MatLAB Answers

1.    Two fair cubes are rolled. The random variable X represents the difference between the values of the two cubes.

a)    Find the mean of this probability distribution. (i.e. Find E[X])

• Do the calculations on Matlab, print it out and then write your answers on the attached answer

sheet.

• Attach your Matlab printout to your answer sheet before you hand in.

1. Find all solutions for each of the following systems of equations (if the system is consistent):

(a) 6.5x − 2y = 7 (b) 3.5x1 + 4.5x2 + 5.5x3 = 11

2x − 0.75y = 1.75 x1 + 4x2 − 7x3 = −16

12x − y = 21 0.5x1 − 0.75x2 + 0.75x3 = 3.5

(c) − 0.75x1 + 0.75x2 = −6 (d) 3.4x1 + 3.4x2 − 15.3x3 = −20.4

2.5x1 + 2x2 − 4.5x3 = 2 0.5x1 + 0.25x2 − 0.75x3 = 1

1.25x1 + 1.25x2 − 2.5x3 = 0 0.75x1 + 0.5x2 − 1.5x3 = 1

Please note: You should use Matlab to write your systems in reduced row echelon form, but have to

interpret the results and give the solution(s) if the system is consistent.

The Gaussian distribution also known as the Normal distribution, is given by the following

equation:

𝑦(𝑥) = 𝑒𝑥𝑝 −(𝑥−𝜇)^2/2𝜎^2

where parameter 𝝁 is the mean and 𝝈 the standard deviation.

(i) Write a MATLAB code to create a 1000 point Gaussian distribution of random numbers

having 𝜇 = 0 and 𝜎 = 1. (20)

(ii) Plot this distribution. (10)

(iii) Prove that the full width–half maximum (FWHM), of the above distribution is given by :

FWHM = 2𝜎√2ln 2 (10)

The Gaussian distribution also known as the Normal distribution, is given by the following

equation:

𝑦(𝑥) = 𝑒𝑥𝑝 −(𝑥−𝜇)^2/2𝜎^2

where parameter 𝝁 is the mean and 𝝈 the standard deviation.

(i) Write a MATLAB code to create a 1000 point Gaussian distribution of random numbers

having 𝜇 = 0 and 𝜎 = 1. (20)

(ii) Plot this distribution. (10)

The Gaussian distribution also known as the Normal distribution, is given by the following

equation:

𝑦(𝑥) = 𝑒𝑥𝑝 −(𝑥−𝜇)^2/2𝜎^2

where parameter 𝝁 is the mean and 𝝈 the standard deviation.

(i) Write a MATLAB code to create a 1000 point Gaussian distribution of random numbers

having 𝜇 = 0 and 𝜎 = 1. (20)

(ii) Plot this distribution. (10)

1.    The manager of a supermarket collected the data of 25 customers on a certain date. Out of them 5 purchased Biscuits, 10 purchased Milk, 8 purchased Fruits, 6 purchased both Milk and Fruits.

Let B represents the randomly selected customer purchased Biscuits, M represents the randomly selected customer purchased Milk and F represents the randomly selected customer purchased Fruits.

Represent the given information in a Venn diagram. Use that Venn diagram to answer the following questions.

a)       Find the probability that a randomly selected customer either purchased Biscuits or Milk.

b)      Show that the events “The randomly selected customer purchased Milk” and “The randomly selected customer purchased Fruits” are independent.

A ball is thrown vertically upwards with an initial velocity of 30 m/s.

Using a time step of 0.02 s up to 6.20 s, write a matlab code to give a plot of the vertical distance versus

time for this ball.

Hint ; Motion under gravity is described by the equation : 𝑣𝑦 = 𝑣𝑜𝑦𝑡 +1/2𝑔𝑡^2

and gravitational acceleration 𝑔 is here taken as negative.

Then use your code to answer the following questions:

(i) To what maximum height does the ball rise?

(ii) What is the index of time at maximum height?

(iii) How long does it take the ball to ascend to maximum height?

(iv) How long does it take the ball to hit the ground?

(v) What happens to the ball if the sign for gravitational acceleration is taken as positive?

A ball is thrown vertically upwards with an initial velocity of 30 m/s.

Using a time step of 0.02 s up to 6.20 s, write a matlab code to give a plot of the vertical distance versus

time for this ball.

Hint ; Motion under gravity is described by the equation : 𝑣𝑦 = 𝑣𝑜𝑦𝑡 +

1

2

𝑔𝑡

2

and gravitational acceleration 𝑔 is here taken as negative.

Then use your code to answer the following questions:

(i) To what maximum height does the ball rise?

(ii) What is the index of time at maximum height?

(iii) How long does it take the ball to ascend to maximum height?

(iv) How long does it take the ball to hit the ground?

(v) What happens to the ball if the sign for gravitational acceleration is taken as positive?

A company manufactures three products: Engines, Pumps and Fans. They give a discount

of 10% on order for engines if the order is for Rs.5,000 or more. The same discount of

10% is given on pump orders of values of Rs.2,000 or more and on fan orders for

Rs.1,000 or more. On all other orders they do not give any discount.

ball is thrown vertically upwards with an initial velocity of 30 m/s.

Using a time step of 0.02 s up to 6.20 s, write a matlab code to give a plot of the vertical distance versus

time for this ball.

Hint ; Motion under gravity is described by the equation : 𝑣𝑦 = 𝑣𝑜𝑦𝑡 +

1

2

𝑔𝑡

2

and gravitational acceleration 𝑔 is here taken as negative.

Then use your code to answer the following questions:

(i) To what maximum height does the ball rise?

(ii) What is the index of time at maximum height?

(iii) How long does it take the ball to ascend to maximum height?

(iv) How long does it take the ball to hit the ground?

(v) What happens to the ball if the sign for gravitational acceleration is taken as positive?