Unit1 - Subjective Questions
PEA305 • Practice Questions with Detailed Answers
Explain the Base Method of multiplication in Vedic Mathematics when both numbers are close to 100 but less than 100. Illustrate with an example of .
Base Method Multiplication (Below Base):
When multiplying two numbers near a base (in this case, Base 100), we calculate the deviation (deficiency) of each number from the base.
Steps:
- Identify the base (100).
- Find the difference of each number from 100.
- LHS (Left Hand Side): Subtract the cross-difference from one number.
- RHS (Right Hand Side): Multiply the two differences.
Example:
- Base = 100
Calculation:
- RHS: (Must occupy 2 digits as base has 2 zeros).
- LHS: (OR ).
Result: Combine LHS and RHS 9024.
Distinguish between Rational and Irrational numbers with suitable examples. How can you classify the number ?
Rational Numbers:
- Definition: A number that can be expressed in the form , where and are integers and $q
eq 0$. - Decimal Expansion: It is either terminating or non-terminating repeating (recurring).
- Examples: , $5$, $0.333...$, .
Irrational Numbers:
- Definition: A number that cannot be written as a simple fraction .
- Decimal Expansion: It is non-terminating and non-repeating.
- Examples: , , .
Classification of :
is an Irrational Number. Although we often use approximations like or $3.14$, the actual value of is a non-terminating, non-repeating decimal ($3.14159...$).
Derive the method to find the number of trailing zeros in (n factorial) and calculate the number of trailing zeros in .
Method Derivation:
Trailing zeros are produced by pairs of prime factors $2$ and $5$ (since ). In any factorial , the frequency of factor $2$ is always higher than the frequency of factor $5$. Therefore, the number of zeros depends on the number of $5$s in the prime factorization of .
Legendre's Formula:
Number of trailing zeros =
Calculation for :
- (Stop here as denominator > numerator)
Total Zeros: .
So, has 24 trailing zeros.
Define the Divisibility Rule of 11 and check if the number $928,345$ is divisible by 11.
Divisibility Rule of 11:
A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places (counting from the right) is either $0$ or a multiple of $11$.
Checking $928,345$:
- Digits: $9, 2, 8, 3, 4, 5$
- Odd places (from right): $5, 3, 2$
- Sum
- Even places (from right): $4, 8, 9$
- Sum
- Difference:
Since the difference is $11$, which is divisible by $11$, the number $928,345$ is divisible by 11.
Explain the concept of Cyclicity in Unit Digit calculations. Find the unit digit of .
Concept of Cyclicity:
The unit digit of a number raised to a power repeats in a specific pattern. The length of this repeating pattern is called cyclicity.
- Cyclicity of 4: Digits $2, 3, 7, 8$.
- Cyclicity of 2: Digits $4, 9$.
- Cyclicity of 1: Digits $0, 1, 5, 6$.
For number 3:
Powers of 3 end in: , , , . The pattern ($3, 9, 7, 1$) repeats every 4 powers. Hence, cyclicity is 4.
Calculation for :
- Divide the power by cyclicity: .
- Find Remainder: . The remainder is $1$.
- The unit digit corresponds to .
Result: The unit digit is 3.
Calculate the Total Number of Factors and the Sum of Factors for the number $360$.
1. Prime Factorization:
Here, ; ; .
2. Total Number of Factors:
Formula:
3. Sum of Factors:
Formula:
- Term 1 (for 2):
- Term 2 (for 3):
- Term 3 (for 5):
Total Sum:
State Fermat's Little Theorem and use it to find the remainder when is divided by $101$.
Fermat's Little Theorem:
If is a prime number and is an integer not divisible by , then:
Application:
Find Remainder of .
- Here, and .
- Check conditions: 101 is a prime number, and 2 is not divisible by 101.
- Apply theorem: .
- According to the theorem: .
Answer: The remainder is 1.
Explain the relationship between HCF (Highest Common Factor) and LCM (Least Common Multiple) of two numbers. If the HCF of two numbers is 12 and their product is 2880, find their LCM.
Relationship:
For any two positive integers and , the product of the numbers is equal to the product of their HCF and LCM.
Problem Solving:
- Given: Product () = 2880
- Given: HCF = 12
- To find: LCM
Calculation:
Answer: The LCM is 240.
Describe the formula for Weighted Average. A class has two sections: Section A with 30 students having an average score of 60, and Section B with 20 students having an average score of 70. Calculate the overall average.
Weighted Average Formula:
The weighted average () is used when different data points have different "weights" (frequencies or importance). It is calculated as:
Where is the weight (number of items) and is the value (average of that group).
Calculation:
- Section A: ,
- Section B: ,
Answer: The overall average score is 64.
Discuss the concept of Inclusion and Exclusion in averages. If the average weight of a group of 10 persons increases by 1.5 kg when a new person replaces one weighing 65 kg, find the weight of the new person.
