Dalkeith High School CfE Higher Mathematics Unit ......9 Qu. Points of Expected Response...
Transcript of Dalkeith High School CfE Higher Mathematics Unit ......9 Qu. Points of Expected Response...
2
Unit 1 – Page 3
Expressions and Functions
Unit 2 – Page 23
Relationships
Unit 3 – Page 36
Applications
3
Expressions and Functions Practice Assessment A
E&F Assessment Standard 1.1
1. (a) Simplify the expression .
(b) Hence solve
(1)
(#2.1
)
(2)
2. Given the equation
find the value of n.
(2)
3. Solve correct to 2 decimal places.
(2)
E&F Assessment Standard 1.2
4. Express in the form given that and .
(4)
5. A kite is shown in the diagram below.
Find the exact value of .
(4)
6. Given that
show that the exact value of is
.
(#2.1
)
(2)
E&F Assessment Standard 1.3
7. A function is given as
20 x
(a) State the maximum and minimum turning points of the function.
(b) Find the point where the graph intersects the y-axis.
(2)
(1)
x°
y° 3 cm
2 cm
4 cm
4
8. The diagram shows the graph of a cubic equation, .
Sketch the graph of showing the new coordinates of the stationary points.
(3)
9. Sketch the graph of for
(3)
10. The graph shows a function in the form .
Find the values of a and b.
(2)
11. Two functions are defined as:
, .
(a) Find an expression for .
(b) Find an expression for .
(c) Which expression has the same domain as ? Justify your answer
(2)
(2)
(1)
(#2.2
)
12. A function is given by .
Find the inverse function .
(2)
4
O –3 2
(2, –2)
(–3, 4)
–2
y
x
(-1, 1)
(1, 9)
y
x O
5
E&F Assessment Standard 1.4
13. On orienteering course is being laid out in a forest.
Relative to suitable axes, the first three “control points” can be represented by the points
A(1, 3, 10), B(2, 1, 9) and C(4, –3, 7).
Show that the three points are collinear.
(3)
14. A canine agility test features a see saw with a pivot point that is slightly off-centre.
The diagram below represents the see saw, relative to a suitable axes.
The points P, Q and R lie in a straight line, as shown. Q should divide PR in the ratio 3:5.
Is Q in the correct position? Justify your answer.
(3)
(#2.2
)
15. ABCDEF is a triangular prism as shown.
The vectors , and are given by:
= –4i + 8j + 4k
= 10i + 4j + 2k
= –i – 4j + 13k
Express in component form.
(3)
Turn over for question 16
A
1
(1, 3, 10) (2, 1, 9) (4, –3, 7)
B
2
C
3
A B
C D
E
F
P(2, 1, 72)
Q(5, 16, 45)
R(10, 41, 0)
6
16. The points S(3, 0, 2), T(7, 1, -5) and U(4, 3, -2) are shown in the diagram below.
Find the size of the shaded angle.
(5)
End of question paper
T(7, 1, -5)
S(3, 0, 2)
U(4, 3, -2)
7
Expressions and Functions, Practice Assessment A Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.1: Logs and Exponentials
1(a) 1 Apply
1
1(b) #2.1
Strategy: Substitute for LHS
2 Start to solve
3 Find m
#2.1
2 OR
3
2 1 Apply
OR
2 Find n
1
OR
2
3 1 Take natural logarithm of both sides
2 Find x (round to 2 d.p.)
1
2
7 + #2.1
E&F Outcome 1.2: Manipulating Trig Expressions
4 1 Expand
2 Match coefficients
3 Find k
4 Find a
1 stated explicitly
2 and stated explicitly
3
4
5 1 Find missing sides
2 Expand and begin subs
3 Complete substitution
4 Process
1 and
2
3
4
or
6 Method 1
1 Square both sides
#2.1
Strategy: Use
2 Simplify
Method 2
#2.1
Strategy: Start to find
1 State
2 Square both sides
Method 1
1
#2.1
2
Method 2
#2.1
Find opposite side in right-angled triangle,
1
2
10 + #2.1
8
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.3: Functions and Graphs
7(a) 1 Maximum turning point
2 Minimum turning point
1
2
7(b) 3 y-intercept
3
or equivalent (0, 2·12)
8 1 Correct horizontal scaling
2 Correct vertical scaling
3 Shape and stationary points coordinates
1 x-coordinates correct
2 y-coordinates correct
3 Correct shape and annotation
9 1 Correct amplitude
2 Correct period
3 Shape and correct vertical translation
1 2
2 4
3 +1
10 1 Find a
2 Find b
1 a = 3
2 b = 1
11(a) 1 Start composite process
2 Complete process
1
2
11(b) 3 Start composite process
4 Complete process
3
4
11(c) 5 State expression
#2.2
Explain solution
5
#2.2
The denominator of a fraction cannot be zero.
12 1 Start inverse process
2 State inverse function
1
2
18 + #2.2
(1, -6)
(-1·5, 12)
y
x
x
3
–1
y
O
1
π
9
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.4: Vectors
13
1 Find vector between two points
2 Find second vector and interpret
multiple
3 Complete proof of collinearity
1 e.g.
2
3 hence the vectors are parallel. B is a
common point. Therefore the points A, B and C are
collinear.
