HL Differentiation Mark Scheme

!!!!WATCH OUT THERE ARE LOTS OF QUESTIONS IN HERE THAT ARE NOT IN THE QUESTIONS –

LOOK ACREFULLY AT THE QUESTION NUMBERS!!!!

1. [5 marks]

Markscheme

attempt to substitute x = −1 or x = 2 or to divide polynomials      (M1)

1 − p − q + 5 = 7, 16 + 8p + 2q + 5 = 1 or equivalent      A1A1

attempt to solve their two equations M1

p = −3, q = 2      A1

[5 marks]

2a. [3 marks]

Markscheme

recognising normal to plane or attempting to find cross product of two vectors lying in the plane     

(M1)

for example,       (A1)

     A1

[3 marks]

2b. [4 marks]

Markscheme

EITHER

     M1A1

1

OR

     M1A1

Note: M1 is for an attempt to find the scalar or vector product of the two normal vectors.

     A1

angle between faces is       A1

[4 marks]

2c. [3 marks]

Markscheme

or       (A1)

     (M1)

     A1

[3 marks]

2d. [4 marks]

Markscheme

METHOD 1

line AD : (r =)      M1A1

intersects  when       M1

so       A1

2

hence P is the midpoint of AD      AG

 

METHOD 2

midpoint of AD is (0.5, 0, 0.5)      (M1)A1

substitute into       M1

0.5 + 0.5 − 0.5 = 0     A1

hence P is the midpoint of AD     AG

[4 marks]

2e. [5 marks]

Markscheme

METHOD 1

      A1A1A1

     A1

area       A1

 

METHOD 2

line BD : (r  =)

     (A1)

3

    A1

area =       M1

    A1

Note: This A1 is dependent on M1.

area =       A1

[5 marks]

3a. [2 marks]

Markscheme

attempt at chain rule or product rule     (M1)

     A1

[2 marks]

3b. [5 marks]

Markscheme

sin θ = 0     (A1)

θ = 0,       A1

obtaining cos θ = sin θ      (M1)

tan θ = 1     (M1)

4

     A1

[5 marks]

4a. [4 marks]

Markscheme

     (M1)(A1)

     A1

     A1

[4 marks]

4b. [2 marks]

Markscheme

    (M1)

= 12     A1

[2 marks]

5. [6 marks]

Markscheme

EITHER

     M1

     A1

OR

     M1A1

5

THEN

or       A1

or          (M1)A1   

Note: (M1) is for an appropriate use of a log law in either case, dependent on the previous M1 being

awarded, A1 for both correct answers.

solution is       A1

[6 marks]

6a. [2 marks]

Markscheme

    M1A1

Note: M1 is for use of the chain rule.

[2 marks]

6b. [7 marks]

Markscheme

attempt at integration by parts     M1

     (A1)

      A1

using integration by substitution or inspection      (M1)

      A1

6

Note: Award A1 for   or equivalent.

Note: Condone lack of limits to this point.

attempt to substitute limits into their integral     M1

     A1

[7 marks]

7. [5 marks]

Markscheme

EITHER

or  or  …      (M1)(A1)

Note: Award M1 for any one of the above, A1 for having final two.

OR

     (M1)(A1)

Note: Award M1 for one of the angles shown with b clearly labelled, A1 for both angles shown. Do

not award A1 if an angle is shown in the second quadrant and subsequent A1 marks not awarded.

THEN

or       (A1)(A1)

     A1

7

[5 marks]

8a. [5 marks]

Markscheme

attempt to differentiate      (M1)

     A1

Note: Award M1 for using quotient or product rule award A1 if correct derivative seen even

in unsimplified form, for example  .

     M1

     A1

     A1

[5 marks]

8b. [5 marks]

Markscheme

     M1

     A1

Note: Award A1 for correct derivative seen even if not simplified.

     A1

hence (at most) one point of inflexion      R1

Note: This mark is independent of the two A1 marks above. If they have shown or stated their equation

has only one solution this mark can be awarded.

 changes sign at        R1

8

so exactly one point of inflexion

[5 marks]

8c. [3 marks]

Markscheme

      A1

     (M1)A1

Note: Award M1 for the substitution of their value for  into  .