Concept:
In average problems, when an element is replaced, the change in the average is distributed equally among all elements. The formula for the new value is:
(Note: Use subtraction if the average decreases)
Calculation:
- Old Value (Replaced person's weight) = 65 kg
- Number of persons () = 10
- Change in Average = kg
Weight of New Person:
Answer: The weight of the new person is 80 kg.
Explain the Concept of Remainder using the basic definition . Find the smallest number which when divided by 12, 15, and 18 leaves a remainder of 7 in each case.
Concept of Remainder:
The Remainder Theorem states that for any integer dividend and divisor , there exist unique integers (quotient) and (remainder) such that:
Where .
Problem Application:
We need a number such that divided by 12, 15, and 18 leaves . This implies that is perfectly divisible by 12, 15, and 18.
- Find LCM of 12, 15, 18:
- .
- Add the Remainder:
Answer: The number is 187.
How can you determine if a number is divisible by Composite Numbers? Explain with the help of the divisibility rule for 72.
Divisibility by Composite Numbers:
To check divisibility by a composite number , we express as the product of two co-prime factors (factors whose HCF is 1), say and (). If a number is divisible by both and , it is divisible by .
Rule for 72:
- Find co-prime factors of 72: The best pair is $8$ and $9$ (since and HCF(8, 9) = 1).
- Check divisibility by 8: The last three digits of the number must be divisible by 8.
- Check divisibility by 9: The sum of the digits of the number must be divisible by 9.
If a number satisfies both conditions, it is divisible by 72.
Define Average Speed and explain why it is distinct from the arithmetic mean of speeds. Derive the formula for Average Speed when covering two equal distances at speeds and .
Definition:
Average Speed is defined as the Total Distance traveled divided by the Total Time taken. It is distinct from the arithmetic mean because time taken for different segments varies based on speed.
Derivation:
Let the total journey consist of two equal parts of distance each.
- Distance 1: , Speed: . Time .
- Distance 2: , Speed: . Time .
This is the harmonic mean of the speeds.
Using Wilson's Theorem, determine the remainder when is divided by $17$.
Wilson's Theorem:
If is a prime number, then:
OR
Calculation:
- Here, divisor (which is a prime number).
- We need to find the remainder of .
- According to the theorem, .
- .
Interpreting Negative Remainder:
In modular arithmetic, a remainder of is equivalent to .
Rem = .
Answer: The remainder is 16.
Explain the method of Successive Division. A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. Find the number.
Successive Division:
In successive division, the quotient of the first division becomes the dividend for the second division, and so on.
Structure:
- divided by Divisor1 () gives Quotient1 () and Remainder1 ().
- divided by Divisor2 () gives Quotient2 () and Remainder2 ().
Calculation:
Let the final quotient be (usually assume for the smallest number).
Answer: The smallest such number is 37.
Describe the Vedic Maths trick for squaring numbers ending in 5. Calculate and .
Technique:
To square a number ending in 5 (Format: ):
- The last two digits of the answer are always 25.
- The leading digits are obtained by multiplying the prefix number () by its successor ().
Calculation for :
- Prefix .
- Multiply .
- Append 25.
- Result: 5625.
Calculation for :
- Prefix .
- Multiply .
- Append 25.
- Result: 13225.
Explain the difference between Prime, Composite, and Co-prime numbers.
1. Prime Numbers:
- Natural numbers greater than 1 that have exactly two factors: 1 and the number itself.
- Examples: 2, 3, 5, 7, 11.
2. Composite Numbers:
- Natural numbers greater than 1 that have more than two factors (i.e., they are not prime).
- Examples: 4 (factors: 1, 2, 4), 6, 8, 9.
- Note: 1 is neither prime nor composite.
3. Co-prime (Relatively Prime) Numbers:
- Two numbers are co-prime if their Highest Common Factor (HCF) is 1.
- They do not need to be prime numbers individually.
- Example: 8 and 15. Factors of 8: {1,2,4,8}, Factors of 15: {1,3,5,15}. Common factor is only 1.
Derive the logic to find the Unit Digit of Perfect Squares. Which digits can never appear at the unit place of a perfect square?
Derivation:
We analyze the square of digits from 0 to 9:
- ,
- ,
- ,
- ,
Possible Unit Digits:
A perfect square can only end in $0, 1, 4, 5, 6,$ or $9$.
Impossible Unit Digits:
The digits 2, 3, 7, and 8 can never appear at the unit place of a perfect square.
Solve the following problem using LCM: Three bells ring at intervals of 12, 15, and 20 minutes respectively. If they started ringing together at 8:00 AM, at what time will they ring together again?
Concept:
The bells will ring together again at a time interval that is a multiple of all three individual intervals. The earliest time corresponds to the Least Common Multiple (LCM).
Calculation:
- Find LCM of 12, 15, 20.
- .
Result:
They will ring together after 60 minutes.
Time:
.
Define Factorials and simplify the expression . Explain its significance in combinations.
Definition:
The factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to .
Simplification:
- Cancel from numerator and denominator.
Significance:
This expression represents (Combinations), which is the number of ways to select 2 items from a set of 12 distinct items without regarding the order.