Dalkeith High School CfE Higher Mathematics Unit ......9 Qu. Points of Expected Response...

1

Dalkeith High School

CfE Higher Mathematics

Unit Assessment Revision

Booklet

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)


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)


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


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


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


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


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


ratio 3:5

Method 3

1

2

3 or equivalent

#2.2


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


1. (a) Simplify fully

.

(b) Express

in the form .

(2)

(2)

2. Given that , find x correct to 2 decimal places.

(#2.1

)

(1)


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

)


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 .


(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)


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.


= –8i + 2j + 2k

= 2i + 10j – 2k

= i + 7j + 7k


(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)


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



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


3 1 Expand

2 Compare coefficients

3 Find k

4 Find a

1 stated explicitly


3

4

4 1 Process missing sides



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



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 .



1

2

11 + #2.2

5

-5

O

x

y

15



11


2 Find second vector and interpret

multiple


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



3 Process and state point N

1

2

OR

3 N = (19, 13, –7)


1

2

3 N = (19, 13, –7)


2 Identify


1

2

3 OR

Do not award 3 for (–9 , –5, –5)

14

1 Find component vectors


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


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)


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)


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:

, .


(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


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.


= –6i –2j + 4k

= 2i + 10j + 8k

= –3i + 5j + 5k


(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)


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



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


4 1 Expand

2 Match coefficients

3 Find k

4 Find a

1 stated explicitly


3 or equivalent

4

5 1 Process missing sides



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



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.



1

2

19 + #2.2

y

x

1

7 O

(9, 1)

O

(-4, -3)

(1, -1)

y

x

22



13

1 Find vector between two points and

take out common factor

2 Find second vector and take out

common factor


#2.2

Explain solution in context

Alternative for 1 and

2


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




3 Process and state point G

1

2

3 G = (10, –9, 10)


1

2

3 G = (10, –9, 10)


2 Identify


1

2 –

3 OR

Do not award 3 for (–5 , –5, –3)

16

1 Find component vectors


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)


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

)


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


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)


x

y y = f(x)

5

O

26

Relationships and Calculus, Practice Assessment A Mark 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


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)


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


1. The graph below shows the function .


(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)


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


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)


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)


30

Relationships and Calculus, Practice Assessment B Mark Scheme




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


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 ;


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)


3

1 leading to

2 leading to

31

10 + #2.1



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)


3 (ms

-2)


2 Find gradient


1

2

3

(leads to )

13 + #2.1


9(a)


2 Integrate


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


4

Substitute point

#2.2

Interpret answer: state

1

2

3

*

4

*

#2.2



3 Substitute limits

4 Evaluate

1

2

3

4 or

Note: 3 and



Substituting into the integrand receives no credit.

Candidates who differentiate gain no credit.

32

Relationships and Calculus Practice Assessment C


1. The graph below shows the function .


(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)


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


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)


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)


34

Relationships and Calculus, Practice Assessment C Mark Scheme




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


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



6(a) 1 Prepare to differentiate

2 Differentiate

1

2

6(b) 3 Prepare to differentiate

4 Differentiate

3

4


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.


2 Find gradient

3 Find y


1

2

3

4

(leads to )

12 + #2.2


9(a)


2 Integrate


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


#2.1

1

2 *



3 Substitute limits

4 Evaluate

1

2

3

4

Note: 3 and



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)


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


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)


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)


5 t

v

O

Vel

oci

ty

(m/

s)

Time (seconds)

2 x

y

O –2

39

Applications Practice Assessment A Mark 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


#2.1

Evidence of and attempt to find

acute angle OR locate point B(3, )

1 or 1·73

2

(leads to )


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


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.


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


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


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

)


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


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)


x

y

6 –6 O

44

Applications Practice Assessment B Mark Scheme



1 1 Find perpendicular gradient


1

2

(leads to

)


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 )


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




1 C(0, 10) and

2

For 2 do not accept


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


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 .


1


3 Process limit

2 OR

3 1·78 (mg/l)

7

45



8 1 Find derivative


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


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.


2 and

3 are available.



10 + #2.1

46

Applications Practice Assessment C


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


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

)


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


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)


x

y

2π π

1

–1

O

49

Applications, Practice Assessment C Mark Scheme



1 1 Find perpendicular gradient


1

2

(leads to )


2 1 Find parallel gradient


1

2

(leads to )


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




1 C(45, 45) and

2

For 2 do not accept


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


6 1 Start to substitute

2 Expression for

3 Solve to find a

1

2

3


1


3 Process limit

4 Interpret limit

2 OR

3 950

4 If the pattern continues the school population

will not exceed 1000.

7

50



8 1 Find derivative


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.


#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


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.


1 and

2 are available.



Finding area between x = 0 and x = 2 and doubling

answer is perfectly acceptable.

13 + #2.1

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Dalkeith High School CfE Higher Mathematics Unit ......9 Qu. Points of Expected Response...

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