🎯 Attachment 1: How to pass — for exam tension

Minimum study · target 30+ / 60

Breathe. You do not need all 6 units, and you do not need 60 out of 60. Passing lives around 27–30 marks out of 60. The plan below puts 35 marks on the table from just the Q1 short skills plus four full-question topics — so even with mistakes, you stay above the pass band. Add the safe set and it becomes 51 marks on the table.

Step 1 — the minimum set (learn these first, in this order)

Learn	Where it pays	Marks on the table
Q1 short skills — eigenvalues shortcut, LPP standard form / dual, Fourier a_n, MGF / moments, remainder, Spearman	Q1, any 5	15
Poisson distribution	Q2a	4
Karl Pearson correlation	Q2b	5
Chinese Remainder Theorem	Q3b	5
Fourier expansion (x² in (0, 2) is the repeat)	Q2c	6
Total on the table		35

Step 2 — the safe set (add these and you cover 51 marks)

Eigenvalues & eigenvectors full question (6) + Cayley–Hamilton (4) + Lagrange multipliers (6) = 16 more marks. 35 + 16 = 51 out of 60 of the paper's usual shape.

Step 3 — how to attempt in the hall (2 hours)

Q1 first (25 min) — the six short skills are the most predictable marks in the paper. Then Q2 (30 min), then Q3 (30 min) — both are templated, see the model paper below. Then your strongest remaining question (25 min), last 10 min to check. Attempt every part you touch, even half an answer — step marks add up, a blank scores zero. If a slot surprises you, fall back to the safe set instead of freezing on one question.

Honest note, not a guarantee. This pattern comes from only two papers, and MU can break a slot any year. Nothing here guarantees a pass — but it is the highest-coverage, lowest-effort path the evidence supports: 35 marks on the table from the minimum set, 51 with the safe set, against a 27–30 pass band.

⚡ Attachment 2: Inside The Papers

Read this first — the cheat code inside the papers

Q1 is identical topic-for-topic in both papers, in the same a–f order: eigenvalues, LPP, Fourier a_n, moments / MGF, remainder, Spearman. Learn those six short-answer skills and Q1 stops being a lottery.

One full question repeats word-for-word: Fourier expansion of x² in (0, 2) is Q2c in both papers.

Exam dates are expected, not confirmed for AI & DS (taken from the AI & ML sheet: 17 / 19 / 21 / 25 / 27 Nov, 2:30–4:30 pm). Treat them as expected until the official sheet is seen.

📋 Attachment 3: Model paper — MU's strict pattern

Predicted shape · practice only

MU sets this paper to a strict template. The table below is that template, read off both papers slot by slot. Below it is a model paper built from the template for practice — it is NOT the real exam paper, and no question in it is claimed to be a leaked or real question. Practise each slot with the Dec / May question listed in the ranked study below.

Question	a (marks)	b (marks)	c (marks)	Evidence
Q1 — any 5	Eigenvalues · LPP · Fourier a_n · MGF / moments · Remainder · Spearman — 3 marks each			Identical topics, same a–f order, both papers
Q2	Poisson (4)	Pearson (5)	Fourier x² in (0, 2) (6)	Identical, both papers
Q3	Statistics: regression / fit (4)	CRT (5)	Eigenvalues & vectors (6)	Same topics, both papers
Q4	Moments / MGF (4)	Diagonalisation or Normal (5)	RSA or Diagonalisation (6)	Topics rotate, marks fixed
Q5	Cayley–Hamilton (4)	Normal or LPP (5)	Simplex or RSA (6)	Q5a fixed, rest rotate
Q6	Fit or Fourier (4)	Fourier or Regression (5)	Lagrange multipliers (6)	Q6c fixed, rest rotate

The model paper to practise (predicted shape, 60 marks)

Q1 (any 5 × 3): a) eigenvalues of a matrix function · b) LPP standard form or dual · c) Fourier a_n · d) MGF / moments · e) remainder by modular arithmetic · f) Spearman rank correlation.

Q2 (4 + 5 + 6): a) Poisson · b) Pearson correlation · c) Fourier expansion of x² in (0, 2).

Q3 (4 + 5 + 6): a) regression line or straight-line fit · b) CRT · c) eigenvalues and eigenvectors.

Q4 (4 + 5 + 6): a) moments / MGF · b) diagonalisation · c) RSA.

Q5 (4 + 5 + 6): a) Cayley–Hamilton · b) normal distribution · c) simplex method.

Q6 (4 + 5 + 6): a) straight-line fit · b) Fourier expansion of x · c) Lagrange multipliers.

If you can attempt Q1 + Q2 + Q3 + any one of Q4–Q6 from this shape, you are attempting 60 marks with the most predictable material first.

📊 Attachment 4: Ranked PYQs & Question Bank

Sorted by frequency and yield. Tick the boxes as you master each topic to watch your marks tracker update live.

Unit I — Linear Algebra

Rank 1 · 8 questions

Eigenvalues run this unit. Every paper asks them twice: once as a 3-mark short, once as a full 6-mark question.

Eigenvalues & eigenvectors

2 / 2 papers min 3 · max 6 marks

Mark topic done

Dec 2025 · Q1a · 3 marks

If A =

3	2	1
0	−1	4
0	0	−2

, find eigenvalues of A⁻¹ + 6I.

Dec 2025 · Q3c · 6 marks

Find eigenvalues and eigenvectors of A =

3	−1	1
−1	5	−1
1	−1	3

May 2026 · Q1a · 3 marks

Find the sum and product of the eigenvalues of A =

−2	−9	5
−5	−10	7
−9	−21	14

. (Shortcut: sum = trace, product = determinant.)

May 2026 · Q3c · 6 marks

Find eigenvalues and eigenvectors of A =

−9	4	4
−8	3	4
−16	8	7

Diagonalisation

2 / 2 papers min 5 · max 6 marks

Mark topic done

Dec 2025 · Q4b · 5 marks

Show that A =

2	2	1
1	3	1
1	2	2

is diagonalisable, hence find its diagonalisable matrix.

May 2026 · Q4c · 6 marks

Check whether A =

8	−8	−2
4	−3	−2
3	−4	1

is diagonalisable; if yes, find the diagonalising matrix P and the diagonal matrix D.

Cayley–Hamilton theorem

2 / 2 papers 4 marks both times

Mark topic done

Dec 2025 · Q5a · 4 marks

Verify Cayley–Hamilton theorem for A =

2	−1	1
−1	2	−1
1	−1	2

May 2026 · Q5a · 4 marks

Verify Cayley–Hamilton theorem for A =

3	2	1
1	2	2
3	2	1

Unit II — LPP & NLPP

Rank 2 · 6 questions

Only unit where the two papers split the short question: standard form in Dec, dual in May. Learn both — they are one skill.

Lagrange multipliers (NLPP)

2 / 2 papers 6 marks both times

Mark topic done

Dec 2025 · Q6c · 6 marks

Using Lagrange's multipliers, solve the following NLPP:

Optimise z = −x₁² − x₂² + 4x₁ + 8x₂

Subject to:

x₁ + x₂ = 4

x₁, x₂ ≥ 0

May 2026 · Q6c · 6 marks

Using Lagrange's multipliers, solve the following NLPP:

Maximise Z = x₁² + x₂² + x₃² − 10x₁ − 6x₂ − 4x₃

Subject to:

x₁ + x₂ + x₃ = 7

x₁, x₂, x₃ ≥ 0

Solving an LPP (Simplex method)

2 / 2 papers min 5 · max 6 marks

Mark topic done

Dec 2025 · Q5c · 6 marks

Applying Simplex method, solve the following LPP:

Maximise Z = 3x₁ + 2x₂

Subject to:

x₁ + x₂ ≤ 4

x₁ − x₃ ≤ 2

x₁, x₂ ≥ 0

May 2026 · Q5b · 5 marks

Maximise Z = 7x₁ + 5x₂

Subject to:

−x₁ − 2x₂ ≥ −6

4x₁ + 3x₂ ≤ 12

x₁, x₂ ≥ 0

LPP standard form & dual

1 paper each 3 marks both times

Mark topic done

Dec 2025 · Q1b · 3 marks

Transform the LPP to standard form:

Maximise Z = 2x₁ + x₂ + 4x₃

Subject to:

x₁ − x₂ + 2x₃ ≤ 5

3x₁ + x₂ + 5x₃ ≥ 14

x₁, x₂ ≥ 0

May 2026 · Q1b · 3 marks

Construct the dual of the following LPP:

Maximise Z = 5x₁ + 4x₂

Subject to:

2x₁ + 6x₂ ≤ 5

7x₁ + 3x₂ ≤ 2

x₁, x₂ ≥ 0

Unit III — Modular Arithmetic

Rank 2 · 6 questions

The most predictable unit after Q1: CRT, RSA and a remainder question, every single paper.

Chinese Remainder Theorem (CRT)

2 / 2 papers 5 marks both times

Mark topic done

Dec 2025 · Q3b · 5 marks

By using Chinese Remainder theorem, solve x ≡ 1 (mod 3), x ≡ 2 (mod 4), x ≡ 3 (mod 5).

May 2026 · Q3b · 5 marks

By using Chinese Remainder theorem, solve x ≡ 3 (mod 4), x ≡ 5 (mod 7).

RSA algorithm

2 / 2 papers 6 marks both times

Mark topic done

Dec 2025 · Q4c · 6 marks

In RSA system the public key (E, N) of user A is (7, 187). Calculate φ(N) and private key D. What is the cipher text for M = 10 using the public key?

May 2026 · Q5c · 6 marks

Perform key generation, encryption and decryption for RSA algorithm with primes p = 3, q = 11, message M = 5.

Remainder by modular arithmetic

2 / 2 papers 3 marks both times

Mark topic done

Dec 2025 · Q1e · 3 marks

What is the remainder when 13⁷⁰ is divided by 21?

May 2026 · Q1e · 3 marks

What is the remainder when 5¹¹ is divided by 7?

Unit IV — Fourier Series

Rank 2 · 6 questions

Only two skills, repeated endlessly: expand the series, and find a_n.

