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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: WEDNESDAY 1, JUNE 2016

COURSE CODE: MST125/D

SECTION A

Question 1

What is the least residue of 3 401 706 × 17 000 018 modulo 17?

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

3 401 706 × 17 000 018 modulo 17 = 6 modulo 17

The least residue = 6

Answer: C

Question 2

Which of the following is a multiplicative inverse of 11²³ modulo 59?

Question 3

Given that 13 is a multiplicative inverse of 71 modulo 922, which of the following is a solution of the linear congruence

71x ≡ 92 (mod 922) ?

Question 4

What is the equation of the directrix of the parabola with equation

3y² − 32x =0?

Question 5

A block of mass 5.2 kg rests on a horizontal surface. The coefficient of static friction between the block and the surface is 0.47. What is the maximum magnitude of the static friction force from the surface on the object, in newtons to two significant figures? Take the acceleration due to gravity to be g = 9.8ms⁻².

Question 6

Which option describes the linear trans formation represented by the matrix   ?

Question 7

Which of the following specifies the glide reflection g formed from the reflection in the y-axis followed by the translation 2 units down?

Question 8

What is the form of the partial fraction expansion of the expression

 ?

In the options, A, B and C represent constants.

Question 9

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

 ?

Question 10

What is the solution of the initial value problem

, where y = 5 when x =0?

Question 11

Let P(n) be the statement

n is a multiple of 6.

Which of the following statements is true?

Question 12

Let P be the following statement.

At least one of m and n is odd.

Which of the following statements is the negation of P?

Question 13

Consider the following statement.

If n is prime, then n + 2 is prime.

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

Question 14

The graph below is the velocity–time graph of an object moving along a straight line, where v is the velocity (in metres per second) of the object at time t (in seconds)

What is the approximate acceleration (in ms⁻²) of the object at time t = 6?

Question 15

A projectile is launched with an initial velocity (in ms⁻¹) of 2i + 3j, where the Cartesian unit vectors i and j point right and up, respectively. Which of the following gives the velocity v (in ms⁻¹) of the projectile at time t (in seconds) after launch? In the options, g represents the magnitude of the acceleration due to gravity (in ms⁻²).

Question 16

What are the eigenvalues of the matrix ?

Question 17

Which of the following vectors is an eigenvector of the matrix , corresponding to the eigenvalue 3?

Question 18

The matrix A = can be expressed in the form PDP⁻¹ , where

P = , D = and P⁻¹ = . What is A⁵?

Question 19

Consider the set S = {1, 2, 3, 4, 5, 6, 7}.

How many subsets of S contain both the elements 3 and 4? Examples of such subsets are {3, 4, 6} and {3, 4, 5, 7}.

Question 20

What is the probability that when two dice are rolled the total is at least 4?

SECTION B

Question 21

Consider the curve with parametrisation

x = 2 + 4 cost, y = −3 + 4 sin t (0 ≤ t ≤ π/2).

(a) Calculate the coordinates of the endpoints of the curve.

(b) Sketch the curve, or describe it geometrically.

(c) Find the y-coordinate of the point on the curve with x-coordinate 4. Give an exact answer.

Question 22

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 forces act. The magnitude of the force R is 20 N.

(a) Find expressions for the component forms of the three forces R, S and T, 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 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 23

(a) Find the affine transformation f that maps the points (0, 0),(1, 0) and (0, 1) to the points (−2, 3),(4, 5) and (2, 6).

(b) Hence find the area of the triangle with vertices (−2, 3),(4, 5) and (2, 6).

(c) Find the image of the point (1, 1) under f.

Question 24

Evaluate the integral .

Give an exact answer

Question 25

(a) Find the integral .

(b) Hence, or otherwise, solve the following initial value problem, giving your answer in implicit form.

  (y > 0), where y = 1 when x = π/ 2

 Question 26

Let P(n) be the following statement:

5ⁿ > 20n.

(a) Show that P(2) is false and P(3) is true.

(b) Use mathematical induction to show that P(n) is true for all n ≥ 3.

SECTION C

Question 27

Consider the function .

(a) Find its domain and intercepts.

(b) Find f’ (x).

(c) Find the coordinates of any stationary points of f. Construct a table of signs for f (x), determine the intervals on which f is increasing or decreasing, and determine the nature(s) of the stationary point(s).

(d) Write down the equations of the asymptotes of f.

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

(f) Sketch the graph of f.

Question 28

A scene in a film involves a motorcyclist riding off a cliff edge and over a river. Assume that at the instant the motorcyclist leaves the cliff edge, he is 8 m above the river, his speed is V m s−1 and he is travelling in an upwards direction, making an angle of 15⁰ with the horizontal. Take the origin to be vertically below the edge of the cliff and on the surface of the river. Take the y-axis to point vertically upwards from the origin and the x-axis to point horizontally from the origin across the river, as shown below.

Let i and j be the Cartesian unit vectors in the positive directions of the x- and y-axes respectively. Assume that the motorcyclist and his motorcycle can be modelled as a particle and that the only force acting on it is its weight. Take the magnitude of the acceleration due to gravity to be g = 9.8ms⁻².

(a) Write down an expression for the acceleration of the motorcyclist. Hence show that the position r of the motorcyclist at time t seconds after he has left the cliff edge is given by

.

(b) Assume that the motorcyclist lands on the opposite riverbank, after travelling a horizontal distance of 45 m. Find the time, in seconds to two significant figures, that it took the motorcyclist to reach the riverbank.

(c) Determine the speed V , in m s⁻¹ to two significant figures, at which the motorcyclist left the cliff edge.

For each positive integer n, let u_n be the number of ways of tiling a 10 cm × (10n) cm strip with these tiles.

(a) Find the values of u₁ and u₂, briefly explaining your answers.

(b) Explain clearly why the sequence (u_n) satisfies the recurrence relation

u_n = 2u_n−1 + 3u_n−2.

(c) Find a closed form for the sequence (u_n).

(d) Find the number of ways of tiling a 10 cm × 50 cm strip.

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