Question Paper 1 Question Paper 2 Marking Instructions Paper 1 Marking Instructions Paper 2

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

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Paper 1

Question 1

1a) Hint 1: recognise that you need to use the product rule and the chain rule

1b) Hint 2: recognise that the quotient rule is needed along with a standard differential from the Formula List

Hint 3: and here is a video of the solution:

Question 2

2a) Hint 1: write out the 2×2 matrix and the 2×3 matrix that will be multiplied together

2a) Hint 2: write out the 6 different calculations for the resulting 2×3 matrix. It's easy to make an arithmetical error if you rush this!

2b) Hint 3: know the standard format for the inverse of a 2×2 matrix

2b) Hint 4: identify the valus of a, b, c, d and calculate the determinant

2b) Hint 5: the inverse matrix is the reciprocal of the determinant multiplied by a 2×2 matrix

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

Question 3

Hint 1: knowing that u = sin(θ), differentiate this to give du/dθ

Hint 2: rearrange this to make dθ the subject, treating the du/dθ as if it were a fraction (but it's not really a fraction!)

Hint 3: write out the integral, replacing sin(θ) with 'u' and dθ with the expression you just obtained

Hint 4: after simplification, you should have an integral that is only in terms of 'u'

Hint 5: after integrating, do not forget the constant of integration

Hint 6: finally, substitute sin(θ) for u

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

Question 4

Hint 1: use a standard Gaussian elimination method to obtain a third row which has two zeros in i

Hint 2: know that no solutions will come from a third row equation which has the form of 0 × z equal to a non-zero value

Hint 3: your coefficient of z ought to be an expression in λ, so solve for this coefficient to be zero

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

Question 5

Hint 1: know the standard integrand required is πy²dx to calculate a volume of revolution about the x-axis

Hint 2: be careful with brackets to ensure that you do (2√x)² and not 2(√x)²

Hint 3: and here is a video of the solution:

Question 6

6a) Hint 1: know that displacement comes from integrating velocity, with respect to time

6a) Hint 2: after integrating, fix the constant of integration knowing that at t = 0, displacement = 0

6b) Hint 3: know that acceleration is the derivative of velocity, with respect to time

6b) Hint 4: evaluate the expression for acceleration when t = 0, remembering to give the units of acceleration

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

Question 7

7a)i) Hint 1: know that a vertical asymptote exists when the denominator of an expression has the value of 0

7a)ii) Hint 2: notice that the expression for f(x) is 'top heavy' in terms of the powers of x

7a)ii) Hint 3: use polynomial long division, synthetic devision or algebraic manipulation to divide x² by (x - 2)

7a)ii) Hint 4: the equation of the non-vertical asymptote will be the part of f(x) that remains when the fractional term tends to zero as x tends to ±∞

7a)ii) Hint 5: be sure to clearly state the equation of the asymptote and why the fractional term can be discarded

7b) Hint 6: draw in dotted lines for the two asymptotes

7b) Hint 7: plot the stationary points (0, 0) and (4, 8)

7b) Hint 8: by considering the locations of the points relative to the asymptotes, decide whether these stationary points are minimums or maximums or inflexions

7b) Hint 9: sketch the two parts of the function, making sure that the four 'end points' converge on the asymptotes

7c)i) Hint 10: know that the |f(x)| returns only positive values

7c)i) Hint 11: sketch in the asymptotes, as before, and reflect everything that was below the x-axis to above the x-axis

7c)ii) Hint 12: imagine a horizontal line with equation y = k being drawn on your diagram, and this line can move up and down the y-axis

7c)ii) Hint 13: if this line is too high, it will intersect the function in 4 places; and if it's too low, it will not intersect the function at all

7c)ii) Hint 14: hence determine from the coordinates of (0, 0) and (4, 8) what the maximum and minimum values of k could be

7c)ii) Hint 15: be very careful when deciding whether to use ≤ or <

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

Question 8

Hint 1: obtain the auxiliary equation

Hint 2: solve the auxiliary quadratic equation to give two distinct roots

Hint 3: write down the complementary function, using constants A and B

Hint 4: decide on the form of the particular integral, knowing that it cannot be the same as either of the terms in the complementary function

Hint 5: differentiate the particular integral twice, substitute it into the differential equation and determine its constant multiplier

Hint 6: write down the general solution = complementary function + particular integral

Hint 7: use the information provided about y and dy/dx to determine the values of the constants A and B

Hint 8: present a final answer for y that has 3 terms and all constants evaluated as numerical values.

