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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: WEDNESDAY 3, JUNE 2015

COURSE CODE: MST125/C

SECTION A

Question 1

What is the least residue of 14 × 15 modulo 17?

ANSWERS(Purchase full paper to get the solutions):

14 × 15 modulo 17 = 210 modulo 17

210 modulo 17 = 6 modulo 17

Therefore, the least residue of 14 × 15 modulo 17 = 6

Answer: B

Question 2

Which of the following is an integer n such that 144ⁿ ≡ 1 (mod 23)?

Question 3

What is the equation of the parabola in standard position with focus (2, 0)?

Question 4

What is the equation of the line with parametric equations

x = t − 1, y = 3t + 2?

Question 5

A paperweight of mass 0.62 kg rests on a horizontal tabletop. What is the normal reaction of the tabletop on the paperweight, in newtons to two significant figures? Take the magnitude of the acceleration due to gravity to be 9.8ms−2.

Question 6

A particle, which remains at rest, is acted on by three forces only. Two of the forces have component forms 2i − 7j and i + 5j, respectively, where i and j are the Cartesian unit vectors. What is the component form of the third force?

Question 7

Which of the following transformations is a linear transformation?

Question 8

Which matrix represents the linear transformation that maps the points (1, 0) and (0, 1) to the points (−3, 5) and (6, 2), respectively?

Question 9

Which of the following is the graph of the function

Question 10

What is the remainder on dividing the polynomial expression 3x² + x + 1 by the polynomial expression x − 1?

Question 11

Which of the following is the general solution of the differential equation

 ?

In the options, c is an arbitrary constant.

Question 12

Which of the following is an integrating factor p(x) for the differential equation

 ?

Question 13

Let P be the following statement:

Every multiple of 8 is even.

Which of the following statements is the negation of P?

Question 14

A football is kicked along the ground and moves along a straight line. It slows down at the rate of 2.5 m s⁻² and comes to a halt after travelling for 20 m. What is the speed with which the ball was kicked, in m s⁻¹?

Question 15

The position x (in metres) of an object moving along a straight line is given in terms of the time t (in seconds) by

x = e^t/2 + 4t.

What is the magnitude of the acceleration of the object after 2 seconds, in m s⁻² to two decimal places?

Question 16

A box is pushed with a force P down a straight, rough slope that is inclined at an angle of 30⁰ to the horizontal. The force P acts on the box at an angle of 45⁰ to the horizontal. If W represents the weight of the box, N the normal reaction from the slope and F the friction force, which of the following diagrams represents the forces acting on the box?

Question 17

What are the eigenvalues of the matrix ?

Question 18

How many ways are there to select three different students from a group of 10, if the order in which they are considered does not matter?

Question 19

How many 3-letter sequences of letters from the 26-letter alphabet A,B,..., Z contain at least one A? Examples of such sequences include CBA and AZA

Question 20

What is the general solution of the recurrence relation

u_n = 6u_n−1 − 8u_n−2 ?

In the options, A and B represent constants.

SECTION B

Question 21

Use Euclid’s algorithm to show that a multiplicative inverse of 32 modulo 77 exists, and find one.

Question 22

The linear transformation f is formed from the composite of the horizontal shear with shear factor 3 followed by the (−3, 2)-scaling.

(a) Find the matrix of f.

(b) Find the images of the points (−4, 2) and (0, 1) under f.

(c) Given that the triangle with vertices (0, 0), (−4, 2) and (0, 1) has area 2, use your answer to part (a) to find the area of the image of this triangle under f.

Question 23

A particle, which remains at rest, is acted on by three forces, F, G and H, and no other forces. The force F acts at an angle of 25⁰ to the horizontal, and the force G acts at an angle of 125⁰ to F, as shown. The force H acts vertically downwards and has magnitude 40 N. Let F = |F| and G = |G|. Take the Cartesian unit vectors i and j to be parallel and perpendicular to F, respectively, as shown.

(a) Find expressions for the component forms of the three forces F, G and H.

(b) Hence or otherwise find the magnitude of the force G, in newtons to two significant figures.

Question 24

Find

Question 25

Solve the initial value problem

, y(1) = 3.

Question 26

Let P(n) be the statement

(n − 1)! > 2ⁿ , where n ∈ N.

(a) Determine whether P(n) is true or false for each of n = 5 and n = 6.

(b) Use mathematical induction to show that P(n) is true for all n ≥ 6.

SECTION C

Question 27

Consider the equation 4x² − 25y² = 16.

(a) Show that this equation represents a hyperbola in standard position.

(b) Find the vertices and asymptotes of the hyperbola.

(c) Sketch the hyperbola, marking the vertices and asymptotes and labelling them with their coordinates or equations as appropriate.

(d) Find the eccentricity, foci and directrices of the hyperbola.

(e) Write down a parametrisation of the part of the hyperbola that lies in the second quadrant, excluding the point that lies on the x-axis.

Question 28

(a) Find the integral dx

(b) Evaluate the integral

Question 29

(a) Express the matrix in the form PDP⁻¹, where D is a 2 × 2 diagonal matrix and P is a 2 × 2 invertible matrix.

Find the matrix P⁻¹ as part of your answer.

(b) Find A¹⁰.

(c) Find the general solution of the following system of differential equations.

x? = 9x − 20y

y? = 4x − 9y.

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