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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: TUESDAY 6, JUNE 2017

COURSE CODE: MST125/C

SECTION A

Question 1

What is the least residue of 82 007 × 41 000 011 modulo 41?

ANSWERS(Purchase full paper to get all the solutions):

The last digit obtained 7, hence 7 modulo 41 = 7

Answer: A

Question 2

What is the least residue of 5⁶⁰⁶ modulo 7?

Question 3

Given that 75 is a multiplicative inverse of 426 modulo 743 and that 743 is prime, which of the following is a solution of the linear congruence

426x ≡ 11 (mod 743) ?

Question 4

What is the equation of the line with parametric equations

x = t + 3, y = 2t − 1 ?

Question 5

A computer rests on a horizontal desk. The normal reaction of the desk on the computer is 26 N vertically upwards. What is the mass of the computer in kilograms to two significant figures? Take the magnitude of the acceleration due to gravity to be 9.8ms⁻².

Question 6

A wooden box of mass 15 kg rests on a horizontal wooden plank. The coefficient of static friction between the box and the plank is 0.43. What is the maximum horizontal force, in newtons to two significant figures, that can be applied to the box without the box moving? Take the magnitude of the acceleration due to gravity to be 9.8ms⁻²,

Question 7

Which of the following specifies the glide-reflection h formed from the reflection in the x-axis followed by the translation 3 units to the right?

Question 8

Let f and g be the linear transformations represented by the matrices and , respectively. What is the image of the point (1, 0) under g⁰f?

Question 9

Which of the following describes the linear transformation represented by the matrix

Question 10

The graph of a function f is shown below.

Question 11

What is the remainder on dividing the polynomial expression 2x² − x − 7 by the polynomial expression x − 2?

Question 12

What is the form of the partial fraction expansion of the expression 

?

In the options, A, B and C represent constants

Question 13

What is the solution of the initial value problem 

, where s = 6 when t =0?

Question 14

What is the general solution of the differential equation  

?

In the options, c is an arbitrary constant and A is an arbitrary positive constant.

Question 15

Which of the following is a direction field for the differential equation

 

Question 16

Which of the following statements is not equivalent to the statement Every multiple of 6 is even ?

Question 17

A car is travelling in a straight line with a constant acceleration of 4ms⁻². Its initial velocity is 10 m s⁻¹. After how many seconds is its velocity 30 m s⁻¹?

Question 18

The position r of a particle is given in terms of the time t by

 

where i, j and k are the Cartesian unit vectors. What is the velocity of the particle when t = 0?

Question 19

A curling stone of mass 12 kg is being pushed along a straight line by a resultant horizontal force of magnitude 48 N. What is the acceleration of the stone in m s⁻²?

Question 20

What is the general solution of the recurrence relation

 ?

In the options, A and B represent constants.

Question 21

Consider the hyperbola in standard position with focus (36, 0) and directrix x = 4.

(a) Find the eccentricity e, the positive x-intercept a, and the positive y-intercept b of this hyperbola.

(b) Hence show that the equation of the hyperbola is

.

(c) Write down the equations of the asymptotes of the hyperbola, giving the exact values of the coefficients.

Question 22

A particle, which remains at rest, is acted on by three forces, R, S and T, and no others. The force diagram below shows the angles at which the forces act. The magnitude of the force R is 50 N.

(a) Find expressions for the component forms of the three forces R, S and T, taking the directions of the Cartesian unit vectors i and j to be as shown in the diagram (j is parallel to R), and denoting the magnitudes of S and T by S and T, respectively.

(b) Hence, or otherwise, find S in newtons to two significant figures.

Question 23

Evaluate the definite integral

Question 24

(a) Let P(n) be the variable proposition

                  n² − 5n + 1 is greater than 0,

where n is an integer. State one value of n for which P(n) is true and one value of n for which P(n) is false.

(b) Use proof by contradiction to show that the following statement is true:

There are no integers m and n with 7m − 21n = 778

Question 25

Express the matrix A = in the form PDP⁻¹, where D and P are 2 × 2 matrices, and D is a diagonal matrix. Evaluate P⁻¹ as part of your answer.

Question 26

Consider the sequences of five letters that can be made using the first nine letters of the alphabet, A, B, C, D, E, F, G, H, I. Examples include HBGIC and DDCDC.

(a) How many such sequences are there altogether?

(b) How many of the sequences start with either AB or BA?

(c) How many of the sequences contain at least one repeat of a letter?

Question 27

(a) Let f be the affine transformation that rotates points anticlockwise through π/2 about the point (−1, 0).

(i) Draw a diagram that shows the effect of f on the unit square.

(ii) By using your diagram, or otherwise, write down the images of the points (0, 0), (1, 0) and (0, 1) under f.

(iii) Hence, or otherwise, find f in the form f(x) = Ax + a.

(b) By finding the linear transformation g that describes the reflection in the line y = −x and using conjugation, find the affine transformation k that describes the reflection in the line y = −x + 7, in the form k(x) = Bx + b.

Question 28

(a) Find the integral  .

(b) Use your answer to part (a) to show that

 

is an integrating factor for the differential equation

 (x > ).

(Alternatively, you may show that the expression p(x) above multiplied by some positive constant is an integrating factor.)

(c) Find the general solution of the differential equation in part (b)

Question 29

A child moves in a straight line down a slide inclined at 30⁰ to the horizontal, as shown below

Denote the coefficient of sliding friction between the child and the slide by μ, and take the magnitude of the acceleration due to gravity to be g = 9.8ms⁻². Model the child as a particle.

(a) State the three forces acting on the child.

Draw a force diagram representing these forces, labelling them clearly.

(b) Find expressions for the component forms of the three forces, in terms of unknown magnitudes where appropriate, taking the Cartesian unit vectors i and j to point parallel and perpendicular to the slide in the directions shown above. Hence, or otherwise,

show that the magnitude of the acceleration a (in m s⁻²) of the child is given by

 .

(c) The child starts from rest and has a speed of 3 m s⁻¹ after sliding 4 m. Calculate the magnitude of the child’s acceleration.

(d) Hence, or otherwise, find the value of μ to two significant figures.

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