[3 marks]

8d. [4 marks]

Markscheme

A1A1A1A1

A1 for shape for x < 0

A1 for shape for x > 0

A1 for maximum at A

A1 for POI at B.

9

Note: Only award last two A1s if A and B are placed in the correct quadrants, allowing for follow

through.

[4 marks]

9. [4 marks]

Markscheme

cos θ =        (M1)

     A1A1

Note: A1 for correct numerator and A1 for correct denominator.

      A1

[4 marks]

 

10a. [5 marks]

Markscheme

attempt to make the subject of        M1

      A1

     A1

     A1

Note: Do not allow  in place of .

     A1

Note: The final A mark is independent.

[5 marks]

10

10b. [2 marks]

Markscheme

     A1A1

[2 marks]

10c. [3 marks]

Markscheme

hyperbola shape, with single curves in second and fourth quadrants and third quadrant blank,

including vertical asymptote       A1

horizontal asymptote       A1

11

intercepts       A1

[3 marks]

10d. [4 marks]

Markscheme

the domain of  is       A1A1

the range of  is       A1A1

[4 marks]

11. [6 marks]

Markscheme

valid attempt to find       M1

      A1A1

attempt to solve       M1

     A1A1

[6 marks]

12a. [2 marks]

Markscheme

attempt at product rule      M1

      A1

[2 marks]

12b. [1 mark]

Markscheme

12

      A1

[1 mark]

12c. [4 marks]

Markscheme

METHOD 1

Attempt to add  and        (M1)

    A1

(or equivalent)      A1

Note: Condone absence of limits.

    A1

 

METHOD 2

OR       M1A1

     A1

    A1

[4 marks]

13a. [7 marks]

Markscheme

differentiating implicitly:       M1

13

     A1A1

Note: Award A1 for each side.

if   then either  or         M1A1

 two solutions for        R1

 not possible (as 0 ≠ 5)     R1

hence exactly two points      AG

Note: For a solution that only refers to the graph giving two solutions at    and no solutions

for  award R1 only.

[7 marks]

13b. [5 marks]

Markscheme

at (2, 1)        M1

     (A1)

gradient of normal is 2       M1

1 = 4 + c       (M1)

equation of normal is       A1

[5 marks]

13c. [3 marks]

Markscheme

substituting      (M1)

or         (A1)

14

      A1

[3 marks]

13d. [7 marks]

Markscheme

recognition of two volumes      (M1)

volume        M1A1A1

Note: Award M1 for attempt to use  , A1 for limits, A1 for   Condone omission of at this

stage.

volume 2

EITHER

     (M1)(A1)

OR

     (M1)(A1)

THEN

total volume = 19.9      A1

[7 marks]

14a. [5 marks]

Markscheme

attempt at implicit differentiation      M1

     A1M1A1

Note: Award A1 for first two terms. Award M1 for an attempt at chain rule A1 for last term.

     A115

     AG

[5 marks]

14b. [4 marks]

Markscheme

EITHER

when       M1

    (A1)

OR

 or equivalent      M1

     (A1)

THEN

therefore       A1

or       A1

[4 marks]

14c. [3 marks]

Markscheme

m1 =      M1A1

m2 =       A1

m1 m2 = 1     AG

16

Note: Award M1A0A0 if decimal approximations are used.

Note: No FT applies.

[3 marks]

14d. [7 marks]

Markscheme

equate derivative to −1    M1

     (A1)

     R1

in the first case, attempt to solve       M1

(0.486,0.486)      A1

in the second case,  and       (M1)

(0,1), (1,0)      A1

[7 marks]

15. [8 marks]

Markscheme

    M1A1

 

Note:     Differentiation wrt is also acceptable.

 

    (A1)

 

Note:     All following marks may be awarded if the denominator is correct, but the numerator incorrect.

17

 

    M1

EITHER

    M1A1

    A1

    A1

OR

    M1

    A1

    A1

    A1

[8 marks]

16a. [5 marks]

Markscheme18

attempt to use quotient rule or product rule     M1

    A1A1

 

Note:     Award A1 for or equivalent and A1 for or equivalent.

 

setting     M1

or equivalent     A1

    AG

[5 marks]

16b. [2 marks]

Markscheme

    A1A1

 

Note:     Award A1 for and A1 for . Accept .