14 Method 1 – Ratio to find Q
1 Find
2 Use correct ratio
3 Find point Q
#2.2
Justify answer
Method 2 – Section Formula to find Q
1 Begin substitution
2 Complete substitution
3 Find point Q
#2.2
Justify answer
Method 3 – Component Vectors
1 Find
2 Find
3 Interpret multiples
#2.2
Justify answer
Method 1
1
2
3 Q = (5, 16, 9)
#2.2
Q is in correct position as it divides PR in the
ratio 3:5
Method 2
1
2
3 Q = (5, 16, 9)
#2.2
Q is in correct position as it divides PR in the
ratio 3:5
Method 3
1
2
3 or equivalent
#2.2
Q is in correct position as it divides PR in the
ratio 3:5
15 1 Find pathway for
2 Identify
3 Complete calculation of
1
2
3 OR
Do not award 3 for (–3 , 12, –9)
16 1 Find component vectors
2 Use scalar product
3 Find scalar product
4 Find magnitudes
5 Find angle
1
and
2
3 31
4 and
5 θ = 35∙6° or 0∙62 (radians)
10
Expressions and Functions Practice Assessment B
E&F Assessment Standard 1.1
1. (a) Simplify fully
.
(b) Express
in the form .
(2)
(2)
2. Given that , find x correct to 2 decimal places.
(#2.1
)
(1)
E&F Assessment Standard 1.2
3. Express in the form where and .
(4)
4. The diagram shows two right-angled triangles.
Find the exact value of .
(4)
5. By substituting into the function , show that .
(2)
(#2.1
)
E&F Assessment Standard 1.3
6. Sketch the graph of
for .
The stationary points and the x-intercepts should be clearly shown.
(3)
7. The graph of a function, , has a maximum turning point at (–3, 19) and a
minimum turning point at (4, –30).
State the coordinates of the stationary points on the graph
.
(2)
y°
x°
16
21
12
11
8. The diagram shows the graph of .
Write down the values of a and b.
(2)
9. Two functions are given by and .
(a) Find an expression for .
(b) Explain why that the range for this function is .
(2)
(#2.2
)
10. A function is defined as
.
Find the inverse function xf 1.
(2)
E&F Assessment Standard 1.4
11. A PE teacher is laying out cones for a fitness test.
Three cones need to be set up to meet the following conditions:
The three cones should be in a straight line.
The distance from the start (S) to point B is 25 metres ± 1 metre.
Cone A divides the line SB in the ratio 3:2.
Relative to a suitable axes, where each unit is 1 metre, the cones can be represented by the
points S(2, 10, 1), A(11, –2, 4) and B(17, –10, 6) respectively.
Has the teacher laid the cones correctly? You must justify your answer.
(5)
(#2.2
)
S(2, 10, 1) A(11, –2, 4) B(17, –10, 6)
(0, 4)
(2, 7)
y
x O
12
12. The points M, N and P lie in a straight line, as shown. P divides MN in the ratio 2:7.
Find the coordinates of N.
(3)
13. CDEFGH is an octahedron.
The vectors , and are given by:
= –8i + 2j + 2k
= 2i + 10j – 2k
= i + 7j + 7k
Express in component form.
(3)
14. Points P, Q and R have coordinates A(10, 5, 2), B(–3, 4, 1) and C(4, –8, 3).
Find the size of the acute angle ABC.
(5)
End of question paper
M(–8, –5, 2)
N
P(–2, –1, 0)
A
B
C
F
G
D
E
H
C
13
Expressions and Functions, Practice Assessment B Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.1: Logs and Exponentials
1(a) 1 Use
2 Simplify fully
1
2
5
1(b) 3 Use OR
4 Simplify to
3
OR
4
2 #2.1
Use
1
#2.1
1
5 + #2.1
E&F Outcome 1.2: Manipulating Trig Expressions
3 1 Expand
2 Compare coefficients
3 Find k
4 Find a
1 stated explicitly
2 and stated explicitly
3
4
4 1 Process missing sides
2 Expand and begin subs
3 Complete substitution
4 Process
1 20 and 29
2
3
4
or equivalent
5 1 Substitution
#2.1
Strategy: Know to use identity
2 Simplify and complete
1
#2.1
2
10 + #2.1
14
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.3: Functions and Graphs
6 1 Max/min correct
2 x-intercepts
3 Correct shape
1 On graph max value 5 and min value –5
2
and
3 Start and finish at 5
7 1 Correct horizontal scaling
2 Correct vertical translation
1 (–6, …) and (8, …)
2 (…, 24) and (…, –25)
Note: Can award 1 for (–6, 24) and
2 for (8, –25)
8 1 Find a
2 Find b
1 a = 1
2 b = 5
9(a) 1 Interpret composite process
2 Complete function
1
2
9(b) #2.2
Explain a solution #2.2
so
OR min TP of function is (–3, –2) so
range is .
10 1 Start inverse process
2 State inverse function
1
2
11 + #2.2
5
-5
O
x
y
15
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.4: Vectors
11
1 Find vector between two points
2 Find second vector and interpret
multiple
3 Complete proof of collinearity
4 Length of
5 Interpret ratio
#2.2
Explain solution in context
1 e.g.
2
3 hence the vectors are parallel. A is
a common point so the points S, A and B are
collinear.
4 (metres)
5 are in the ratio 3:2
#2.2
The cones have been laid out correctly as all
conditions have been met.
12
1 Find component vector
2 Use correct ratio
3 Process vectors and find point N
Alternative Method: Section formula
1 Begin substitution
2 Complete substitution
3 Process and state point N
1
2
OR
3 N = (19, 13, –7)
Alternative Method: Section formula
1
2
3 N = (19, 13, –7)
13 1 Find pathway for
2 Identify
3 Complete calculation of
1
2
3 OR
Do not award 3 for (–9 , –5, –5)
14
1 Find component vectors
2 Use scalar product
3 Process scalar product
4 Process magnitudes
5 Find angle
1
and
2
3 90
4 and
5 θ = 61∙3° or 1∙07 (radians)
16 + #2.2
16
Expressions and Functions Practice Assessment C
E&F Assessment Standard 1.1
1. (a) Simplify the expression .
(b) Hence solve
(1)
(#2.1
)
(2)
2. Given the equation
find the value of n.
