Hints offered by N Hopley

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Paper 1

Question 1

1a) Hint 1: recognise that the quotient rule is needed

1b) Hint 2: recognise that the chain rule is needed

1b) Hint 3: also consider rewriting cosec(5x) and cot(5x) in terms of sin(5x) and cos(5x), to help with the simplification of terms

Question 2

Hint 1: use the standard method of gaussian elimination

Hint 2: the final row of your augmented matrix that should be 2z = 0

Hint 3: solve for z, and calculate the corresponding values for y and x

Question 3

Hint 1: write down what z̄2 is

Hint 2: multiply the two complex numbers together, taking care with the signs for the i² term

Hint 3: check that you give your final answer in the requested form

Question 4

4a) Hint 1: know that implicitly differentiating y³ will involve the chain rule, to give 3y².y'

4a) Hint 2: know that implicitly differentiating x.y will involve the product rule to give 1.y + x.y'

4a) Hint 3: gather the y' terms on one side of the equation, and the non-y' terms on the other side

4a) Hint 4: factorise out y' from all terms, and then divide to make y'(x) the subject

4b) Hint 5: realise that you need both x and y values in order to evaluate y'(x) at the point of interest

4b) Hint 6: substitute y = -1 into the original equation and solve for x

4b) Hint 7: substitute the values of x and y into the expression for y'(x) from part (a)

4c) Hint 8: know that a stationary point happens with y'(x) = 0

4c) Hint 9: set the expression from part (a) to equal zero

4c) Hint 10: know that this will only be solved when the numerator is equal to zero (and the denominator is a finite value)

4c) Hint 11: now substitute the obtained value of y into the original equation to see what happens

4c) Hint 12: make an appropriate conclusion about the validity of the value of y from what you have just discovered

4c) Hint 13: interpret what this means for the existance of a stationary value

Question 5

5a) Hint 1: use the standard method for calculating each term of a Maclaurin Series, using the chain rule at each stage

5b) Hint 2: rewrite the given expression as (3 + 2x)e-4x

5b) Hint 3: replace e-4x with the expression from part (a)

5b) Hint 4: multiply out the two brackets, discarding any terms that have order x4 or greater

Question 6

6a) Hint 1: think of an odd number and substitute it into the expression, to see if it gives a prime number, or not

6a) Hint 2: when you find an odd number that generates a prime number, then that is the counterexample you needed to find

6a) Hint 3: clearly demonstrate that substitution of your found odd number into the expression does not give a prime number

6a) Hint 4: clearly communicate that this is the desired counter-example

6b) Hint 5: write down 'consider a general pair of consecutive integers to be n and (n + 1)'

6b) Hint 6: calculate the difference between their cubes, which is (n + 1)³ - n³

6b) Hint 7: expand out the brackets and simplify

6b) Hint 8: factorise out 3 from as many terms as you can, leaving a '+1' at the end

6b) Hint 9: clearly communicate that this is an expression of the general form that is 1 more than a multiple of 3

6b) Hint 10: clearly communicate a final statement about what you have just proven

Question 7

7a) Hint 1: use a standard method of integration by substitution to simplify the integral to a simple polynomial in u, with u limits

7b) Hint 2: carefully read the information in the question about symmetry about the y-axis

7b) Hint 3: realise that the area of the full cross-section will be 2 times the definite integral that was calculated in part (a)

7c) Hint 4: re-write the numerator of y² as y² + 1 - 1

7c) Hint 5: now expand the single fraction into the addition of two fractions: one has numerator of y² + 1, the other has numerator -1

7c) Hint 6: simplify the first fraction to give the value of 'a'

7d) Hint 7: know that the volume of revolution is the definite integral between 0 and 5 of π x² dy

7d) Hint 8: simplify the integrand so that it aligns with the structure of part (c)'s original expression

7d) Hint 9: use the fomula sheet, if required, to correctly integrate 1/(y² + 1) to give an inverse trigonometric function

7d) Hint 10: evaluate the integral leaving it in exact value form

Paper 2

Question 1

Hint 1: the denominators of the two fractions will be x and (x² + 5)

Hint 2: the numerators of the two fractions with be A and (Bx + C)

Hint 3: proceed with a standard method for partial fractions, clearly presenting your final answer

Question 2

Hint 1: recognise that this integral will introduce natural logs during the integration process

Hint 2: check your answer by differentiating back, remembering to use the chain rule

Hint 3: consider whether modulus signs are really needed, given the expression inside the log and the limits of x

Hint 4: remember that ln(1) is equal to zero

Question 3

Hint 1: use the standard method of the extended Euclidean Algorithm

Hint 2: the first line is 634 = 7 × 87 + 25

Hint 3: at the end, clearly communicate the values of integers 'a' and 'b'

Question 4

Hint 1: use the standard method of integration by parts

Hint 2: the expression (x + 2) is what you will differentiate, whilst (2x + 7)1/2 is what will be integrated

Hint 3: don't forget the constant of integration, and the 'dx' terms in each integrand!