 

[2 marks]

16c. [3 marks]

Markscheme

19

concave up curve over correct domain with one minimum point above the -axis.     A1

approaches asymptotically     A1

approaches asymptotically     A1

 

Note:     For the final A1 an asymptote must be seen, and must be seen on the -axis or in an equation.

 

[3 marks]

16d. [4 marks]

Markscheme

    (A1)

attempt to solve for     (M1)

    A120

    A1

[4 marks]

16e. [3 marks]

Markscheme

    (M1)(A1)

 

Note:     M1 is for an integral of the correct squared function (with or without limits and/or ).

 

    A1

[3 marks]

17a. [2 marks]

Markscheme

(or equivalent)     (M1)A1

[2 marks]

17b. [4 marks]

Markscheme

21

     A1A1A1A1

 

Note:     Award A1 for correct behaviour at , A1 for correct domain and correct behaviour for

, A1 for two clear intersections with -axis and minimum point, A1 for clear maximum point.

 

[4 marks]

17c. [2 marks]

Markscheme

    A1

    A1

[2 marks]

17d. [2 marks]

Markscheme

attempt to write in terms of only     (M1)

    A1

[2 marks]

22

17e. [3 marks]

Markscheme

    (A1)

attempt to use     (M1)

    A1

[3 marks]

17f. [2 marks]

Markscheme

    M1

(or equivalent)     A1

    AG

[2 marks]

17g. [3 marks]

Markscheme

or     (M1)

    A1

    A1

 

Note:     Only accept answers given the required form.

 

23

[3 marks]

18a. [4 marks]

Markscheme

the width of the rectangle is and let the height of the rectangle be

     (A1)

     (A1)

     M1A1

[4 marks]

18b. [5 marks]

Markscheme

     A1

     M1

     (A1)

hence the width is      A1

     R1

hence maximum     AG

[5 marks]

18c. [2 marks]

Markscheme

EITHER

24

     M1

    A1

     AG

OR

     M1

     A1

     AG

[2 marks]

19a. [2 marks]

Markscheme

A1 for correct shape

25

A1 for correct and intercepts and minimum point

[2 marks]

19b. [4 marks]

Markscheme

A1 for correct shape

A1 for correct vertical asymptotes

A1 for correct implied horizontal asymptote

A1 for correct maximum point

[??? marks]

19c. [2 marks]

Markscheme

26

A1 for reflecting negative branch from (ii) in the -axis

A1 for correctly labelled minimum point

[2 marks]

19d. [5 marks]

Markscheme

EITHER

attempt at integration by parts     (M1)

     A1A1

     A1

     A1

OR

attempt at integration by parts     (M1)

     A1A1

     A1

27

     A1

[5 marks]

19e. [4 marks]

Markscheme

    M1A1A1

 

Note:     Method mark is for differentiating the product. Award A1 for each correct term.

 

both parts of the expression are positive hence is positive     R1

and therefore is an increasing function (for )     AG

[4 marks]

20a. [2 marks]

Markscheme

     (M1)

or      A1

[2 marks]

20b. [3 marks]

Markscheme

28

shape     A1

and      A1

-intercepts     A1

[3 marks]

20c. [1 mark]

Markscheme

EITHER

is symmetrical about the -axis     R1

OR

     R1

[1 mark]

20d. [1 mark]

Markscheme29

EITHER

is not one-to-one function     R1

OR

horizontal line cuts twice     R1

 

Note:     Accept any equivalent correct statement.

 

[1 mark]

20e. [4 marks]

Markscheme

     M1

     M1

     A1A1

[4 marks]

20f. [3 marks]

Markscheme

     M1A1

     A1

[3 marks]

20g. [2 marks]

Markscheme

     M1

30

which is not in the domain of (hence no solutions to )     R1

 

[2 marks]

20h. [2 marks]

Markscheme

     M1

as so no solutions to      R1

 

Note:     Accept: equation has no solutions.

