Hints offered by I Mackie, with video solutions by 'DLBmaths'

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Paper 1

Question 1

Hint 1: find the midpoint of PQ

Hint 2: establish the gradient of the median

Hint 3: and here is a video of the solution:

Question 2

Hint 1: use a standard method to work out the inverse of a given function

Hint 2: and here is a video of the solution:

Question 3

Hint 1: differentiate cos(2x) using formulae sheet to help

Hint 2: complete differentiation

Hint 3: substitute π/6

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

Question 4

Hint 1: Use formula sheet to write down the centre of the circle

Hint 2: calculate the gradient of the radius

Hint 3: establish the perpendicular gradient

Hint 4: write the equation of the tangent

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

Question 5

Hint 1: and here is a video of the solution:

Question 6

Hint 1: apply laws of logs eg. 2log(n) = log(n²)

Hint 2: apply laws of logs e.g. log(x) - log(y) = log(x/y)

Hint 3: and here is a video of the solution:

Question 7

7a) Hint 1: y-axis intercept is when x = 0

7b) Hint 2: differentiate given function

7b) Hint 3: substitute in the x coordinate to calculate the gradient

7b) Hint 4: write the equation of the tangent

7c) Hint 5: equate the line and curve

7c) Hint 6: write in standard quadratic form

7c) Hint 7: factorise by removing a common factor

7c) Hint 8: establish the x coordinate

7c) Hint 9: substitute the x coordinate into the equation of the line to
establish the y coordinate

Hint 10: and here is a video of the solution:

Question 8

Hint 1: use m = tan(θ)

Hint 2: use exact values to calculate the angle

Hint 3: and here is a video of the solution:

Question 9

9a) Hint 1: identify vector pathway in terms of u and t

9b) Hint 2: identify vector pathway

9b) Hint 3: express pathway in terms of t, u and v

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

Question 10

Hint 1: integrate each term

Hint 2: remember the +c

Hint 3: calculate c by substituting in values for x and y

Hint 4: write the equation y = …

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

Question 11

11a) Hint 1: reflect curve in the x axis

11a) Hint 2: translate the curve vertically by 1 unit

11b) Hint 3: equate log equations

11b) Hint 4: apply laws of logs e.g. 2log(n) = log(n²)

11b) Hint 5: apply law of logs to calculate the x coordinate

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

Question 12

12b) Hint 1: write an expression for the magnitude

12b) Hint 2: start to solve, by squaring both sides

12b) Hint 3: write in standard quadratic from

12b) Hint 4: factorise to find values for p

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

Question 13

13a) Hint 1: use formula sheet to expand sin(2x)

13a) Hint 2: use right angled triangle to calculate the value of cos x

13a) Hint 3: substitute into forumula

13a) Hint 4: multiply fractions and remember to simplify

13b) Hint 5: use the compound angle formula for sin(2x+x) written in terms of
sin(2x), cos(2x), sin(x) and cos(x)

13b) Hint 6: substitute in values

13b) Hint 7: remember to simplify and write as a single fraction

Hint 8: and here is a video of the solution:

Question 14

Hint 1: Prepare to integrate by changing the ∛ to a power and moving the
integratable terms to the numerator

Hint 2: start to integrate: increase power by 1, divide by the new power

Hint 3: complete integration

Hint 4: substitute the limits into the integrated function

Hint 5: complete evaluation, taking care with the negatives

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

Question 15

Hint 1: identify the roots on the graph

Hint 2: identify the stationary points of the graph

Hint 3: decide on the orientation of the cubic.

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

Paper 2

Question 1

Hint 1: write an integral for the shaded area: remember the 'dx'

Hint 2: integrate each term: increase power by 1, divide by the new power

Hint 3: substitute the limits into the integrated function

Hint 4: complete evaluation, remember to simplify

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

Question 2

2a) Hint 1: calculate the scalar product using the formula from the formula sheet

2b) Hint 2: calculate the magnitude of __u__

2b) Hint 3: calculate the magnitude of __v__

2b) Hint 4: substitute into the alternative formula for the scalar product: you
can find this on the formula sheet

2b) Hint 5: calculate the angle: remember cos^{-1}

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

Question 3

Hint 1: for increasing or decreasing functions, differentiate

Hint 2: evaluate the differential at the given value

Hint 3: make a decision: f'(x) < 0 means the function is decreasing, f'(x)
> 0 means the function is increasing

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

Question 4

Hint 1: remove the common factor

Hint 2: complete the square using a standard method

Hint 3: it is worth taking a minute to multiply out to check your answer

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

Question 5

5a) Hint 1: calculate the gradient of PQ

5a) Hint 2: calculate the perpendicular gradient

5a) Hint 3: find the midpoint of PQ

5a) Hint 4: use the midpoint and the perpendicular gradient to form the
equation of the line L₁

5b) Hint 5: solve the two equations simultaneously

5c) Hint 6: calculate the radius of the circle

5c) Hint 7: write the equation of the circle

Hint 8: and here is a video of the solution:

Question 6

6a) Hint 1: replace g(x) with 2x in the composite function.

6a) Hint 2: substitute 2x into the expression

6b) Hint 3: equate expressions from part (a)

6b) Hint 4: use formula sheet to substitute for cos(2x)

6b) Hint 5: arrange in standard quadratic form

6b) Hint 6: factorise quadratic

6b) Hint 7: solve for cos(x)

6b) Hint 8: solve for x: remember the question is in radians

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

Question 7

7a)i) Hint 1: use synthetic division or evaluate the cubic at 2

7a)i) Hint 2: complete division or evaluation, and interpret result

7a)ii) Hint 3: establish quadratic factor

7a)ii) Hint 4: complete factorisation remember to include the given factor

7b) Hint 5: substitute the given term into the recurrence relation

7c) Hint 6: equate u₅ and u₇

7c) Hint 7: arrange in standard form

7c) Hint 8: link to part (a)

7c) Hint 9: solve the cubic

7c) Hint 10: remember to discard invalid solutions: -1 < a < -1

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

Question 8

8a) Hint 1: expand k cos(x - α)

8a) Hint 2: compare that expression with the one given in the question

8a) Hint 3: identify what k cos(α) must be equal to, and what k
sin(α) must be equal to

8a) Hint 4: use your standard method to obtain the values of k and α

8b)i) Hint 5: state the minimum value

8b)ii) Hint 6: equate and start to solve

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

Question 9

Hint 1: write P in differentiable form

Hint 2: differentiate

Hint 3: equate derivative to 0

Hint 4: solve to find x

Hint 5: verify nature using table of signs, or second derivative

Hint 6: evaluate P

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

Question 10

Hint 1: use the discriminant > 0

Hint 2: find the zeros of the quadratic equation

Hint 3: create a sketch of the quadratic

Hint 4: from the sketch identify the range

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

Question 11

11a) Hint 1: substitute in the values of p and t

11a) Hint 2: arrange in the form A = e^{kt}

11a) Hint 3: take logs of both sides

11a) Hint 4: solve equation for k

11b) Hint 5: substitute k and t to evaluate P

11b) Hint 6: this gives the percentage who wait less than 5 minutes, so
subtract from 100

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

Question 12

12a)i) Hint 1: write down the centre, using the formula sheet for guidance

12a)ii) Hint 2: substitute the coordinates into the equation of C₂

12b)i) Hint 3: calculate distances

12b)i) Hint 4: use a standard method to determine the ratio

12b)ii) Hint 5: use a standard method to determine the coordinates from a
ratio

12c) Hint 6: calculate the radius of C₃, and therefore the equation of
C₃

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

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