1. Define GCD. (1 Mark)

GCD (Greatest Common Divisor) is the largest positive integer that divides two or more integers exactly.

2. Find the prime factorization of 1330. (1 Mark)

1330=2×5×7×19​

3. Define Residue. (1 Mark)

A residue modulo mmm is any integer equivalent to another integer modulo mmm.

4. State the Chinese Remainder Theorem. (1 Mark)

If the moduli are pairwise coprime, a system of congruences has a unique solution modulo the product of the moduli.

5. Define Fermat Numbers.

9.  A researcher wants to know the intelligence of students in a school. He selected two groups of students. In the first group there 150 students having mean IQ of 75 with a S.D. of 15 in the second group there are 250 students having mean IQ of 70 with S.D. of 20. Is there a significant difference between the means of two groups ?

Solution

Given

For Group I:

• Sample size,

n1=150n1​=150

• Sample mean,

xˉ1=75xˉ1​=75

• Standard deviation,

s1=15s1​=15

For Group II:

• Sample size,

The marks obtained in Statistics in a certain examination found to be normally distributed. If 15% of the students ≥ 60 marks, 40% < 30 marks, find the mean and standard deviation?

where μμ is the mean and σσ is the standard deviation.

Given

• P(X≥60)=0.15P(X≥60)=0.15

• P(X<30)=0.40P(X<30)=0.40

We need to find μμ and σσ.

Step 1: Convert to Z-values

(i) P(X≥60)=0.15P(X≥60)=0.15

Hence,

P(X<60)=0.85P(X<60)=0.85

From the standard normal table,

P(Z<1.04)

11. Calculate the coefficient of correlation from the following data

Step 3: Substitute into the for

To calculate the Karl Pearson's coefficient of correlation (r):

The formula is:

r=∑dx dy∑dx2×∑dy2r=∑dx2×∑dy2​∑dxdy​

where:

• dx=X−Xˉdx=X−Xˉ

• dy=Y−Yˉdy=Y−Yˉ

Step 1: Find the means

Xˉ=12+9+8+10+11+13+77=707=10Xˉ=712+9+8+10+11+13+7​=770​=10Yˉ=14+8+6+9+11+12+37=637=9Yˉ=714+8+6+

1. Pleural effusion -

Interpretation-

Quality- RIPE- good 

1. Airways- trachea deviated to the right (pushing effect- because of effusion); cannot visualise the carina or bronchus

2. Breathing- effusion in the left lower and middle zones which end in the upper zone with a meniscus sign; lung markings present throughout the right lung 

3. Cardiac- cannot assess the size, difficult to visualise left heart borders 

4. Diaphragm- right hemidiaphragm can be seen; left - cannot be seen, blunting of

Find the GCD OF (143,227)?

The Euclidean Algorithm is used to find the Greatest Common Divisor (GCD) of two integers by repeatedly dividing the larger number by the smaller number until the remainder becomes 0.

Step 1:

227=143×1+84227 = 143 \times 1 + 84227=143×1+84

Step 2:

143=84×1+59143 = 84 \times 1 + 59143=84×1+59

Step 3:

84=59×1+2584 = 59 \times 1 + 2584=59×1+25

Step 4:

59=25×2+959 = 25 \times 2 + 959=25×2+9

Step 5:

25=9×2+725 = 9 \times 2 + 725=9×2+7

Step 6:

9=7×1+29 = 7 \times

2. Technical Feasibility Checks: ◉Machinery ◉Technology ◉Skilled workers ◉Raw materials

3. Financial Feasibility Checks: ◉Investment ◉Profit ◉Cash flow ◉ROI

4. Social Feasibility Checks whether project benefits society. Examples: ◉Employment ◉Environment ◉Community welfare

Project Implementation Stages ◉Idea generation ◉Feasibility study ◉Project report ◉Finance arrangement ◉Registration ◉Purchase machinery ◉Production ◉Marketing

MODULE 3: Financial Analysis

1. Entrepreneur An entrepreneur is a person who starts, organizes and manages a business by taking risks to earn profit. Example: Ratan Tata, Narayana Murthy, Kiran Mazumdar-Shaw.

2. Qualities/Characteristics of an Entrepreneur ◉Risk-taking ability ◉Leadership ◉Creativity and innovation ◉Self-confidence ◉Decision-making ability ◉Hard-working ◉Problem-solving ability ◉Vision and planning Example: Elon Musk invested in electric cars when many believed they wouldn't succeed.

3. Venture