Question Papers Marking Instructions

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

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Paper 1

Question 1

Hint 1: know to differentiate f(x) first

Hint 2: replace x with the value of 2 in your expression for f'(x)

Question 2

Hint 1: rearrange 2y = 3x + 5 in the form y = mx + c

Hint 2: determine the value of the gradient, m

Hint 3: know that the perpendicular gradient is -1/m, and this will be the value of k

Question 3

Hint 1: know that if (x-2) is a factor, then you will use the value of x = 2

Hint 2: in the given polynomial, replace x with 2

Question 4

Hint 1: notice that all options involve cosine, so sketch out the standard y = cos(x) graph

Hint 2: notice that this graph goes from -3 to 3, instead of from -1 to 1

Hint 3: notice that this graph starts on the y-axis at -3 instead of starting a 1 …. hence it will be y = -3cos(…)

Hint 4: know that the graph y = cos(x) has a period of 2π radians

Hint 5: notice that this graph has a period of π radians, which is half of what it is for y = cos(x)

Hint 6: hence it will be y = -3cos(2x)

Question 5

Hint 1: to use the value of u₅ = 11, notice that we shall let n = 4

Hint 2: substitute in the value of u₅ and rearrange to get u₄

Hint 3: repeat the same process, letting n = 3

Question 6

Hint 1: know that we want to compare the magnitude of vector PQ with vector QR

Hint 2: work out vector PQ, using the coordinates for P and Q

Hint 3: calculate the magnitude of vector PQ (written as |PQ|) and the magnitude of vector QR (written as |QR|)

Hint 4: the ratio will be |PQ| : |QR|

Question 7

Hint 1: know that the brackets must be expanded first, and then integration can happen

Hint 2: multiply out the brackets to obtain x² - 16

Hint 3: integrate this term by term, and remember the constant of integration

Question 8

Hint 1: know that m = tan(θ)

Hint 2: know that tan(60) can be found without a calculator using 'exact value triangles'

Question 9

Hint 1: know that the y = sin(x) graph has a minimum value of -1 when x = 3π/2

Hint 2: know that the new minimum value will be 3×(-1) + 5

Hint 3: know that the x-coordinate of this minimum will be half of 3π/2, because the graph has '2x' in it

Question 10

Hint 1: rearrange the equation so it reads cos(x) = ….

Hint 2: know that cos^-1(-½) is related to the value for cos^-1(½)

Hint 3: know that the value of cos^-1(½) can come from an exact value triangle

Hint 4: sketch the graph of y = cos(x) between 0 and 2π and connect together all of the information

Question 11

Hint 1: realise that we want to deduce f(x) from f'(x), so that involves integration

Hint 2: integrate f'(x) to obtain f(x) and remember to include a constant of integration

Hint 3: determine the constant by making x = 2 and f(x) = 9

Question 12

Hint 1: work out vector RT knowing that it is 3 times vector RS

Hint 2: the coordinates of T will come from starting at the coordinates of R and moving in the direction given by vector RT

Question 13

Hint 1: know that a quadratic that has a minimum turning point has a positive co-efficient of x²

Hint 2: know that a quadratic that does not intercept the x-axis has no real roots

Hint 3: know that a quadratic with no real roots has a discriminant that is less than zero

Question 14

Hint 1: look at the formula sheet and locate the formula for cos(2x) that is only in terms of cos(x), and no sin(x) terms

Hint 2: use this formula, replacing cos(x) with -2/5, making sure to use brackets to square the whole negative fraction.

Question 15

Hint 1: notice that the cubic has roots at -2, -1 and 3

Hint 2: know that this means the cubic has factors of (x + 2), (x + 1) and (x - 3)

Hint 3: the curve cuts the y-axis at (0, -3) so this will help calculate any constant multiplier, k, in the expression y = k (x + 2)(x + 1)(x -3)

Hint 4: replace x with 0 and y with -3 in the previous line, to determine the value of k

Question 16

Hint 1: know that you can re-write an exponential expression as a logarithmic expression

Hint 2: take the natural logarithms of both sides of the equation

Hint 3: once you have 4t = log_e(6), rearrange to make t the subject

Question 17

Hint 1: know that a unit vector is vector with magnitude of 1 unit

Hint 2: calculate the magnitude of u, when t = 4/5

Hint 3: know that parallel vectors have the same direction and thus one can be written as a multiple of the other

Hint 4: let t = 1 and note that multiplying the z-component of u by -10 would give the z-component of v. Does that work for the x-components?