(2)
3. Solve correct to 2 decimal places.
(2)
E&F Assessment Standard 1.2
4. Express in the form where and .
(4)
5. The diagram below shows two right-angled triangles.
Find the exact value of yx sin .
(4)
6. Given that
show that the exact value of is
.
(#2.1
)
(2)
E&F Assessment Standard 1.3
7. A function is given as
for .
(a) State the maximum and minimum turning points of the function.
(b) Find the point where the graph intersects the y-axis.
(2)
(1)
x°
y°
20
8
15
17
8. The diagram shows the graph of with a maximum turning point at (1, 2) and a
minimum turning point at (–3, –4).
Sketch the graph of
.
(3)
9. The diagram shows the graph of cbxay )sin( .
Write down the values of a, b and c.
(3)
10. Sketch the graph of .
(3)
11. Two functions are defined as:
, .
(a) Find an expression for .
(b) Find an expression for .
(c) Which expression has the domain ? Justify your answer.
(2)
(2)
(1)
(#2.2
)
12. A function is given by
Find the inverse function xf 1 .
(2)
4
–2
y
O x π
2
π
4
1
O 1 –3
2 (1, 2)
(–3, –4) –4
y
x
18
E&F Assessment Standard 1.4
13. A mechanical engineer is designing a cable car for a tourist resort so that visitors can get to
a restaurant high up a hillside.
The base station can be represented as A(1, 1, 4) and the top of each pylon can be
represented by the points B(5, –1, 6) and C(13, –7, 10).
To help minimise cost the base stations and pylons should be collinear.
Have the pylons been positioned correctly to reduce cost? You must justify your answer.
(3)
(#2.2
)
14. The points F, G and H lie in a straight line, as shown. G divides FH in the ratio 5:2.
Find the coordinates of G.
(3)
Turn over for questions 15 and 16
A(1, 1, 4)
B(5, –1, 6)
C(13, –7, 10)
F(–20, 1, 5)
H(22, –13, 12) G
19
15. ABCDEFGH is a cuboid with centre P.
The vectors , and are given by:
= –6i –2j + 4k
= 2i + 10j + 8k
= –3i + 5j + 5k
Express in component form.
(3)
16. Points P, Q and R have coordinates A(1, 5, –2), B(–5, 7, –1) and C(4, 7, 1).
Find the size of the obtuse angle BAC.
(5)
End of question paper
A B
C D
E
F
H
G
P
A(1, 5, –2)
B(–5, 7, –1)
C(4, 7, 1)
20
Expressions and Functions, Practice Assessment C Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.1: Logs and Exponentials
1(a) 1 Apply
1
1(b) #2.1
Strategy: Substitute for LHS
2 Start to solve
3 Find a
#2.1
2 OR
3
2
1 Apply
OR
2 Find n
1
OR
2
3 1 Take logarithm of both sides
2 Find x (round to 2 d.p.)
1 OR
2
7 + #2.1
E&F Outcome 1.2: Manipulating Trig Expressions
4 1 Expand
2 Match coefficients
3 Find k
4 Find a
1 stated explicitly
2 and stated explicitly
3 or equivalent
4
5 1 Process missing sides
2 Expand and begin subs
3 Complete substitution
4 Process
1 17 and 25
2
3
4
or equivalent
6 Method 1
1 Square both sides
#2.1
Strategy: Use
2 Simplify
Method 2
#2.1
Strategy: Start to find
1 State
2 Square both sides
Method 1
1
#2.1
2
Method 2
#2.1
Find adjacent side in right-angled triangle,
1
2
10 + #2.1
21
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.3: Functions and Graphs
7(a) 1 Maximum turning point
2 Minimum turning point
1
2
7(b) 3 y-intercept
3 or equivalent (= 6·93)
8 1 Correct vertical scaling
2 Correct vertical translation
3 Shape and stationary points coordinates
1 y-coordinates halved
2 y-coordinates translated down two units
3 Correct shape and annotation
9 1 Find a
2 Find b
3 Find c
1 a = 3
2 b = 4
3 c = 1
10 1 x-intercept
2 Point (a, 1)
3 Shape of graph
1 (7, 0) annotated
2 (9, 1) annotated
3 Suitable curve that doesn’t cross x = 6
11(a) 1 Start composite process
2 Complete process
1
2
(Do not penalise further working)
11(b) 3 Start composite process
4 Complete process
3
4
11(c) 5 State expression
#2.2
Explain solution
5
#2.2
The denominator of a fraction cannot be zero.
12 1 Start inverse process
2 State inverse function
1
2
19 + #2.2
y
x
1
7 O
(9, 1)
O
(-4, -3)
(1, -1)
y
x
22
Qu. Points of Expected Response Illustrative Scheme
E&F Outcome 1.4: Vectors
13
1 Find vector between two points and
take out common factor
2 Find second vector and take out
common factor
3 Complete proof of collinearity
#2.2
Explain solution in context
Alternative for 1 and
2
1 Find vector between two points
2 Find second vector and scale to
compare (shown explicitly)
1 e.g.
2
OR
3
The vectors are not parallel so the points are
not collinear.
#2.2
The pylons are not positioned correctly to
reduce cost.
Alternative for 1 and
2
1 e.g.
2 e.g.
and
14
1 Find component vector
2 Use correct ratio
3 Process vectors and find point G
Alternative Method: Section formula
1 Begin substitution
2 Complete substitution
3 Process and state point G
1
2
3 G = (10, –9, 10)
Alternative Method: Section formula
1
2
3 G = (10, –9, 10)
15 1 Find pathway for
2 Identify
3 Complete calculation of
1
2 –
3 OR
Do not award 3 for (–5 , –5, –3)
16
1 Find component vectors
2 Use scalar product
3 Process scalar product
4 Process magnitudes
5 Find angle
1
and
2
3 –11
4 and
5 θ = 111∙5° or 1·95 (radians)
14 + #2.2
23
Relationships and Calculus Practice Assessment A
R&C Assessment Standard 1.1
1. A polynomial is given by the formula .
(a) Show that is a root of .
