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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: TUESDAY 17, SEPTEMBER 2019

COURSE CODE: MST125/C

SECTION A

Question 1

What is the least residue of  modulo 11?

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

 

Using Fermat’s theorem

 

 

Recall,

 

 

 

The least residue of

Answer: C

Question 2

What is the equation of the parabola in standard position with directrix x = −4?

Question 3

A hyperbola in standard position has equation

 .

What are the equations of its asymptotes?

Question 4

What is the y-intercept of the line with parametric equations

x = t + 4, y = 2t +3 ?.

Question 5

A sign hangs from an inextensible string. The tension in the string is 6.6 N vertically upwards. What is the mass of the sign, in kilograms to two significant figures? Take the magnitude of the acceleration due to gravity to be 9.8m^s-2 .

Question 6

Which option describes the linear transformation represented by the

matrix ?

Question 7

 An affine transformation f consists of the reflection in the x-axis, followed by the translation

t(x, y)=(x + 3, y).

What is the image of the point (−4, 3) under f?

Question 8

The graph of a function f is shown below.

 

Which of the following could be the rule for f?

Question 9

What is the remainder on dividing the polynomial expression  by the polynomial expression ?

Question 10

Which of the following is the solution to the initial value problem

 

Question 11

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

 

Question 12

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

 (y > 0)?

In the options, c is an arbitrary constant.

Question 13

Let P(n) be the statement

12 is a multiple of n.

Which of the following statements is true?

Question 14

An object moving along a straight line slows down at the rate of 5 m ^s-2  and comes to a halt after 10 seconds. What is the distance in metres travelled by the object during this time?

Question 15

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

r = −3t²  i + 2tj − cos(4t) k,

where i, j and k are the Cartesian unit vectors.

What is the velocity of the particle when t = 0?

Question 16

How many four-digit positive integers are there in which no digit is repeated? Examples of such integers are 6942, 8137, 4239.

Question 17

Two dice are rolled. What is the probability that the total obtained is greater than 4?

Question 18

What is the general solution of the recurrence relation  

 

Question 19

For each of the following linear congruences, determine whether it has a solution, and find a solution if it exists. Give your solution as a least residue.

(a) 10x ≡ 5 (mod 126)

(b) 10x ≡ 6 (mod 126)

Question 20

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 three forces act. The magnitude of the force R is 10 N.

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

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

Question 21

Evaluate the definite integra

 

Question 22

In each of the following statements, n represents a positive integer. One of the statements is true and the other is false.

(A) For all n, the integer  is a multiple of 3.

(B) For all n, the integer  is a multiple of 3.

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

(b) Prove the other statement by mathematical induction.

Question 23

(a) Show that    and    are eigenvectors of the matrix

A =  .

For each eigenvector write down the corresponding eigenvalue.

(b) Express A in the form  , where D is a diagonal matrix and P is an invertible matrix, and hence find .

Question 24

(a) (i) Write down the matrix of the linear transformation

f(x, y) = (3x + 2y, −x). 

(ii) Show that f is invertible and find the matrix of .

(iii) Let E be the ellipse with equation

   .

Fine the equation of the image of E under the linear transformation f. Write your answer in the form , where a, b, c and d are integers whose values you should find.

(iv) The area of the ellipse E is . Find the area of the image of E under the linear transformation f.

(b) (i) Write down the matrix that represents the reflection g in the line through the origin with angle of inclination  

(ii) The matrix that represents the rotation h through  about the origin is

 

By using this matrix, show that the composite transformation formed from the reflection g followed by the rotation h is the reflection in the line y = x 3 .

Question 25

(a) Find the integral  , where x > 3. 

(b) Use a hyperbolic substitution to find the integral .

Question 26

(a) A skier of mass 75 kg accelerates from rest down a straight slope. Her distance x (in metres) down the slope from her starting point in terms of the time t (in seconds) since she left her starting point is given by

 

(i) Find the velocity (in ) of the skier after 4 seconds. Give your answer to two significant figures.

(ii) Find the resultant force (in newtons) on the skier in terms of the time t.

(b) On a second run down the slope, the skier hits a patch of wet snow when her speed is 15 m . The wet snow slows her down. While she is still moving, her acceleration is , where x is now the distance in metres down the slope measured from the start of the wet snow.

(i) Show that, during the period the skier is slowing down,

   

(ii) Hence show that the velocity of the skier during the period of slowing down is

 

(iii) Find the distance of the skier from the start of the wet snow when she comes to a halt. Give your answer to two significant figures.

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Last updated: Sep 02, 2021 12:37 PM

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