Linear independence

Let k = number of vectors
Let r=rank(A)

Consistency

A system of equations is consistent if it has at least one solution. It is inconsistent if no solution exists.

Using rank: the system is consistent when the rank of the coefficient matrix equals the rank of the augmented matrix; otherwise it is inconsistent.

Let A = coefficient matrix
Let [A∣B] = augmented matrix
Let r(A) = rank of coefficient matrix
Let r([A∣B]) = rank of augmented matrix

Case 1: Consistent System

r(A)=r([A∣B])

Then:

• If this common rank = number of unknowns → unique solution

• If this rank < number of unknowns → infinite solutions

Case 2: Inconsistent System

r(A)≠r([A∣B])​

Meaning: the augmented matrix introduces a contradiction.

Eigen values

det(A−λI)=0 → λ = a, b

eigen vectors

(A−aI)v=0; with v=(x,y) → x+y=0⟹y=−x →

Lets choose x=1 $$v = egin{bmatrix} 1 -1 end{bmatrix}$$ also, $$egin{bmatrix} 1 -1 end{bmatrix} = egin{bmatrix} -1 1 end{bmatrix}$$ in terms of eigen vectors.

(A−bI)v=0; similar.

Maximization using lagrange's multiplier

Step 1: Form the Lagrangian

$L = 4u_1^2 + 3u_1u_2 + 6u_2^2 + lambda(56 - u_1 - u_2)$$

Step 2: Take partial derivatives and set them to zero

(i) Derivative w.r.t. $u_1$​:

$$rac{partial mathcalL}{partial u_1} = 8u_1 + 3u_2 - lambda = 0$$

(ii) Derivative w.r.t. $u_2$​:

$$rac{partial mathcalL}}{partial u_2} = 3u_1 + 12u_2 - lambda = 0$$

(iii) Derivative w.r.t. constraint:

$$u_1 + u_2 = 56$$

Step 3: Use the first two equations to eliminate λ and substitute into the constraint

$$oxed{u_1 = 36,quad u_2 = 20}$$

These values maximize the function under the given constraint

Cramer's rule to solve equations

Taylor's series and Maclaurin’s series expansion

Taylor's polynomial

same formular f(x) = ....

A question

Concave, Convex, Inflection

Convex flexes. Concave caves.

Hessian of a function

Jacobian

Mean value theorem

Theory

Open and Close Set

• A set is open if it does not include its boundary points.
  • A set S is open if every point in S has a small neighborhood (a ball around it) that is also completely inside S.
  • (2,5), R, {(x,y):x2+y2<1}, ∅ (empty set)
• A set is closed if it contains all its limit points.
  • A set is closed if it contains its boundary or if its complement is open.
  • [2,5], {(x,y):x2+y2≤1}, {2,7,10},
  • Complement of an Open Set
    • Example: The complement of the open interval (0,1) is: $(-infty, 0] cup [1, infty)$
• A set can be neither open nor closed
  • [2, 5)

How is Euler’s theorem used in product exhaustion theorem ?

Advantages of Sample Survey + Explain Sampling Design (12 marks)

⭐ Advantages of Sample Survey

1. Lower Cost:
   Sampling is cheaper than a census because fewer observations are collected.

2. Less Time:
   Results can be obtained quickly since only part of the population is studied.

3. Greater Accuracy (in many cases):
   With trained investigators and smaller data size, errors are often lower than full census.

4. Operational Feasibility:
   Some studies (e.g., destructive testing, case studies) cannot be done via census.

5. Better Quality Control:
   Supervising small samples is easier and more reliable.

6. Useful when population is infinite:
   For example, quality control in industrial production.

7. More detailed information:
   Researchers can collect high-quality data from fewer units.

⭐ Explain Sampling Design

Sampling design refers to the method and plan used to select the sample from the population.

A good sampling design includes:

1. Target Population:
   Define clearly who or what is being studied.

2. Sampling Frame:
   A list or representation from which samples are drawn.

3. Sampling Method:

   • Probability sampling: simple random, stratified, systematic, cluster.

   • Non-probability sampling: convenience, quota, judgement.

4. Sample Size:
   Decide number of units based on cost, accuracy, and variability.

5. Selection Procedure:
   How units will be chosen (random numbers, systematic rule, etc.)

6. Execution:
   Collecting data according to the design while avoiding bias.

Sampling design ensures:

• Representativeness

• Minimization of bias

• Reliability of conclusions

Sampling methods in breif

1. Probability Sampling

In probability sampling, the selection process is random, ensuring that the sample is statistically representative of the population and minimizing bias.

2. Non-Probability Sampling

In non-probability sampling, the selection process is non-random and relies on the researcher's subjective judgment or convenience, which can introduce selection bias. These methods are often used in qualitative or exploratory research.

Types of Biases in Sample Survey (6 marks)

Bias = systematic error in data collection or sampling.

1. Selection Bias

Occurs when the sample is not representative of the population.
Example: surveying only urban households for national consumption.

2. Non-response Bias

Some selected units refuse to respond or cannot be contacted.
The final sample differs from intended one.

3. Response Bias

Respondents give inaccurate answers intentionally or unintentionally.
Example: understating income, overstating charitable donations.

4. Sampling Bias

Arises due to poor sampling method, e.g., convenience sampling.

5. Interviewer Bias

Interviewer influences responses through tone, wording, or behaviour.

6. Measurement Bias

Errors introduced by faulty instruments, ambiguous questions, or poor questionnaire design.

7. Recall Bias

Respondents cannot remember past events correctly (common in surveys on spending, health, etc.).

8. Processing Bias

Mistakes in coding, entering, or processing data.

Properties of a Continuous Function (4 marks)

A function f(x)f(x)f(x) is continuous at a point aaa if:

$$lim_{x o a} f(x) = f(a)$$

Key properties:

Homogeneous vs. Homothetic Functions

Examples of Different Types of Sequences

(a) Finitely Oscillatory Sequence

A sequence that oscillates only a finite number of times and then settles.

$$1,−1,1,−1,1,1,1,1,1,…$$

(b) Sequence Divergent to $+infty$ $$a_n = n$$(c) Sequence Divergent to $-infty$ $$a_n = -n$$

(d) Infinitely Oscillatory Sequence

A sequence that keeps oscillating without settling.

Example:

$$a_n = (-1)^n$$ or $$a_n = sin(n)$$

Terms

(a) Critical region

(b) One-tailed and two-tailed tests

(c) Standard error

(d) P-value method of hypothesis testing

(e) Orthogonal Matrix

(f) Idempotent Matrix

(g) Eigenvalue, Eigenvector, Characteristic Equation

(h) norm, inner product, linear independence of vectors

• norm = magnitude of vector
• inner product = dot product
• 

(i) Differences

a. Parameter vs. Statistic

b. Type I and Type II errors

c. Normal distribution and Standard normal distribution