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

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Paper 1

Question 1

Hint 1: we need to find the gradient of the line y + 4x = 7

Hint 2: rearrange the given equation into the form y = mx + c

Hint 3: you now have the point (-2, 3) and a gradient, so use a standard method to obtain the equation of the line required

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

Question 2

Hint 1: realise that the function is not yet ready to differentiate as it stands

Hint 2: rewrite the 8√x term as x to the power of a fraction

Hint 3: differentiate each term

Hint 4: why not use this opportunity to practise converting your answer back into fractions with roots, rather than leave it with fractional powers?

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

Question 3

3a) Hint 1: know that u_{4} can be calculated from knowing u_{3}, by replacing n with 3 in the given formula

3b) Hint 2: know the condition that the 1/3 must satisfy, and state it clearly

3c) Hint 3: use a standard method to work out the limit of the recurrence relation

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

Question 4

Hint 1: know that we need the centre of the circle and its radius

Hint 2: if AB is a diameter, the centre of the circle will be the midpoint of AB

Hint 3: if AB is a diameter, the radius will be half of the distance from A to B

Hint 4: when writing the equation in the form (x - a)² + (y - b)² = r² be sure to evaluate r² and not leave it as a number squared

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

Question 5

Hint 1: recognise that this is a 'chain rule in reverse' situation

Hint 2: after integrating, check your answer works by differentiating it back, using the chain rule, to see if it gives 8cos(4x + 1)

Hint 3: don't forget the constant of integration

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

Question 6

6a) Hint 1: use a standard method to work out the inverse of a function

6b) Hint 2: know that 2 is the 'input' and 7 is the 'output'

6b) Hint 3: know that the inverse function takes what was the 'output' and returns the 'input'

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

Question 7

7a) Hint 1: draw a sketch of how the three vectors FG, GH and EH might link together

7a) Hint 2: realise that vector FH = vector FG + vector GH

7b) Hint 3: realise that vector FE = vector FH + vector HE

7b) Hint 4: know that vector HE is the negative of vector EH

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

Question 8

Hint 1: know that a line is a tangent to a circle if they intersect at the same point, twice (in other words, a double root)

Hint 2: substitute y = 3x -5 into the circle's equation, replacing all the y terms with (3x - 5)

Hint 3: remember to use brackets around (3x - 5) so that the y² and -4y terms are worked out correctly

Hint 4: after simplifying, you should have a quadratic equation in x

Hint 5: we need to show that this quadratic equation has a double root

Hint 6: factorise the quadratic to give something of the form (x - a)² = 0

Hint 7: clearly communicate that this double root means that you have shown that the line is a tangent

Hint 8: to find the point of contact, know that you now have the x coordinate, which you can substitute into the equation of the line

Hint 9: be sure to write your final answer as a pair of coordinates: two numbers in a bracket, with a comma between them

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

Question 9

9a) Hint 1: know that stationary points are found from differentiating the function and making the derivative equal to zero

9a) Hint 2: be sure to state clearly that you are setting f'(x) = 0

9a) Hint 3: after differentiating and equating to zero, you should have a quadratic equation in terms of x, that you can factorise and solve for values of x

9b) Hint 4: know that 'strictly increasing' means that f'(x) > 0

9b) Hint 5: sketch a graph of y = f'(x), showing clearly the x-axis intercepts and whether the quadratic has a maximum or minimum turning point

9b) Hint 6: look at the graph and see where the curve of the function is above the x-axis, and is thus > 0

9b) Hint 7: clearly state the region(s) for x that mean f'(x) > 0, using inequality signs

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

Question 10

Hint 1: know that the graph of an inverse function is geometrically related somehow to the graph of the original function

Hint 2: know that the graph of an inverse function is the reflection of the original function in the line y = x

Hint 3: draw in the line y = x on your sketch and use it to reflect over the points (1, 0) and (4, 1)

Hint 4: draw in a smooth curve for f^{-1}(x) and label it as y = f^{-1}(x)

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

Question 11

11a) Hint 1: draw a sketch of a line segment, label the ends as A and C and put point B between them. Decide whether B will be closer to either A or C

11a) Hint 2: knowing that C divides AC in the ratio 1:2, add in these numbers to your sketch

11a) Hint 3: the coordinates of B can be obtained from knowing the position vector OB

11a) Hint 4: vector OB = vector OA + vector AB

11a) Hint 5: know that vector AB is a fraction of vector AC, using your sketch to work out what the fraction is

11a) Hint 6: be sure to write your final answer as a pair of coordinates: three numbers in a bracket, with commas between them

11b) Hint 7: work out the magnitude of vector AC. You should find that it is a number larger than 1

11b) Hint 8: know that for k×(vector AC) to have a magnitude of 1, then k must be a unitary fraction