Question 5

Hint 1: know that a square matrix is singular if it does NOT have an inverse

Hint 2: know that a matrix that does not have an inverse, is a matrix whose determinant is equal to zero

Hint 3: use a standard method to calculate the determinant of matrix A

Hint 4: set the quadratic expression for the determinant in terms of k to be equal to zero

Hint 5: factorise and solve the quadratic equation to give two values of k

Question 6

6a) Hint 1: know that an arithmetic sequence has a constant difference between successive terms

6a) Hint 2: define u1 = x+ 5, u2 = 3x+ 2 and u3 = 5x - 1

6a) Hint 3: calculate u2 - u1

6a) Hint 4: calculate u3 - u2

6a) Hint 5: clearly communicate that the two expressions for the differences are equal and so the sequence is arithemetic

6b) Hint 6: use the standard formulae for Sn, substituting in a = x + 5, d = 2x - 3 and n = 15

6c) Hint 7: use the standard formulae for Sn, substituting in n = 20, S20 = 1130, a = x + 5, and d = 2x - 3

6c) Hint 8: solve the resulting equation for a value of x

Question 7

7a) Hint 1: know that a polynomial with real coefficients will have roots that come in complex conjugate pairs

7a) Hint 2: write down the complex conjugate of 3 + i

7b) Hint 3: know that if z1 is a root, then z1² - 6z1 + a = 0

7b) Hint 4: substitute 3 + i into the quadratic expression, and solve for 'a'

7c) Hint 5: know that if the quadratic expression is a factor of the cubic expression, then it will have a remainder of zero after polynomial division

7c) Hint 6: perform polynomial long division to solve the remainder to be zero, and obtain the value of 'b'

Question 8

8a) Hint 1: know that the product rule will be needed

8b) Hint 2: recognise the format of the differential equation will require solving using an integrating factor

8b) Hint 3: set P(x) = ln(x) and Q(x) = x-x

8b) Hint 4: use the calculations from part (a) when integrating P(x)

8b) Hint 5: complete a standard integrating factor method

Question 9

Hint 1: verify the statement with a base case, where n = 1

Hint 2: write down what the statement would look like for the (n+1) case, as this helps you know what you are aiming for

Hint 3: proceed with a standard method of proof by induction

Hint 4: write in words at the end the clear logic behind how the base case and the induction step show that the statement is true for all positive natural numbers

Question 10

Hint 1: obtain the auxiliary equation

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

Hint 3: know the format of the complementary function when there are two equal roots

Hint 4: know the format of the particular integral that matches the form of the right hand side of the original differential equation

Hint 5: fix the values of the constants in the particular integral by calculating y' and y'', and substituting into the original differential equation

Hint 6: construct a final solution of the complementary function (and its two undetermined constants) plus the particular integral

Hint 7: use the initial conditions to fix the values of the two constants in the complementary function part of the function

Hint 8: present a final solution, with all constants determined

Question 11

Hint 1: know that dy/dx = (dy/dt).(dt/dx)

Hint 2: calculate dx/dt through using the chain rule

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

Hint 4: substitute both dy/dx and dt/dx into dy/dx = (dy/dt).(dt/dx)

Hint 5: solve for dy/dt

Hint 6: solve for y, remembering the constant of integration

Hint 7: use the initial condition of t = 1 and y = 5 to fix the value of the unknown constant

Hint 8: present the final function of y in terms of t

Question 12

12a) Hint 1: know how use De Moivre's Theorem

12b) Hint 2: write out the binomial expansion for (a + b)4 and then replace 'a' with (cos(θ)) and replace 'b' with (isin(θ))

12b) Hint 3: take care when simplifying the terms involving powers of i

12c)i) Hint 4: put the two answers from part (a) and part (b) together

12c)i) Hint 5: notice that the cos(4θ) terms are not imaginary

12c)i) Hint 6: take the real parts of both sides of the equation that you formed

12c)i) Hint 7: replace all of the sin²θ terms with (1 - cos²θ)

12c)ii) Hint 8: know that cot(4θ) = cos(4θ)/sin(4θ)

12c)ii) Hint 9: know that you can obtain an expression for sin(4θ) by taking imaginary parts of equation, in a similar way to that already done in part (c)(i)

12c)ii) Hint 10: again, replace all of the sin²θ terms with (1 - cos²θ)

12c)ii) Hint 11: gather and simplify the expressions for the numerator and denominator

Question 13

13a)i) Hint 1: recognise that dθ/dt is the rate of change of radians per second

13a)i) Hint 2: know that completing 1 revolution in 12 seconds is the same as completing 2π radians in 12 seconds

13a)i) Hint 3: proportionally scale this down to give the number of radians turned in just 1 second

13a)ii) Hint 4: from the diagram, notice that tan(θ) = x/10

13a)ii) Hint 5: rearrange this to give x in terms of &theta

13a)ii) Hint 6: calculate dx/dθ

13a)ii) Hint 7: know that dx/dt = (dx/dθ).(dθ/dt)

13a)ii) Hint 8: substitute in your answer from (a)(i) and your last calculation to obtain dx/dt

13b) Hint 9: write down the identity that sin²θ + cos²θ ≡ 1

13b) Hint 10: divide this whole identity through by cos²θ

13c) Hint 11: notice that knowing the value of x allows you to know the value of tan(θ)

13c) Hint 12: this in turn allows you to know tan²θ

13c) Hint 13: this in turn allows you to know sec²θ

13c) Hint 14: this can finally be used to evaluate the expression from (a)(ii)


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