Practice MCQ

Unit 1 - Notes

PEA307 6 min read

Unit 1: Speed maths, Advanced numeration and Mean

1. Speed Maths: Fast Calculation Techniques

Speed mathematics involves mental calculation techniques and shortcuts that drastically reduce the time taken for basic arithmetic operations. These are highly useful in competitive exams and data analysis.

1.1 Fast Multiplication Tricks

Base Method (Numbers close to 100, 1000, etc.):
- Example: $96 \times 94$
- Both are close to 100. $96$ is $(-4)$ from $100$, and $94$ is $(-6)$ from $100$.
- Step 1: Cross add/subtract: $96 - 6 = 90$ (or $94 - 4 = 90$ ). This is the first part of the answer.
- Step 2: Multiply the deviations: $(-4) \times (-6) = 24$ . This is the second part.
- Answer: $9024$.
Multiplication by 11:
- Write the first digit, add adjacent digits, write the last digit.
- Example: $435 \times 11 \rightarrow 4 \mid (4+3) \mid (3+5) \mid 5 \rightarrow 4785$ .

1.2 Fast Squaring Techniques

Numbers ending in 5 (e.g., 65):
- Multiply the tens digit by the next integer: $6 \times (6+1) = 6 \times 7 = 42$ .
- Append $25$ to the result.
- Answer: $4225$.
Numbers between 40 and 60 (Base 50):
- Example: $54^2$
- Deviation from $50$ is $+4$ .
- Step 1: Add/Subtract deviation to $25$: $25 + 4 = 29$ .
- Step 2: Square the deviation: $4^2 = 16$ .
- Answer: $2916$.

1.3 Fraction to Percentage Conversions

Memorizing base fractions is crucial for speed:

$1/2 = 50\%$
$1/3 = 33.33\%$
$1/4 = 25\%$
$1/5 = 20\%$
$1/6 = 16.66\%$
$1/7 = 14.28\%$
$1/8 = 12.5\%$
$1/9 = 11.11\%$
$1/10 = 10\%$
$1/11 = 9.09\%$
$1/12 = 8.33\%$

2. Advanced Numeration

2.1 Classification of Numbers

Real Numbers: Any number that can be plotted on a number line.
Rational Numbers: Can be expressed as $p/q$ where $q \neq 0$ . Terminating or non-terminating repeating decimals.
Irrational Numbers: Non-terminating and non-repeating decimals (e.g., $\sqrt{2}$ , $\pi$ ).
Integers: $\{\dots, -2, -1, 0, 1, 2, \dots\}$ .
Whole Numbers: $\{0, 1, 2, 3, \dots\}$ .
Natural Numbers: Counting numbers $\{1, 2, 3, \dots\}$ .
Prime Numbers: Have exactly two distinct factors: 1 and itself (e.g., 2, 3, 5, 7). Note: 1 is neither prime nor composite.
Composite Numbers: Have more than two factors.
Co-primes (Relatively Prime): Two numbers whose HCF is 1 (e.g., 8 and 15).

2.2 Divisibility Rules

2: Last digit is even.
3: Sum of digits is divisible by 3.
4: Last two digits form a number divisible by 4.
5: Last digit is 0 or 5.
6: Divisible by both 2 and 3.
8: Last three digits form a number divisible by 8.
9: Sum of digits is divisible by 9.
10: Last digit is 0.
11: Difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.
7, 11, 13 (Grouping rule): For large numbers, group digits in threes from right to left. The alternating difference of these groups must be divisible by 7, 11, or 13 respectively.

2.3 Factors

Let a number $N$ be represented by its prime factorization: $N = p^a \cdot q^b \cdot r^c \dots$

Total Number of Factors: $(a+1)(b+1)(c+1)\dots$
Sum of Factors: $\left(\frac{p^{a+1}-1}{p-1}\right) \cdot \left(\frac{q^{b+1}-1}{q-1}\right) \cdot \dots$
Product of Factors: $N^{\text{Total Factors} / 2}$

2.4 Factorials ( $n!$ )

$n! = n \times (n-1) \times (n-2) \times \dots \times 1$ .

Highest power of a prime $p$ in $n!$ :
Formula: $\lfloor \frac{n}{p} \rfloor + \lfloor \frac{n}{p^2} \rfloor + \lfloor \frac{n}{p^3} \rfloor + \dots$ (Stop when $p^k > n$ ).
Trailing Zeroes in $n!$ :
A trailing zero is formed by a pair of $(2 \times 5)$ . In any factorial, the power of 5 is always less than the power of 2. Thus, the number of trailing zeroes is determined by the highest power of 5 in $n!$ .
Formula: $\lfloor \frac{n}{5} \rfloor + \lfloor \frac{n}{25} \rfloor + \lfloor \frac{n}{125} \rfloor + \dots$

2.5 Unit Digit and Cyclicity

To find the unit digit of $x^y$ , we look at the unit digit of the base $x$ and the cyclicity of its powers.