Question 18

Hint 1: either complete the square in the x terms, and the y terms, or use the circle equation formula to determine the centre and the radius

Hint 2: from the previous step, you should know the coordinates of the centre of the circle

Hint 3: sketch a set of axes and draw in a circle that would meet the coordinate axes at exactly 3 points

Hint 4: your sketch should be a circle going through the origin, plus one point on the y-axis and one point on the x-axis

Hint 5: realise that the radius of the circle will be the distance from the origin to the centre of the circle

Hint 6: match this value of the radius with your expression for the radius from the first step, and work out the value of k

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

Question 19

Hint 1: know that the graph of a cos(bx) is symmetrical about the x-axis

Hint 2: therefore the shaded region above the x-axis is connected to the non-shaded region below the x-axis

Hint 3: know that the non-shaded region would give an integral value that is negative

Hint 4: realise that the non-shaded region is -2 lots of the shaded region

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

Question 20

Hint 1: recognise that this is a graph transformation question, not a calculus question

Hint 2: know that the graph of y = -f(x) reflects a graph of y = f(x) in the x-axis

Hint 3: know that the graph of y = f(2x) compresses the graph of y = f(x) towards the y-axis by a factor of 2

Question 21

21a) Hint 1: know that showing that an expression is a factor is the same as evaluating the function at the value of the root, to show it is equal to 0.

21a) Hint 2: evaluate f(1) to show that it is zero, and clearly communicate that this means that (x - 1) is a factor

21a) Hint 3: using polynomial long division (or synthetic division) to determine the quadratic function that comes from the cubic divided by (x - 1)

21a) Hint 4: factorise the quadratic function and present the full factorised cubic

21b)i) Hint 5: differentiate y(x) to obtain y'(x)

21b)i) Hint 6: evaluate y'(x) when x = 1 to obtain the gradient at point A(1, 3)

21b)i) Hint 7: use the gradient and the coordinates (1, 3) to obtain the equation of the straight line that is the tangent

21b)i) Hint 8: write the equation in the form y = ax + b or ax + by + c = 0

21b)ii) Hint 9: know that the intersection point comes from equating the equation of the tangent with the original cubic equation

21b)ii) Hint 10: starting with 2x + 1 = x³ - 6x² + 11x - 3, rearrange to make the equation 0 = ….

21b)ii) Hint 11: use the workings from part (a) to help factorise your cubic equation

21b)ii) Hint 12: know that the 3 solutions to this equation represent the 3 intersection points of the tangent with the cubic function

21b)ii) Hint 13: identify which solution relates to point B and then determine the y coordinate using the equation of the tangent

Question 22

Hint 1: know that the end points of the interval require to be checked

Hint 2: evaluate f(1) and f(4)

Hint 3: to find other stationary points, we need to differentiate f(x) and find where f'(x) = 0

Hint 4: re-write f(x) as 4x^-2 + x, so that you can differentiate it

Hint 5: after differentiating f(x), re-write it back with fractions and positive indices

Hint 6: set f'(x) = 0 and solve for x to obtain a single value

Hint 7: calculate f(x) for this value, to determine the y co-ordinate of the stationary point

Hint 8: determine the nature of the stationary point using either a nature table, or the second derivative

Hint 9: summarise all information gathered so far, using a graph if that helps, to help answer the question

Question 23

Hint 1: know to use laws of logs to rearrange equation to have a single log term OR use the fact that 3 can be written as 3log₂2

Hint 2: rewrite equation without logs to create a linear equation in terms of x

Question 24

Hint 1: know that a quadratic expression has real roots if the discriminant is greater than or equal to zero

Hint 2: determine the values of a, b and c to substitute into b² - 4ac ≥ 0

Hint 3: simplify and fully factorise the algebraic side of this inequality

Hint 4: sketch a graph of the quadratic that is in terms of k, noting where the roots are

Hint 5: interpret your graph to determine the values of k that make the discriminant inequality true.

Question 25

25a) Hint 1: use Pythagoras' Theorem on the co-ordinates of (2t - 5, 0) and (0, t - 10)

25b) Hint 2: know that increasing or decreasing can be determined from looking at the sign of the gradient of D when t = 5

25b) Hint 3: know that differentiating D(t) is what is required to be done

25b) Hint 4: write D(t) with a fractional power, instead of the square root

25b) Hint 5: differentiate D(t), remembering to use the chain rule

25b) Hint 6: evaluate D'(t) when t = 5 but note that we're only interested in whether the answer is either positive or negative

Paper 2

Question 1

1a) Hint 1: the altitude through C has a gradient that is perpendicular to the gradient of AB

1a) Hint 2: work out m_AB and then take the negative reciprocal to give the gradient of the altitude

1a) Hint 3: use a standard method to work out the equation of the line through T with the required gradient

1b) Hint 4: know that the median from B will go through the midpoint of AC

1b) Hint 5: calculate the coordinates of M, the mid-point of AC

1b) Hint 6: calculate the gradient of line BM

1b) Hint 7: use a standard method to work out the equation of the line through B with the gradient m_BM

1c) Hint 8: realise that you have simultaneous equations, using your answers from parts (a) and (b)

1c) Hint 9: be sure to state the coordinates in brackets, with a comma between them.

