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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: MONDAY 12, SEPTEMBER 2016

COURSE CODE: MST125/B

SECTION A

Question 1

What is the least residue of 3⁴² modulo 5?

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

using Fermat’s rule

 

Since,

Therefore,

 

3⁴² modulo 5 =

Hence,

The least residue = 4

Answer: E

Question 2

What is the equation of the parabola in standard position with directrix x = −3 ?

Question 3

What is the gradient of the line with parametric equations

x = 1 − 2t, y =4+3t ?

Question 4

A ball of mass 1.2 kg is suspended by a chain from a ceiling, and a ball of mass 0.6 kg is suspended by a chain from the first ball, as shown.

Which of the following is the magnitude, in newtons to two significant figures, of the tension in the chain attached to the ceiling? Take the magnitude of the acceleration due to gravity to be g = 9.8ms⁻².

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 F i + G j − G i − 18 j = 0, where i and j are the Cartesian unit vectors. What is the value of F (in newtons)?

Question 6

A metal cube is placed on a horizontal sheet of glass. The coefficient of static friction between the cube and the glass is 0.62. If the angle of inclination of the glass sheet is gradually increased, at what angle (to two significant figures) will the cube start slipping?

Question 7

Which of the following describes the linear transformation with matrix

Question 8

Consider the following two-stage diagram.

Which of the composite transformations below is illustrated by this diagram,

where f(x) =  x, g(x) =  x, h(x) = x ?

Question 9

What is the inverse transformation of the affine transformation

 

Question 10

What is the quotient on dividing the polynomial expression 3x² − 7x + 5 by the polynomial expression x − 2?

Question 11

Which of the following is the graph of the function  ?

Question 12

What is the solution of the initial value problem

, where y = 4 when t =0?

Question 13

What is the general solution of the differential equation

(y > 0, x ≥ 0) ?

In the options, A and c are arbitrary constants

Question 14

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

 (x > 0) ?

Question 15

You know that:

If Tigger is a cat, then Tigger has two tails.

Tigger has two tails.

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

Question 16

What is the negation of the following statement?

m is even and n is odd.

Question 17

Consider the following statements, where n ∈ N.

P(n) means n is a multiple of 12.

Q(n) means n is a multiple of 3.

Which of the following statements is true?

Question 18

A train is travelling at 45 m s⁻¹ along a length of straight track. The driver receives an instruction to reduce the speed to 10 m s−1 by the time the train reaches a point 750 m away. What is the magnitude, in m s⁻² to three significant figures, of the constant acceleration (the slowing down) needed?

Question 19

Which of the following vectors is an eigenvector of the matrix   corresponding to the eigen value −5?

Question 20

What is the general solution of the recurrence relation

 (n = 2, 3, 4,...) ?

In the options, A and B represent constants.

Question 21

(a) Use Euclid’s algorithm to find a multiplicative inverse of 7 modulo 44.

(b) Explain why the linear congruence

14x ≡ 8 (mod 88) has solutions,

and use your answer to part (a) to find a solution.

Question 22

Consider the hyperbola in standard position with focus (18, 0) and directrix x =  .

(a) Find the positive x-intercept a, and the eccentricity e of the hyperbola.

(b) Find the equations of the asymptotes. Give exact answers.

(c) Write down the equation of the hyperbola.

Question 23

Evaluate the integral

Question 24

A curling stone of mass 18 kg lies on smooth horizontal ice, as shown below. It is pushed by a force of magnitude 12 N, acting at an angle of 20⁰ to the horizontal, which causes the stone to move along the ice.

(a) Draw a force diagram representing the pushing force P, the weight W and the normal reaction N acting on the stone, and find expressions for the component forms of these three forces. In doing this, model the stone as a particle, take the Cartesian unit vectors i and j to point in the directions shown, and denote the magnitude (in ms⁻²) of the acceleration due to gravity by g and the magnitude (in N) of the normal reaction by N.

b) Find the magnitude of the acceleration of the stone, in m s⁻² to two significant figures.

Question 25

(a) Show that  and  are eigenvectors of the matrix A = . For each eigenvector write down the corresponding eigenvalue. 

(b) Express A in the form PDP⁻¹, where D is a diagonal matrix and P is an invertible matrix, and hence find A⁶.

Question 26

A scientific publisher has published 8 new physics books and 6 new chemistry books and needs to choose a selection of 5 of these books for a display. The chosen books must all be different, and the order of the books is not important.

(a) (i) How many ways are there to select 2 physics books?

(ii) How many ways are there to select 3 chemistry books?

(iii) How many ways are there to choose the selection of 5 books if it is to contain 2 physics books and 3 chemistry books?

(b) Determine the number of ways to choose the selection of 5 books if it must contain at least 2 physics books and at least 2 chemistry books.

Question 27

A full crate of mass 55 kg is held at rest on a rough ramp inclined at 26⁰ to the horizontal by a rope parallel to the ramp, as shown below. The crate is on the point of slipping down the ramp. The coefficient of static friction between the crate and the ramp is 0.45. The rope passes over a pulley at the top of the ramp, and a metal weight hangs from the other end of the rope. Take the magnitude of the acceleration due to gravity to be g = 9.8ms⁻².

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

(b) Find expressions for the component forms of the four forces, in terms of unknown magnitudes where appropriate. In doing this, take the Cartesian unit vectors i and j to point parallel and perpendicular to the slope in the directions shown above.

(c) Hence or otherwise find the magnitude of the tension in the rope, in newtons to two significant figures.

(d) State the forces acting on the metal weight, and draw a force diagram representing these forces, labelling them clearly. Hence find the mass of the metal weight, in kilograms to two significant figures.

Question 28

(a) Find the integral   

(b) Use a hyperbolic substitution to find the integral

Question 29

(a) Prove that the following statement is true for all integers n:

2n² + 7n + 3 is odd if and only if n is even.

(b) Use mathematical induction to prove that

), for all n ∈ N.

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