Full Fourier expansion

2 / 2 papers min 4 · max 6 marks

Mark topic done

Dec 2025 · Q2c · 6 marks

Expand Fourier series for f(x) = x² in (0, 2).

Dec 2025 · Q6b · 5 marks

Find Fourier series for f(x) = x, 0 < x < 2π.

May 2026 · Q2c · 6 marks

Expand Fourier series for f(x) = x² in (0, 2). — the exact repeat.

May 2026 · Q6a · 4 marks

Expand Fourier series for f(x) = x in (−π, π).

Finding a_n in a Fourier expansion

2 / 2 papers 3 marks both times

Mark topic done

Dec 2025 · Q1c · 3 marks

Find the value of a_n in the expansion of Fourier series for f(x) = 4 − x² in (0, 2π).

May 2026 · Q1c · 3 marks

Find value of a_n in the expansion of Fourier series for f(x) = x² in (−π, π).

Unit V — Statistics

Rank 1 · 8 questions

Four topics, two questions each, no surprises. The data tables change; the method never does.

Spearman rank correlation

2 / 2 papers 3 marks both times

Mark topic done

Dec 2025 · Q1f · 3 marks

Obtain Spearman's rank correlation coefficient for following data

X	24	24	24	55	25
Y	24	31	51	40	38

May 2026 · Q1f · 3 marks

Obtain Spearman's rank correlation coefficient for following data

X	18	17	13	17	19	21
Y	21	22	14	17	18	13

Karl Pearson correlation coefficient

2 / 2 papers 5 marks both times

Mark topic done

Dec 2025 · Q2b · 5 marks

Five students got the following percentage of marks in mathematics and statistics
Calculate the coefficient of correlation.

Maths	28	34	18	25	75	23	24	33	37	43
Stats	24	31	11	34	65	24	23	53	53	42

May 2026 · Q2b · 5 marks

Calculate the correlation coefficient for the following heights of fathers (X) and their sons (Y)

X	65	66	67	67	68	69	70	72
Y	67	68	65	68	72	72	69	71

Regression lines

2 / 2 papers min 4 · max 5 marks

Mark topic done

Dec 2025 · Q3a · 4 marks

Obtain equations of line of regression of x on y for following data.

X	31	44	65	49	41
Y	48	32	21	66	43

May 2026 · Q6b · 5 marks

Obtain the equation of line of regression of X on Y

X	21	23	25	27	28	20
Y	8	12	15	17	18	20

Fitting a straight line y = a + bx

2 / 2 papers 4 marks both times

Mark topic done

Dec 2025 · Q6a · 4 marks

Fit a straight line of the form y = a + bx to the following data

X	1	4	7	9	8	10
Y	18	14	19	27	18	28

May 2026 · Q3a · 4 marks

Fit a straight line of the form y = a + bx to the following data

X	1	3	5	7	8	10
Y	8	4	5	7	8	4

Unit VI — Probability

Rank 1 · 8 questions

Moments and MGF overlap in the same questions — learn them together, not as two topics.

Moments (raw moments, moments about origin / mean)

2 / 2 papers min 3 · max 4 marks

Mark topic done

Dec 2025 · Q4a · 4 marks

A random variable X has probability density function
f(x) = x²e^−x x ≥ 0
Find 2 moments about origin and mean.

May 2026 · Q1d · 3 marks

Calculate first 2 raw moments for probability density function
f(x) = {

xe^−x	x ≥ 0
0	Otherwise

May 2026 · Q4a · 4 marks

A random variable X has the following probability distribution

X	1	2	3	4
P(X = x)	0.2	0.3	0.3	0.2

Find moment generating function about origin of x, hence find first two raw moments.

Moment generating function (MGF)

2 / 2 papers min 3 · max 4 marks

Mark topic done

Dec 2025 · Q1d · 3 marks

A random variable x defines the possible outcomes of tossing a fair die, find moment generating function of x about origin

May 2026 · Q4a · 4 marks

A random variable X has the following probability distribution

X	1	2	3	4
P(X = x)	0.2	0.3	0.3	0.2

Find moment generating function about origin of x, hence find first two raw moments.

Poisson distribution

2 / 2 papers 4 marks both times

Mark topic done

Dec 2025 · Q2a · 4 marks

It is known that the probability of an item produced by a certain machine will be defective is 0.001. If the produced items are sent to the market in packets of 100, find the number of packets containing at least 2 are defective in the consignment of 2000 Packets.

May 2026 · Q2a · 4 marks

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7. Find the probability of observing exactly three accidents on this stretch of road next month.

Normal distribution

2 / 2 papers 5 marks both times

Mark topic done

Dec 2025 · Q5b · 5 marks

The height of school children of one school is normally distributed with mean of 54 inches and standard deviation 12 inches. What proportion of students have height between i. 46 and 56 inches ii. More than 56

May 2026 · Q4b · 5 marks

The distribution of wait times at a drive-through restaurant is approximately normal with mean 185 seconds and standard deviation 15 seconds. Find the percentage of wait times that are
i. between 155 and 215 seconds.
ii. more than 170 seconds.