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

Question 2

2a) Hint 1: know that f(g(x)) means to do g(x) first, then f(…) second

2a) Hint 2: replace g(x) in f(g(x)) with what g(x) is equal to

2a) Hint 3: apply function f(…) to what its input is, then expand and simplify the quadratic expression

2b) Hint 4: use a standard method to complete the square for your answer from part (a)

2c) Hint 5: know that h(x) will not be defined if its denominator has the value of zero.

2c) Hint 6: use your answer from part (b) to determine the values of x that would make f(g(x)) equal to zero.

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

Question 3

3a) Hint 1: use the formula for the t_n sequence, replacing n with 1

3b) Hint 2: work out the limit of each sequence by a standard method

3b) Hint 3: compare each limit to the number 50 to decide if either the frog or toad can escape

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

Question 4

4a) Hint 1: set f(x) = g(x) and solve for x

4b) Hint 2: realise that integration will be used

4b) Hint 3: realise that the shield is symmetrical, so we can work out the area between f(x) and h(x), and then double it to obtain the total area

4b) Hint 4: treat f(x) as the 'top function' and h(x) as the 'bottom' function

4b) Hint 5: when subtracting h(x) from f(x), be sure to include brackets so that the correct terms are all subtracted

4b) Hint 6: the limits of integration are from zero to the value you found in part(a)

4b) Hint 7: remember to double the integral answer at the end to get the total area!

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

Question 5

5a) Hint 1: complete the square in the x and y terms, or use the circle formula, to obtain the centre and radius of circle C₁

5a) Hint 2: work out the distance from the centre of C₁ to the centre of C₂

5a) Hint 3: this distance is the sum of the two circles' radii, and you know the radius of C₁, so you can work out the radius of C₂

5b) Hint 4: copy the given diagram of two circles and draw C₃ such that it has its centre on the line joining the other two circles' centres

5b) Hint 5: know that the radius of C₃ will be the sum of the diameters of C₁ and C₂

5b) Hint 6: draw another sketch of the line joining the centres of C₁ and C₂, and extend it out to show the diameters of C₁ and C₂

5b) Hint 7: know that the centre of C₃ will be in the middle of this line that the length of the two diameters

5b) Hint 8: work out the ratio of where the centre point for C₃ is relative to line segment joining the centres of C₁ and C₂

5b) Hint 9: use your ratio technique to obtain the coordinates of the centre of C₃ and therefore the equation of C₃

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

Question 6

6a) Hint 1: know that p.(q + r) = p.q + p.r

6a) Hint 2: know that the angle between vectors p and q is 60°

6a) Hint 3: use exact value triangle knowledge to give cos(60°) and know what cos(0°) is as well.

6a) Hint 4: know that |q| = |p| = 3

6b) Hint 5: look for a route from point E to point A that goes along segments that you know about

6b) Hint 6: vector EC = vector EA + vector AB + vector BC

6b) Hint 7: know that vector EA is the negative of vector AE, and that vector BC is the same as vector ED (because it is a parallelogram)

6c) Hint 8: use your answer from part (b) and a similar technique to part (a) to expand and simplify a series of scalar products involving p, q and r

6c) Hint 9: you will need to know the angle between q and r, when the vectors are placed 'tail to tail'

6c) Hint 10: you should find that you want the exact value for cos(30°)

6c) Hint 11: substituting all values into the given equation should allow you to solve an equation in |r|

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

Question 7

7a) Hint 1: realise that integrating the 3cos(2x) term will involve the 'opposite' of the chain rule

7a) Hint 2: remember to include the constant of integration

7b) Hint 3: work with the left hand side and chose an identity for cos(2x) that includes terms like those on the right hand side

7c) Hint 4: notice that the integrand is a linear multiple of the right hand side of part (b)

7c) Hint 5: rewrite the integrand in terms of the left hand side of part (b)

7c) Hint 6: use your workings from part (a) to complete the integration

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

Question 8

Hint 1: use a standard method for re-writing the given expression in the stated form

Hint 2: set the value of h in the first equation to be h = 100

Hint 3: subtract 65 from both sides of the equation and use your earlier work to re-write the equation in terms of a single sine function

Hint 4: check you have your calculator in radian mode, and take inverse sine to obtain two values for (1.5t - 0.395)

Hint 5: add 0.395 to each of your values

Hint 6: divide each of your answers by 1.5, to give the values of t required.

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