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Paper 1

Question 1

1a) Hint 1: notice that you shall have to use the chain rule for differentiation

1a) Hint 2: refer to the formula sheet, if you need to, in order to remind yourself of what the derivative of cot(x) is

1b) Hint 3: notice that you shall have to use both the product rule, and the chain rule, for differentiation

1b) Hint 4: after you have differentiated the terms, it's often a good idea to re-write each term in its simplest form, using fractions and either positive powers, or re-write terms with roots

Question 2

2a) Hint 1: consider drawing an argand diagram and plot the number 1 + i on it

2a) Hint 2: draw a right angled triangle using the origin and the point, so that you can see the argument and modulus of the number

2a) Hint 3: use Pythagoras' theorem and exact value triangles to get the values required

2a) Hint 4: write the complex number in the form r(cos(θ) + isin(θ))

2b) Hint 5: know that De Moivres' theorem says that [r(cos(θ) + isin(θ))]ⁿ = rⁿ(cos(nθ) + isin(nθ))

2b) Hint 6: carefully evaluate (√2)⁸ and decide what cos(8π/4) and sin(8π/4) are each equal to

2b) Hint 7: if successful, your final answer will be a real integer, with no imaginary part

Question 3

3a) Hint 1: know that to go from u₃ to u₅, we multiply by the common ratio, r, twice

3a) Hint 2: set up an equation in r²

3a) Hint 3: when solving for r, you ought to obtain two values (one positive, and one negative)

3a) Hint 4: notice in the question that all terms in the sequence were positive, so r itself must therefore also be positive

3b) Hint 5: know that to go from u₁ to u₃, we multiply by the common ratio, r, twice

3b) Hint 6: use the value of the ratio calculated in part (a) and the value of u₃ to create and solve an equation in u₁

3c) Hint 7: know that a sum to infinity only exists when -1 < r < 1

3c) Hint 8: clearly communicate that the value for r, from part (a), satisfies this inequality

3d) Hint 9: know the formula for S_∞ = u₁ / (1 - r)

3d) Hint 10: evaluate this formula using the values from parts (a) and (b)

Question 4

4a) Hint 1: know that the determinant of a 2×2 matrix is 'ad - bc'

4a) Hint 2: know that the inverse of a matrix involves the reciprocal of the determinant and a matrix with first row [d -b] and a second row [-c a]

4a) Hint 3: substitute in all of the values, taking care not to make any arithmetical errors

4b) Hint 4: we are told that A.M = B

4b) Hint 5: pre-multiply both sides of the equation by the inverse of A

4b) Hint 6: this gives A^-1.A.M = A^-1.B

4b) Hint 7: know that A^-1.A is equal to the identity matrix, I.

4b) Hint 8: hence M = A^-1.B and you know the matrices A^-1 and B to then substitute in

4b) Hint 9: remember when multiplying two matrices together, take care with the arithmetic involving multiplications and additions

Question 5

5a) Hint 1: know that an odd function is when f(-x) = -f(x)

5a) Hint 2: to test if the given function is odd, evaluate f(-x) and see if it can be re-written as -(x³ - x)

5a) Hint 3: you should find that it can be re-written in that way, so clearly communicate that the function is indeed odd.

5b) Hint 4: know that a point of inflexion is where f''(x) = 0, and it has a change of sign of f''(x) on either side of the inflexion point

5b) Hint 5: calculate f'(x) and f''(x) and find the value of x when f''(x) is equal to zero

5b) Hint 6: you should find that x = 0 is a possible point of inflexion

5b) Hint 7: now consider x = 0^-, which is a point just to the left of x = 0. What would the sign of f''(0^-) be?

5b) Hint 8: repreat for x = 0⁺, which is a point just to the right of x = 0

5b) Hint 9: clearly communicate that f(0^-) has a different sign to f(0⁺) and therefore x = 0 is a point of inflexion

Question 6

6a) Hint 1: know that a reflection in the x-axis means that the point (x, y) would transform to the point (x, -y)

6a) Hint 2: carefully construct a 2×2 matrix which contains 0, 1 and -1, so that when it pre-multiplies the vector with components (x y), it gives the vector with components (x -y)

6b) Hint 3: with the given matrix, pre-multiply it to the vector with components (x y) to see what vector you obtain

6b) Hint 4: you should find that the point (x, y) transforms to the point (y, x)

6b) Hint 5: to help you decide what this transformation is doing, consider drawing a diagram and map a select of random points to see where they go

6b) Hint 6: if matrix A is from part (a) and matrix B is from part (b), then applying matrix A followed by matrix B is written as C = B.A

