Hints offered by N Hopley, with video solutions by 'DLBmaths'
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Paper 1
Question 1
Hint 1: realise that f(g(0)) will involve working out the value of g(0) first
Hint 2: work out g(0) to obtain a 'number', then evaluate f('number')
Hint 3: realise that g(f(x)) involves working out f(x) first
Hint 4: using substitution for f(x), write g(f(x)) as g(5x)
Hint 5: and here is a video of the solution:
Question 2
Hint 1: realise that you need to know the centre of the circle of the given equation
Hint 2: sketch a diagram with a circle that has the centre that you have found, that goes through P(-2, 1)
Hint 3: draw in the tangent to the circle that goes through P(-2, 1)
Hint 4: draw in the radius of the circle from the centre to point P
Hint 5: realise that in order to work out the equation of the tangent, you will need a point (which you have) and a gradient
Hint 6: work out the the gradient of the radius of the circle
Hint 7: obtain the gradient of the tangent that is perpendicular to the gradient of the radius
Hint 8: and here is a video of the solution:
Question 3
Hint 1: know that you will use the chain rule here
Hint 2: think of (4x -1) as the 'inside function' and (…)12 as the 'outside function'
Hint 3: after differentiating, tidy up the expression as much as you can, but do not expand the brackets!
Hint 4: and here is a video of the solution:
Question 4
Hint 1: know that equal roots means that the discriminant has a certain value
Hint 2: correctly identify the values of a, b and c to substitute into b² - 4ac
Hint 3: solve the equation in the variable k, that you will have formed
Hint 4: and here is a video of the solution:
Question 5
5a) Hint 1: your calculation will involve three multiplications, all being added together
5b) Hint 2: note that the vectors are 'tail-to-tail' and so the scalar product is appropriate to use
5b) Hint 3: realise that you need to work out the magnitude of vector u which is |u|
5b) Hint 4: know the value of cos(π/3) from exact value triangles
5b) Hint 5: replace the values of |u|, |w| and cos(π/3) into u.w = |u||w|cos(π/3)
Hint 6: and here is a video of the solution:
Question 6
Hint 1: use a standard method for working out the inverse of a function
Hint 2: make sure to present your final answer as h-1(x) = …
Hint 3: and here is a video of the solution:
Question 7
Hint 1: draw a clear diagram of the given information, labelling each vertex with both its label and coordinates
Hint 2: know that a median through C is a line that goes through C and through the mid-point of the opposite side, AB
Hint 3: work out the coordinates of the midpoint of AB, and call the point M ... for midpoint!
Hint 4: look carefully at the coordinates of C and the coordinates of M and think about what type of straight line CM will be
Hint 5: realise that the equation of the median that you want does NOT have an equation of the form y = mx + c
Hint 6: and here is a video of the solution:
Question 8
Hint 1: know that working out the 'rate of change' will require differentiating the function
Hint 2: realise that the function is not in a form that is ready to be differentiated
Hint 3: split the 1/(2t) up into ½×(1/t)
Hint 4: re-write (1/t) as t to a power
Hint 5: proceed to differentiate, but note that the question wants you to do something with d'(t) when you have it
Hint 6: evaluate d'(t) when t = 5
Hint 7: and here is a video of the solution:
Question 9
9a) Hint 1: write out the given recurrence relation, replacing n with 1, so that you have u2 = m×u1 + 6
9a) Hint 2: know that we have the values of u2 and u1, so you can substitute in and calculate the value of m
9b)i) Hint 3: know the condition on m that will mean that the recurrence relation has a finite limit
9b)ii) Hint 4: use a standard method for working out the limit
Hint 5: and here is a video of the solution:
Question 10
10a) Hint 1: know that we need to have an expression that is the (top function) - (bottom function)
10a) Hint 2: simplify this expression down to three terms
10a) Hint 3: know that we need to integrate this expression from its start at zero, to its end at 2.
10b) Hint 4: repeat part (a) but this time replace either the top or the bottom function, with the straight line equation of y = 1 - x
10b) Hint 5: with the numerical answer of the whole shaded area that you have, you can now calculate the value of the other part of the shaded region, on the other side of the straight line
10b) Hint 6: compare the numerical values of the two shaded parts to decide how the line split the shaded region
Hint 7: and here is a video of the solution:
Question 11
Hint 1: know that we need the gradient of the line 3y - 2x = 4, and that this gradient is NOT -2
Hint 2: after rearranging 3y - 2x = 4 into the form y = mx + c, you can read off the value for the gradient, m
Hint 3: one possible method is now to work out the equation of the line through (-7, 2) with gradient m (that you just found)
Hint 4: then, with that equation, replace x with 5 and work out the value of y, which will be the value of a
Hint 5: and here is a video of the solution:
Question 12
Hint 1: realise that the left side of the equation can be re-written as a single loga term, using laws of logarithms
Hint 2: know that lognx = y can be re-written as ny = x
Hint 3: and here is a video of the solution:
Question 13
Hint 1: recognise that the integrand needs to be re-written before it can be integrated
Hint 2: the integrand should now have a negative power
Hint 3: know that you are integrating to obtain an expression that would need the chain rule to differentiate it back, so do this check to ensure that it works
Hint 4: don't forget the constant of integration!
Hint 5: and here is a video of the solution:
Question 14
14a) Hint 1: use a standard method to work out the values of k and a
14b) Hint 2: using your answer from part (a), sketch a graph of the transformed sine wave, marking clearly the coordinates of where the turning points are located
14b) Hint 3: you also need to note on your diagram the values of where the function intercepts the x-axis
14b) Hint 4: in addition, work out the value of where it intercepts the y-axis, by substituting in the value of x = 0, and note this clearly on your diagram
Hint 5: and here is a video of the solution:
Question 15
15a) Hint 1: know that the (x + a) part will capture how far left or right the graph has moved
15a) Hint 2: know that the … + b part will capture how far up or down the graph has moved
15b) Hint 3: look at Diagram 1 and Diagram 2 and spot where you can see the region from Diagram 1 within Diagram 2
15b) Hint 4: realise that the only difference is the move to the right and the move up
15b) Hint 5: the move to the right has not changed the area, but the move up has increased the area by the amount that is surrounded by the dotted lines
15b) Hint 6: work out the area of the rectangle in Diagram 2 and use the information given in part (b) of the question to deduce the value of the integral being asked about
15c) Hint 7: f'(1) = 6 tells you about the gradient of the function f(x) when x = 1
15c) Hint 8: in the very first sentence of the question, we are told that f(x) is a quadratic function, so this means that it is symmetrical
15c) Hint 9: using an argument of symmetry, you should be able to state the gradient of f(x) when x = 3. In other words, f'(3)
15c) Hint 10: now look to see how f'(3) relates to h'(8) - are they the equivalent point on the curve?
Hint 11: and here is a video of the solution:
Paper 2
Question 1
1a) Hint 1: know that a perpendicular bisector of BC is one which cuts BC in half and meets BC at right angles
1a) Hint 2: using the coordinates of B and C, work out the midpoint, M
1a) Hint 3: using the coordinates of B and C, work out the gradient mBC
1a) Hint 4: knowing mBC, work out the perpendicular gradient, that should be positive (from referring to the diagram's dotted line)
1a) Hint 5: combine the perpendicular gradient and the coordinates of point M to give the equation of the line required
1b) Hint 6: know how to use the equation m = tan(θ) and your knowledge of exact values to work out the gradient of AB
1b) Hint 7: use the gradient and the coordinates of point B to work out the equation of AB
1c) Hint 8: you should have two equations from parts (a) and (b) that now need to be solved simultaneously
1c) Hint 9: make sure that you state the answer as a pair of coordinates, as two numbers inside brackets, with a comma between them.
Hint 10: and here is a video of the solution:
Question 2
2a) Hint 1: know that if (x - 1) is a factor, then x = 1 will be a root
2a) Hint 2: know that if x = 1 is a root, then f(1) should be equal to zero ... so proceed to verify this.
2b) Hint 3: use your preferred, standard method to factorise a cubic, knowing that (x - 1) is a factor
2b) Hint 4: be sure to fully factorise the final answer into three distinct brackets, if possible
2b) Hint 5: finally, equate the expression for f(x) = 0 and read off the values of x that make it true.
Hint 6: and here is a video of the solution:
Question 3
Hint 1: substitute the y = 3x into the equation for the circle to obtain an expression that is only in terms of x
Hint 2: expand out the brackets and gather terms
Hint 3: rearrange the equation into the form … = 0. Do you notice what type of equation you have, in terms of x?
Hint 4: factorise out any common numerical factor, then factorise the remaining algebra
Hint 5: solve the equation for values of x
Hint 6: work out the corresponding value of y, for each value of x
Hint 7: make sure to write your two intersection points as coordinates.
