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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: THURSDAY 20, SEPTEMBER 2018

COURSE CODE: MST125/B

SECTION A

Question 1

What is the least residue of 69 000 007 × 46 000 004 modulo 23?

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

69 000 007 × 46 000 004

The last possible digit = 28

28 modulo 23 = 5

Answer: B

Question 2

Which of the following is a multiplicative inverse of  modulo 61?

Question 3

Which of the following is a value of b for which the linear congruence 

18x ≡ b (mod 30)

has a solution?

Question 4

A parabola has parametric equations

.

What is the equation of the parabola?

Question 5

The magnitude F (in newtons) of a force F and the magnitude G (in newtons) of a force G satisfy the vector equation

,

where i and j are the Cartesian unit vectors. What is the value of F (in newtons)?

Question 6

Which matrix represents the linear transformation that maps the points (1, 0) and (0, 1) to the points (5, 4) and (−7, 3), respectively?

Question 7

What is the area of the image of the unit circle under the linear transformation represented by the matrix  ?

Question 8

What is the partial fraction expansion of 

?

Question 9

What is the quotient on dividing the polynomial expression 

 by the polynomial expression ?

Question 10

The half-life of a radioactive substance is 6 days. What is the value of the decay constant, in day−1 to three significant figures?

Question 11

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

 ?

Question 12

What is the solution of the initial value problem

 where y = 3 when x =0?

Question 13

What is the contrapositive of the following statement?

if  is a multiple of 7, then n is a multiple of 7.

Question 14

A crate of mass 16 kg, initially at rest, is being pushed along a straight line by a resultant horizontal force of 8 N. What is the speed of the crate after 6 seconds, in ms^-1?

Question 15

What are the eigenvalues of the matrix ?

Question 16

Which of the following is an eigenvector of the matrix  corresponding to the eigenvalue 3?

Question 17

The matrix A =  can be expressed in the form  , where P =  , D =  and  =  . What is ?

Question 18

What is the general solution of the recurrence relation 

 ?

In the options, A and B represent arbitrary constants.

Question 19

An ellipse in standard position has vertices (±4, 0) and (0, ±1).

(a) Find the equation of the ellipse.

(b) Calculate the eccentricity of the ellipse. Give your answer in exact form.

(c) Find the foci and the directrices of the ellipse.

(d) The ellipse is translated 2 units to the right and 1 unit down. Write down a parametrisation of the translated ellipse.

Question 20

Evaluate the definite integral

 

Question 21

In each of the following statements, x and y represent positive real numbers. One of the statements is true, and the other is false.

.

(a) Write down which statement is false, give a counterexample to show that it is false and write down its negation.

(b) Give a proof that the other statement is true.

Question 22

The velocity v of a particle is given in terms of the time t by

 

Its position at time t = 0 is i + 2 j.

(a) Find the position r of the particle in terms of the time t.

(b) Find the position of the particle at time t = 3. Give an exact answer.

Question 23

A caterer’s sandwich menu has 5 meat options and 4 vegetarian ones. A customer requests a tray of 6 sandwiches and would like all the sandwiches to be different.

(a) (i) How many different ways are there to select 3 meat options?

(ii) How many different ways are there to select 3 vegetarian options?

(iii) In how many different ways can the customer choose the selection of 6 different sandwiches if the tray must contain 3 meat options and 3 vegetarian options?

(b) Determine the number of different ways in which the customer can choose the selection of sandwiches if the tray must contain at least 2 meat options and at least 2 vegetarian options.

Question 24

A shipping crate of mass m (in kg) is at rest on horizontal ground. A rope is attached from the crate to a truck. The rope is taut and makes an acute angle θ with the horizontal as shown below.

Model the crate as a particle. Denote the magnitude of the acceleration due to gravity by g.

(a) State the four forces acting on the crate. Draw a force diagram representing these forces, labelling the forces clearly.

(b) Take the unit vector i to point horizontally and the unit vector j to point vertically upwards as shown above. Find expressions for the component forms of the four forces acting on the crate in terms of m, g and any unknown magnitudes of the forces (for example, let the magnitude of an unknown force A be A).

(c) Let the coefficient of static friction between the crate and the ground be μ. When the crate is on the point of slipping, the magnitude of the force in the rope is 7000 N. Use the expressions for the forces that you obtained in part (b) to show that

 .

(d) The crate has mass 1300 kg and the rope makes an angle of 30? with the horizontal. Take g = 9.8ms^-2. What is the coefficient of static friction between the crate and the ground? Give your answer to two significant figures.

Question 25

Let g be the rotation through π about the origin, and let f be the rotation by π about the point (3, 1).

(a) Write down the matrix that represents g.

(b) Write down the rule of the translation that maps the point (3, 1) to the origin.

(c) Use your solutions to parts (a) and (b) to show that f is given by

 .

(d) Draw a diagram showing the unit square and its images under g and under f on the same coordinate axes. Label each vertex of the square and each vertex of each image with its coordinates.

Question 26

Consider the function

   

The y-intercept is 4 and the two x-intercepts are approximately −2.4 and 0.4. The function is neither even nor odd.

(a) Write down the domain of the function and state the equation of the vertical asymptote.

(b) (i) Find By constructing a table of signs for f (x), or otherwise, find the intervals on which f is increasing or decreasing.

(ii) Find any stationary points and state their natures.

(c) The function can be expressed in the form f(x)=2x +5+ 1 2x − 1 . Find the asymptotic behaviour as x → ∞ and x → −∞.

(d) Use the information given in the question and your answers to parts (a) to (c) to sketch the graph of the function. Label any stationary points and any asymptotes.

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