Hints offered by N Hopley

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Paper 1

Question 1

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

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

1a) Hint 3: recognise that a standard differential for the inverse of tan is needed

1b) Hint 4: recognise that the quotient rule is needed

1b) Hint 5: don't expand the denominator. Simplify the numerator to a single term

1c) Hint 6: determine dx/dt and dy/dt

1c) Hint 7: know that dy/dx = (dy/dt) × (dt/dx)

1c) Hint 8: know that dt/dx = 1/(dx/dt)

Question 2

2a) Hint 1: identify that u2 = 108 and u5 = 4 from the question

2a) Hint 2: determine how many times u2 is multiplied by r to arrive at u5

2b) Hint 3: know the condition for the sum to infinity to exist is that -1 < r < 1

2c) Hint 4: know the formula for the sum to infinity is a/(1-r)

2c) Hint 5: from knowing r and u2, work out what the first term, a, is

2c) Hint 6: substitute in the values of a and r into the sum to infinity formula

Question 3

Hint 1: consider writing down the general rth term for (a+b)^13

Hint 2: replace 'a' with (3/x) and 'b' with (-2x)

Hint 3: simplify the general term to obtain an expression that has a single x term to the power of a linear function of r

Hint 4: equate the linear function of r to be 9, to determine the value of r

Hint 5: evaluate the coefficient of x^9 for the found value of r

Question 4

Hint 1: use the standard method of Gaussian Elimination

Hint 2: know that 'redundancy' means that the system of equations has an infinite number of solutions

Hint 3: think what the value of λ must be in order to create an equation in z which would mean that z could be any real number

Question 5

Hint 1: verify the statement for the base case of n = 1

Hint 2: consider writing down the statement for Σ summing from 1 to (n+1), to know what you are aiming for

Hint 3: proceed with standard proof by induction process

Hint 4: be sure to write a clear final sentence that communicates the correct logic behind the induction process

Question 6

Hint 1: repeatedly differentiate sin(3x) to obtain first, second and third derivatives, using the chain rule

Hint 2: repeatedly differentiate exp(4x) to obtain first, second and third derivatives, using the chain rule

Hint 3: evaluate each of these derivatives when x = 0

Hint 4: use the Macluarin expansion formula from page 2 of the question paper to generate two separate series for the two functions

Hint 5: multiply the two series together, discarding terms that generate any term with power greater than 3

Hint 6: your final answer should have three distinct terms

Question 7

7a) Hint 1: know that the determinant of a standard 2x2 matrix is (ad - bc)

7b) Hint 2: calculate A² by A × A

7b) Hint 3: create the equation pA + qI = A²

7b) Hint 4: equate corresponding components of the matrix equation to determine p and q

7c) Hint 5: recognise that you can use A² × A²

7c) Hint 6: make use of the answer from part (b) for A²

7c) Hint 7: simplify the final answer into the form rA + sI, where r and s are integers

Question 8

8a) Hint 1: ensure your diagram has its axes fully labelled and it's clear what coordinates have been used

8b) Hint 2: know that polar form requires calculating the magnitude and the argument

8b) Hint 3: use Pythagoras' Theorem to calculate the magnitude of z and thus deduce the magnitude of w

8b) Hint 4: use trigonometry to calculate the argument of z and thus deduce the argument of w

8b) Hint 5: check that the argument of the complex number z is negative, as arguments are measured anti-clockwise from the real axis

8c) Hint 6: knowing the polar expression for w, use De Moivre's Theorem to work out w^8

8c) Hint 7: after using the 8 within the polar expression, use trigonometric exact values to convert polar form to cartesian form

Question 9

Hint 1: know to use integration by parts

Hint 2: choose the strategy where ln(x) is differentiated, rather than integrated

Hint 3: remember to use the chain rule to differentiate [ln(x)]²

Hint 4: you will need to use integration by parts a second time when the integrand again has ln(x)

Hint 5: again, choose to differentiate ln(x), rather than integrate that term

Hint 6: don't forget the constant of integration

Question 10

10A) Hint 1: recognise that this statement is likely to be false

10A) Hint 2: test the statement for p = 2, 3, 5, 7, 11, ... , until you find a counterexample

10B) Hint 3: anticipate that this statement is likely to be true

10B) Hint 4: know that a positive integer with remainder 1 when divided by 3 can be written as 3k+1, where k is an integer

