Hints offered by N Hopley, with video solutions by 'DLBmaths'.
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Paper 1
Question 1
Hint 1: rearrange the equation into the form y = mx + c
Hint 2: read off the gradient
Hint 3: calculate the perpendicular gradient (by taking the negative reciprocal)
Hint 4: use a standard method to calculate the equation of a line through (-1,
6), with
the gradient you just calculated
Hint 5: write your final answer in simplified, expanded form (either y = mx +
c or ax
+ by = c)
Hint 6: and here is a video of the solution:
Question 2
Hint 1: know that the logs cannot be combined whilst one of them has a coefficient
of 2 in
front of it
Hint 2: use laws of logarithms to move the 2 inside the log that will change the
6 to
6²
Hint 3: know that one log subtracting another log can be re-written as the log
of a
fraction (when have the same base, which they do here)
Hint 4: simplify the fraction that is 6²/4
Hint 5: know that you can simplify the final expression as the base of the log
has a
connection with the value inside the log(...) function
Hint 6: and here is a video of the solution:
Question 3
Hint 1: start with the general definition of inverse functions that
h(h-1(x)) =
x
Hint 2: use composition of function skills to re-write the left side of this
equation
Hint 3: you should now have 4 + (1/3)h-1(x) = x
Hint 4: rearrange to make h-1(x) the subject
Hint 5: and here is a video of the solution:
Question 4
Hint 1: know that the sqaure root term needs to be re-written before you can
differentiate
Hint 2: know that √(x³) can be re-written as x3/2
Hint 3: clearly communicate that you now have y'(x) when you differentiate both
terms
.... be careful with all the negatives and fractions!
Hint 4: consider re-writing the answer back in terms of square roots and
fractions,
given that the question had them in that form
Hint 5: and here is a video of the solution:
Question 5
Hint 1: recognise that this question will make use of m = tan(θ)
Hint 2: know that θ is the angle measured anti-clockwise from the positive
direction of the x-axis
Hint 3: recognise that the given angle of π/3 radians is not the angle you
need, but
rather you need to use its complementary angle
Hint 4: so, using θ = π/2 - π/3 which equals π/6, work out the
gradient,
m ...
Hint 5: ... using the appropriate exact value triangle
Hint 6: now use a standard method to obtain the equation of a line through
(-2, 0)
with the gradient you have just calculated.
Hint 7: write your final answer in simplified, expanded form (either y = mx
+ c or
ax + by = c)
Hint 8: and here is a video of the solution:
Question 6
Hint 1: use your chosen standard method for integrating a compound function,
making sure
you check your answer by differentiating back and using the chain rule
Hint 2: before substituting in the limits, re-write the expression in terms of
roots, to
remove the fractional powers
Hint 3: take care when substituting in the limits to the expression, especially
with all
the negative signs
Hint 4: take care with the arithmetic of simplifying each term ... there are no
rewards
for getting it wrong quickly!
Hint 5: and here is a video of the solution:
Question 7
7a)i) Hint 1: focus on right-angled triangle ADE
7a)i) Hint 2: use Pythagoras' Theorem to calculate the length of side AD
7a)ii) Hint 3: focus on right-angled triangle ABC
7a)ii) Hint 4: use Pythagoras' Theorem to calculate the length of side BC
7b) Hint 5: know that sin(q - r) can be expanded to give four trigonometric
terms
7b) Hint 6: use your triangles from part (a) to substitute in values for each
of the
four trigonometric terms
7b) Hint 7: simplify your answer into a single fraction
Hint 8: and here is a video of the solution:
Question 8
Hint 1: know that the two log(...) expressions on the left side can be combined to
give a
single log(...) term
Hint 2: re-write the equation in terms of exponents
Hint 3: you should now have x(x + 5) = 6²
Hint 4: expand out the brackets and think what type of equation you have here
to solve
Hint 5: rearrange the equation to give ... = 0
Hint 6: fully factorise the expression
Hint 7: write down two solutions for x from this equation
Hint 8: note that the question stated that x > 0, so one of these solutions
needs
to be rejected
Hint 9: clearly communicate your final answer, along with your reason for
rejecting one solution
Hint 10: and here is a video of the solution:
Question 9
Hint 1: know that cos(2x) can be re-written in one of three different ways
Hint 2: look at the rest of the equation to see that there is a cos(x) term
Hint 3: choose the cos(2x) expansion that also only has cos(x) terms
Hint 4: rearrange the equation to give ... = 0
Hint 5: recognise what type of equation you are having to solve here
Hint 6: fully factorise the expression
Hint 7: obtain two values for what cos(x) must equal
Hint 8: know that one of the values means that x cannot be calculated, so
its
solution is rejected
Hint 9: use the appropriate exact value triangle to calculate a possible
value of
x for the other, allowed value
Hint 10: use your knowledge of the cosine function, or its graph, to
obtain a
second possible value for x
Hint 11: clearly communicate your final answers, along with your reason
for
rejecting cos(x) = 2
Hint 12: and here is a video of the solution:
Question 10
10a) Hint 1: know that y = 2f(x) + 1 involves two different transformations of y =
f(x)
10a) Hint 2: on one set of axes, sketch the unchanged y = f(x) recording the
points (0,
3) and (4, 0)
10a) Hint 3: on a new set of axes, sketch the graph y = 2f(x) and record which
one of
the two points needs their coordinates changed.
10a) Hint 4: on a new set of axes, sketch the graph y = 2f(x) + 1 and record
which one
of the two points now needs their coordinates changed.
10b) Hint 5: know that y = f(1/2x) will affect the
horizontal
scaling of the function y = f(x)
10b) Hint 6: realise that the stationary point of (4, 0) on y = f(x) will
equate to
the stationary point of (8, 0) on y = f(1/2x)
10b) Hint 7: clearly write down the coordinates of both stationary points
Hint 8: and here is a video of the solution:
Question 11
Hint 1: realise that the 2x² term means that it has to be dealt with before
embarking
upon completing the square
Hint 2: factorise 2 out from the first two terms of the expression
Hint 3: use a standard method to complete the square of the expression inside
the
brackets you just created
Hint 4: expand out the 'outer brackets' and simplify terms to obtain the format
of the
required answer
Hint 5: for completeness, you could also clearly state the values of p, q and
r
Hint 6: and here is a video of the solution:
Question 12
Hint 1: recognise that this differentiation will require the chain rule
Hint 2: after differentiating, simplify the expression ready for evaluating with
π/6
Hint 3: after substituting in π/6, simplify the expression inside the
cos(...) term
Hint 4: use the appropriate exact value triangle to calculate cos(π/6)
Hint 5: present a final answer, as an exact value
Hint 6: and here is a video of the solution:
Question 13
13a)i) Hint 1: know that if (x + 2) is a factor then x = -2 will be a root
13a)i) Hint 2: proceed to show that f(-2) equals zero, thus confirming that x =
-2 is a
root
13a)ii) Hint 3: use synthetic division, or polynomial long division, to
factorise the
cubic using the knowledge that (x + 2) is a factor
13a)ii) Hint 4: once fully factorised, clearly list the values of x that are
the roots
of the original cubic function
13b) Hint 5: realise that we know one of the existing roots is x = -2
13b) Hint 6: notice that we now need this root to be located at x = 1
13b) Hint 7: realise that this is a total shift of 3 (from -2 to +1)
13b) Hint 8: think carefully whether this would be y = f(x - 3) or y = f(x
+3)
13b) Hint 9: see if your thinking matches the constraint that k > 0 for y
= f(x -
k), as stated in the question
Hint 10: and here is a video of the solution:
Question 14
14a)i) Hint 1: know that you can simply read off the circle's centre coordinates
from the
given, factorised equation
14a)i) Hint 2: know that the radius can also be read off, once you process the
number 100
in the appropriate way.
14a)ii) Hint 3: sketch a diagram showing the centre point of the circle (7, -5)
and the
point P(-2, 7)
14a)ii) Hint 4: calculate the distance between these two points
14a)ii) Hint 5: determine whether this distance is greater than, or less than,
the
radius you stated in part (a)(i)
14b) Hint 6: sketch an accurate diagram showing the circle with centre (7,
-5) and
radius 5
14b) Hint 7: add to your diagram the point P(-2, 7)
14b) Hint 8: consider circles of different size radii that have centre
P(-2, 7)
14b) Hint 9: you should notice that there are two possible values for the
radius -
one for a small circle that just touches C1, and one for a larger centre that encompasses
C1
14b) Hint 10: use your diagram, and all of its labelled lengths, to
logically
deduce the two radius values
Hint 11: and here is a video of the solution:
Paper 2
Question 1
1a) Hint 1: know that the altitude through C will require a gradient that is
perpendicular
to mAB
1a) Hint 2: calculate gradient mAB
1a) Hint 3: calculate the gradient that is perpendicular to mAB, by
taking
the negative reciprocal
1a) Hint 4: use your chosen standard method for calculating the equation of a
line
through point C with the required gradient
1a) Hint 5: clearly write your equation in the format y = mx + c, or ax + by =
c, or
ax + by + c = 0
1b) Hint 6: calculate the coordinates of the midpoint of segment AC, and call
it
point D
1b) Hint 7: calculate the gradient of BD
1b) Hint 8: use your chosen standard method for calculating the equation of
a line
through point D with the required gradient
1b) Hint 9: clearly write your equation in the format y = mx + c, or ax +
by = c,
or ax + by + c = 0
1c) Hint 10: you should have y1 = x - 4 and y2 = 5x
- 14
from parts (a) and (b)
1c) Hint 11: know that the intersection point will happen when
y1 =
y2
1c) Hint 12: solve the equation x - 4 = 5x - 14
1c) Hint 13: use the x-coordinate to calculate the y-coordinate, by
using
either y1 or y2
1c) Hint 14: clearly state your final answer as a set of coordinates,
with
brackets around both numbers and a comma between them
Hint 15: and here is a video of the solution:
Question 2
Hint 1: know that the discriminant, b² - 4ac, will be needed here
Hint 2: clearly identify the values of a, b and c from the given quadratic
equation
Hint 3: know that two real and distinct roots happen when the discriminant is
greater
than zero
Hint 4: substitute values of a, b and c into b² - 4ac > 0
Hint 5: carefully simplify and solve the inequation, watching out for all the
negatives
Hint 6: present a final answer as an inequality in terms of the variable, p
Hint 7: and here is a video of the solution:
Question 3
3a) Hint 1: expand k sin(x + a)
3a) Hint 2: compare that expression with the one given in the question
3a) Hint 3: identify what k cos(a) must be equal to, and what k sin(a) must be
equal to
3a) Hint 4: use your standard method to obtain the values of k and a
3b) Hint 5: use your answer from (a) to re-write the given equation
3b) Hint 6: rearrange the trig equation into the form sin(x + a) = some
number
3b) Hint 7: take the inverse sine and list all possible values for (x + a)
between 0
and 2π
3b) Hint 8: note that we have to work in radians that are not the usual
fractional
multiples of π, so your final values for x should be decimals between 0 and 6.28
Hint 9: and here is a video of the solution:
Question 4
4a) Hint 1: notice that for the area above the x-axis, the limits of -1 and 2 will
be used
4a) Hint 2: calculate the definite integral of y from -1 to 2, making sure to put
brackets around the expression, as well as 'dx' on the end
4a) Hint 3: take care with the substitution of negative values into the
integrated
expression!
4b) Hint 4: notice that some of your work from part (a) can be 'recycled' in
part (b),
as they share the limit value of 2
4b) Hint 5: you can calculate the definite integral of y from 2 to 4, knowing
that
your answer will come out to be negative
4b) Hint 6: the total area will be the answer from part (a) plus the positive
of the
answer from part (b)
Hint 7: and here is a video of the solution:
Question 5
5a)i) Hint 1: know that f(g(x)) will become f(3x + 5)
5a)i) Hint 2: evaluate f(3x + 5), expand the resulting brackets, and simplify
5a)ii) Hint 3: know that g(f(x)) will become g(x² - 2)
5a)ii) Hint 4: evaluate g(x² - 2), expand the resulting brackets, and
simplify
5b) Hint 5: replace each side of the inequation with your expressions from
part (a)
5b) Hint 6: realise that you have a quadratic inequation to solve
5b) Hint 7: rearrange so that the inequation takes the form ... < 0
5b) Hint 8: notice the expression on the left side of the
inequation has a
common factor that can be factorised out
5b) Hint 9: fully factorise the expression on the left side of the
inequation
5b) Hint 10: sketch a graph of the factorised quadratic
expression to
help determine when it is less than zero
5b) Hint 11: clearly present your answer in the form ... < x <
...
Hint 12: and here is a video of the
solution:
Question 6
Hint 1: realise that you will have to integrate the given
expression to
determine y(x)
Hint 2: notice that the given expression is not in a form ready
to be
integrated
Hint 3: after rewriting 3/x² as 3x-2, integrate
the
expression
Hint 4: don't forget the constant of integration!
Hint 5: use the information that when x = 3 then y = 6 in
order to fix
the value of the constant of integration
Hint 6: clearly present your final answer in the form y =
....
Hint 7: and here is a video of the solution:
Question 7
Hint 1: recognise that the graph gives you an equation of the form
Y = mX
+ c
Hint 2: calculate the value of m, and read off the value of c
Hint 3: replace Y with log5y, and replace X with
log5x
Hint 4: notice that all of the terms involve logs, except for
the 3 on
the end
Hint 5: re-write 3 as 3log55, because
log55 is
equal to the number 1
Hint 6: use the laws of logarithms to move the coefficients
of the
log terms inside the log functions, turning the coefficients into powers
Hint 7: use the laws of logarithms to combine the two log
terms that
are being added, into a single log term of a product
Hint 8: you should have an equation of the form
log5y =
log5(...)
Hint 9: taking 'inverse logs' of both sides, gives you an
equation
of the form y = (...)
Hint 10: read off the values of k and n, and clearly
state them
Hint 11: and here is a video of the solution:
Question 8
8a) Hint 1: realise that we need the width and length of the
rectangular
pond
8a) Hint 2: note that the pond width is (y - 2) and the pond
length is (x
- 3)
8a) Hint 3: calculate the area of the pond, A(x), by expanding
out (x -
3)(y - 2)
8a) Hint 4: in the expansion, you will see an 'xy' term, which
represents the area of the entire rectangular plot
8a) Hint 5: note from the question that xy = 150
8a) Hint 6: and also note, therefore, that y = 150/x
8a) Hint 7: substitute all of this information into your
expression
for A(x), and simplify to give the desired version of function A(x)
8b) Hint 8: realise that you will have to differentiate
A(x), but
that it is not yet in a form ready to be differentiated
8b) Hint 9: re-write the -450/x term as -450x-1
8b) Hint 10: differentiate the expression for A(x)
8b) Hint 11: communicate that you will find stationary
points
when A'(x) = 0
8b) Hint 12: solve the equation A'(x) = 0 to obtain two
values
of x
8b) Hint 13: note from the context of the question
that x must
be positive, so the negative value of x can be discarded
8b) Hint 14: check the nature of the remaining
stationary
point by using either a nature table, or the second derivative, A''(x)
8b) Hint 15: after determining that the stationary
point is
a maximum AND clearly stating this, calculate A(x) for when x = 15
Hint 16: and here is a video of the
solution:
Question 9
9a) Hint 1: know to substitute the equation of the line into the
equation
of the circle
9a) Hint 2: solve the resulting equation in x. There will be two
solutions for x, as it is a quadratic equation.
9a) Hint 3: calculate the corresponding values of y for each of
the
values of x, using y = 3x + 7
9a) Hint 4: clearly state the coordinates of both P and Q,
using
brackets and commas for each.
9b) Hint 5: know that we will need to find out the coordinates
of the
centre of the circle, as well as the mid-point of PQ
9b) Hint 6: process the equation of the circle, either by
formula or
by completing the squares, to obtain the coordinates of its centre
9b) Hint 7: calculate the midpoint of P and Q, using your
coordinates from part (a)
9b) Hint 8: calculate the distance between the midpoint and
the
centre of the circle, as this will be the radius of the smaller circle
9b) Hint 9: assemble the centre and radius information
into a new
equation of the smaller circle
9b) Hint 10: make sure to simplify the constant value,
and not to
leave it as (√10)²
Hint 11: and here is a video of the solution:
Question 10
10a) Hint 1: realise that this question is only asking to
substitute one
value into the given formula
10a) Hint 2: evaluate P when T = 24.55
10b) Hint 3: realise that the given equation has three unknowns,
P, D
and k, but that we are given values for both P and D already
10b) Hint 4: substitute in P = 850 and D = 600
10b) Hint 5: simplify each term to give an equation of the
form
390k = ...
10b) Hint 6: know that to obtain the value of k, as it's in a
power,
then logarithms will need to be used.
10b) Hint 7: take either log10 or loge
of both
sides of the equation
10b) Hint 8: use the laws of logarithms to bring the power
of k
down and outside of its log term
10b) Hint 9: solve for k, giving your answer to at least 2
decimal
places.