Hint 8: and here is a video of the solution:
Question 4
4a) Hint 1: use your preferred, standard method to re-write the given quadratic in completed square form
4b) Hint 2: after differentiating, take a moment to think how part (b) is linked to part (a)
4c) Hint 3: know that strictly increasing means that the gradient is always positive, and never zero or negative
4c) Hint 4: know that an expression squared, such as (x + b)² can never be negative, but it could be zero or positive
4c) Hint 5: think about the difference between a(x + b)² and a(x + b)² + c, and the effect that the '+c' term has
4c) Hint 6: construct a clearly written argument, using words and equations, that says why you can now be sure that f'(x) > 0
Hint 7: and here is a video of the solution:
Question 5
5a) Hint 1: know that vector PQ = vector PR + vector RQ
5a) Hint 2: after working out vector PQ, write it as: something times i + something times j + something times k
5b) Hint 3: use your preferred, standard method to work out vector PS knowing that S is two thirds along vector QR
5c) Hint 4: know that to work out angle QPS, we need vector PQ and vector PS
5c) Hint 5: use the scalar product to work out the angle, using the magnitute of vector PQ and the magnitude of vector PS
Hint 6: and here is a video of the solution:
Question 6
Hint 1: notice that you have an equation with sin(x) and cos(2x), and that this 'mixture of trigonometric functions' is not ideal
Hint 2: realise that if the equation could be written entirely in sine functions, or entirely in cosine functions, then it would be better
Hint 3: decide to rewrite cos(2x) in terms of an expression that only has sin(x) terms in it
Hint 4: now gather the sin(x) terms together and make the expression equal to zero, and you should have a quadratic equation in terms of sin(x)
Hint 5: after factorising the quadratic, you will have two linear trigonometric equations, both in terms of sin(x)
Hint 6: solve both of these equations, where possible, and if not possible, be sure to explain why it can't be done
Hint 7: make sure that you present your answers in radians, that will just be decimal values between 0 and 6.28
Hint 8: and here is a video of the solution:
Question 7
7a) Hint 1: know that stationary point location will involve differentiation
7a) Hint 2: recognise that the function provided is not ready to differentiate as it stands
7a) Hint 3: convert the square root term into one where it is x to the power of a fraction (put the 2 and the 3 in the correct places in that fraction!)
7a) Hint 4: after differentiating, clearly state that you are trying to find when y'(x) = 0 in order to determine the stationary point(s)
7a) Hint 5: proceed to solve y'(x) = 0 for a value in x
7b) Hint 6: determine whether you have a local maximum or a local minimum at the value of x from part (a), using either a nature table, or y''(x)
7b) Hint 7: know that you have to also check the values of y at both x = 1 and x = 9 (the ends of the interval)
7b) Hint 8: from the assembled information, determine what the highest and lowest values of the function are
Hint 9: and here is a video of the solution:
Question 8
8a) Hint 1: knowing the value of u0, you can work out the value of u1 in terms of k.
8a) Hint 2: knowing the expression for u1, you can work out an expression for u2 in terms of k.
8b) Hint 3: replace u2 and u0, with the expression and values that you know already from part (a)
8b) Hint 4: rearrange the inequality into the form ... < 0
8b) Hint 5: you should have a quadratic expression in terms of k on the left side of the inequality
8b) Hint 6: factorise out any common numerical factor, then factorise the remaining algebra
8b) Hint 7: use your preferred, standard method for solving a factorised quadratic inequality
Hint 8: and here is a video of the solution:
Question 9
Hint 1: calculate the gradient of the straight line
Hint 2: think of Y = m X + c, where Y = log2y and X = log2x
Hint 3: replace Y, m, X and c with what they are equal to
Hint 4: you now have a logarithmic equation to rearrange to obtain y = ...
Hint 5: use your preferred, standard method for processing the log2 terms
Hint 6: and here is a video of the solution:
Question 10
10a) Hint 1: one possible method is to work out mAB and mBC, to show that they are the same
10a) Hint 2: you need to clearly state that this means that AB and BC are parallel
10a) Hint 3: you also need to clearly state that the line segments AB and BC have a common point B which, together with the previous parallel statement, means that A, B and C are collinear.
10b) Hint 4: look at the distance from A to C on the diagram. How many lots of rA is it?
10b) Hint 5: look at the distance from A to D on the diagram. How many lots of rA is it?
10b) Hint 6: you can now calculate the ratio that the point D divides AC in.
10b) Hint 7: you know the coordinates of A and C from part (a), so work out the coordinates of point D using a standard ratio method
10b) Hint 8: notice that the radius of the circle with centre D will be two lots of DC, and you know what DC is in terms of rA
10b) Hint 9: pull all the information together to give the equation of the circle in the form (x - a)² + (y - b)² = r²
10b) Hint 10: remember to simplify the value of r² and not leave it as a number squared.
Hint 11: and here is a video of the solution:
Question 11
11a) Hint 1: notice that this is a 'show that', or proof question, and not an equation to solve
11a) Hint 2: choose to work with the expression on the left hand side as there is more that can be done with it
11a) Hint 3: use the formula for sin(2x) to replace the numerator, and then simplify the fraction term
11a) Hint 4: look to see if there is a common factor that can be factorised out
11a) Hint 5: recognise that you have inside the brackets a rearranged version of sin²(x) + cos²(x) = 1
11a) Hint 6: substitute the contents of the bracket and simplify out to give sin³(x)
11b) Hint 7: realise that you will use part (a) to help with the differentiation, as it's easier to differentiate sin³(x)
11b) Hint 8: know that sin³(x) is the same as [sin(x)]³
11b) Hint 9: remember to use the chain rule to differentiate
Hint 10: and here is a video of the solution: