Solve your IGNOU Doubts

Question:

A sugar manufacture has two production processes. In one hour, Process I makes 100 kg of Grade I (high quality) sugar and also produces 140 kg of Grade II sugar as a byproduct. Process II makes in one hour 60 kg of Grade I (high quality) sugar and also produces as a by-product 40 kg of Grade II sugar. The manufacture is confident that during the festival season all the sugar that is made can be sold. He has committed to selling at least 6000 kg of Grade I and 5600 kg of Grade II for the season. The revenue earned by selling one kilogram of Grade I sugar is ₹ 4 (irrespective of the process used) and the revenue earned by selling one kilogram of Grade II sugar is ₹ 2 (irrespective of the process used). Formulate the problem of maximizing the total revenue earned as an LPP. Solve the problem by graphical-method.

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Question:

For a Queuing Model (M/M/1): (GD/∞/∞) with one server, if the service rate µ increases, the expected number of customers in the system decreases.

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Question:

In the inventory model with finite replenishment rate, if the replenishment rate is equal to consumption rate, the holding cost is 0.

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Question:

An assignment problem can be considered as a special case of transportation problem.

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Question:

In an optimal solution $(x_{1}^{*}x_{2}^{*})$ of the LPP

$Max.$ $4x_1+3x_2$

s.t.

$x_1+x_2=2$

$x_1+x_2\geq 0$

$x_{1}^{*},x_{2}^{*}$ both cannot be positive.

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Question:

Which of the following statements are true or false? Give a short proof or a counter example in support of your answers.

a) There is chance of occurrence of the premature convergence in Roulett-wheel selection shceme used in GA.

b) Gradient based optimization methods are used when the objective function is not smooth and one needs efficient local optimization.

c) The $\alpha -$ cut of a fuzzy set A in U is defined as A $\alpha _{0}=\left \{ x\epsilon \bigcup | \right.\mu _{A}(x)\leq \alpha _{0}\left. \right \}$

d) A single perception with preprocessing is neither an auto associative network nor a multiple layer neural network.

e) If W $(k_{0})=W(K_{0}+2),$ then perception is non-linear separable.

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Question:

The optimal solution of any integer linear programming problem can be obtained by rounding off the optimal solution of its LP relaxation.

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Question:

b) Consider the following single layer perception as shown in the following figure.

Image ignouassignments-ignouacademy-com--p-solve-12649

and the activation function of each unit is defined as $\Phi (v)=\left \{ _{0, otherwise.}^{1, if v \geq 0} \right \}$

Calculate the output y of the unit for each of the following input patterns:

Patterns	$P_{1}$	$P_{2}$	$P_{3}$	$P_{4}$
$X_{1}$	1	0	1	1
$X_{2}$	0	1	0	1
$X_{3}$	0	1	1	1

Also, find the modified weights after one iteration.

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Question:

Alice wants to use the Digital Signature algorithm for signing messages. She chooses p = 83, q = 41, g = 2 and a = 3. Alice wants to sign the message M = 20. She chooses the secret value k = 8. Explain the procedure that Alice will use for computing the signature. What information will she send Bob?

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Question:

Alice wants to use the ElGamal digital signature scheme with public parameters p = 47, α = 2, secret value $\alpha =7$ and $\beta =34$ . She wants to sign the message M = 20 and send it to Bob. She chooses k = 5 as the secret value. Explain the procedure that Alice will use for computing the signature of the message. What information will she send Bob?

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Question:

Solve the discrete logarithm problem $5^{x}\equiv 22$ (mod 47) using Baby-Step, Giant-Step algorithm.

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Question:

i) Bob uses ElGamal cyrptosystem with parameters p = 47, g = 5 and the secret value
x = 3. What values will he make public?
ii) Alice wants to send Bob the message M = 15. She chooses k = 5. How will she compute
the cipher text? What information does she send to Bob?
iii) Explain how Bob will decrypt the message.

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Question:

a) Consider the following travelling salesman problem involving 9 cities

Parent 1	G	J	H	F	E	D	B	I	C
Parent 2	D	C	H	J	I	G	E	F	B

Determine the children solution using.

i) Order crossover #1, assuming $4^{th}$ and $7^{th}$ sites as the crossover sites

ii) Order crossover #2, assuming $3^{rd},$ $5^{th},$ and $7^{th}$ as the key positions.

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Question:

Decrypt the message c = 23 that was encrypted using RSA algorithm with e = 43 and n = 77.

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Question:

a) Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance : $\alpha =0.05$

011001110000110010011100.

b) Apply poker test to the following sequence with level of significane $\alpha =0.05$
1001101000010000101111011
01110100101101100100110.

c) Apply runs test to the following sequence:
1001101000010000101111011
0111010010110110010011010
0110011100001100100111000
1100001101010111101001110
0010001111000001101010010
1000110100000110100101101
1110001001

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Question:

a) Considering the bytes 10001001 and 10101010 as elements of the field $F_2[X]/\left \langle g(X) \right \rangle,$ where $g(X)$ is the polynomial $X^{8}+X^{4}+X^{3}+X+1,$ find their product and quotient.

b) Find a recurrence that generates the sequence 110110110110110.

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Question:

b) Let an activation function be defined as

$\Phi (V)=\frac{1}{1+e^{-av}},a> 0$

Show that $\frac{d\Phi }{dV}=a\Phi (v)[1-\Phi (V]).$ What is the value of $\Phi (V)$ at the origin? Also, find the value of $\Phi (V)$ as v approaches $+\infty$ and $-\infty .$

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Question:

a) Decrypt the ciphertext 101000111001 which was encrypted with the Toy block cipher once
using the key 101010010. Show all the steps. (5)
b) A 64 bit key for the DES is given below
11000111 10000101
11110111 11000001
11111011 10101011
10011101 10010001
i) Check whether the key is error free using the parity bits.
ii) Find the keys for the second round.

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Question:

In this exercise, we introduce you to Hill cipher. In this cipher, we convert our message to numbers, just as in affine cipher. However, instead of encrypting character by character, we encrypt pairs of characters by multiplying them with an invertible matrix with co-efficients in $Z_{26}$ .

Here is an example: Suppose we want to ENCRYPT "ALLISWELL". Since we require the plaintext to have even number of characters, we pad the message with the character ‘X’. We break up the message into pairs of characters AL, LI, SW, EL and LX. We convert each pair of characters into a pair elements in $Z_{26}$ as follows:

Image ignouassignments-ignouacademy-com--p-your-61738

Next, we choose an inveritble 2×2 matrix with coefficients in $Z_{26}$ , for example, $A=\left [ \frac{3}{7}\, \frac{1}{4} \right ].$

This matrix has determinant $\bar{3},\bar{4}-\bar{7},\bar{1}=\bar{5}\, and\, \bar{5}$ and 5 is a unit in $Z_{26}$ with inverse $\bar{21}.$ We write each pair of elements in $Z_{26}$ as a column vector and multiply it by A:

$A\left [ \frac{\bar{0}}{11} \right ]=\left [ \frac{\bar{11}}{18} \right ].A\left [ \frac{\bar{11}}{8} \right ]=\left [ \frac{\bar{15}}{5} \right ]....$

We then convert each pair of numbers to a pair of characters and write them down. In this example, we get the cipher text "LSPFYGXUEN" corresponding to the plain text "ALLWELL". To decrypt, we convert pairs of characters to pairs of numbers and multiply by $A^{1}=\bar{5}^{-1}$ $\left [ \frac{\bar{4}}{-7}\:\, \: \frac{\bar{-1}}{3} \right ]=\bar{21}\left [ \frac{\bar{4}}{-7}\, \, \, \frac{\bar{-1}}{3} \right ]=\left [ \frac{\bar{6}}{9}\: \, \, \frac{\bar{5}}{11} \right ]$ and we have

$\left [ \frac{\bar{6}}{9}\, \, \bar{\frac{5}{11}} \right ]\left [ \frac{\bar{11}}{18} \right ]=\left [ \frac{\bar{0}}{18} \right ]....$

Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill’s cipher with the matrix $\left [ \frac{\bar{3}}{0} \, \, \, \frac{\bar1}{9}\right ]$ as the encryption matrix