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Practice MCQ Subjective

Unit 1 - Notes

PEA305

Unit 1: Vedic maths, Number system, Average

1. Vedic Maths and Fast Calculation Techniques

Vedic Mathematics provides arithmetic shortcuts to increase speed and accuracy in calculations.

1.1 Multiplication Techniques

The Base Method (Nikhilam Sutra)

Used when numbers are close to a power of 10 (10, 100, 1000, etc.).

Case 1: Numbers below the base.
- Example: $96 \times 94$ (Base 100)
- Deviation: $96 (-4)$ , $94 (-6)$
- Left side: $96 - 6 = 90$ (or $94 - 4$ )
- Right side: $(-4) \times (-6) = 24$
- Result: 9024
Case 2: Numbers above the base.
- Example: $103 \times 105$ (Base 100)
- Deviation: $103 (+3)$ , $105 (+5)$
- Left side: $103 + 5 = 108$
- Right side: $3 \times 5 = 15$
- Result: 10815

Criss-Cross Method (Urdhva Tiryakbhyam)

A general method for multiplication applicable to any two numbers.

A step-by-step diagram illustrating the "Criss-Cross" multiplication method for two 2-digit numbers ... — AI-generated image — may contain inaccuracies

1.2 Squaring Techniques

Numbers ending in 5:
- Formula: For number $n5$ , result is $(n \times (n+1)) \, | \, 25$ .
- Example: $65^2 \rightarrow 6 \times 7 = 42$ , append 25 $\rightarrow 4225$ .
Numbers near 50:
- Compare with 25.
- Example: $54^2$ . Excess is 4. Result starts with $25 + 4 = 29$ . Ends with $4^2 = 16$ . Result: $2916$.
- Example: $46^2$ . Deficit is 4. Result starts with $25 - 4 = 21$ . Ends with $4^2 = 16$ . Result: $2116$.

2. Number System

2.1 Classification of Numbers

Understanding the hierarchy of numbers is fundamental to analytical skills.

Complex Numbers: $a + ib$ (Real + Imaginary).
Real Numbers: All numbers on the number line.
- Rational Numbers: Can be written as $p/q$ where $q \neq 0$ (e.g., 5, 2/3, 0.5).
- Irrational Numbers: Non-terminating, non-recurring decimals (e.g., $\sqrt{2}, \pi$ ).
Integers: $\{..., -2, -1, 0, 1, 2, ...\}$
Whole Numbers: $\{0, 1, 2, ...\}$
Natural Numbers: $\{1, 2, 3, ...\}$

A detailed hierarchical tree diagram showing the Classification of Numbers. The top box should say "... — AI-generated image — may contain inaccuracies

2.2 Divisibility Rules

2: Last digit is even (0, 2, 4, 6, 8).
3: Sum of digits is divisible by 3.
4: Last two digits are divisible by 4.
5: Last digit is 0 or 5.
6: Divisible by both 2 and 3.
8: Last three digits are divisible by 8.
9: Sum of digits is divisible by 9.
11: Difference between sum of digits at odd places and sum of digits at even places is 0 or divisible by 11.
- Example: 1331 $\rightarrow (1+3) - (3+1) = 0$ . Divisible.

2.3 Unit Digit Calculation (Cyclicity)

To find the unit digit of $x^n$ , we look at the cyclicity of the base digit.

Digits	Cyclicity	Cycle Pattern
0, 1, 5, 6	1	Same as the number
4, 9	2	4: (4, 6), 9: (9, 1)
2, 3, 7, 8	4	Calculate $n \pmod 4$

Method for Cyclicity 4:
1. Divide power by 4.
2. Find remainder $R$ .
3. If $R=1$ , unit digit is base $^1$ . If $R=2$ , base $^2$ . If $R=3$ , base $^3$ .
4. If $R=0$ (perfectly divisible), use base $^4$ .

2.4 Factorials and Trailing Zeros

Factorial ( $n!$ ): Product of first $n$ natural numbers. $n! = n \times (n-1) \times ... \times 1$ .
Trailing Zeros: Number of zeros at the end of .
- Count the number of pairs of $(2 \times 5)$ . Since 5s are rarer than 2s, we count powers of 5.
- Formula: Number of zeros = $\lfloor \frac{n}{5} \rfloor + \lfloor \frac{n}{25} \rfloor + \lfloor \frac{n}{125} \rfloor + ...$

2.5 Factors

Let $N = p^a \times q^b \times r^c$ where $p, q, r$ are prime numbers.

Total Number of Factors: $(a+1)(b+1)(c+1)$
Sum of Factors: $\left(\frac{p^{a+1}-1}{p-1}\right) \times \left(\frac{q^{b+1}-1}{q-1}\right) \times ...$
Number of Prime Factors: $a + b + c$

3. Remainder Theorems, HCF & LCM

3.1 Remainder Theorem

Dividend = (Divisor \times Quotient) + Remainder

Negative Remainders: Useful for calculation.
- $23 \div 5$ : Remainder is 3 (Positive) or -2 (Negative because $25-23=2$ ).
- Final answer must always be positive. If calculation yields -2, add divisor: $-2 + 5 = 3$ .

3.2 HCF and LCM

HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD.
LCM (Least Common Multiple): The smallest number divisible by two or more numbers.
Relationship: For two numbers $A$ and $B$ :
$HCF(A, B) \times LCM(A, B) = A \times B$
Fractions:
- HCF of fractions = $\frac{\text{HCF of numerators}}{\text{LCM of denominators}}$
- LCM of fractions = $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$

4. Average

Average (Arithmetic Mean) is the central value of a set of data.

Average = \frac{\text{Sum of Observations}}{\text{Number of Observations}}

4.1 Deviation Method (Assumed Mean)

Instead of summing large numbers, assume an average ( $A$ ) and find deviations.

Actual Average = A + \frac{\text{Sum of Deviations}}{n}

4.2 Weighted Average

Used when groups have different weights or frequencies.
Let group 1 have $n_1$ items with average $A_1$ , and group 2 have $n_2$ items with average $A_2$ .

Weighted Average = \frac{n_1 A_1 + n_2 A_2}{n_1 + n_2}

A conceptual illustration of Weighted Average using a beam balance scale. On the left side of the fu... — AI-generated image — may contain inaccuracies

4.3 Inclusion and Exclusion Principles

Shortcuts for when a person enters or leaves a group.

Inclusion (Person Joins):
- If average increases, sign is $+$ . If decreases, sign is $-$ .
Exclusion (Person Leaves):
- Note: Pay close attention to signs. If the person leaving caused the average to drop, they must have been heavier/older than the average.
Replacement:

$\text{New Person} = \text{Old Person} + (\text{Change in Avg} \times \text{Number of Members})$