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

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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

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

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

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

Question 3

Hint 1: write down what z̄_{2} 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

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

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

Hint 14: and here is a video of the solution:

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
x^{4} or greater

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

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

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

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

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

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

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

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

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

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'

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

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!

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

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

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

Question 6

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

6a) Hint 2: define u_{1} = x+ 5, u_{2} = 3x+ 2 and u_{3}
= 5x - 1

6a) Hint 3: calculate u_{2} - u_{1}

6a) Hint 4: calculate u_{3} - u_{2}

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 S_{n}, substituting in a =
x + 5, d
= 2x - 3 and n = 15

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

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

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

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 z_{1} is a root, then z_{1}² -
6z_{1} + 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'

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

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

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

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

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

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

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

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

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

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

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

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)

Hint 15: and here is a video of the solution:

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