 

[2 marks]

21a. [3 marks]

Markscheme

area of segment      M1A1

     A1

[3 marks]

21b. [4 marks]

Markscheme

METHOD 1

     M1A1

     (M1)31

     A1

METHOD 2

     (M1)

     A1

     (M1)

     A1

[4 marks]

22a. [6 marks]

Markscheme

METHOD 1

     (M1)(A1)

 

Note:     Award M1A1 for finding using any alternative method.

 

hence gradient of normal      (M1)

hence gradient of normal at is      (A1)

hence equation of normal is      (M1)A1

32

METHOD 2

     (M1)

     (A1)

 

Note:     Award M1A1 for finding using any alternative method.

 

hence gradient of normal      (M1)

hence gradient of normal at is      (A1)

hence equation of normal is      (M1)A1

[6 marks]

22b. [3 marks]

Markscheme

Use of

    (M1)(A1)

 

33

Note:     Condone absence of limits or incorrect limits for M mark.

Do not condone absence of or multiples of .

 

     A1

[3 marks]

23a. [5 marks]

Markscheme

attempt to differentiate implicitly     M1

   A1A1A1

 

Note: Award A1 for correctly differentiating each term.

 

   A1

 

Note: This final answer may be expressed in a number of different ways.

 

[5 marks]

23b. [4 marks]

Markscheme

   A1

   M1

34

at  the tangent is  and     A1

at  the tangent is      A1

 

Note: These equations simplify to .

 

Note: Award A0M1A1A0 if just the positive value of is considered and just one tangent is found.

 

[4 marks]

24a. [2 marks]

Markscheme

attempting to solve either or for     (M1)

(or equivalent eg  )     A1

 

Note: Accept or equivalent eg .

 

[2 marks]

24b. [5 marks]

Markscheme

considering      (M1)

   A1

considering one of  or      M1

35

   A1

   A1

 

Note: Award A0A0 for  and  stated without any justification.

 

[5 marks]

24c. [3 marks]

Markscheme

   M1A1A1

   AG

[3 marks]

24d. [4 marks]

Markscheme

is (strictly) decreasing     R1

 

Note: Award R1 for a statement such as and so the graph of has no turning points.

 

one branch is above the upper horizontal asymptote and the other branch is below the lower horizontal

asymptote     R1

has an inverse     AG

   A2

 36

Note: Award A2 if the domain of the inverse is seen in either part (d) or in part (e).

 

[4 marks]

24e. [4 marks]

Markscheme

   M1

 

Note: Award M1 for interchanging and (can be done at a later stage).

 

   M1

   A1

   A1

[4 marks]

24f. [4 marks]

Markscheme

use of      (M1)

   (A1)(A1)

 

Note: Award (A1) for the correct integrand and (A1) for the limits.

 

   A1

[4 marks]

37

25. [6 marks]

Markscheme

METHOD 1

substituting for and attempting to solve for (or vice versa)     (M1)

   (A1)

EITHER

   M1A1

OR

   M1A1

THEN

attempting to find      (M1)

   A1

 

Note: Award all marks except the final A1 to candidates who do not consider ±.

 

METHOD 2

   M1A1

   (M1)(A1)

   (M1)

   A1

38

 

Note: Award all marks except the final A1 to candidates who do not consider ±.

 

[6 marks]

26. [7 marks]

Markscheme

   A1

when      A1

   M1

 is a factor     A1

   (M1)

Note:     M1 is for attempting to find the quadratic factor.

   (M1)A1

Note:     M1 is for an attempt to solve their quadratic factor.

[7 marks]

27. [7 marks]

Markscheme

   M1A1

 

Note:     Award follow through marks below if their answer is a multiple of the correct answer.

39

 

considering either or      (M1)

   A1

   M1

 

Note:     Condone absence of .

   A1

   A1

[7 marks]

28a. [8 marks]

Markscheme

use of      (M1)

Note:     Condone any or missing limits.

   (A1)

   A1

   (M1)

   M1A1

   (A1)

40

   A1

Note:     If the coefficient “ ” is absent, or eg, “ ” is used, only M marks are available.

[8 marks]

28b. [4 marks]

Markscheme

(i)     attempting to use with      M1

   A1

(ii)     substituting into      (M1)

   A1

Note:     Do not allow FT marks for (b)(ii).

[4 marks]

28c. [7 marks]

Markscheme

(i)          (M1)

   M1A1

Note:     Award M1 for attempting to find .

   A1

(ii)          A1

Note:     Award A1 for  from an incorrect .

41

(iii)     METHOD 1

 is a minimum at and the container is widest at these values     R1

is a maximum at  and the container is narrowest at this value     R1

[7 marks]

29. [6 marks]

Markscheme

    (M1)

    A1

    (A1)

    A1

    M1

    A1

so Hayley’s conjecture is correct     AG

[6 marks]

30a. [3 marks]

Markscheme

valid method eg, sketch of curve or critical values found     (M1)

     A1

    A1

 

42

Note:     Award M1A1A0 for correct intervals but with inclusive inequalities.

 

[3 marks]

30b. [5 marks]

Markscheme

(i)         A1A1

Note:     Award A1A0 for any two correct terms.

(ii)    

    (M1)

    A1A1

Note:     M1 should be awarded if graphical method to find zeros of or turning points of is

shown.

[5 marks]

30c. [2 marks]

Markscheme

1.67     A1

[2 marks]

30d. [2 marks]

Markscheme

43

     M1A1A1

 

Note:     Award M1 for reflection of their in the line provided their is one-one.

A1 for , (Accept axis intercept values) A1 for the other two sets of coordinates of other end

points

[2 marks]

30e. [2 marks]

Markscheme

    M1

     A1

[2 marks]

30f. [4 marks]

Markscheme

    (M1)

44

    A1

     A1A1

Note:     Award A1 for −5 and −1, and A1 for correct inequalities if numbers are reasonable.

[8 marks]

30g. [4 marks]

Markscheme

     (M1)

    (A1)

Note:     Accept = in the above.

    A1A1

Note:     A1 for (allow ≥) and A1 for .

[4 marks]

31a. [2 marks]

Markscheme

    (M1)

    A1

[2 marks]

31b. [4 marks]

Markscheme

     M1A1A1A1

Note:     Award M1 for an attempt at implicit differentiation, A1 for each part.

45

    AG

[4 marks]

31c. [3 marks]

Markscheme

at     (A1)

finding the negative reciprocal of a number     (M1)

gradient of normal is

     A1

[3 marks]

31d. [4 marks]

Markscheme

substituting linear expression     (M1)

or equivalent

    (M1)A1

     A1

[4 marks]

31e. [3 marks]

Markscheme

   M1A1

   A1

46

[3 marks]

32a. [4 marks]

Markscheme

EITHER

     (M1)(A1)(A1)

Note:     Award (M1) for , (A1) for a correct  and (A1) for a correct .

OR

     (M1)(A1)(A1)

Note:     Award (M1) for , (A1) for a correct  and (A1) for a correct .

OR

     (M1)(A1)(A1)

Note:     Award (M1) for use of cosine rule, (A1) for a correct numerator and (A1) for a correct

denominator.

THEN

   A1

[4 marks]

32b. [4 marks]

Markscheme

EITHER

   M1A1A1

Note:     Award M1 for use of , A1 for a correct numerator and A1 for a correct

denominator.

47

   M1

OR

   M1A1A1

Note:     Award M1 for use of xxx, A1 for a correct numerator and A1 for a correct denominator.

       M1

OR

   M1A1

Note:     Award M1 for either use of the cosine rule or use of .

   A1

   M1

THEN

   AG

[4 marks]

32c. [11 marks]

Markscheme

(i)          M1A1A1

48

Note:     Award M1 for attempting product or quotient rule differentiation, A1 for a correct numerator

and A1 for a correct denominator.

(ii)     METHOD 1

EITHER

   (M1)

   A1

   (A1)

OR

attempting to locate the stationary point on the graph of

   (M1)

   A1

   (A1)

THEN

   A1

METHOD 2

EITHER

   M1

   A1

OR

attempting to locate the stationary point on the graph of

   (M1)

49

   A1

THEN

   (A1)

   A1

(iii)          M1A1

substituting into      M1

 and so is the maximum value of      R1

never exceeds 10°     AG

[11 marks]

32d. [3 marks]

Markscheme

attempting to solve      (M1)

Note:     Award (M1) for attempting to solve .

and      (A1)

   A1

[3 marks]

Printed for International School of Monza

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