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

Paper 2

Question 1

Hint 1: recognise that you need to use the chain rule

Hint 2: use exact value triangles to evaluate sec(π/4) and tan(π/4)

Hint 3: and here is a video of the solution:

Question 2

2a) Hint 1: perform the standard, extended Euclidean Algorithm, starting with 105 = 1 × 72 + 33

2b) Hint 2: recognise that this equation is the equation from part (a) multipled by 120

Hint 3: and here is a video of the solution:

Question 3

Hint 1: perform a standard integration by parts process, once

Hint 2: do not forget the constant of integration!

Hint 3: and here is a video of the solution:

Question 4

4a) Hint 1: calculate dx/dt and dy/dt using the chain rule and standard differentials from the formula sheet

4b) Hint 2: know that dy/dx = dy/dt × dt/dx

4b) Hint 3: know that dt/dx is the reciprocal of dx/dt

4b) Hint 4: evaluate dy/dx when t = 0 to obtain the gradient of the function at that point

4b) Hint 5: evaluate x(0) and y(0) to obtain the coordinates of the point that the tangent goes through

4b) Hint 6: calculate the equation of the line through the calculated point, with the calculated gradient

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

Question 5

5a) Hint 1: notice that A⁴ = A².A²

5a) Hint 2: replace A² with 2A + 5I, and expand brackets

5a) Hint 3: repeat, replacing A² with 2A + 5I

5b) Hint 4: starting with A² = 2A + 5I, pre-multiply each term by A^-1

5b) Hint 5: rearrange equation to make A^-1 the subject

5b) Hint 6: the values of 'r' and 's' will be fractions of different signs

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

Question 6

Hint 1: recognise that this standard differential equation will require an integrating factor

Hint 2: identify P(x), integrate it and take the exponential of it

Hint 3: multiply the differential equation through by the integrating factor

Hint 4: proceed to rearrange and gather terms, and then integrate, including a constant of integration

Hint 5: use the information provided about x = 0 and y = 3 to fix the value of the constant

Hint 6: now rearrange to make y the subject

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

Question 7

7a) Hint 1: know that (x + y)³ = x³ + 3x²y + 3xy² + y³

7a) Hint 2: take care when evaluating the i² and i³ terms

7b) Hint 3: replace z with a + 2i in the given equation

7b) Hint 4: use the answer from part (a) to help with the simplification

7b) Hint 5: evaluate the real parts of the equation, and evaluate the imaginary parts of the equation

7b) Hint 6: one should give you the value of a, which can then be used to evaluate b in the second equation

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

Question 8

8a) Hint 1: recognise that you will have to differentiate implicitly

8a) Hint 2: know that y is really y(x), i.e. y is a function of x.

8a) Hint 3: after differentiating implicitly, gather y' terms together on one side of the equation

8a) Hint 4: factorise out y' from the terms and then divide by the coefficient of y'

8b) Hint 5: write down that a stationary point is when y'(x) = 0

8b) Hint 6: realise that the answer from part (a) can only equal zero when its numerator equals zero

8b) Hint 7: solving the numerator = 0 equation, this gives a possible value for x and a possible value for y

8b) Hint 8: explore each of these values by substituting them back into the original equation, to determine their corresponding solutions

8b) Hint 9: you should find that one route gives a solution for (x, y) whilst the other does not

8b) Hint 10: make sure that you clearly communicate the reason for rejecting the 'non-solution'

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

Question 9

9a) Hint 1: use a standard method to express the given rational expression in partial fractions

9b) Hint 2: recognise that the differential equation can be solved by the method of separation of variables

9b) Hint 3: recognise that the workings from part (a) can help when integrating one side of your equation

9b) Hint 4: use the information about t and P to fix the value of the constant of integration. Note that P is a value between 0 and 5.

