(-2, 7)

14b) Hint 8: consider circles of different size radii that have centre P(-2, 7)

14b) Hint 9: you should notice that there are two possible values for the
radius -
one for a small circle that just touches C_{1}, and one for a larger centre that encompasses
C_{1}

14b) Hint 10: use your diagram, and all of its labelled lengths, to logically deduce the two radius values

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

Paper 2

Question 1

1a) Hint 1: know that the altitude through C will require a gradient that is
perpendicular
to m_{AB}

1a) Hint 2: calculate gradient m_{AB}

1a) Hint 3: calculate the gradient that is perpendicular to m_{AB}, by
taking
the negative reciprocal

1a) Hint 4: use your chosen standard method for calculating the equation of a line through point C with the required gradient

1a) Hint 5: clearly write your equation in the format y = mx + c, or ax + by = c, or ax + by + c = 0

1b) Hint 6: calculate the coordinates of the midpoint of segment AC, and call it point D

1b) Hint 7: calculate the gradient of BD

1b) Hint 8: use your chosen standard method for calculating the equation of a line through point D with the required gradient

1b) Hint 9: clearly write your equation in the format y = mx + c, or ax + by = c, or ax + by + c = 0

1c) Hint 10: you should have y_{1} = x - 4 and y_{2} = 5x
- 14
from parts (a) and (b)

1c) Hint 11: know that the intersection point will happen when
y_{1} =
y_{2}

1c) Hint 12: solve the equation x - 4 = 5x - 14

1c) Hint 13: use the x-coordinate to calculate the y-coordinate, by
using
either y_{1} or y_{2}

1c) Hint 14: clearly state your final answer as a set of coordinates, with brackets around both numbers and a comma between them

Hint 15: and here is a video of the solution:

Question 2

Hint 1: know that the discriminant, b² - 4ac, will be needed here

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

Hint 3: know that two real and distinct roots happen when the discriminant is greater than zero

Hint 4: substitute values of a, b and c into b² - 4ac > 0

Hint 5: carefully simplify and solve the inequation, watching out for all the negatives

Hint 6: present a final answer as an inequality in terms of the variable, p

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

Question 3

3a) Hint 1: expand k sin(x + a)

3a) Hint 2: compare that expression with the one given in the question

3a) Hint 3: identify what k cos(a) must be equal to, and what k sin(a) must be equal to

3a) Hint 4: use your standard method to obtain the values of k and a

3b) Hint 5: use your answer from (a) to re-write the given equation

3b) Hint 6: rearrange the trig equation into the form sin(x + a) = some number

3b) Hint 7: take the inverse sine and list all possible values for (x + a) between 0 and 2π

3b) Hint 8: note that we have to work in radians that are not the usual fractional multiples of π, so your final values for x should be decimals between 0 and 6.28

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

Question 4

4a) Hint 1: notice that for the area above the x-axis, the limits of -1 and 2 will be used

4a) Hint 2: calculate the definite integral of y from -1 to 2, making sure to put brackets around the expression, as well as 'dx' on the end

4a) Hint 3: take care with the substitution of negative values into the integrated expression!

4b) Hint 4: notice that some of your work from part (a) can be 'recycled' in part (b), as they share the limit value of 2

4b) Hint 5: you can calculate the definite integral of y from 2 to 4, knowing that your answer will come out to be negative

4b) Hint 6: the total area will be the answer from part (a) plus the positive of the answer from part (b)

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

Question 5

5a)i) Hint 1: know that f(g(x)) will become f(3x + 5)

5a)i) Hint 2: evaluate f(3x + 5), expand the resulting brackets, and simplify

5a)ii) Hint 3: know that g(f(x)) will become g(x² - 2)

5a)ii) Hint 4: evaluate g(x² - 2), expand the resulting brackets, and simplify

5b) Hint 5: replace each side of the inequation with your expressions from part (a)

5b) Hint 6: realise that you have a quadratic inequation to solve

5b) Hint 7: rearrange so that the inequation takes the form ... < 0

5b) Hint 8: notice the expression on the left side of the inequation has a common factor that can be factorised out

5b) Hint 9: fully factorise the expression on the left side of the inequation

5b) Hint 10: sketch a graph of the factorised quadratic expression to help determine when it is less than zero

5b) Hint 11: clearly present your answer in the form ... < x < ...

