CST294 D

H1 02000CST294062202 Pages: 4
Reg No.:_______________ Name:__________________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
Fourth Semester B.Tech (Honours) Degree Examination June 2023 (2021 Admission)
Course Code: CST294

Course Name: Computational Fundamentals for Machine Learning
Max. Marks: 100 Duration: 3 Hours
PART A
(Answer all questions; each question carries 3 marks) Marks
1 Show that the vector 𝑤 = (9,2,7) is a linear combination of the vectors 𝑢 = 3

(1,2, −1)and𝑣 = (6,4,2).
2 Find T(x1,x2), where T:R2→R3 is defined by T(1,2)=(3,-1,5) and T(0,1) = (2,1,-1). 3
3 Find the norm of the vector u = (2, −2, 3, −4). 3
4 5 4 3
Find the eigen values of the matrix [ ]
1 2
5 Confirm that Ø(x, y, z) = x2-3y2+4z3 is a potential function for 𝐹⃗ = 2xi -6yj+12z2 k 3
6 What is back propagation? 3

7 Two cards are drawn in succession from a pack of 52 cards. Find the chance that the first 3
is a king and the second a queen if the first card is (i) replaced, (ii) not replaced.
8 A continuous Random Variable X has a probability density function = kx2e-x, x>0. Find 3
k, mean and variance.
9 Explain Gradient Descent algorithm. 3
10 Convert the following L.P.P. to the standard form: 3

Maximize 𝑧 = 3𝑥1 + 5𝑥2 + 7𝑥3 subject to
6𝑥1 − 4𝑥2 ≤ 5
3𝑥1 + 2𝑥2 + 5𝑥3 ≥ 11
4𝑥1 + 3𝑥3 ≤ 2
𝑥1 , 𝑥2 ≥ 0
PART B
(Answer one full question from each module, each question carries 14 marks)
Module -1
11 a) Solve the system of equations: 7
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02000CST294062202
𝑥1 − 𝑥2 + 𝑥3 + 𝑥4 = 2
𝑥1 + 𝑥2 − 𝑥3 + 𝑥4 = −4
𝑥1 + 𝑥2 + 𝑥3 − 𝑥4 = 4
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 0
b) Prove that the vector(1, −2,5)is a linear combination of the vectors 𝑢 = (1,1,1), 𝑣 = 3
(2, −1,1)and 𝑤 = (1,2,3).
c) Consider the linear mapping defined as 4
𝐹(𝑥, 𝑦, 𝑧, 𝑡) = (𝑥 − 𝑦 + 𝑧 + 𝑡, 2𝑥 − 2𝑦 + 3𝑧 + 4𝑡, 3𝑥 − 3𝑦 + 4𝑧 + 5𝑡)
Find a basis and dimension for the image of F.
12 a) Show that the set of all pairs of real numbers of the form(1, 𝑥)with the operations: 10
(1, 𝑦) + (1, 𝑦′) = (1, 𝑦 + 𝑦′), 𝑘(1, 𝑦) = (1, 𝑘𝑦)
is a vector space.
b) Show that the linear mapping defined by𝐹((𝑥, 𝑦, 𝑧)) = (𝑥, 𝑦, 0)is linear. 4
Module -2
13 a) Verify triangle inequality for x = (1, 1, 1) and y= (1, 2, 3) in V3(R) with standard
inner product. 7
Apply Gram-Schmidt orthogonalization process to construct an orthonormal

b) basis for V3(R) with standard inner product for the basis{V1,V2,V3},Where
V1=(1,0,1), V2=(1,3,1) and V3=(3,2,1). 7
14 a) −2 2 −3
Find the Eigen values and Eigen vectors of the matrix 𝐴 = [ 2 1 −6].
−1 −2 0 7
b) 0 1
Compute the Singular Value Decomposition of A=[1 1].
1 0 7
Module -3
15 a) Use the Chain Rule to differentiate 𝑅(𝑧) = √5𝑧 − 8 4
b) i) Compute the Hessian of 𝑓(𝑥, 𝑦) = 𝑥 3 − 2𝑥𝑦 − 𝑦 6 at the point (1,2) 10
ii) Find the nth derivative of 𝑥𝑒 𝑥 𝑎𝑡 𝑥 = 0
16 a) Consider the function 𝑓(𝑥) = cos (𝑥) seek Taylor series expansion of f at x0=0 6
b) Explain automatic differentiation. Draw a computation graph by selecting 9