(b) Hence, or otherwise, solve
(2)
(4)
2. The equation of the graph shown below is of the form .
[Not to scale]
Find the values of a, b, c and k.
(3)
3. The line touches the curve at only one point.
Find the value of k.
(#2.1
)
(3)
R&C Assessment Standard 1.2
4. Solve
for and where sinx
(4)
5. Solve for .
(2)
x
y y = f(x)
6 1 -3
(3, -12)
O
24
6. An engineer has set up an experiment in a water tank to test some new equipment. The
tank has been set up to produce waves with height , where t is the time in
seconds since the start of the experiment and h is the wave height in millimetres.
(a) Find the point marked A, the first time that the wave height is 100 mm.
(b) Find the time when the wave height is next 100 mm.
(c) Write down the next 2 times when the wave height will be 100 mm.
(4)
(1)
(#2.2
)
R&C Assessment Standard 1.3
7. Differentiate the expression below with respect to x.
,
(3)
8. A parabola has equation .
Find the equation of the tangent to the curve at the point where .
(4)
9. A child’s toy has a football attached to an elasticated chord which is tied to a pole.
When the child kicks the football it’s horizontal distance from the pole in centmetres is
given by the expression , where t is the time is seconds.
(a) How far from the pole is the ball initially?
(b) Find x when t = 3. Explain what this means in terms of the balls position.
(c) Find the velocity of the ball after 2 seconds.
(1)
(1)
(#2.2
)
(2)
10. Given that find . (1)
t 3
200
-200
h
O 1
A
2
100
25
R&C Assessment Standard 1.4
11. Integrate with respect to x.
(3)
12. Find .
(1)
13. The graph of the function is shown.
The gradient of is given by the function .
Find .
(#2.1
)
(2)
14. Find
.
(4)
End of question paper
x
y y = f(x)
5
O
26
Relationships and Calculus, Practice Assessment A Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
R&C Outcome 1.1: Quadratics and Polynomials
1(a) 1 Direct substitution OR strategy to start
factorising (e.g. synthetic division or
algebraic long division)
2 Show remainder = 0 and communicate
result.
1
OR –4
1 1
–22
–40
2 therefore is a root
OR –4
1 1
–4
–22
12
–40
40
1 –3 –10 0
Remainder = 0 so therefore is a root
1(b) 3 First linear factor
4 Quadratic factor
5 Complete factorisation and set equal to
zero
6 State roots
3
4
5
6
2 1 State values of a, b, c OR sub values
into function.
2 Substitute point
3 State value of k
1 , , (in any order) OR
2
3
3 1 Equate expressions
2 Rearrange and set equal to zero
#2.1
Strategy: Know to use discriminant and
substitute OR deduce repeated factors
OR complete square
3 Find k
1
2
#2.1
OR OR
3
12 + #2.1
R&C Outcome 1.2: Solving Trig Equations
4 1 Correct substitution for
2 Simplify to
3 Solve for both values of x
1
2
3 and
5 1 Rearrange
2 Solve for
3 Solve for x
1
or equivalent
2 and
3 and
Note: 2 and
3 can be marked vertically or
horizontally
27
Qu. Points of Expected Response Illustrative Scheme
6(a) 1 Equate expressions and rearrange
2 First solution for 2t
3 Find t
1
2
3 (seconds)
6(b) 4 Find second solution
4 (seconds)
6(c) #2.2
Interpret answer: Find next pair of
solutions #
2.2 (seconds)
Note: Where candidate has worked in degrees do
not award 2; t = 30 can be awarded
3;
4 for 150
and #2.2
for 210, 330.
Do not penalise variations in rounding.
10 + #2.2
R&C Outcome 1.3: Differentiation
7 1 Prepare to differentiate
2 Differentiate first term
3 Differentiate second term
1
2
3
8 1 Differentiate correctly
2 Find gradient
3 Find y
4 Equation of tangent
1
2
3
4 (leads to )
9(a) 1 Evaluate for t = 0
1 75 (cm)
9(b) 2 Substitute and evaluate for t = 3
#2.2
Interpret answer
2 0
#2.2
The ball has hit the pole.
9(c) 3 Differentiate
4
Substitute and evaluate for t = 2
3
4 (cm/s)
10 1 Differentiate correctly
1
12 + #2.2
R&C Outcome 1.4: Integration
11
1 Prepare to integrate
2 Integrate
3 Constant of integration*
1
2
3 *
12 1 Integrate correctly
1 *
13 #2.1
Strategy: Know to integrate
1 Integrate
2 Find c and state equation
#2.1
1 *
2
14 1 Start integration
2 Complete integration
3 Substitute limits
4 Evaluate
1
2
3
4 or
Note: 3 and
4 are only available as a result of an
attempt to integrate.
10 + #2.1
* awarded for the constant
appearing at least once in
Q11, 12 or 13.
28
Relationships and Calculus Practice Assessment B
R&C Assessment Standard 1.1
1. The graph below shows the function .
(a) Show that is a root of .
(b) Find algebraically the coordinates of points A and B.
(2)
(2)
(#2.2
)
2. The function has equal roots.
Find the possible values for .
(3)
R&C Assessment Standard 1.2
3. Solve the equation in the domain . (3)
4. Solve for . (5)
5. When a doorbell is pressed a solenoid rapidly fires a plunger out and in to ring the bell.
The position of the tip of the plunger is given by the equation , where x is
the distance of the tip of the plunger from its normal position, in millimetres, and t is the
time in seconds since the doorbell was pressed.