Hint 12: and here is a video of the
solution:

Question 6

Hint 1: realise that you will have to integrate the given expression to determine y(x)

Hint 2: notice that the given expression is not in a form ready to be integrated

Hint 3: after rewriting 3/x² as 3x^{-2}, integrate
the
expression

Hint 4: don't forget the constant of integration!

Hint 5: use the information that when x = 3 then y = 6 in order to fix the value of the constant of integration

Hint 6: clearly present your final answer in the form y = ....

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

Question 7

Hint 1: recognise that the graph gives you an equation of the form Y = mX + c

Hint 2: calculate the value of m, and read off the value of c

Hint 3: replace Y with log_{5}y, and replace X with
log_{5}x

Hint 4: notice that all of the terms involve logs, except for the 3 on the end

Hint 5: re-write 3 as 3log_{5}5, because
log_{5}5 is
equal to the number 1

Hint 6: use the laws of logarithms to move the coefficients of the log terms inside the log functions, turning the coefficients into powers

Hint 7: use the laws of logarithms to combine the two log terms that are being added, into a single log term of a product

Hint 8: you should have an equation of the form
log_{5}y =
log_{5}(...)

Hint 9: taking 'inverse logs' of both sides, gives you an equation of the form y = (...)

Hint 10: read off the values of k and n, and clearly state them

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

Question 8

8a) Hint 1: realise that we need the width and length of the rectangular pond

8a) Hint 2: note that the pond width is (y - 2) and the pond length is (x - 3)

8a) Hint 3: calculate the area of the pond, A(x), by expanding out (x - 3)(y - 2)

8a) Hint 4: in the expansion, you will see an 'xy' term, which represents the area of the entire rectangular plot

8a) Hint 5: note from the question that xy = 150

8a) Hint 6: and also note, therefore, that y = 150/x

8a) Hint 7: substitute all of this information into your expression for A(x), and simplify to give the desired version of function A(x)

8b) Hint 8: realise that you will have to differentiate A(x), but that it is not yet in a form ready to be differentiated

8b) Hint 9: re-write the -450/x term as -450x^{-1}

8b) Hint 10: differentiate the expression for A(x)

8b) Hint 11: communicate that you will find stationary points when A'(x) = 0

8b) Hint 12: solve the equation A'(x) = 0 to obtain two values of x

8b) Hint 13: note from the context of the question that x must be positive, so the negative value of x can be discarded

8b) Hint 14: check the nature of the remaining stationary point by using either a nature table, or the second derivative, A''(x)

8b) Hint 15: after determining that the stationary point is a maximum AND clearly stating this, calculate A(x) for when x = 15

Hint 16: and here is a video of the
solution:

Question 9

9a) Hint 1: know to substitute the equation of the line into the equation of the circle

9a) Hint 2: solve the resulting equation in x. There will be two solutions for x, as it is a quadratic equation.

9a) Hint 3: calculate the corresponding values of y for each of the values of x, using y = 3x + 7

9a) Hint 4: clearly state the coordinates of both P and Q, using brackets and commas for each.

9b) Hint 5: know that we will need to find out the coordinates of the centre of the circle, as well as the mid-point of PQ

9b) Hint 6: process the equation of the circle, either by formula or by completing the squares, to obtain the coordinates of its centre

9b) Hint 7: calculate the midpoint of P and Q, using your coordinates from part (a)

9b) Hint 8: calculate the distance between the midpoint and the centre of the circle, as this will be the radius of the smaller circle

9b) Hint 9: assemble the centre and radius information into a new equation of the smaller circle

9b) Hint 10: make sure to simplify the constant value, and not to leave it as (√10)²

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

Question 10

10a) Hint 1: realise that this question is only asking to substitute one value into the given formula

10a) Hint 2: evaluate P when T = 24.55

10b) Hint 3: realise that the given equation has three unknowns, P, D and k, but that we are given values for both P and D already

10b) Hint 4: substitute in P = 850 and D = 600

10b) Hint 5: simplify each term to give an equation of the
form
390^{k} = ...

10b) Hint 6: know that to obtain the value of k, as it's in a power, then logarithms will need to be used.

10b) Hint 7: take either log_{10} or log_{e}
of both
sides of the equation

10b) Hint 8: use the laws of logarithms to bring the power of k down and outside of its log term

10b) Hint 9: solve for k, giving your answer to at least 2 decimal places.

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

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