6b) Hint 7: replace matrices B and A, then multiply them and simplify to obtain matrix C

Question 7

7a) Hint 1: consider re-writing the given expression to make clear that y is actually a function of x by writing it as y(x)

7a) Hint 2: hence the equation is now x².y(x) + 4.x.[y(x)]² = -32

7a) Hint 3: implicitly differentiate this equation, using the product rule and knowing that [y(x)]² will become 2[y(x)]¹.y'(x)

7a) Hint 4: now simplify writing the y(x) back to just y, and y'(x) back to just y', and rearrange to make y' the subject

7b) Hint 5: know that a stationary point is likely when y'(x) equals zero

7b) Hint 6: know that this will happen when the numerator of the answer to part (a) is zero

7b) Hint 7: using factorisation, obtain two different expressions that can lead to the numerator being zero

7b) Hint 8: one solution is y = 0. See if you can use the original equation to obtain a value for x.

7b) Hint 9: you should find that y = 0 does not have a corresponding solution for x, so we can discard it.

7b) Hint 10: the other solution is x + 2y = 0, so rearrange this to make x the subject and substitute it into the original equation

7b) Hint 11: you should have a cubic expression in terms of y, that can be solved for y

7b) Hint 12: knowing the value for y, use x + 2y = 0 to obtain the value for x

7b) Hint 13: clearly state the coordinates of the stationary point that you have found

Question 8

Hint 1: know that when you use a substitution, the limits will change value as well

Hint 2: evaluate u(0) and u(π/8) to obtain the two new limits

Hint 3: differentiate u with respect to x and then rearrange to made 'dx' the subject

Hint 4: proceed with the substitution of both the integrand and the limits

Hint 5: if successful, you should find yourself integrating (1/2)u^1/2, with respect to u

Hint 6: proceed with the integration and evaluation of both limits in order to obtain a final answer of a single fraction

Paper 2

Question 1

Hint 1: notice that you shall have to use both the quotient rule, and the chain rule, for differentiation

Question 2

Hint 1: perform the standard, extended Euclidean algorithm

Hint 2: your first line will read 533 = 1 × 455 + 78

Hint 3: when you start the reverse process, your first line should be 13 = 78 - 1 × 65

Hint 4: be sure to clearly state the values of integers 'a' and 'b'

Question 3

3a) Hint 1: perform the standard Gaussian elimination algorithm

3a) Hint 2: the final line of the final grid should read 0 0 λ+6 | 10

3a) Hint 3: this means that (λ + 6)z = 10, so that it can be rearranged to make z the subject

3b) Hint 4: know that an inconsistent system is one where there is no solution able to be found

3b) Hint 5: consider what value of λ would mean that z would be undefined

3c) Hint 6: for the given value of λ evaluate first z, then y, then x, using your lines of working from part (a)

3c) Hint 7: state your final solution as a set of (x, y, z) coordinates

Question 4

Hint 1: perform a standard solution of a second order differential equation

Hint 2: your auxiliary equation should give two solutions of -2 and 4

Hint 3: to fix the values of the constants that you've likely called A and B, you will need to evaluate y(0) and y'(0)

Hint 4: you will obtain two simulteanous linear equations in A and B that can be solved.

Hint 5: be sure to write the full solution function out at the end

Question 5

5a) Hint 1: write down that (a + b)ⁿ has a general term of ⁿC_r.a^r.b^n-r

5a) Hint 2: replace the n with 16, replace the a with (2x²) and replace the b with (-x^-3)

5a) Hint 3: carefully expand the brackets so that you can separate out a single x term that has a power that's in terms of r

5b) Hint 4: know that 1/x¹⁸ is x^-18

5b) Hint 5: equate the -18 to the expression in terms of r that is the power of x from your part (a)

5b) Hint 6: solve for r

5b) Hint 7: substitute the value for r back into the full expression from part (a)

5b) Hint 8: for the final answer, write down just the numercial value as the coefficient of the requested term, and do not include the variable x

Question 6

6a) Hint 1: differentiate each of x(t) and y(t) with respect to t, using the product rule where required

6a) Hint 2: know that dy/dx = dy/dt × dt/dx

6a) Hint 3: know that dt/dx is the reciprocal of dx/dt

6a) Hint 4: substitute in the expressions for dy/dt and dx/dt and tidy up

6b) Hint 5: know that d²y/dx² = d/dx of dy/dx

6b) Hint 6: and that d²y/dx² = d/dt [ dy/dx ] × dt/dx

6b) Hint 7: carefully substitute in all the expressions from part (a), taking care with all of the fractional terms