suitable example
Page 2 of 4
02000CST294062202
Module -4
17 a) A lot consists of 10 good articles, 4 articles with minor defects and 2 with major defects. 6
Two articles are chosen at random from the lot (without replacement) Find the
probability that:
i. atleast one of the chosen articles is good.
ii. atmost one of the chosen articles is good.
iii. neither of the chosen article is good.
b) Establish the Beta-Binomial Conjugacy. 4
c) A random variable X takes the value 1,2,3 and 4 such that 4
2𝑃(𝑋 = 1) = 3𝑃(𝑋 = 2) = 𝑃(𝑋 = 3) = 5𝑃(𝑋 = 4)
Find the probability distribution.
18 a) A continuous random variable X has the probability density function given by 𝑓(𝑥) = 5
𝑘𝑥 2 𝑒 −𝑥 , 𝑥 ≥ 0.Find k, mean and variance.
b) Suppose that 3% of bolts made by a machine are defective, the defectives occurring at 5
random during production. If the bolts are packaged 50 per box, what is the Poisson
approximation of the probability that a given box will contain x = 0, 1, 2, 3 defectives?
c) There are 3 fair coins and 1 false coin with ‘head’ on both sides. A coin is chosen at 4
random and tossed 4 times. If ‘head’ occurs all the 4 times, what is the probability that
the false coin has been chosen for tossing?
Module -5
19 a) Three grades of coal A, B and C contains phosphorus and ash as impurities. In a 10

particular industrial process, fuel up to 100 ton (maximum) is required which should
contain ash not more than 3% and phosphorus not more than 0.03%. It is desired to
maximize the profit while satisfying these conditions. There is an unlimited supply of
each grade The percentage of impurities and the profits of each grades are as follows:
Formulate this as an LPP and then use simplex method to determine the proportions in
which the three grades be used. Also find the maximum profit.
b) Give an overview of stochastic gradient descent with relevant equations. 4
20 a) Give the definition of quadratic programming. Cite an example. 3
b) Solve the following LPP using graphical method: 6
Maximize 𝑧 = 5𝑥1 + 3𝑥2 subject to
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02000CST294062202
4𝑥1 + 5𝑥2 ≤ 1000

5𝑥1 + 2𝑥2 ≤ 1000
3𝑥1 + 8𝑥2 ≤ 1200
𝑥1 , 𝑥2 ≥ 0
c) Construct the dual of the following LPP: 5
Maximize 𝑧 = 3𝑥1 − 2𝑥2 + 4𝑥3 subject to
3𝑥1 + 5𝑥2 + 4𝑥3 ≥ 7
6𝑥1 + 𝑥2 + 3𝑥3 ≥ 4
7𝑥1 − 2𝑥2 − 𝑥3 ≤ 10
𝑥1 − 2𝑥2 + 5𝑥3 ≥ 3
4𝑥1 + 7𝑥2 − 2𝑥3 ≥ 2
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
*****
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H1 02000CST294062202 Pages: 4

Reg No.:_ Name:____________

Course Code: CST294

1 Show that the vector 𝑤 = (9,2,7) is a linear combination of the vectors 𝑢 = 3

6 What is back propagation? 3

9 Explain Gradient Descent algorithm. 3

10 Convert the following L.P.P. to the standard form: 3

Apply Gram-Schmidt orthogonalization process to construct an orthonormal

b) i) Compute the Hessian of 𝑓(𝑥, 𝑦) = 𝑥 3 − 2𝑥𝑦 − 𝑦 6 at the point (1,2) 10

ii) Find the nth derivative of 𝑥𝑒 𝑥 𝑎𝑡 𝑥 = 0

b) Explain automatic differentiation. Draw a computation graph by selecting 9

19 a) Three grades of coal A, B and C contains phosphorus and ash as impurities. In a 10

4𝑥1 + 5𝑥2 ≤ 1000

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