IB Math SL Year 2| Mock Final Exam Mathematics

Standard Level

Mock Final Exam

Tuesday/Wednesday, 30 April, 1 May 2019

1 class period

Instructions to Candidates

Do not open this test until instructed to do so.

A graphic display calculator is permitted for part 1 of this paper.

A graphic display calculator is not permitted for part 2 of this paper.

Show all of your work for each question.

Unless otherwise stated in the question, all numerical answers should be given exactly or correct to

three significant figures.

A clean copy of the Mathematics SL formula booklet is required for this paper (printed on yellow

paper *Do not write in)

Name:

Paper 1: NON-CALCULATOR 1. The position vector of point A is 2i + 3 j + k and the position vector of point B is 4i − 5 j + 21k.

(a) (i) Show that = 2i −8 j + 20k.

(ii) Find the unit vector u in the direction of .

(iii) Show that u is perpendicular to .

Let S be the midpoint of [AB]. The line L1 passes through S and is parallel to .

(b) (i) Find the position vector of S.

(ii) Write down the equation of L1.

The line L2 has equation r = (5i +10 j +10k) + s (−2i + 5 j − 3k).

(c) Explain why L1 and L2 are not parallel.

(d) The lines L1 and L2 intersect at the point P. Find the position vector of P.

AB

AB

OA

OA

2. (a) Factorize x2 – 3x – 10.

(b) Solve the equation x2 – 3x – 10 = 0.

3. A) The line y = 2x+1 and y = (x-1)2 intersect at two points.

Find the x-coordinates of both of those points

B) Determine the area between the curves shown below:

4. The diagram shows a circle of radius 5 cm. Find the perimeter of the shaded region.

5. Find the equation of the normal to the curve with equation: y = x3 + 1 at the point (1, 2).

6. Show that: cos 2𝜃 + 2𝑠𝑖𝑛2𝜃 = 1

1 radian

7. (a) Express 2 cos2 x + sin x in terms of sin x only.

(b) Solve the equation 2 cos2 x + sin x = 2 for x in the interval 0 x , giving your answers exactly.

8. Solve the following equation for 0 ≤ x ≤ 𝜋

6

sin(6x) = √3

2

9. The velocity v of a particle at time t is given by v = e−2t

+ 12t. The displacement of the particle at time t is s.

Given that s = 2 when t = 0, express s in terms of t.

10. The function f is given by f (x) = 2sin (5x – 3).

(a) Find f " (x).

(b) Write down .

11. ABCD is a rectangle and O is the midpoint of AB. Express the following vectors in terms of and

(a)

(b)

(c)

12. Given that f (x) = 2e3x

, find the inverse function f –1

(x).

xxf d)(

OC OD

CD

OA

AD A B

CD

O

13. The graph represents the function f ( x)= p cos x, find

(a) the value of p;

(b) the area of the shaded region.

14. Solve the equation 9

x–1 =

15. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for

(a) log2 5; (b) loga 20.

16. A discrete random variable X has a probability distribution

as shown in the table below.

(a) Find the value of a + b.

(b) Given that E(X) =1.5, find the value of a and of b.

.31

2x

x 0 1 2 3

P(X = x) 0.1 a 0.3 b

3

–3

x

y

17. Let

(a) and .

(i) Find AB. (ii) Find |AB|.

(b) The point C is such that . Show that the coordinates of C are .

(c) The following diagram shows triangle ABC. Let D be a point on [BC], with acute angle .

Write down an expression in terms of for

(i) angle ADB; (ii) area of triangle ABD.

18. Let , for .

Find .

19. Let .

(a) Find .

(b) Find the gradient of the graph of g at .

20. A function f has its first derivative given by . Find the second derivative.

21. A quadratic function can be written in the form . The graph of has axis of

symmetry and -intercept at .

(a) Find the value of .

(b) Find the value of .

(c) The line is a tangent to the curve of . Find an expression to represent all values of .

22. The following diagram shows the graphs of the displacement, velocity and acceleration of a moving object as

functions of time, t.

(a) Complete the following table by noting which graph A, B or C corresponds to each function.

(b) Write down the value of t when the velocity is greatest.

Paper 2: CALCULATOR

23. Let . Part of the graph of f is shown below.

The shaded region is enclosed by the curve of f , the x-axis, and the y-axis.

(a) Write down the x-intercept of f , for .

(b) The area of the shaded region is k . Find the value of k , giving your answer rounded to 3SF.

(c) Let . The graph of f is transformed to the graph of g. Give a full geometric

description of this transformation.

24. The standard deviation of masses of loaves of bread is 20g. Only 1% of loves weight less that 500g. Find the mean mass of the loaves.

25. The following diagram shows the graph of , for .

(a) There is a minimum point at P(2, − 3) and a

maximum point at Q(4, 3) .

(i) Write down the value of a .

(ii) Find the value of b .

(b) Write down the gradient of the curve at P.

(c) Find f’(x).

26. The population below is listed in ascending order. 5, 6, 7, 7, 9, 9, r, 10, s, 13, 13, t

The median of the population is 9.5. The upper quartile Q3 is 13.

(a) Write down the value of

(i) r;

(ii) s.

(b) The mean of the population is 10. Find the value of t.

27. A factory makes switches. The probability that a switch is defective is 0.04.The factory tests a random sample

of 100 switches.

(a) Find the mean number of defective switches in the sample.

(b) Find the probability that there are exactly six defective switches in the sample.

(c) Find the probability that there is at least one defective switch in the sample.

28. Find the volume of revolution when y = x2 + 3 is revolved through 2𝜋 about the x-axis from 0 to 1.

29. The following diagram shows a pentagon ABCDE, with AB = 9.2 cm, BC = 3.2 cm, BD = 7.1 cm,

=110, = 52 and = 60. Find AD.

DEA EDA DBA

30. Today Philip intends to go walking. The probability of good weather (G) is . If the weather is good, the probability

he will go walking (W) is . If the weather forecast is not good (NG) the probability he will go walking is .

(a) Complete the probability tree diagram to illustrate this information.

(b) What is the probability that Philip will go walking?

(c) What is the probability that Philip will go walking given

that it is good weather?

31. The velocity v ms-1, of a particle at time t seconds is given by:𝑣 = 3𝑠𝑖𝑛2𝑡, 𝑓𝑜𝑟 𝑡 ≥ 0

(a) Find the velocity of the particle when t = 5.

(b) Determine the total distance of the particle in the first 3 seconds.

(c) Find the function to represent the accelaration of the particle at any time t.

4

3

20

17

5

1

34

G

NG

NW

NW

W

W

1720

32. The following graph shows the depth of water, y metres , at a point P, during one day. The time t is given in hours,

from midnight to noon.

(a) Use the graph to write down an estimate of the value of t when

(i) the depth of water is minimum;

(ii) the depth of water is maximum;

(iii) the depth of the water is increasing most rapidly.

(b) The depth of water can be modelled by the function = Acos(𝐵(𝑡 − 1)) + 𝐶 .

(i) Show that .

(iii) Write down the value of C.

(iv) Find the value of B.

(c) A sailor knows that he cannot sail past P when the depth of the water is less than 12 m . Calculate the values of t

between which he cannot sail past P.

33. The following diagram shows a triangle ABC, where BC = 5 cm, = 60°, = 40°.

(a) Calculate AB.

(b) Find the area of the triangle.

B C

60° 40°

A

B C5 cm