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

Question 12

12a) Hint 1: know that f(g(x)) means that g(x) is the 'inside function' and f(x) is the 'outside function'

12a) Hint 2: replace g(x) with (3 - x) in the expression f(g(x))

12a) Hint 3: write out f(x), but instead of x, write (3 - x) in __brackets__ for each term, to help you process the 2x² and -4x terms correctly

12a) Hint 4: simplify your expression into a quadratic in terms of x. It should match the expression for h(x) that is given in the question

12b) Hint 5: use a standard method to 'complete the square'

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

Question 13

Hint 1: know that cos(q - p) can be written as an expression in terms of cos(q), cos(p), sin(q) and sin(p)

Hint 2: from the diagram, know that you can work out the lengths of AC and AD using Pythagoras' Theorem

Hint 3: use your updated diagram to write down fractions for each of cos(q) and sin(q) from looking at right-angled triangle ABD

Hint 4: use your updated diagram to write down fractions for each of cos(p) and sin(p) from looking at right-angled triangle ABC

Hint 5: substitute all the fractions into your expression for cos(q - p)

Hint 6: after correctly multiplying and adding the fractions, you should have a fraction that has a square root in its denominator

Hint 7: if you rationalise the denominator of this fraction, you should obtain the value given in the question

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

Question 14

14a) Hint 1: know that log_{n}q = p can be written as n^{p} = q

14a) Hint 2: know that 25 is the same as 5²

14b) Hint 3: on the left side of the equation, use laws of logarithms to combine into a single log term

14b) Hint 4: change the right hand side to the value you obtained from part (a)

14b) Hint 5: know (again!) that log_{n}q = p can be written as n^{p} = q

14b) Hint 6: you should obtain a quadratic equation in terms of x

14b) Hint 7: after rearranging the equation to equal zero, factorise and solve for two values of x

14b) Hint 8: note that from the question, x must be greater than 6, so you need to discard one of the solutions obtained

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

Question 15

15a) Hint 1: know that the values of a and b will come from the roots of the function and that k will be determined from the point (1, 9)

15a) Hint 2: notice that the root at x = -5 happens twice, whilst the root at x = 4 happens only once

15a) Hint 3: write out f(x), replacing a and b with the values that you have decided they will be

15a) Hint 4: now replace x with 1 and f(x) with 9, to give a single equation in terms of k

15a) Hint 5: evaluate all the pieces of this equation, and solve for k

15b) Hint 6: realise that g(x) is the function f(x) that is moved down by a distance of 'd' units

15b) Hint 7: notice that, at present, the function f(x) has 3 real roots

15b) Hint 8: imagine the graph of f(x) being moved downward by 9 units. You would have this new function also with 3 real roots

15b) Hint 9: now imagine moving it down just a tiny bit more than 9 units. The point that was (1, 9) is now below the x-axis, but there is still an intersection point with the x-axis off on the left side

15b) Hint 10: state clearly your conclusion for the range of values that d could take

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

Paper 2

Question 1

1a)i) Hint 1: sketch a copy of the diagram, with all the coordinates and labels on it, ready to add information to it as we progress through this question

1a)i) Hint 2: after working out the coordinates of the midpoint, M, add these coordinates to your diagram

1a)ii) Hint 3: you have two points (M and P) so use a standard method to work out the equation of the line through them

1b) Hint 4: draw on your diagram the line L

1b) Hint 5: realise that we need to first work out the gradient of PR to then work out the perpendicular gradient

1b) Hint 6: you now have the coordinates of M (from part (a)(i)) and the gradient of the line L that goes through it, so work out the equation

1c) Hint 7: state the midpoint of PR, and add the information to your diagram

1c) Hint 8: use the equation of the line, L, to show that if you put in the x-coordinate from the midpoint of PR, then you obtain the y-coordinate of the midpoint of PR

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

Question 2

Hint 1: know that 'no real roots' means that the discriminant < 0

Hint 2: correctly identify the values of a, b and c that you will put into b² - 4ac

Hint 3: when substituting in the values of a, b and c, be careful to include brackets around negative values and any expressions that you have

Hint 4: after simplifying, you should have a linear inequation in terms of p, that you can solve

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

Question 3

3a)i) Hint 1: know that if (x + 1) is a factor, then f(-1) is a root

3a)i) Hint 2: work out f(-1) and show that it is zero, and clearly state that this means that x = -1 is a root and thus (x + 1) is a factor

3a)ii) Hint 3: use either synthetic division, or polynomial long division, to divide the given cubic expression by (x + 1)

3a)ii) Hint 4: make sure you write the factorised cubic equal to zero, and then solve for three values of x

3b)i) Hint 5: use your answer from part (a)(ii) to help determine the coordinates of A and B, writing them as coordinates

3b)ii) Hint 6: realise that this is now an integration question with the limits coming from the x-coordinates of points A and B

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

Question 4

4a) Hint 1: for C_{1} you can just read off the coordinates of the centre, and quickly determine the radius

4a) Hint 2: for C_{2} you can either complete the square in the x-terms, or use a formula provided on the formula sheet to obtain the centre coordinates and radius

4b) Hint 3: know that two circles will not intersect if the distance between their centres is greater than the sum of their radii

4b) Hint 4: work out the distance between the centres of the circles.

