Applied Maths Class 12 Unit 1: Introduction to Numbers, Quantification & Numerical Applications
Applied maths class 12 unit 1 focuses on important concepts like Numbers, Quantification, and Numerical Applications that are widely used in real-life problem solving. This chapter builds a strong foundation for understanding practical mathematics in Class 12.
In this complete guide on applied maths class 12 unit 1, you will learn detailed notes, formulas, solved examples, and important questions for exam preparation.
Unit 1: Numbers, Quantification & Numerical Applications
This unit focuses on real-life numerical systems and logical number operations. It helps students understand how numbers behave in cyclic patterns and modular systems.
1. Modulo Arithmetic
Definition: Modulo arithmetic is a system where numbers repeat after a fixed value called modulus.
a mod n = remainder when a is divided by n
Example 1:
17 mod 5 = 2
Example 2:
29 mod 7 = 1
Clock Concept:
15 mod 12 = 3
So, 15 o’clock = 3 o’clock
Properties:
(a + b) mod n = [(a mod n) + (b mod n)] mod n
(a × b) mod n = [(a mod n) × (b mod n)] mod n
Applications:
- Digital clocks
- Computer algorithms
- Cryptography
- Coding systems
2. Congruence Modulo
Definition: Two numbers are said to be congruent modulo n if they leave the same remainder when divided by n.
a ≡ b (mod n)
Example 1:
17 ≡ 2 (mod 5)
Example 2:
23 ≡ 3 (mod 10)
Important Rule:
If a ≡ b (mod n), then n divides (a − b)
Example:
20 ≡ 8 (mod 6)
20 − 8 = 12 (divisible by 6)
Properties:
a ≡ b (mod n), c ≡ d (mod n)
⇒ a + c ≡ b + d (mod n)
⇒ a × c ≡ b × d (mod n)
The chapter applied maths class 12 unit 1 is very important for students preparing for board exams. A clear understanding of applied maths class 12 unit 1 helps in solving real-life numerical problems easily.
Congruence Modulo – Important Questions with Solutions
Q1. Show that 35 ≡ 5 (mod 10)
We know that:
a ≡ b (mod n) ⇔ n divides (a − b)
Here,
a = 35, b = 5, n = 10
Calculate:
35 − 5 = 30
30 is divisible by 10
Therefore, 35 ≡ 5 (mod 10) (Verified)
Q2. Show that 48 ≡ 3 (mod 9)
a = 48, b = 3, n = 9
Calculate:
48 − 3 = 45
45 is divisible by 9
Therefore, 48 ≡ 3 (mod 9) (Verified)
Q3. Find the remainder when 87 is divided by 8 using congruence
Divide:
87 ÷ 8 = 10 remainder 7
So,
87 ≡ 7 (mod 8)
Remainder = 7
Q4. If a ≡ 4 (mod 7) and b ≡ 3 (mod 7), find (a + b) mod 7
Using property:
a + b ≡ (4 + 3) (mod 7)
a + b ≡ 7 (mod 7)
7 mod 7 = 0
Therefore, (a + b) ≡ 0 (mod 7)
Q5. If a ≡ 2 (mod 5), find the value of a² (mod 5)
Using property:
a² ≡ (2)² (mod 5)
a² ≡ 4 (mod 5)
Therefore, a² ≡ 4 (mod 5)
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Difference Between Modulo and Congruence – Important Questions with Solutions
Modulo → Gives remainder
Congruence → Shows relation between numbers
Q1. Find 23 mod 5 and express it in congruence form.
Modulo form:
23 ÷ 5 = 4 remainder 3
23 mod 5 = 3
Congruence form:
23 ≡ 3 (mod 5)
Final Answer: Modulo gives remainder 3, and congruence shows relation 23 ≡ 3 (mod 5)
Q2. Verify whether 42 ≡ 2 (mod 10) and find the modulo value.
Check congruence:
42 − 2 = 40
40 is divisible by 10 ✔
So, congruence is correct.
Modulo form:
42 mod 10 = 2
Final Answer: Congruence verified and modulo value = 2
Q3. Convert 57 ≡ 1 (mod 8) into modulo form and verify.
Modulo form:
57 ÷ 8 = 7 remainder 1
57 mod 8 = 1
Verification:
57 − 1 = 56 (divisible by 8)
Final Answer: Modulo form is correct and congruence is verified
Q4. Find 35 mod 6 and write the congruence statement. Explain the difference.
Modulo:
35 ÷ 6 = 5 remainder 5
35 mod 6 = 5
Congruence:
35 ≡ 5 (mod 6)
Difference:
Modulo gives remainder, while congruence shows relation between numbers.
Final Answer: 35 mod 6 = 5 and 35 ≡ 5 (mod 6)
Q5. Explain with example how modulo and congruence are related.
Example:
29 mod 7 = 1
Congruence form:
29 ≡ 1 (mod 7)
Explanation:
Modulo gives the remainder (1), while congruence shows that 29 and 1 behave the same under modulus 7.
Final Answer: Modulo and congruence are directly related; congruence is the extended form of modulo.
For official concepts, refer to the NCERT Class 12 Maths textbook.
Exam Important Points
- Modulo = remainder based
- Congruence = relation based
- Questions include finding remainder and proving congruence
Conclusion
Modulo Arithmetic helps in finding remainders, while Congruence helps in comparing numbers under a modulus. These concepts are widely used in coding, cryptography, and logical computations.