Hints offered by N Hopley

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Paper 1

Question 1

1a) Hint 1: recognise that you need to use the chain rule and a standard differential from the Formula List

1b) Hint 2: recognise that you need to use the quotient rule with the chain rule

1c) Hint 3: consider writing y as y(x) to emphasise that y is a function of x, and then complete the implicit differentiation

Question 2

Hint 1: factorise the denominator into two linear factors

Hint 2: use the standard method of partial fractions on the integrand

Hint 3: integrate each fraction on its own, bringing in natural logarithms

Question 3

3a) Hint 1: consider writing out the general term to the expansion of (a+b)^n first

3a) Hint 2: then substitute the a for 2x and the b for 5/x² and the n for 9

3a) Hint 3: simplify each term to obtain a term with factorials, numerical terms to powers of r, and x to the power of a linear expression in r

3b) Hint 4: know that the term independent of x is the term whose power of x is zero

3b) Hint 5: set the linear expression in r to be equal to zero, and solve for r

3b) Hint 6: evaluate your answer from part (a) with r taking on the value that you just obtained

Question 4

4a) Hint 1: know that the bar over z means the complex conjugate of z

4a) Hint 2: multiply the two complex numbers together and gather the real terms and the imaginary terms

4a) Hint 3: factorise i out of the imaginary terms

4b) Hint 4: know that a real number has an imaginary term of 0

Question 5

Hint 1: use the standard method of the euclidean algorithm

Hint 2: the first line is 306 = 119 × 2 + 68

Question 6

Hint 1: recognise that we want dy/dx when t = -1/3

Hint 2: work out dy/dt and dx/dt

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

Hint 4: know that dy/dx is (dy/dt)×(dt/dx)

Hint 5: evaluate dy/dx when t = -1/3, to obtain the gradient of the curve at that point

Hint 6: evaluate x and y when t = -1/3 to obtain the x and y coordinates

Hint 7: use these coordinates and the gradient to obtain the equation of the line

Question 7

7a) Hint 1: obtain the resulting 3×3 matrix that has an expression in terms of k in row 2, column 1

7b) Hint 2: use a standard method to obtain the determinant of D, in terms of k

7b) Hint 3: know that the inverse of D does not exist if the determinant of D is equal to zero

Question 8

Hint 1: work out du/dθ

Hint 2: work out the values of u for both of the values of θ from the limits of the integral

Hint 3: use a standard method of integration by substitution to simplify the integral to a simple polynomial in u, with u limits

Question 9

9a) Hint 1: know that consecutive integers can be written a n, n+1 and n+2, where n is an integer

9a) Hint 2: write words to explain the logic behind what your algebraic terms mean in the context of divisibility

9b) Hint 3: know that an odd integer can be written as 2n+1 where n is an integer

9b) Hint 4: write words to explain the meaning of your algebraic terms

Question 10

Hint 1: know that z = x + iy

Hint 2: replace z with x + iy in the given modulus equation

Hint 3: square both sides of the modulus equation to prevent square roots appearing

Hint 4: know that |a+ib|² = a² + b²

Hint 5: simplify the expression for y in terms of x to obtain the equation of a straight line, which is the locus required

Hint 6: sketch the locus, noting points of intercept with the axis, and plotting the numbers 0 [at (0,0)] and 2-2i [at (2,-2)]

Question 11

11a) Hint 1: refer to the Formula List for the 2x2 matrix that represents a rotation of θ anticlockwise around the origin

11a) Hint 2: use exact value triangles to evaluate each term when θ= π/3

11b) Hint 3: know that a reflection in the x-axis transforms the point (x,y) to the point (x,-y)

11c) Hint 4: know that P = B A, and not P = A B

11d) Hint 5: refer to the general matrix for a rotation, noting which elements have to be the same sign

Question 12

Hint 1: evaluate a base case when n = 1, to verify the statement is true

Hint 2: use a standard method for proof by induction, for the inductive step

Hint 3: write words to draw together how the base case and the inductive step together mean the statement is true for all positive integers

Question 13

13a) Hint 1: use pythagoras' theorem on one of the right angled triangles

13a) Hint 2: solve for h, giving a reason for rejecting the possible negative solution

13b) Hint 3: recognise that a decreasing rate means that it will be negative

13b) Hint 4: deduce that dx/dt = -0.3

13b) Hint 5: recognise that we want dh/dt when x = 30

13b) Hint 6: implicitly differentiate with respect to t, the relationship h² + x² = 2500, from part (a)'s workings

13b) Hint 7: calculate the value for h, when x = 30, using part(a)

13b) Hint 8: replace in your implicitly differentiated equation the values for x, h and dx/dt

13b) Hint 9: rearrange to make dh/dt the subject

Question 14

14a) Hint 1: use standard formulae to calculate each of u7 and S∞

14b)i) Hint 2: use the standard formula for Sn and replace the values for Sn, n and a, to then solve for d

14b)ii) Hint 3: use a standard formula, now that a and d are known

14c) Hint 4: use a similar approach to part (b), but this time substituting in values for Sn, a and d to then obtain an equation in n

14c) Hint 5: rearrange to make the quadratic in n equal to zero

14c) Hint 6: factorise the quadratic by first taking out a common factor of 16 from all terms

Question 15

15a) Hint 1: use a standard method for integration by parts

15a) Hint 2: remember to include the constant of integration

15b) Hint 3: recognise that this equation requires an integrating factor, or...

15b) Hint 4: ...alternatively, in order to obtain part (a)'s integrand in part (b), just divide all terms through by x

15b) Hint 5: integrate by a standard method and use the initial conditions to fix the value of the constant of integration

15b) Hint 6: present your final answer in the required form

Question 16

16a) Hint 1: use the standard method of gaussian elimination

16a) Hint 2: carefully interpret the final row of your augmented matrix that should be (a-8)z = 0

16a) Hint 3: consider which values of a would give an infinite number of solutions (that would give the intersection line)

16b) Hint 4: work out expressions for y and x, each in terms of z

16b) Hint 5: write the x, y and z components in vector form

16b) Hint 6: extract the constant vector and a parametric multiple of a direction vector, introducing a parameter, instead of z

16c) Hint 7: know that the angle between two planes is the angle between their two normal vectors

16c) Hint 8: know that we need the acute angle, so careful consideration with the help of an angle diagram should help

16d) Hint 9: compare the two planes' direction vectors

16d) Hint 10: know that one vector being the multiple of another means that they are parallel vectors

16d) Hint 11: know what parallel normal vectors mean in terms of the planes themselves

16d) Hint 12: know how to check that the two planes are not coincident with each other (i.e. they are the same plane in the same space)

Question 17

17a) Hint 1: use the standard method for calculating each term of a Maclaurin Series

17b)i) Hint 2: recognise that this will require repeated use of both the chain rule and the product rule

17b)ii) Hint 3: evaluate the answers from (b)(i) when x = 0

17c) Hint 4: write down the two series up to and including their cubic powers of x

17c) Hint 5: multiply the two polynomials together to obtain all 8 terms

17c) Hint 6: discard all the terms that have powers higher than 3

17d) Hint 7: recognise that the expression given is the derivative of that from part (c)

17d) Hint 8: differentiate the series answer from part (c), term by term.

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