6b) Hint 8: use the quotient rule when evaluating d/dt [ dy/dx ]

Question 7

7a)i) Hint 1: know that to obtain the terms up to x³, you will need f(x), f'(x), f''(x) and f'''(x)

7a)i) Hint 2: obtain expressions for each of the derivatives (using the chain rule) and evaluate each of them when x = 0

7a)i) Hint 3: refer to the formula sheet to then construct the Maclaurin expansion

7a)ii) Hint 4: repeat the same procees as done in (a)(i)

7b) Hint 5: notice that what's being asked for is the same as in part (a)(i) but the 'x' has been replaced with sin(3x)

7b) Hint 6: so, write out the answer from part (a)(i) but write sin(3x) instead of each x

7b) Hint 7: now replace each sin(3x) with the answer from part (a)(ii)

7b) Hint 8: when you expand each set of brackets, you can discard any terms that have x⁴ or higher powers

7b) Hint 9: simplify the remaining terms into a cubic in terms of x

Question 8

Hint 1: know that a volume of revolution involves integrating πy²

Hint 2: write out the integral with limits from 0 to 'a', and equate it to π²/3

Hint 3: refer to the formula sheet, if you need to, in order to remind yourself of what the integral of 1/(1 + x²) is

Hint 4: know that tan^-1(0) = 0

Hint 5: draw out an exact value triangle to help determine 'a' where tan^-1(a) = π/3

Question 9

9a) Hint 1: know that u_n = a + (n - 1)d

9b) Hint 2: use the previous hint to write down expressions in terms of 'd' for u₃ and u₈

9b) Hint 3: now substitute those expressions into the equation u₈ = 5 u₃

9b) Hint 4: solve for d

9c) Hint 5: refer to the formula sheet, if you need to, in order to remind yourself of the formula for the sum of the first n terms in an arithmetic sequence

9c) Hint 6: replace 'a' and 'd' in this formula with the values that you already know

9c) Hint 7: construct an inequality for this formula, which is now in terms of 'n', to be > 500

9c) Hint 8: expand and rearranged this inequality to be in the form: .. .. .. > 0

9c) Hint 9: this quadratic does not easily factorise, so use the quadratic formula to find two roots

9c) Hint 10: notice that we are interested in positive values of n, so we can discard the negative solution

9c) Hint 11: the value for n is not an integer, so try an appropriate integer value of n that would lead to the inequality being satisfied

9c) Hint 12: it would be good to then check whether that integer value does indeed satisfy the inequality

Question 10

Hint 1: in the formula given, know that V is actually V(t) and r is actually r(t) as they each depend upon time

Hint 2: write out the formula emphasising this: V(t) = 5 π [r(t)]³

Hint 3: implicitly differentiate this formula with respect to 't', making use of the chain rule

Hint 4: check that [r(t)]³ is differentiated to give 3[r(t)]²r'(t)

Hint 5: know that we are after the value of dr/dt when r = 10

Hint 6: we know that dV/dt = 12

Hint 7: replace dV/dt and r with their values, and rearrange to make dr/dt the subject

Hint 8: put in the units for dr/dt which are mm/min, as this shows you are aware of the context and the units used for each of r and t.

Question 11

Hint 1: to prove that something is 'always prime' is going to be very challenging

Hint 2: hence, we already suspect that statement A is false

Hint 3: therefore, look for a counter-example with some relatively small numbers

Hint 4: when you find the counterexample, make sure you clearly state that A is false by counterexample

Hint 5: know that the general form of two consecutive integers is n and (n + 1), where n is an integer

Hint 6: after squaring each of these and adding them together, consider what you are aiming for

Hint 7: know the general form of an odd number is 2m + 1, where m is an integer.

Hint 8: try to reorganise your expression that's in terms of n, into the form of 2(..) + 1

Hint 9: be sure to communicate that what you have in the brackets is also an integer

Hint 10: clearly communicate that you have proven B to be true

Hint 11: make sure that all of your variable letters have been defined as integers and that you've use the phrase 'general form of' where appropriate

Question 12

Hint 1: know that z̄ = x - iy

Hint 2: in the given equation, substitute the terms in z and z̄ for x + iy and x-iy, respectively

Hint 3: after expanding brackets and simplifying you should have an equation in x and y with both real and imaginary terms

Hint 4: create a new equation by equating the real parts

Hint 5: create a new equation by equating the imaginary parts

Hint 6: look at these two equations and decide which one you might be able to solve most easily