10B) Hint 5: expand and simplify (3k + 1)³

10B) Hint 6: try re-writing the simplifed expression in the form of 3(… … …) + 1

Question 11

Hint 1: know that a cube of side length h has volume, V = h³

Hint 2: implicitly differentiate V = h³ with respect to time, t

Hint 3: recognise that the question told us that dh/dt = 5

Hint 4: evaluate dV/dt when h = 3

Question 12

12a) Hint 1: note that the value of c is positive, from the diagram

12a) Hint 2: know that y = f(x) - c has translated f(x) down by distance c

12a) Hint 3: know that the absolute value function |…| only returns positive values

12a) Hint 4: realise that the graph of |f(x) - c| must only exist above the x-axis

12a) Hint 5: reflect the graph of y = f(x) - c in the x-axis

12b) Hint 6: know that 2f(x) has 'stretched' the graph of y = f(x) away from the x-axis by a factor of 2

12b) Hint 7: realise that the graph of |2f(x)| must only exist above the x-axis

12b) Hint 8: reflect the graph of y = 2f(x) in the x-axis

Question 13

Hint 1: notice that the denominator has two distinct linear terms

Hint 2: know that the rational function should be written in the form A/(x+4) + B/(6-x)

Hint 3: perform standard partial fractions method to obtain values of A and B

Hint 4: as a check, in this case, both A and B work out to be integers

Hint 5: recognise that the integrand can now be processed as the integral of two single, simple rational functions, using your earlier result

Hint 6: when integrating the term with (6-x) in the denominator, be sure to check back by using differentiation that you have correct signs

Hint 7: carefully substitute in the limits into the expression that should have natural logs in it

Hint 8: use laws of logarithms to combine and simplify the expression to give the form requested by the question

Question 14

14a) Hint 1: write both lines' equations in parametric form, using vector notation

14a) Hint 2: know that if lines meet, corresponding components will be equal

14a) Hint 3: create three equations in the two parameters of λ and μ (where μ is the chosen parameter for line L2)

14a) Hint 4: simultaneously solve the equations and check that the values of the parameters obtained, work in all three equations

14a) Hint 5: use one of the line's equations to obtain the (x, y, z) coordinates of the intersection

14b) Hint 6: know that the angle between two lines is based upon the angle between their direction vectors

14b) Hint 7: use the scalar product of the direction vectors to obtain the angle between them

14b) Hint 8: check that the angle obtained is an obtuse angle, as requested by part (b)

Question 15

Hint 1: recognise that this is a non-homogeneous second order differential equation

Hint 2: construct the auxiliary equation

Hint 3: solve the auxiliary equation to obtain two distinct roots

Hint 4: construct the complementary equation, with constants A and B

Hint 5: know that the particular solution will be of the form of a quadratic expression

Hint 6: write the quadratic expression with unknown coefficients P, Q and R

Hint 7: differentiate this particular solution twice and subsitute the relevant terms back into the original differential equation

Hint 8: compare coefficients of quadratic, linear and constant terms in order to determine values of P, Q and R

Hint 9: assemble the solution that still contains unknown constants A and B

Hint 10: use the given initial conditions to determine the values of A and B

Hint 11: present the final solution that will have 5 distinct terms, two of which have exponential expressions in them

Question 16

Hint 1: recognise that T(0) = 25

Hint 2: recognise that T(q) = 9.8, where q is the time in minutes before 12 noon, when it was first placed in the fridge

Hint 3: recognise that T(q+15) = 6.5

Hint 4: recognise that TF, the temperature of the fridge, is 4, so this constant can be replaced in the differential equation

Hint 5: note the form of the differential equation is one with a single derivative on one side equalling a linear expression

Hint 6: recognise that this is a case for integration by separation of variables

Hint 7: integrate by the method of separation of variables, remembering to include a constant of integration

Hint 8: rearrange the equation from the form of ln(T - 4) into the form T = ….

Hint 9: use the fact that T(0) = 25 to fix the first constant

Hint 10: use T(q) = 9.8 and T(q+15) = 6.5 to form two equations in terms of exp(-kq)

Hint 11: split up the exp(-k(q+15)) term into exp(-kq) × exp(-15k)

Hint 12: divide one equation by the other to eliminate exp(-kq)

Hint 13: solve the remaining equation for k

Hint 14: re-substitute the value of k into an equation to obtain the value of q

Hint 15: interpret the value of q as a clock time, to the nearest minute

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