paper 2 most likely questions May 2018 [327 marks]

1a. [2 marks]

Let , for .

Find the -intercept of the graph of .

f(x) = 6x2−4ex 0 ⩽ x ⩽ 7

x f

1b. [2 marks]The graph of has a maximum at the point A. Write down the coordinates of A.f

1c. [3 marks]On the following grid, sketch the graph of .f

2a. [2 marks]

Let , for . The graph of passes through the point , where .

Find the value of .

f(x) = 6 − ln(x2 + 2) x ∈ R f (p, 4) p > 0

p

2b. [3 marks]The following diagram shows part of the graph of .

The region enclosed by the graph of , the -axis and the lines and is rotated 360° about the -axis. Find the volume of thesolid formed.

f

f x x = −p x = p x

3. [6 marks]In the expansion of , the coefficient of the term in is 11880. Find the value of .ax3(2 + ax)11 x5 a

4. [7 marks]The heights of adult males in a country are normally distributed with a mean of 180 cm and a standard deviation of . 17% ofthese men are shorter than 168 cm. 80% of them have heights between and 192 cm.

Find the value of .

σ cm(192 −h) cm

h

ex

5a. [2 marks]

Consider the graph of , for .

Find the -intercept.

f(x) = + 3ex

5x−10x ≠ 2

y

5b. [2 marks]Find the equation of the vertical asymptote.

5c. [2 marks]Find the minimum value of for .f(x) x > 2

6a. [2 marks]

In a large university the probability that a student is left handed is 0.08. A sample of 150 students is randomly selected from theuniversity. Let be the expected number of left-handed students in this sample.

Find .

k

k

6b. [2 marks]Hence, find the probability that exactly students are left handed;k

6c. [2 marks]Hence, find the probability that fewer than students are left handed.k

7. [7 marks]The following diagram shows the chord [AB] in a circle of radius 8 cm, where .

Find the area of the shaded segment.

AB = 12 cm

8. [7 marks]Let . Find the term in in the expansion of the derivative, .f(x) = (x2 + 3)7 x5 f ′(x)

9a. [1 mark]

A particle P moves along a straight line. Its velocity after seconds is given by , for . The following

diagram shows the graph of .

Write down the first value of at which P changes direction.

vP ms−1 t vP = √t sin( t)π

20 ⩽ t ⩽ 8

vP

t

9b. [2 marks]Find the total distance travelled by P, for .0 ⩽ t ⩽ 8

9c. [4 marks]A second particle Q also moves along a straight line. Its velocity, after seconds is given by for . After seconds Q has travelled the same total distance as P.

Find .

vQ ms−1 t vQ = √t 0 ⩽ t ⩽ 8k

k

10. [2 marks]

The following table shows a probability distribution for the random variable , where .

A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement.The number of blue marbles drawn is given by the random variable .

A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if threewhite marbles are drawn.

Jill plays the game nine times. Find the probability that she wins exactly two prizes.

X E(X) = 1.2

X

11a. [1 mark]

The following diagram shows a circle, centre O and radius mm. The circle is divided into five equal sectors.

One sector is OAB, and .

Write down the exact value of in radians.

r

AOB = θ

θ

11b. [3 marks]

The area of sector AOB is .

Find the value of .

20π mm2

r

11c. [3 marks]Find AB.

12a. [3 marks]

Let and .

The graphs of and intersect at and , where .

Find the value of and of.

f(x) = xe−x g(x) = −3f(x) + 1

f g x = p x = q p < q

p

q

12b. [3 marks]Hence, find the area of the region enclosed by the graphs of and.

f

g

13a. [3 marks]

The weights, , of newborn babies in Australia are normally distributed with a mean 3.41 kg and standard deviation 0.57 kg. Anewborn baby has a low birth weight if it weighs less than kg.

Given that 5.3% of newborn babies have a low birth weight, find .

W

w

w

13b. [3 marks]A newborn baby has a low birth weight.

Find the probability that the baby weighs at least 2.15 kg.

14a. [3 marks]

A population of rare birds, , can be modelled by the equation , where is the initial population, and is measured in

decades. After one decade, it is estimated that .

(i) Find the value of .

(ii) Interpret the meaning of the value of .

Pt Pt = P0ekt P0 t

= 0.9P1

P0

k

k

14b. [5 marks]Find the least number of whole years for which .< 0.75Pt

P0

15a. [1 mark]

Consider the expansion of .

Write down the number of terms of this expansion.

(x2 + )102x

15b. [5 marks]Find the coefficient of .x8

16. [7 marks]A particle moves in a straight line. Its velocity after seconds is given by

After seconds, the particle is 2 m from its initial position. Find the possible values of .

v ms−1 t

v = 6t− 6, for 0 ⩽ t ⩽ 2.

p p

17a. [2 marks]

Let and be independent events, with and , where .

Write down an expression for in terms of.

C

D P(C) = 2k P(D) = 3k2 0 < k < 0.5

P(C∩ D)k

17b. [3 marks]Find .P(C′|D)

18. [7 marks]Let for .

Points and are on the curve of . The tangent to the curve of at is perpendicular to the tangent at . Find the coordinatesof .

f(x) = ln(4x)x 0 < x ≤ 5

P(0.25, 0) Q f f P Q

Q

19a. [5 marks]

Let and .

Find

(i) ;

(ii) ;

(iii) .

u = 6i + 3j + 6k v = 2i + 2j + k

u ∙ v

|u|

|v|

19b. [2 marks]Find the angle between and .u v

20. [8 marks]Let and

. The graphs of and intersect at two distinct points.

Find the possible values of .

f(x) = kx2 + kx

g(x) = x− 0.8 f g

k

[3 marks]21a.

The population of deer in an enclosed game reserve is modelled by the function

, where

is in months, and

corresponds to 1 January 2014.

Find the number of deer in the reserve on 1 May 2014.

P(t) = 210 sin(0.5t− 2.6) + 990

t

t = 1

[2 marks]21b. Find the rate of change of the deer population on 1 May 2014.

[1 mark]21c. Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.

22. [8 marks]Ramiro and Lautaro are travelling from Buenos Aires to El Moro.

Ramiro travels in a vehicle whose velocity in

is given by

, where

is in seconds.

Lautaro travels in a vehicle whose displacement from Buenos Aires in metres is given by

.

When

, both vehicles are at the same point.

Find Ramiro’s displacement from Buenos Aires when

.

ms−1

VR = 40 − t2

t

SL = 2t2 + 60

t = 0

t = 10

23a. [2 marks]

The following table shows the Diploma score and university entrance mark for seven IB Diploma students.

Find the correlation coefficient.

x y

23b. [2 marks]The relationship can be modelled by the regression line with equation .

Write down the value of and of .

y = ax+ b

a b

23c. [2 marks]Rita scored a total of in her IB Diploma.

Use your regression line to estimate Rita’s university entrance mark.

26

24. [2 marks]

A particle starts from point and moves along a straight line. Its velocity, , after seconds is given by , for . The particle is at rest when .

The following diagram shows the graph of .

Find the distance travelled by the particle for .

A v ms−1 t v(t) = e cos t − 112

0 ≤ t ≤ 4 t = π

2

v

0 ≤ t ≤ π2

[3 marks]25a.

The following diagram shows triangle ABC.

Find AC.

25b. [3 marks]Find.BCA

26a. [4 marks]

The weights of fish in a lake are normally distributed with a mean of g and standard deviation . It is known that of the fishhave weights between g and g.

(i) Write down the probability that a fish weighs more than g.

(ii) Find the probability that a fish weighs less than g.

760 σ 78.87%705 815

760

815

26b. [4 marks](i) Write down the standardized value for g.

(ii) Hence or otherwise, find.

815

σ

26c. [2 marks]A fishing contest takes place in the lake. Small fish, called tiddlers, are thrown back into the lake. The maximum weight of a tiddleris standard deviations below the mean.

Find the maximum weight of a tiddler.

1.5

26d. [2 marks]A fish is caught at random. Find the probability that it is a tiddler.

26e. [2 marks] of the fish in the lake are salmon. of the salmon are tiddlers. Given that a fish caught at random is a tiddler, find theprobability that it is a salmon.25% 10%

27a. [5 marks]

The first two terms of a geometric sequence are and .

(i) Find the common ratio.

(ii) Hence or otherwise, find .

un u1 = 4 u2 = 4.2

u5

R Z+

27b. [5 marks]Another sequence is defined by , where , and , such that and .

(i) Find the value of .

(ii) Find the value of .

vn vn = ank a, k ∈ R n ∈ Z+ v1 = 0.05 v2 = 0.25

a

k

27c. [5 marks]Find the smallest value of for which .n vn > un

28a. [3 marks]

Adam is a beekeeper who collected data about monthly honey production in his bee hives. The data for six of his hives is shown in thefollowing table.

The relationship between the variables is modelled by the regression line with equation .

Write down the value of and of .

P = aN + b

a b

28b. [2 marks]Use this regression line to estimate the monthly honey production from a hive that has 270 bees.

28c. [1 mark]

Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the followingcumulative frequency graph.

Adam’s hives are labelled as low, regular or high production, as defined in the following table.

Write down the number of low production hives.

28d. [3 marks]

Adam knows that 128 of his hives have a regular production.

Find the value of ;k

28e. [2 marks]Find the number of hives that have a high production.

28f. [3 marks]Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regularproduction hives.

29a. [2 marks]

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time is given by , for .

Write down the values of when .

t a = 3t2 − 14t+ 8 0 ⩽ t ⩽ 5

t a = 0

29b. [2 marks]Hence or otherwise, find all possible values of for which the velocity of P is decreasing.t

29c. [6 marks]

When , the velocity of P is .

Find an expression for the velocity of P at time .

t = 0 3 ms−1

t

29d. [4 marks]Find the total distance travelled by P when its velocity is increasing.

30a. [1 mark]

Let and , for .

The graph of can be obtained from the graph of by two transformations:

Write down the value of ;

f(x) = lnx g(x) = 3 + ln( )x2

x > 0

g f

a horizontal stretch of scale factor q followed by

a translation of (h

k) .

q

30b. [1 mark]Write down the value of ;h

30c. [1 mark]Write down the value of .k

30d. [2 marks]

Let , for . The following diagram shows the graph of and the line .

The graph of intersects the graph of at two points. These points have coordinates 0.111 and 3.31 correct to three significantfigures.

Find .

h(x) = g(x) × cos(0.1x) 0 < x < 4 h y = x

h h−1 x

∫ 3.310.111 (h(x) −x)dx

30e. [3 marks]Hence, find the area of the region enclosed by the graphs of and .h h−1

30f. [7 marks]Let be the vertical distance from a point on the graph of to the line . There is a point on the graph of where isa maximum.

Find the coordinates of P, where .

d h y = x P(a, b) h d

0.111 < a < 3.31

31a. [2 marks]

A random variable is normally distributed with mean, . In the following diagram, the shaded region between 9 and represents 30%of the distribution.

Find .

X μ μ

P(X < 9)

31b. [3 marks]

The standard deviation of is 2.1.

Find the value of .

X

μ

31c. [5 marks]

The random variable is normally distributed with mean and standard deviation 3.5. The events and are independent,and .

Find .

Y λ X > 9 Y > 9P ((X > 9) ∩ (Y > 9)) = 0.4

λ

31d. [5 marks]Given that , find .Y > 9 P(Y < 13)

The following diagram shows the graph of , for .f(x) = asin bx+ c 0 ⩽ x ⩽ 12

32a. [6 marks]

The graph of has a minimum point at and a maximum point at .

(i) Find the value of .

(ii) Show that .

(iii) Find the value of .

f (3, 5) (9, 17)

c

b = π

6

a

32b. [3 marks]

The graph of is obtained from the graph of by a translation of . The maximum point on the graph of has coordinates

.

(i) Write down the value of .

(ii) Find .

g f (k

0) g

(11.5, 17)

k

g(x)

32c. [6 marks]

The graph of changes from concave-up to concave-down when .

(i) Find .

(ii) Hence or otherwise, find the maximum positive rate of change of .

g x = w

w

g

33a. [2 marks]

A particle P starts from a point A and moves along a horizontal straight line. Its velocity after seconds is given by

The following diagram shows the graph of .

Find the initial velocity of .

v cms−1 t

v(t) = { −2t+ 2, for 0 ⩽ t ⩽ 1

3√t+ − 7, for 1 ⩽ t ⩽ 124t2

v

P

33b. [2 marks]

P is at rest when and .

Find the value of .

t = 1 t = p

p

33c. [4 marks]

When , the acceleration of P is zero.

(i) Find the value of .

(ii) Hence, find the speed of P when .

t = q

q

t = q

33d. [6 marks](i) Find the total distance travelled by P between and .

(ii) Hence or otherwise, find the displacement of P from A when .

t = 1 t = p

t = p

34a. [2 marks]

The points A and B lie on a line

, and have position vectors and respectively. Let O be the origin. This is shown on the following diagram.

Find .

L⎛⎜⎝

−3−22

⎞⎟⎠⎛⎜⎝

64

−1

⎞⎟⎠

−−→AB

34b.

The point C also lies on

, such that .

Show that .

L−−→AC = 2

−−→CB

−−→OC =

⎛⎜⎝320

⎞⎟⎠

34c. [5 marks]

Let

be the angle between and .

Find .

θ−−→AB

−−→OC

θ

34d. [6 marks]

Let D be a point such that

, where . Let E be a point on such that is a right angle. This is shown on the following diagram.

(i) Show that .

(ii) The distance from D to line is less than 3 units. Find the possible values of .

−−→OD = k

−−→OC k > 1 L

CED

∣∣∣−−→DE∣∣∣ = (k− 1)∣

∣∣−−→OC∣

∣∣sin θ

L k

35a. [2 marks]

A factory has two machines, A and B. The number of breakdowns of each machine is independent from day to day.

Let be the number of breakdowns of Machine A on any given day. The probability distribution for can be modelled by the followingtable.

Find .

A A

k

35b. [3 marks](i) A day is chosen at random. Write down the probability that Machine A has no breakdowns.

(ii) Five days are chosen at random. Find the probability that Machine A has no breakdowns on exactly four of these days.

35c. [2 marks]

Let be the number of breakdowns of Machine B on any given day. The probability distribution for can be modelled by the followingtable.

Find .

B B

E(B)

35d. [8 marks]

On Tuesday, the factory uses both Machine A and Machine B. The variables and are independent.

(i) Find the probability that there are exactly two breakdowns on Tuesday.

(ii) Given that there are exactly two breakdowns on Tuesday, find the probability that both breakdowns are of Machine A.

A B

36a. [5 marks]

A particle P moves along a straight line so that its velocity, , after seconds, is given by , for. The initial displacement of P from a fixed point O is 4 metres.

Find the displacement of P from O after 5 seconds.

vms−1 t v = cos3t− 2 sin t− 0.50 ⩽ t ⩽ 5

36b. [2 marks]

The following sketch shows the graph of .

Find when P is first at rest.

v

36c. [2 marks]Write down the number of times P changes direction.

36d. [2 marks]Find the acceleration of P after 3 seconds.

36e. [3 marks]Find the maximum speed of P.

37a. [2 marks]

Let , for .

Write down the equation of the horizontal asymptote of the graph of .

f(x) = + 21x−1

x > 1

f

37b. [2 marks]Find .f ′(x)

37c. [2 marks]

Let , for . The graphs of and have the same horizontal asymptote.

Write down the value of .

g(x) = ae−x + b x ⩾ 1 f g

b

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37d. [4 marks]Given that , find the value of .g′(1) = −e a

37e. [4 marks]There is a value of , for, for which the graphs of and have the same gradient. Find this gradient.

x

1 < x < 4 f g