9b) Hint 5: there are several lines of working now required to make P the subject. Take it slowly and carefully.

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

Question 10

Hint 1: perform a standard proof by induction

Hint 2: and here is a video of the solution:

Question 11

11a)i) Hint 1: work out the difference, d₁ = u₂ - u₁

11a)ii) Hint 2: work out the difference, d₂ = u₃ - u₂

11a)ii) Hint 3: if d₁ = d₂ then this allows you to work out the value of x

11b)i) Hint 4: know that u_n = a + (n -1)d

11b)i) Hint 5: rewrite this formula with n = 21

11b)i) Hint 6: replace u₂₁, n and d with their values, to then solve for a

11b)ii) Hint 7: use u_n = a + (n -1)d, using the values for a and d that are now known

11c) Hint 8: work out r₁ = u₂/u₁ and r₂ = u₃/u₂

11c) Hint 9: equating r₁ = r₂ will create an equation in terms of y

11c) Hint 10: solve this quadratic equation to obtain two values for y

11c) Hint 11: for each value of y, calculate the common ratio, r

11d)i) Hint 12: know that -1 < r < 1 for the sum to infinity to exist

11d)ii) Hint 13: with the value of r from part (d)(i), use the formula for S_∞ to calculate the value for a

11d)ii) Hint 14: using this value of a, along with the value for r, generate the first four or five terms of the geometric sequence

11d)ii) Hint 15: realise that the value for r that has been used has come from a value of y, in part (c)

11d)ii) Hint 16: using that value of y, generate the first three terms of the geometric sequence, using y - 1, y - 7, and 2y - 9

11d)ii) Hint 17: compare these two sequences of numbers to determine whether 64/3 is a possible value, or not

Hint 18: and here is a video of the solution:

Question 12

12a) Hint 1: work out vector AB and vector AC

12a) Hint 2: perform the vector cross product on AB and AC to obtain the normal vector of plane π₁

12a) Hint 3: use any of the coordinates of points A, B or C to determine the constant for the equation of plane π₁

12b) Hint 4: know that if planes are parallel, then they will have the same normal vector

12b) Hint 5: use the coordinates of the origin to determine the constant for the equation of plane π₂

12c)i) Hint 6: to hel visualise the situation, sketch a diagram of a circle with a diameter and points A and Q at either end of that diameter (a 2D sketch will be okay to use here)

12c)i) Hint 7: draw tangent lines to the circle through points A and Q

12c)i) Hint 8: realise that line AQ has direction vector equal to the normal vector(s) of the planes

12c)i) Hint 9: chose either point A or point Q for the parametric equation of line AQ to go through

12c)ii) Hint 10: point Q lies on the intersection of line AQ and plane π₂

12c)ii) Hint 11: substitute in the parametric expressions for x, y and z into the equation of plane π₂, and solve for the parameter variable

12c)ii) Hint 12: evaluate the coordinates for Q using the equation of line AQ and the parameter value

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

Question 13

13a) Hint 1: consider sketching an Argand diagram, plotting the real number -1 on it, and thinking about the argument of that location

13b) Hint 2: either just substitute z₁ into the equation z⁵ + 1 = 0 to show that it works or . . .

13b) Hint 3: . . . anticipate the remainder of the question and do a full solution for all five roots of -1, which will support the working for parts (b), (c) and (d) all at once!

13c) Hint 4: use either a full solution of all roots, or use the geometry of the diagram knowing that all roots will be π/5 radians apart, around the unit circle

13d) Hint 5: use either a full solution of all roots, or use the geometry of the diagram knowing that all roots will be π/5 radians apart, around the unit circle

13e) Hint 6: re-write any of the values for z₃, z₄ or z₅ so that they have positive angles inside the trigonometric terms

13e) Hint 7: notice that some of the solutions for z have equivalent imaginary terms, but they only differ by a sign

13e) Hint 8: hence adding the five solutions together will result in the imaginary terms cancelling out

13e) Hint 9: also note that sin(π) equals 0, and cos(π) equals -1

13e) Hint 10: carefully gather and rearrange the remaining numerical and real terms to give the required equation

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