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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 = ekt
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: