IB Math SL Year 2 | Cumulative Year 2 Booklet 2 of 2

CALCULATOR 1. a) The following diagram shows the graphs of and , for

.

Let A be the area of the region enclosed by the curves of f and g.

(i) Find an expression for A.

(ii) Calculate the value of A.

b)

(i) Find .

(ii) Find .

c) There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x.

2. Let . Part of the graph of is shown in the following diagram. The graph crosses the -axis at

the points and .

a) Find the -coordinate of and of .

b) The region enclosed by the graph of and the -axis is revolved

about the -axis.Find the volume of the solid formed.

Due

MONDAY

4/29

3. 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.

a) Find the displacement of P from O after 5 seconds.

b) The following sketch shows the graph of .Find when P is first at rest.

c) Write down the number of times P changes direction.

d) Find the acceleration of P after 3 seconds.

e) Find the maximum speed of P.

4. The velocity of a particle after seconds is given by

, for

The following diagram shows the graph of .

a) Find the value of when the particle is at rest.

b) Find the value of when the acceleration of the particle is .

5. 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 .

6. A particle moves in a straight line with velocity , for , where v is in centimetres per

second and t is in seconds.

a) Find the acceleration of the particle after 2.7 seconds.

b) Find the displacement of the particle after 1.3 seconds.

7. Let and for .

a) On the same diagram, sketch the graphs of f and g .

b) Consider the graph of . Write down

(i) the x-intercept that lies between and ;

(ii) the period;

(iii) the amplitude.

c) Consider the graph of g . Write down

(i) the two x-intercepts;

(ii) the equation of the axis of symmetry.

d) Let R be the region enclosed by the graphs of f and g . Find the area of R.

8) Consider the curve with equation , where p and q are constants. The point lies on

the curve. The tangent to the curve at A has gradient . Find the value of p and of q .

9) Let , for . The graph of f is shown below. The graph of f crosses the x-axis at

, and .

a) Find the value of a and of b .

b) The graph of f has a maximum value when .Find the value of c

.

c) The region under the graph of f from to is rotated about the x-axis. Find the volume of

the solid formed.

d) Let R be the region enclosed by the curve, the x-axis and the line , between and . Find

the area of R .

10) A particle moves along a straight line such that its velocity, , is given by , for .

a) On the grid below, sketch the graph of , for .

b) Find the distance travelled by the particle in the first three seconds.

c) Find the velocity of the particle when its acceleration is zero.

11) 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 .

a) Find the initial velocity of .

b) P is at rest when and .Find the value of .

c) When , the acceleration of P is zero.

(i) Find the value of .

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

d) (i) Find the total distance travelled by P between and .

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

12) The position vectors of points P and Q are i 2 j k and 7i 3j 4k respectively.

a) Find a vector equation of the line that passes through P and Q.

b) The line through P and Q is perpendicular to the vector 2i nk. Find the value of .

13) Let , for , and , for .

Let . Write in the form , where .

14) Note: In this question, distance is in metres and time is in seconds. Two particles and start moving from

a point A at the same time, along different straight lines.

After seconds, the position of is given by r = .

a) Find the coordinates of A.

b) Two seconds after leaving A, is at point B. Find ;

c) Find .

d) Two seconds after leaving A, is at point C, where .

Find .

e) Hence or otherwise, find the distance between and two seconds after they leave A.

15)

a) Consider the points and .

Find .

b) Let C be a point such that . Find the coordinates of C.

c) The line passes through B and is parallel to (AC). Write down a vector equation for .

d) Given that , find .

16) The diagram below shows part of the graph of

, where .

The point is a maximum point and the point

is a minimum point.

a) Find the value of a.

b) (i) Show that the period of f is .

(ii) Hence, find the value of b .

c) Given that , write down the value of c .

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

, where is in months, and corresponds to 1 January 2014.

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

b) Find the rate of change of the deer population on 1 May 2014.

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

18) At Grande Anse Beach the height of the water in metres is modelled by the function ,

where is the number of hours after 21:00 hours on 10 December 2017. The following diagram shows the

graph of , for .

The point represents the first low tide and

represents the next high tide.

a) How much time is there between the first low tide and the

next high tide?

b) Find the difference in height between low tide and high tide.

c) Find the value of ;

d) Find the value of ;

e) Find the value of .

19)

20)

21)

(b) Find the value of Cos𝜃.

22)

23)

24)

(b) The graph of f is translated by the vector ( 3−1

) to

obtain the graph of a function g. Find an

expressions for g(x)

25)

26)

27)

29)

NON-CALCULATOR 30) Part of the graph of is shown below.

The point P lies on the graph of . At P, x = 1.

a) Find .

b) The graph of has a gradient of at the point P. Find the value of .

31) Let . The line L is the tangent to the curve of f at (4, 6) .Find the equation of L .

32) A. Let . Use the quotient rule to show that .

b. Find .

33) Let , for . The following diagram shows the graph of g .

a) Find the area of the region enclosed by the curve of g , the x-axis, and the lines and . Give

your answer in the form , where .

34) Let , for .

a) Use the quotient rule to show that .

b) The graph of g has a maximum point at A. Find the x-coordinate of A.

35) Consider f(x) =

a) Find the value of k .

b) Consider f(x) = ∫2

2𝑥+5𝑑𝑥, the graph of passes through ( , ) . Find .

36) Let (𝑥) = 1

3𝑥3 − 𝑥2 − 3𝑥 . Part of the graph of f is shown below.

There is a maximum point at A and a minimum point at

B(3, − 9) .

a) Find the coordinates of A.

b) Write down the coordinates of

(i) the image of B after reflection in the y-axis;

(ii) the image of B after translation by the vector ;

(iii) the image of B after reflection in the x-axis followed by a horizontal stretch with scale

factor .

37)

38)

a) Find the period of f.

b) Write down the amplitude of f

c) Hence, write down the value of a

d) Hence find the value of b

39) The following cumulative frequency diagram shows the lengths of 160 fish in cm.

40)

41)

42)

43)

44)

45)

46)