Each time the plunger makes contact with the bell.
Find the first two times when the plunger will hit the bell after the doorbell is pressed.
(#2.1
)
(2)
x
y y = f(x)
5 B A O
Solenoid Plunger Bell
29
R&C Assessment Standard 1.3
6. Differentiate the expressions below with respect to x.
(a) , (b)
,
(c) (d)
(4)
(2)
7. A student is opening a bottle of champagne to celebrate her birthday. The cork pops and
is fired upwards. The velocity, in metres per second, of the cork t seconds after it pops can
be represented by the formula .
(a) At what velocity is the cork travelling immediately after it pops?
(b) How long does it take the cork to reach its maximum height?
(c) What is the acceleration of the cork when ?
(1)
(#2.1
)
(1)
(1)
8. A curve has equation .
Find the equation of the tangent to the curve at the point (1, –6).
(3)
R&C Assessment Standard 1.4
9. Complete the integrals below.
(a)
(b)
(c) (d)
(4)
(2)
10. A function exists such that .
The function goes through the point (7, 0).
Find .
(4)
(#2.2
)
11. Find
.
(4)
End of question paper
30
Relationships and Calculus, Practice Assessment B Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
R&C Outcome 1.1: Quadratics and Polynomials
1(a) 1 Direct substitution OR strategy to start
factorising (e.g. synthetic division)
2 Show remainder = 0 and clear
communication of result.
1
OR 5
1 –3 –13 15
2 therefore is a root
OR 5
1 –3
5
–13
10
15
-15
1 2 –3 0
Remainder = 0 so is a root.
1(b) 3 Quadratic factor
4 Complete factorisation
#2.2
Interpret answer: point A and B
3
4 …
#2.2
A(–3, 0) and B(1, 0)
Note: Trial and error to find A and B receives no
credit.
2 1 Substitute into the discriminant
2 Simplify and start to solve equation
3 Solve equation
1
2
3
7 + #2.2
R&C Outcome 1.2: Solving Trig Equations
3 1 Rearrange to sin…
2 Solve for
3 Solve for x
1
2
3
Note: Alternative for 1 and
2
2 leading to
3 leading to
4 1 Correct substitution for
2
Simplify
3 Factorise
4 Solve for
5 Solve for
1
2
3
4 and
5 ;
Note: Alternative for 4 and
5
4 leading to
5
leading to
5 #2.1
Strategy: equate and start to solve
1 Find values for
2 Find t
#2.1
1
2 and (seconds)
Note: Alternative for 2 and
3
1 leading to
2 leading to
31
10 + #2.1
Qu. Points of Expected Response Illustrative Scheme
R&C Outcome 1.3: Differentiation
6(a) 1 Prepare to differentiate
2 Differentiate
1
2
6(b) 3 Prepare to differentiate
4 Differentiate
3
4
6(c) 5 Differentiate correctly
5
6(d) 6 Differentiate correctly
6
7(a) 1 Differentiate
1 16 (m/s)
7(b) #2.1
Strategy: Know to solve
2 Solve
#2.1
2 (s)
7(c) 3 Differentiate
3 (ms
-2)
8 1 Differentiate correctly
2 Find gradient
3 Equation of tangent
1
2
3
(leads to )
13 + #2.1
R&C Outcome 1.4: Integration
9(a)
1 Prepare to integrate
2 Integrate
3 Constant of integration*
1
2
3 *
9(b)
4 Integrate
4
*
9(c)
5 Integrate
5 *
9(d) 6 Integrate
* this mark is awarded for the constant
appearing at least once in Q9 or 10
6
*
10 1 Know to integrate
2 Start integration
3 Complete integration
4
Substitute point
#2.2
Interpret answer: state
1
2
3
*
4
*
#2.2
11 1 Start integration
2 Complete integration
3 Substitute limits
4 Evaluate
1
2
3
4 or
Note: 3 and
4 are only available as a result of an
attempt to integrate.
Substituting into the integrand receives no credit.
Candidates who differentiate gain no credit.
32
Relationships and Calculus Practice Assessment C
R&C Assessment Standard 1.1
1. The graph below shows the function .
(a) Show that is a root of .
(b) Solve algebraically .
(2)
(3)
2. A quadratic equation goes through the points (0, –6) and (7, 0).
One of the factors of the equation is
State the roots of the quadratic equation.
(#2.2
)
3. The function has two distinct real roots.
What are the possible values for ?
(3)
R&C Assessment Standard 1.2
4. Solve
for and where cosx .
(3)
5. 1. In a child’s bedroom a toy hot air balloon is suspended from the ceiling by
a spring.
2. The balloon is pulled down and then released.
3. Its height in centimetres, t seconds after release, is given by the equation
.
Find the first two times when the balloon will reach 205 cm.
(#2.1
)
(4)
x
y y = f(x)
O
33
R&C Assessment Standard 1.3
6. Differentiate the expressions below with respect to x.
(a)
; (b)
;
(c) (d)
(4)
(2)
7. A bottle rocket is fired vertically upwards. The height (in metres) of the bottle rocket t
seconds after it is fired can be represented by the formula .
The velocity of the bottle rocket at time t is given by
.
(a) At what times is the bottle at ground level?
(b) Find an expression for the velocity of the bottle rocket.
(c) A student claims that after 10 seconds the bottle rocket will be travelling at –112 m/s.
Comment on the validity of the students’ claim.
(1)
(1)
(#2.2
)
8. A curve has equation .
Find the equation of the tangent that intersects the curve when .
(4)
R&C Assessment Standard 1.4
9. Integrate each expression below.
(a)
, (b)
(c) (d)
(4)
(2)
10. The rate of change of the function is given by the expression ,
. Find .
(#2.1
)
(2)
11. Find
.