4b) Hint 5: add the two radii from part (a) together

4b) Hint 6: compare and comment clearly on what you have shown

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

Question 5

5a) Hint 1: know that vector AB = vector OB - vector OA = __b__ - __a__

5a) Hint 2: know that vector AC = vector OC - vector OA = __c__ - __a__

5b) Hint 3: realise that you will use the scalar product to work out the angle

5b) Hint 4: check that the two vectors you will be using are aligned 'tail-to-tail' (which they are)

5b) Hint 5: work out the magnitudes of vector AB and vector AC

5b) Hint 6: work out (vector AB).(vector AC)

5b) Hint 7: substitute all the calculated values into the scalar product and rearrange to make cos(θ) the subject

5b) Hint 8: use inverse cosine to obtain angle BAC

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

Question 6

6a) Hint 1: know that at the start of the study, t = 0

6b) Hint 2: double your answer from part (a)

6b) Hint 3: in the formula, replace B(t) with that doubled answer

6b) Hint 4: solve this exponential equation in t, by using natural logarithms

6b) Hint 5: it is always nice to turn the decimal time for hours that you obtain into hours and minutes.

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

Question 7

7a) Hint 1: write down a formula for the total area in terms of x and y

7a) Hint 2: this area is given as 108, so substitute this into your formula and rearrange to make y the subject, in terms of x

7a) Hint 3: write down a formula for the total length of fencing, L, in terms of x and y

7a) Hint 4: substitute the expression for y (from the area) into this formula for L, and simplify to obtain the given function for L(x)

7b) Hint 5: know that we will need to differentiate to obtain the minimum turning point

7b) Hint 6: realise that L(x) is not ready to differentiate as it stands

7b) Hint 7: re-write the 144/x term as something multiplied by a power of x

7b) Hint 8: after differentiating, clearly state that you are setting L'(x) = 0 to find its stationary points

7b) Hint 9: solve the equation in x, to obtain __two__ values for x

7b) Hint 10: consider the context of the question, and then reject one of the values for x as a result, clearly stating what you have done

7b) Hint 11: you need to check that for the remaining value of x, this corresponds to a minimum turning point

7b) Hint 12: use a nature table, or L''(x) to show clearly that it is a minimum

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

Question 8

8a) Hint 1: use a standard method to process the wave function into the required form

8b) Hint 2: set the equation of the line equal to the curve: 12 = 10 + 5cos(x) - 2sin(x)

8b) Hint 3: look to see how you can use the result from part (a) to re-write this equation so that it has a single cosine term

8b) Hint 4: solve the trigonometric equation for x, remembering that x is in radians not degrees, so find all values between 0 and 2π (=6.28)

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

Question 9

Hint 1: notice that we have been given f'(x) and we want f(x), so this will involve integration

Hint 2: realise that f'(x) is not in a form ready to be integrated

Hint 3: re-write f'(x) so that it no longer has √x terms in it, but rather powers of x (the powers will be positive or negative fractions)

Hint 4: after integrating f'(x), remember to include the constant of integration

Hint 5: it will be helpful if you can re-write the terms of f(x) that involve fractional powers as square roots

Hint 6: use the f(9) = 40 statement to help you fix the value of the constant of integration

Hint 7: state a final answer for f(x)

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

Question 10

10a) Hint 1: notice that this differentiation will involve the chain rule

10a) Hint 2: it is always nice to write your answer in terms of square roots and fractions (as it will then more clearly link with part (b))

10b) Hint 3: notice how the integrand is related to the answer from part (a)

10b ) Hint 4: remember to include the constant of integration

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

Question 11

11a) Hint 1: notice that the left hand side is more 'complicated' and therefore there is greater scope to simplify it

11a) Hint 2: using a standard identity, rewrite sin(2x) in terms of sin(x) and cos(x)

11a) Hint 3: using a standard identity, rewrite tan(x) in terms of sin(x) and cos(x)

11a) Hint 4: substitute in to the left hand side these two identities and simplify as far as possible

11a) Hint 5: notice that the right hand side has a cos(2x) term, so locate which standard identity for cos(2x) is the one that links best with what you have so far obtained

11b) Hint 6: realise that you should use part (a) to replace f(x), as differentiating 1 - cos(2x) will be more straightforward to do

11b) Hint 7: remember that differentiating 1 - cos(2x) will involve using the chain rule

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

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