Hint 7: factorise your chosen equation and write down the equations that ought to solve it

Hint 8: re-read the question to check on any conditions on x or y that must be taken into consideration

Hint 9: if everything has gone well, you ought to have a single solution of x = 10

Hint 10: substitute this value into the equation that arose from the real parts, to create a quadratic equation in terms of y

Hint 11: carefully solve this equation to give two integer values for y

Hint 12: present a final answer of the two values of z that satisfy the original equation in the question

Question 13

13a) Hint 1: use your standard partial fraction process to obtain numerators of the two fractions that have the denominators of x and (x + 1)

13b) Hint 2: use your standard integration by parts process, and don't forget the constant of integration

13c) Hint 3: recognise that solving this equation will require an integrating factor

13c) Hint 4: identify the P(x) and Q(x) functions

13c) Hint 5: integrating P(x) will require use of your workings from part (a)

13c) Hint 6: arrange the answer of this integral to be a natural logarithm of a single expression

13c) Hint 7: hence state the integrating factor to be e to the power of the natural logarithm

13c) Hint 8: this can then be simplified to give an integrating factor of [ (x + 1) / x]²

13c) Hint 9: multiply the original differential equation through by the integrating factor

13c) Hint 10: the right hand side of the differential equation should simplify to something similar to part (b)

13c) Hint 11: the left hand side of the differential equation will be the derivative of a product of terms involving x and y

13c) Hint 12: integrate both sides of the differential equation, allowing you to use your answer from part (b)

13c) Hint 13: rearrange to make y the subject, and make sure your answer still has a constant of integration in it

Question 14

14a)i) Hint 1: know that vector AB = vector OB - vector OA, and similarly for vector AC

14a)ii) Hint 2: know that we want the vector that is perpendicular to vector AB and vector AC

14a)ii) Hint 3: realise that a vector cross product will deliver this for us

14a)ii) Hint 4: calculate AB × AC, by your preferred method

14a)ii) Hint 5: factorise the final vector answer as we shall only need to use the vector part, and not the scalar multiple

14a)ii) Hint 6: know that a cartesian equation will come from using this simplified normal vector, and the coordinates of any point in the plane

14a)ii) Hint 7: decide whether to use point A, point B or point C, as they all lie in the plane

14a)ii) Hint 8: know that the scalar product of the normal with vector (x y z) will be the same as the scalar product of the normal with any of vectors OA, OB, or OC

14a)ii) Hint 9: calculate these two scalar products and equate them, to give the cartesian equation of the plane

14b) Hint 10: know that a more useful form of the equation of the line is to have it in parametric form, not cartesian form

14b) Hint 11: after writing it in parametric form, extract expressions for each of x, y and z in terms of your chosen parameter letter

14b) Hint 12: substitute these expressions into the cartesian equation of the plane, from part (a)(ii)

14b) Hint 13: attempt to solve for your parameter letter, but you will find that it doesn't work

14b) Hint 14: this non-solution is evidence that the line does not intersect the plane

14b) Hint 15: (this means that the line's direction is parallel to the plane)

14b) Hint 16: make sure that you clearly communicate your conclusion about the line and the plane

Question 15

15a) Hint 1: recognise that that differential equation will involve the technique of separation of variables

15a) Hint 2: after separating variables, integrate each side with respect to its variable

15a) Hint 3: remember to introduce a constant of integration to one side

15a) Hint 4: it is often best to fix the value of this constant as soon as it appears

15a) Hint 5: use the information that at t = 0, W = 8 in order to fix the value of the constant

15a) Hint 6: with the constant value discovered, substitute back into your general solution and rearrange to make W the subject

15a) Hint 7: this will involve re-writing logarithmic expressions as exponential expressions

15b) Hint 8: know that the phrase 'the rate of' means the derivative of W with respect to time

15b) Hint 9: the equation that we were given for dW/dt was in terms of W

15b) Hint 10: use your answer from part (a) to calculate W when t = 67

15b) Hint 11: substitute that value into the original equation to calculate dW/dt at t = 67

15b) Hint 12: put in the units of dW/dt which are kg/min, based on the units of W and the units of t

15c) Hint 13: know that a limit occurs when the value of time, t, becomes very large

15c) Hint 14: look at the answer from part (a), and consider what will happen to the term in t, when t becomes infinitely large

15c) Hint 15: use this knowledge to think what the value of W will therefore be, when t tends to infinity.

15c) Hint 16: note, however, in the question it was stated that W < 36, so it can never actually take on the value of the limit