(4)
End of question paper
34
Relationships and Calculus, Practice Assessment C Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
R&C Outcome 1.1: Quadratics and Polynomials
1(a) 1 Direct substitution OR strategy to start
factorising (e.g. synthetic division)
2 Show remainder = 0 and clear
communication of result.
1 OR
–2
1 3
–10
–24
2 therefore is a factor (or is a
root) OR
–2
1 3
–10
–24
1 1 –12 0
Remainder = 0 so therefore is a factor (or
is a root).
1(b) 3 Quadratic factor
4 Complete factorisation
5 State roots
3
4
5
2 #2.2
Communicate solution from the context #2.2
3 1 Know to use discriminant and substitute
2 Simplify and start to solve inequality
3 Solve inequality
1
2
3
7 + #2.2
R&C Outcome 1.2: Solving Trig Equations
4 1 Substitute for
2 Simplify
3 Solve for x
1
2
3
5 #2.1
Strategy: Start to solve
1 Solve for
2 Solve for
3 Solve for
4 Solve for
#2.1
1
2 and
3 and
4 and (seconds)
35
Qu. Points of Expected Response Illustrative Scheme
R&C Outcome 1.3: Differentiation
6(a) 1 Prepare to differentiate
2 Differentiate
1
2
6(b) 3 Prepare to differentiate
4 Differentiate
3
4
6(c) 5 Differentiate
5
6(d) 6 Differentiate
6
7(a) 1 Solve for
1 0 and 6 (seconds)
7(b) 2 Differentiate correctly
2
7(c) #2.2
Interpret students’ claim
(any indication that the equations are only
valid during the first 6 seconds of flight)
#2.2
The claim is not valid. The rocket hits the
ground after 6 seconds and after that the velocity
will not continue to change in the same manner.
8 1 Differentiate correctly
2 Find gradient
3 Find y
4 Equation of tangent
1
2
3
4
(leads to )
12 + #2.2
R&C Outcome 1.4: Integration
9(a)
1 Prepare to integrate
2 Integrate
3 Constant of integration*
1
2
3 *
9(b)
4 Integrate
4
*
9(c)
5 Integrate
5 *
9(d)
6 Integrate
6
*
*Note: 3 is awarded for constant of integration
appearing at least once in Q9 or Q10.
10 #2.1
Know to integrate
1 Start integration
2 Complete integration
#2.1
1
2 *
11 1 Start integration
2 Complete integration
3 Substitute limits
4 Evaluate
1
2
3
4
Note: 3 and
4 are only available as a result of an
attempt to integrate.
Substituting into the integrand receives no credit.
Candidates who differentiate gain no credit.
12 + #2.1
36
Applications Practice Assessment A
Applications Assessment Standard 1.1
1. Find the equation of the line perpendicular to that passes through the point
(1, –3).
(3)
2. OABCDE is a regular hexagon.
Find the equation of the line going through EB.
(#2.1
)
(2)
Applications Assessment Standard 1.2
3. A graphic designer is designing a logo for a company called “Cooper’s”. He’s basing the
design on seven congruent circles. Each circle touches the next one and their centres lie on
the x-axis.
The diagram shows his first draft.
The “c” circle is centred at the origin with a diameter of 8 units.
Find the equation of the “s” circle.
(2)
4. Determine algebraically how many times the line intersects the circle with
equation .
(3)
(#2.2
)
y
x
A
D
C
B
y
x O
E(1, )
120°
37
Applications Assessment Standard 1.3
5. A sequence is defined by the recurrence relation .
(a) Given that find .
(b) Explain why this sequence does not have a limit.
(2)
(1)
6. A gamekeeper rears pheasants to release onto a country estate for hunting. Each year he
releases 2000 birds for the start of the hunting season.
The gamekeeper expects that 80% of the birds will leave the estate or be hunted.
(a) Set up a recurrence relation which shows the number of pheasant on the estate
immediately after the birds are released.
(b) In the long term, if the gamekeeper continues in this way, how many birds will be on
the estate for the start of each hunting season?
(1)
(2)
Applications Assessment Standard 1.4
7. A company is investigating different cylindrical containers for their fruit juice.
A one litre cylinder has the surface area , where x is the radius of
the base in centimetres.
The containers are to be manufactured from a material that costs 0·02 pence per square
centimetre. If the cost exceeds 10 pence per carton then a new material will be
investigated.
By minimising the surface area deduce whether the company will have to investigate a new
packaging material. You must justify your answer fully.
(5)
(#2.2
)
38
8. On a velocity-time graph the area between a curve and the time axis is equal to the
distance that the object has travelled.
A satellite is being launched aboard a rocket.
The velocity-time graph below shows the change in the velocity of the rocket for the first 5
seconds after launch.
Calculate the distance travelled by the rocket in the first 5 seconds.
(#2.1
)
(4)
9. The line with equation and the curve with equation are shown below.
The line meets the curve at the points where , and .
The two shaded areas are equal in size.
Calculate the total shaded area.
(5)
End of question paper
5 t
v
O
Vel
oci
ty
(m/
s)
Time (seconds)
2 x
y
O –2
39
Applications Practice Assessment A Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
Applications Outcome 1.1: The Straight Line
1 1 Find gradient of perpendicular line
2 Equation of straight line
1
2
(leads to
)
Do not penalise error subsequent to 2.
2 #2.1
Strategy
1 Gradient of line
2 Equation of straight line
#2.1
Evidence of and attempt to find
acute angle OR locate point B(3, )
1 or 1·73
2
(leads to )
Do not penalise error subsequent to 2.
4 + #2.1
Applications Outcome 1.2: Circles
3 1 Find centre and radius
2 Equation of circle
1 C(48, 0) and
2
For 2 do not accept
4 1 Substitute for y
2 Simplify
3 Discriminant OR find roots
#2.2
Interpret solution
1
2
3 OR (repeated)
#2.2
Discriminant = 0/repeated root/only one root so
the line is a tangent to the circle /so the line
intersects the circle once.
