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PAPER TITLE: ESSENTIAL MATHEMATICS 2

EXAM DATE: MONDAY 11, SEPTEMBER 2017

COURSE CODE: MST125/B

SECTION A

 

Question 1

What is the least residue of 732 modulo 11?

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

 modulo 11

Using  theorem

 

 

 

 modulo 11 = 

 

 

 modulo 11 =

Answer: C

Question 2

An ellipse in standard position crosses the x-axis at (±4, 0) and has eccentricity 0.5. Where does it cross the y-axis?

Question 3

A hyperbola in standard position has equation

.

What are the equations of its asymptotes?

Question 4

What is the equation of the curve described by the parametric equations 

 

Question 5

A calculator of mass 0.095 kg rests on a horizontal desk. What is the magnitude of the normal reaction of the desk on the calculator, in newtons to two significant figures? Take the magnitude of the acceleration due to gravity to be .

Question 6

A box rests on a rough plane that is inclined at 40⁰ to the horizontal. Which of the following diagrams represents the forces acting on the box?

Question 7

If f and g are the linear transformations represented by the matrices and , respectively, which of the following matrices represents the linear

Question 8

Which of the following affine transformations maps the points (0, 0), (1, 0) and (0, 1) to the points (1, 1), (2, −1) and (−2, 1), respectively?

Question 9

The graph of a function f is shown below.

Which of the following could be the rule for f?

Question 10

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

Question 11

What is the general solution of the differential equation 

?

In the options, c is an arbitrary constant and A is an arbitrary positive constant.

Question 12

Let P be the statement

If  is odd, then n is even.

Which of the following statements is the contrapositive of P?

Question 13

Consider the following statement.

For every natural number n, the number 2n + 1 is prime.

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

Question 14

An object of mass 15 kg, initially at rest, is subject to a resultant force of magnitude 45 N in a constant direction. At what time (in seconds) is the magnitude of its velocity ?

Question 15

The position x (in metres) of an object moving along a straight line is given in terms of the time t (in seconds) by

 .

What is the magnitude of its velocity (in ms^-1) when t = 0?

Question 16

What are the eigenvalues of the matrix  ?

Question 17

Which of the following is an eigenvector of the matrix  corresponding to the eigenvalue λ =5?

Question 18

What is the characteristic equation of the matrix  ?

Question 19

Consider the 5-character sequences that can be made by using the 7 letters A, B, C, D, E, F, G at most once each. How many of these sequences end with the letter E? Examples of such sequences are AFGDE and BGCFE.

Question 20

What is the general solution of the recurrence relation

 ?

Question 21

(a) Use Euclid’s algorithm to find a multiplicative inverse of 5 modulo 27. 

(b) Explain why the linear congruence

15x ≡ 18 (mod 81)

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

Question 22

Two of the following three transformations of the plane are linear transformations, and one is not.

f(x, y)=(x + 2y, x − 2)

g(x, y) = (2x, −x + 2y)

h(x, y) = (2x − 2y, −x + y)

(a) Which of the transformations is not a linear transformation? Justify your answer.

(b) Write down the matrix of each of the two linear transformations.

(c) For each of the two linear transformations, either show that it is not invertible, or find the rule of its inverse transformation, in the same form as the rules of f, g and h given above.

Question 23

Find the integral

Question 24

Solve the initial value problem

(x > 0), where y = 5 when x = 2

Question 25

Let P(n) be the statement

 is divisible by 3.

Use mathematical induction to show that P(n) is true for all n ∈ N.

Question 26

A football is kicked into the air from the ground. When it leaves the ground, it is travelling at a speed of 6 m s⁻¹ at an angle of 30⁰ to the horizontal. Take the x-axis to point horizontally in the direction of motion and the y-axis to point vertically upwards, with the origin at the starting point of the ball. Assume that there are no forces from the air on the ball.

(a) Find the velocity of the ball when it leaves the ground, in component form.

(b) Find an expression in component form for the position x (in metres) of the ball in terms of the time t (in seconds) since it left the ground. Denote the magnitude of the acceleration due to gravity by g.

Question 27

A crate of mass m (in kg) lies on horizontal ground. A chain is attached from the crate to a truck. The chain is taut and makes an acute angle θ with the horizontal as shown below.

Model the crate as a particle. Denote the magnitude of the acceleration due to gravity by g.

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

(b) Take the unit vector i to point horizontally and the unit vector j to point vertically upwards as shown above. Find expressions for the four forces acting on the crate in terms of m, g and any unknown magnitudes of the forces. (For example, let the magnitude of an unknown force A be A.)

(c) Let the coefficient of static friction between the crate and the ground be μ. Using the expressions for the forces that you obtained in part (b), show that when the crate is on the point of slipping, the magnitude of the force in the chain is

 .

d) The crate has mass 1200kg and the chain makes an angle of 40⁰ with the horizontal. The coefficient of static friction between the crate and the ground is 0.7. Take g = 9.8ms⁻². What is the magnitude of the force in the chain when the crate is on the point of slipping? Give your answer in newtons to two significant figures.

Question 28

(a) Find the integral

(b) Find the integral

Question 29

In a lottery game, players buy tickets on which they have to specify 4 different numbers from 1, 2,..., 40. In the draw, the winning 4 numbers are decided by randomly taking 4 different numbered balls from a drum containing a set of 40 balls numbered 1 to 40. Prizes are awarded for tickets that match 3 or 4 of the winning numbers, in whatever order. In your answers, give all probabilities as fractions.

(a) Show that there are 91 390 ways to choose a ticket to play this lottery.

(b) Find the probability that a ticket matches all 4 winning numbers.

(c) (i) Determine how many ways there are to select

(A) 3 numbers from the 4 winning numbers, in whatever order;

(B) one number from the non-winning numbers.

(ii) Hence show that there are 144 ways to choose a ticket that matches exactly 3 of the four winning numbers.

(iii) Hence, or otherwise, find the probability that a single ticket wins a prize.

(d) Suppose that the lottery organisers decide to award prizes for tickets matching 2, 3 or 4 of the winning numbers, and the lottery is unchanged otherwise. Determine the probability that a single ticket wins a prize.

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