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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: THURSDAY 17, SEPTEMBER 2015

COURSE CODE: MST125/D

SECTION A

Question 1

What is the least residue of 900 002 × 450 002 modulo 45?

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

900 002 × 450 002 modulo 45 = 4 modulo  45

The least residue of 900 002 × 450 002 modulo 45 = 4

Answer: B

Question 2

Which of the following is a multiplicative inverse of 10²¹modulo 43?

Question 3

What is the equation of the ellipse that crosses the x-axis at (±5, 0) and the y-axis at (0, ±3)?

Question 4

Which option describes the graph given by the parametrisation

x =1+3t, y = 4 − t (−1 ≤ t ≤ 2) ?

Question 5

A crate rests on horizontal ground. The normal reaction of the ground on the crate is 25 N vertically upwards. What is the mass of the crate in kilograms to two significant figures? Take the magnitude of the acceleration due to gravity to be 9.8 m s⁻².

Question 6

Which option describes the linear transformation represented by the matrix

Question 7

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

Question 8

Let f(x) = Px and g(x) = Q_x, where

            P = and Q =

Which of the following matrices represents the composite transformation g⁰f?

Question 9

What is the quotient on dividing the polynomial expression x³ − 3 by the polynomial expression x − 3?

Question 10

The graph of a function f is shown below.

Which of the following could be the rule for f?

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(n) be the statement

n > 3.

Which of the following statements is true?

Question 14

What is the contrapositive of the following statement?

If n is odd, then n² + 1 is even.

Question 15

Consider the following statement.

For each integer n, the integer 6n + 1 is prime.

Which of the following integers is a counter-example to this statement?

Question 16

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

r = 3t² i + (4t − 7)j − sin t k,

where i, j and k are Cartesian unit vectors. What is the acceleration of the particle in terms of t?

Question 17

An object of mass 20 kg, initially at rest, is subject to a resultant force of magnitude 25 N in a constant direction. What is its speed in m s⁻¹ after 4 seconds?

Question 18

There are 18 students in a class, taught by four teachers. How many ways are there to choose a team of representatives consisting of two students and a teacher?

Question 19

A die is rolled twice. What is the probability of getting at least one 4

Question 20

What is the general solution of the recurrence relation

u_n = u_n−1 + 6u_n−2 ?

In the options, A and B represent constants.

SECTION B

Question 21

For each of the following linear congruences, determine whether it has a solution and find a solution if one exists.

(a) 11x ≡ 15 (mod 170)

(b) 15x ≡ 11 (mod 35)

Question 22

The equation x² − 4y² = 8 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.

(c) Find the distance between a focus of the conic and the nearest vertex of the conic. Give an exact answer.

Question 23

A particle, which remains at rest, is acted on by three forces, P, Q and R, and no others. The force P acts at an angle of 40⁰ to the horizontal, the force Q acts at an angle of 50⁰ to the horizontal and the force R acts horizontally to the right, as shown. The force P has magnitude 55 N. Let Q = |Q| and R = |R|. Take the Cartesian unit vectors i and j to be parallel to P and Q, respectively, in the directions shown.

 

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

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

Question 24

Find the integral

Question 25

Solve the initial value problem

 , where y(0) = 0.

Leave your answer in implicit form

Question 26

Find the eigenvalue(s) of the matrix   . For each eigenvalue find a corresponding eigenvector.

SECTION C

Question 27

(a) Evaluate the integral  .

(b) Find the integral

Question 28

(a) Prove the following statement, where n is a natural number.

If 7n + 2 is odd, then n is odd.

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

, for all integers n ≥ 2.

Question 29

A tile slides, under gravity, down a flat rough roof that is inclined at 35⁰ to the horizontal. The coefficient of sliding friction between the tile and the roof is 0.2. Take the magnitude of the acceleration due to gravity to be g = 9.8ms⁻². Model the tile as a particle.

(a) State the three forces acting on the tile 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 tile and any unknown magnitude(s) where appropriate. Take the x- and y-axes 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 tile, in m s⁻² to two significant figures.

(d) The tile starts from rest and slides a distance of 3 metres to the edge of the roof. Find the speed of the tile when it reaches the edge of the roof. Give your answer in m s−1 to two significant figures.

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