5 + #2.2
Applications Outcome 1.3: Recurrence Relations
5(a) 1 Evidence of substitution
2 Calculate
1 (= –2)
2
5(b) 3 Understand conditions for a limit to
exist
3 This sequence does not have a limit because
the coefficient of is greater than 1.
Do not accept unless a is clearly identified as
the coefficient of .
6(a) 1 State recurrence relation
1
6(b) 2 Know how to find limit
3 Process limit
2 OR
3 2500
6
40
Qu. Points of Expected Response Illustrative Scheme
Applications Outcome 1.4: Applications of Calculus
7 1 Find derivative
2 Set derivative equal to zero
3 Solve for x
4 Justify nature (minimum)
5 Find cost of carton
#2.2
Explain answer in context
1
2
3
4 Nature table OR 2
nd derivative
x → 5·42 → OR
A’(x) – 0 +
Shape \ _ /
5 11·07 p (allow variations in rounding)
#2.2
e.g. A different material should be investigated
because the carton is too expensive to produce.
8 #2.1
Know to integrate and include limits
1 Include dt *
2 Integrate
3 Substitute limits
4 Evaluate
#2.1
1 *
2
3
4 219∙3 (metres)
Notes
* Include dt in Q9 or dx in Q10.
Candidates who substitute without integrating, only
#2.1
and 1
are available.
Candidates who differentiate at 2, only #
2.1 and
1
are available.
As area is above x-axis 4 is only available for a
positive answer.
9 1 Know to integrate and include limits
2 Use “upper – lower”
3 Integrate
4 Substitute limits
5 Double to find total area
1
*
2
3
4
5 Total Area = 8 sq. units
Notes: Candidates who integrate between -2 and 2 lose
1
(incorrect limits) and also 5 if resulting area is 0.
Candidates who differentiate at 3, only
1 and
2
are available.
Candidates who substitute without integrating, only
1 and
2 are available.
Candidates who integrate “lower – upper” lose 2
and also 5 unless area is stated as positive.
14 + #2.1
+ #2.2
41
Applications Practice Assessment B
Applications Assessment Standard 1.1
1. ABC is a triangle.
Side AB has equation
and point C has
coordinates (–1, 7).
Find the equation of the line CD, that is
perpendicular to AB.
(2)
2. A rectangle is drawn as shown.
Calculate the equation of BC.
(#2.1
)
(2)
3. In the building industry roofs are categorised based on the steepness of their slope.
Roof Category Flat Low-Sloped High-Sloped Mansard
Gradient
Which category does the roof in the picture below belong to?
Justify your answer mathematically.
(1)
(#2.2
)
A
C(–1, 7)
D
B
y
x O
32°
A
B(6, 1)
C
D
y
O x
149°
42
Applications Assessment Standard 1.2
4. A popular cartoon character’s head and body are based on two overlapping circles.
On suitable axes the larger circle, representing the body, is centred at the origin with
diameter 20 units. The smaller circle, representing the head, lies on the y-axis and has half
the diameter of the body.
Find the equation of the circle representing the head whose centre lies on the intersection of
the y-axis and the circumference of the large circle.
(2)
5. Determine algebraically how many times the line intersects the circle with
equation .
(3)
(#2.2
)
Applications Assessment Standard 1.3
6. Two sequences are defined as
and .
(a) Given that and find the next two terms of each sequence.
(b) Explain why only one of these sequences has a limit.
(3)
(1)
7. Swimming pools need to be treated regularly to control the growth of microorganisms.
Tony uses a chlorine based liquid to treat his pool on the same day every week. The
treatment increases the concentration of chlorine by 0∙8 mg/l.
Every week the concentration of chlorine in the pool gradually decreases to 55% of its
original amount.
(a) Set up a recurrence relation which shows the concentration of chlorine in the pool
immediately after a treatment.
(b) In the long term, if Tony continues this schedule, what is the maximum concentration
of chlorine that will be in the pool?
(1)
(2)
y
x
43
Applications Assessment Standard 1.4
8. The height of a section of rollercoaster is defined by the equation
, as
shown in the graph.
Find the minimum height where .
(4)
9. The cross-section of a fan blade is defined by the curve with equation
and the line .
(a) Find the x-coordinates of the points where the lines intersect.
(b) Hence find the area of the cross-section.
(2)
(#2.1
)
(4)
End of question paper
x
y
6 –6 O
44
Applications Practice Assessment B Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
Applications Outcome 1.1: The Straight Line
1 1 Find perpendicular gradient
2 Equation of straight line
1
2
(leads to
)
Do not penalise error subsequent to 2.
2 #2.1
Strategy
1 Find gradient of BC
2 Equation of BC
#2.1
Evidence of and use acute angle
1
2
(leads to )
Do not penalise error subsequent to 2.
3 1 Use
#2.2
Interpret gradient
1 (may be implied by #
2.2)
#2.2
Gradient is 0·62 so the roof is “High Sloped”.
5 + #2.1
+ #2.2
Applications Outcome 1.2: Circles
4 1 Find centre and radius
2 Equation of circle
1 C(0, 10) and
2
For 2 do not accept
5 1 Substitute for y
2 Simplify
3 Discriminant
#2.2
Interpret solution
1
2
3
#2.2
Discriminant < 0 so the line does not intersect
the circle
5 + #2.2
Applications Outcome 1.3: Recurrence Relations
6(a) 1 Start to substitute
2 Find and
3 Find and
1
or
2 and
3 and
6(b) 4 Understand conditions for a limit to
exist
4 Sequence u has a limit because the coefficient
of un is between –1 and 1.
Do not accept unless a is clearly
identified as the coefficient of .
7(a) 1 State recurrence relation
1
7(b) 2 Know how to find limit
3 Process limit
2 OR
3 1·78 (mg/l)
7
45
Qu. Points of Expected Response Illustrative Scheme
Applications Outcome 1.4: Applications of Calculus
8 1 Find derivative
2 Set derivative equal to zero
3 Solve for x and select negative value
4 Evaluate height
1
2
3
4
9(a) #2.1
Strategy: start to find intersections
1 Find x-coordinates
#2.1
1 and
9(b) 2 Know to integrate and include limits
3 Use “upper – lower”
4 Integrate
5 Substitute limits
6 Evaluate
2
*
3
4
5
6 square units
Notes: * dx must be present to gain
2
Candidates who differentiate at 4, only
2 and
3
are available.
Candidates who substitute without integrating, only
2 and
3 are available.
Candidates who integrate “lower – upper” lose 3
and also 6 unless area is stated as positive.
10 + #2.1
46
Applications Practice Assessment C
Applications Assessment Standard 1.1
1. Find the equation of the line perpendicular to
that passes through the
point (–2, 1).
(2)
2. ABCD is a square.
Side AD has equation and point B has coordinates (4, –1).
Find the equation of the side BC.
(2)
3. The diagram shows two lines that intersect at point P.
Find whether the two lines are perpendicular.
You must justify your answer mathematically.
(#2.1
)
(3)
(#2.2
)
A
B(4, –1)
C
D
–3x + y – 7 = 0
y
O x
y
x 160°
P O
2y = 12 – 5x
47
Applications Assessment Standard 1.2
4. The design for a new board game consists of a 5 congruent circles inside a square that is 60
cm wide. The circles are positioned as shown.
The board is shown relative to suitable axes.
Write down the equation of the circle in the upper right corner of the board.
(2)
5. Describe the intersection, if any, between the line and the circle given by the
equation .
(3)
(#2.2
)
Applications Assessment Standard 1.3
6. A sequence is defined by the recurrence relation .
Given that and , find the value of a.
(3)
7. A school population fluctuates each year as pupils join and leave.
One school admits 190 new pupils at the beginning of each school year.
At the end of the year the population will have decreased to 80% due to pupils leaving.
(a) Set up a recurrence relation which shows the size of the school population just after
they have admitted a new batch of pupils.
(b) If this pattern continues long term, will the school population ever exceed 1000?
(1)
(3)
y
x O 60
60
48
Applications Assessment Standard 1.4
8. The volume of gas produced by a particular chemical reaction can be varied by changing
the temperature.
The volume of gas produced by the reaction is given by the function
,
where T is the temperature in Kelvin.
Find the temperature that maximises the volume of gas produced.
(4)
9. The curve with equation is shown below for .
Calculate the shaded area.
(#2.1
)
(4)
10. An area is enclosed above and below by the curves
and respectively.
The two curves intersect when and .
Calculate the enclosed area.
(5)
End of question paper
x
y
2π π
1
–1
O
49
Applications, Practice Assessment C Mark Scheme
Qu. Points of Expected Response Illustrative Scheme
Applications Outcome 1.1: The Straight Line
1 1 Find perpendicular gradient
2 Equation of straight line
1
2
(leads to )
Do not penalise error subsequent to 2.
2 1 Find parallel gradient
2 Equation of straight line
1
2
(leads to )
Do not penalise error subsequent to 2.
3 #2.1
Strategy
1 Find gradient of first line
2 Use
3 Find gradient of second line
#2.2
Interpret answers
#2.1
Evidence of attempt to find gradient of each
line and find
1
2
3
#2.2
so the two lines are not
perpendicular 7 + #
2.1 + #
2.2
Applications Outcome 1.2: Circles
4 1 Find centre and radius
2 Equation of circle
1 C(45, 45) and
2
For 2 do not accept
5 1 Substitute for y
2 Simplify
3 Discriminant OR find roots
#2.2
Interpret solution
1
2
3 OR (repeated)
#2.2
Discriminant = 0/repeated root/only one root so
the line is a tangent to the circle /so the line
intersects the circle once.
5 + #2.2
Applications Outcome 1.3: Recurrence Relations
6 1 Start to substitute
2 Expression for
3 Solve to find a
1
2
3
7(a) 1 State recurrence relation
1
7(b) 2 Know how to find limit
3 Process limit
4 Interpret limit
2 OR
3 950
4 If the pattern continues the school population
will not exceed 1000.
7
50
Qu. Points of Expected Response Illustrative Scheme
Applications Outcome 1.4: Applications of Calculus
8 1 Find derivative
2 Set derivative equal to zero
3 Solve for K
4 Justify nature (maximum)
1
2
3 (Kelvin)
4 Nature table OR 2
nd derivative
x → 337 → OR
– 0 +
Shape / - \
9 #2.1
Know to integrate and include limits
1 Set up integral *
2 Integrate
3 Substitute limits
4 Evaluate and state area (positive)
#2.1
1
*
2
3
4
(square units)
Notes
* Include dx at least once in Q9 or Q10.
Candidates who substitute without integrating, only
#2.1
and 1 are available.
Candidates who differentiate at 2, only #
2.1 and
1
are available.
As area is above x-axis 4 is only available for a
positive answer.
10 1 Know to integrate and include limits
2 Use “upper – lower”
3 Integrate
4 Substitute limits
5 Evaluate
1
*
2
3
4
5
or
Notes: Candidates who differentiate at
3, only
1 and
2
are available.
Candidates who substitute without integrating, only
1 and
2 are available.
Candidates who integrate “lower – upper” lose 2
and also 5 unless area is stated as positive.
Finding area between x = 0 and x = 2 and doubling
answer is perfectly acceptable.
13 + #2.1