Cyclicity 1: Base ending in 0, 1, 5, 6. The unit digit remains the same for any power.
Cyclicity 2: Base ending in 4 or 9.
- $4^{\text{odd}} \rightarrow 4$ , $4^{\text{even}} \rightarrow 6$ .
- $9^{\text{odd}} \rightarrow 9$ , $9^{\text{even}} \rightarrow 1$ .
Cyclicity 4: Base ending in 2, 3, 7, 8. The unit digits repeat every 4th power.
- Divide the power $y$ by 4 and find the remainder $R$ .
- If $R=1, 2, 3$ , the unit digit is $x^R$ .
- If $R=0$ , the unit digit is $x^4$ .

2.6 Remainder Theorems

Basic Remainder Property: $(A \times B \times C) \mod N = [(A \mod N) \times (B \mod N) \times (C \mod N)] \mod N$ .
Negative Remainders: If $25 \div 26$ , remainder is $25$. Alternatively, it can be seen as $-1$ . This is highly useful for powers: $25^{100} \mod 26 \rightarrow (-1)^{100} = 1$ .
Fermat’s Little Theorem: If $P$ is a prime number and $A$ is not a multiple of $P$ , then:
$A^{P-1} \equiv 1 \pmod P$ .
Wilson’s Theorem: If $P$ is a prime number, then:
$(P-1)! \equiv -1 \pmod P$ or $(P-1)! \equiv (P-1) \pmod P$ .
Euler's Totient Theorem: $a^{\phi(n)} \equiv 1 \pmod n$ , where $a$ and $n$ are coprime, and $\phi(n)$ is Euler's totient function (count of numbers up to $n$ that are coprime to $n$ ).

2.7 HCF and LCM

HCF (Highest Common Factor): The greatest number that divides exactly into two or more numbers.
LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers.
Important Properties:
- $\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b$ .
- HCF of fractions = $\frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$ .
- LCM of fractions = $\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$ .
Application: LCM is used in problems involving simultaneous events (e.g., bells ringing together, traffic lights changing). HCF is used for dividing items into the largest possible equal groups.

3. Mean (Averages)

3.1 Advanced Methods of Mean Calculation

The basic formula for mean is $\frac{\text{Sum of observations}}{\text{Number of observations}}$ . However, the Assumed Mean Method (Deviation Method) is much faster for large numbers.

Concept: Choose an arbitrary number (Assumed Mean, $A$ ) close to the values.
Calculate the deviation of each observation from $A$ : $d_i = x_i - A$ .
Find the average of these deviations.
True Mean: $X = A + \left( \frac{\sum d_i}{n} \right)$ .
Example: Find the average of $82, 85, 89, 78, 81$.
- Let $A = 80$ .
- Deviations: $+2, +5, +9, -2, +1$ .
- Sum of deviations = $15$. Average of deviations = $15 / 5 = 3$ .
- True Mean = $80 + 3 = 83$ .

3.2 Combined Mean

When you have multiple groups with known means and known sizes, you can calculate the overall mean.

Formula: Let Group 1 have $n_1$ items with mean $\bar{x}_1$ , and Group 2 have $n_2$ items with mean $\bar{x}_2$ .
Combined Mean ( $\bar{X}$ ) = $\frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$ .
Note: This is a form of weighted average. You can also use the ratio of $n_1:n_2$ instead of absolute values to simplify calculations.

3.3 Inclusion and Exclusion Related Problems

These problems involve finding the value of a newly added, removed, or replaced observation based on how the average changes.

Case 1: Inclusion (A new person joins the group)
- If the average increases, the new value is greater than the old average.
- If the average decreases, the new value is less than the old average.
- Formula: Value of New Item = Old Average + (Change in Average × New Total Number of Items)
Case 2: Exclusion (A person leaves the group)
- If the average increases, the leaving person's value was less than the old average.
- If the average decreases, the leaving person's value was greater than the old average.
- Formula: Value of Excluded Item = Old Average - (Change in Average × New Total Number of Items)
Case 3: Replacement (One leaves, one joins; total count remains the same)
- Formula: Value of New Item = Value of Replaced Item + (Change in Average × Total Number of Items)
- Note: Use a plus sign if the average increases, and a minus sign if the average decreases.

Unit 2