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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: TUESDAY 11, JUNE 2019

COURSE CODE: MST125/D

SECTION A

Question 1

Which is a multiplicative inverse of 12 modulo 37?

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

1 ≡ 38≡75≡852 (mod 126)

 

 

The multiplicative inverse of 12 modulo 37 = 34

Answer: E

Question 2

What is the least residue of  modulo 13?

Question 3

Given that 237 is a multiplicative inverse of 320 modulo 419, and that 419 is prime, which is the solution of

320x ≡ 3 (mod 419) ?

Question 4

A line has parametric equations x =1+ t, y = 3t − 4. What is the y-intercept of the line?

Question 5

A bag of mass 5 kg is suspended by a rope from a branch of a tree and remains at rest. What is the tension in the rope, in newtons, to two significant figures? Take the magnitude of the acceleration due to gravity to be 9.8ms^-2 .

Question 6

Which of the following specifies the glide reflection g formed from the reflection in the y-axis, followed by the translation 4 units down?

Question 7

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

Question 8

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

Question 9

The graph of a function f is shown below.

Question 10

What is the general solution in explicit form of the differential equation  

                                                    (x > 0,y > 0)?

In the options c is an arbitrary constant.

Question 11

Let P(n) be the statement

n is a multiple of 7.

Which of the following statements is true?

Question 12

What is the negation of the following statement?

For all-natural numbers n, the number is an integer.

Question 13

A ball is thrown vertically upwards in a straight line and travels under the influence of gravity alone. The initial speed of the ball is 10 m s-¹ . What is the maximum height (in metres, to two significant figures) reached by the ball, measured from the point of projection? Take the magnitude of the acceleration due to gravity to be 9.8ms-² .

Question 14

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

r = sin(2t)i − 5tj.

What is its acceleration when t = π/4?

Question 15

A matrix A can be written in the form A =  where

P =

What is the trace of A?

Question 16

Consider the set S = {1, 2, 3, 4, 5, 6, 7, 8}. How many subsets of S contain the element 3, but do not contain the element 7? Examples of such subsets are {3}, {1, 2, 3, 4} and {3, 4, 5, 6, 8}.

Question 17

Three dice are rolled. What is the probability that at least one shows a 4?

Question 18

The general solution of the recurrence relation

 

is given by

 

where A and B represent arbitrary constants. Which is the closed form for the sequence defined by this recurrence relation with  = 3 and  = 1?

Question 19

A hyperbola in standard position has equation .

(a) Find the eccentricity of the hyperbola.

(b) Find the foci and the directrices of the hyperbola.

(c) Find the equations of the asymptotes.

(d) Is this hyperbola rectangular? Justify your answer.

Question 20

A particle, which remains at rest, is acted on by three forces, F, N and W. The force diagram below shows the angles at which the forces act. The magnitude of the force W is 30 N.

(a) Find expressions for the component forms of the three forces, F, N and W, taking the directions of the Cartesian unit vectors i and j to be as shown in the diagram (i is in the opposite direction to F and j is in the same direction as N). Denote the magnitudes of F and N by F and N respectively.

(b) Hence, or otherwise, find F and N in newtons to two significant figures.

Question 21

Use a trigonometric substitution to find the integral

 where 0 ≤ x ≤ 5.

Question 22

Solve the initial value problem

 (x > 0), where y = 3 when x = 1.

Question 23

Find the eigenvalues of the matrix .

Find an eigenvector of the matrix that corresponds to the larger eigenvalue.

Question 24

Consider the function  .

(a) Find the domain and intercepts of f.

(b) Show that    .

(c) Find the coordinates of any stationary points of f.

(d) Construct a table of signs for f (x).

(e) Determine the intervals on which f is increasing and the intervals on which it is decreasing.

(f) Determine the nature(s) of the stationary point(s).

(g) Find the equations of the asymptotes of f.

(h) Determine whether f is an even or odd function, or neither.

(i) Sketch the graph of f.

Question 25

(a) Prove that the following statement is true for all-natural numbers n by using a sequence of equivalences:

                    .

(b) Prove the following statement by mathematical induction. 

 , for all integers n ≥ 2.

Question 26

A child pulls a toy of mass 0.5 kg in a straight line, along a horizontal table that is 0.4 m above the floor. The pulling force acts horizontally to the right and the toy has an acceleration of 2 ms^-2 , as shown in the diagram below.

The coefficient of sliding friction is 0.2. Take the magnitude of the acceleration due to gravity to be g = 9.8 ms-2 . Model the toy as a particle.

(a) (i) Four forces act on the toy. One of these is the pulling force P. State the three other forces. Draw a force diagram that represents the four forces and label the forces clearly.

(ii) Take the unit vector i to point horizontally in the direction of motion and the unit vector j to point vertically upwards. Find expressions for the component forms of the four forces acting on the toy.

(iii) Write down Newton’s second law of motion for the toy and hence calculate the magnitude of the pulling force.

(b) The child stops pulling the toy when it reaches the edge of the table. The toy then moves freely under gravity alone. Measure time from the instant the toy leaves the table and take the origin to be on the floor vertically below the point where the toy leaves the table, so that at time 0, the position of the toy is 0.4 j. Assume that at time 0, the velocity is 5 i.

(i) The acceleration due to gravity is −9.8 j. Show that the position of the toy is given by

                       

(ii) Hence, or otherwise, find the horizontal distance that the toy travels before it hits the ground. Give your answer to two significant figures.

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