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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: FRIDAY 8, JUNE 2018

COURSE CODE: MST125/G

SECTION A

Question 1

Which of the following is a multiplicative inverse of 9 modulo 11?

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

1 ≡ 12 ≡ 23 ≡ 34≡ 45 (mod 11)

 

The multiplicative inverse of 9 modulo 11 = 5

Answer: C

Question 2

What is the highest common factor of 322 and 56?

Question 3

What is the least residue of  modulo 17?

Question 4

An ellipse in standard position has a focus at (2, 0) and the equation of one of the directrices is x = 18. Where does this ellipse cross the x-axis?

Question 5

A circle has parametric equations

x = 3 + cost, y = −2 + sin t.

What is its equation?

Question 6

 A box rests on horizontal ground. The normal reaction of the ground on the box is 490 N vertically upwards. What is the mass of the box in kilograms? Take the magnitude of the acceleration due to gravity to be 9.8.

Question 7

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

Question 8

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

Question 9

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

Question 10

Let f be the rotation anti-clockwise through the angle π/2 about the origin. What is the image of the point (2, 4) under f?

Question 11

The graph of a function f is shown below.

Which of the following could be the rule for f?

 

Question 12

What is the solution of the initial value ?

 where y = 1 when x =0?

Question 13

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

 

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

Question 14

You know that:

If Molly is a mouse, then Molly is not grey.

Which of the following statements can you be certain is true?

Question 16

The velocity 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 acceleration of the particle when t = 0?

Question 17

A crate of mass 20 kg is being pushed along a straight line by a resultant horizontal force of magnitude 40 N. What is the acceleration of the crate in  ?

Question 18

A van is travelling at a constant speed of . It then accelerates with a constant acceleration of 3 along a straight road. What is its speed (in ) after 6 seconds?

Question 19

The matrix A =  has eigenvalues 4 and 16 and corresponding eigenvectors and  respectively. What is the general solution of the following system of differential equations?

Question 20

A company consists of 7 junior staff and 4 senior staff. How many ways are there to choose a committee of 4 of the staff, if it must consist of 2 junior staff and 2 senior staff?

Question 21

The equation  represents a conic in standard position.

(a) By rearranging the equation in an appropriate way, identify the type of conic.

(b) Find the eccentricity of the conic, the coordinates of any foci and the equations of any directrices. Give exact answers.

(c) The conic is translated two units to the left. Write down a parametrisation for the translated conic.

Question 22

A particle, which remains at rest, is acted on by three forces, P, Q and R, and no others. The force diagram below shows the angles at which the forces act. The magnitude of the force P is 20 N.

(a) Find expressions for the component forms of the three forces P, Q and R, 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 Q and R by Q and R, respectively. 

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

Question 23

Find the integral .

Question 24

Solve the initial value problem

, where y = 2 when x = 1.

Question 25

Let A = . Find the eigenvalues of A. For each eigenvalue, find a corresponding eigenvector.

Question 26

Consider the first-order recurrence system

 (n = 2, 3, 4,...).

(a) Find .

(b) Find a closed form for the sequence given by the recurrence system.

(c) Find  to the nearest integer

Question 27

(a) Find  .

(b) Use the substitution x = 5 cosh u to find the integral   (x > 5).

(c) Hence find the general solution in implicit form of the differential equation

 

Question 28

 (a) Prove the following statement by using mathematical induction.

 for all n ∈ N.

(b) Show that the following statement is false.

 for all n ∈ N.

Question 29

A block slides, under gravity, down a flat rough slope that is inclined at  to the horizontal. The coefficient of sliding friction between the block and the slope is 0.44. Take the magnitude of the acceleration due to gravity to be g = 9.8ms-2. Model the block as a particle.

(a) State the three forces acting on the block during its motion, and draw a force diagram representing these forces, labelling them clearly.

(b) Find expressions for the component forms of the three forces, in terms of the mass m (in kg) of the block and any unknown magnitude(s) where appropriate. Take the Cartesian unit vectors i and j to point parallel and perpendicular to the slope, respectively, in the directions shown in the diagram above.

(c) Hence or otherwise find the magnitude of the acceleration of the block, in  to two significant figures.

(d) The block starts from rest and slides a distance of 3 metres down to the end of the slope. Find the speed of the block when it reaches the end of the slope. Give your answer in  to two significant figures.

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