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(For NTA UGC NET Paper-I)

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🧠 1. Introduction

Mathematical Reasoning and Aptitude form the logical and quantitative base of human thinking.
This unit tests your ability to think logically, recognize patterns, and solve problems using mathematical concepts.

It is not about complex mathematics, but about how numbers and logic work together in reasoning, argument, and decision-making.

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🎯 2. Key Components of the Unit

The unit has two parts:

1. Mathematical Reasoning → Deals with logic, arguments, statements, validity, and reasoning patterns.

2. Mathematical Aptitude → Deals with numbers, arithmetic operations, ratios, percentages, averages, profit-loss, interest, time-work, etc.

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⚙️ 3. MATHEMATICAL REASONING

🔹 3.1. Statements

A statement is a declarative sentence that is either True (T) or False (F), but not both.

Examples:

• “2 + 2 = 4” → True statement

• “All squares are circles” → False statement

• “Please close the door” → Not a statement (imperative sentence)

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🔹 3.2. Logical Connectives

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🔹 3.3. Truth Tables

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🔹 3.4. Types of Reasoning

1. Deductive Reasoning:

   • From general to specific.

   • If premises are true, conclusion must be true.

   • Example:
     All teachers are educated.
     Anshuman is a teacher.
     ∴ Anshuman is educated.

2. Inductive Reasoning:

   • From specific to general.

   • Based on observation and probability.

   • Example:
     The sun rose every day so far → it will rise tomorrow. (probable)

3. Analogical Reasoning:

   • Drawing comparison between two situations.

   • Example:
     “Hand is to glove as foot is to sock.”

4. Abductive Reasoning:

   • Inferring the most likely explanation.

   • Example:
     Wet roads → it probably rained.

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🔹 3.5. Arguments and Validity

An argument consists of premises and a conclusion.
It is valid if the conclusion logically follows from the premises.

Example:
Premise 1: If it rains, the match is canceled.
Premise 2: It is raining.
Conclusion: Therefore, the match is canceled. ✅ (Valid)

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🔹 3.6. Syllogism

A syllogism is a form of logical reasoning with two premises and a conclusion.

Example:
All roses are flowers.
All flowers are plants.
∴ All roses are plants.

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🔹 3.7. Fallacies (Logical Errors)

• Affirming the consequent:
  If it rains, road is wet. Road is wet → therefore it rained ❌ (not necessarily)

• Denying the antecedent:
  If it rains, road is wet. It did not rain → road not wet ❌

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🧮 4. MATHEMATICAL APTITUDE

Focus: Numerical ability, calculation, and quantitative logic.

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🔹 4.1. Numbers and Operations

• Even numbers: divisible by 2

• Odd numbers: not divisible by 2

• Prime numbers: divisible only by 1 and itself

• Composite numbers: have more than two factors

• LCM (Least Common Multiple): smallest common multiple

• HCF (Highest Common Factor): greatest common divisor

🧩 Relationship:
LCM × HCF = Product of two numbers

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🔹 4.2. Ratio and Proportion

• Ratio: comparison of two quantities.
  $a:b=\frac{a}{b}$
  

• Proportion: equality of two ratios.
  $\frac{a}{b}=\frac{c}{d}$
  

Example:
If 2 pens cost ₹10, then 6 pens cost ₹30.

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🔹 4.3. Percentage

$$\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times 100$$

• Increase/Decrease:
  New Value = Old Value × (1 ± Rate/100)

Example: Price ₹100 increased by 10% = ₹110.

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🔹 4.4. Profit and Loss

$$\text{Profit}=\text{SP}-\text{CP},\text{Loss}=\text{CP}-\text{SP}$$
$$\mathrm{\%}\text{Profit}=\frac{\text{Profit}}{\text{CP}}\times 100$$
$$\mathrm{\%}\text{Loss}=\frac{\text{Loss}}{\text{CP}}\times 100$$

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🔹 4.5. Simple and Compound Interest

• Simple Interest (SI):

  $$SI=\frac{P\times R\times T}{100}$$

• Compound Interest (CI):

  $$CI=P{(1+\frac{R}{100})}^T-P$$

Example: ₹1000 at 10% for 2 years
→ SI = 200; CI = 210.

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🔹 4.6. Average

$$\text{Average}=\frac{\text{Sum of items}}{\text{Number of items}}$$

Example:
Marks in 5 subjects = 60, 70, 80, 90, 100
Average = 400 ÷ 5 = 80

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🔹 4.7. Time and Work

If A can do a work in x days → 1 day’s work = 1/x
If B can do it in y days → 1 day’s work = 1/y

Together:

$$\text{Time}=\frac{xy}{x+y}$$

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🔹 4.8. Speed, Distance, and Time

$$\text{Speed}=\frac{\text{Distance}}{\text{Time}},\text{Time}=\frac{\text{Distance}}{\text{Speed}}$$

If object travels at different speeds for equal distances:
Average Speed = $\frac{2xy}{x+y}$

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🔹 4.9. Permutation and Combination (Basic)

• Permutation: Arrangement → $nPr=\frac{n!}{(n-r)!}$
  

• Combination: Selection → $nCr=\frac{n!}{r!(n-r)!}$
  

Example:
From 5 persons, select 2 → 5C2 = 10.

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🔹 4.10. Probability (Elementary)

$$P(E)=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$$

P(E)=Total outcomesFavorable outcomes​

Example:
A dice rolled → P(getting even number) = 3/6 = 1/2.

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📚 5. TIPS TO SOLVE QUESTIONS IN EXAM

1. Read carefully – watch for “not”, “all”, “some”.

2. Simplify logically – reduce long statements using truth rules.

3. Make short tables for ratio, percentage, etc.

4. Avoid lengthy calculations – use mental math and common fractions.

5. Check for units (e.g., km/hr vs m/s).

6. Eliminate impossible options in MCQs.

7. Practise previous-year patterns – NET repeats logic types (syllogism, implication, numerical pattern).

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🧩 6. SHORTCUT MEMORY CHART

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🎓 7. EXAM STRATEGY

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💡 8. COMMON QUESTION AREAS IN UGC NET

1. Statement–Conclusion reasoning (2–3 questions)

2. Truth table evaluation (1–2 questions)

3. Number/letter series or coding (1–2 questions)

4. Basic arithmetic: ratio, percentage, average, profit–loss (2–3 questions)

5. Time, work, distance (1 question)

6. Simple interest/compound interest (1 question)

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✅ 9. Quick Summary

• Reasoning: Logic, arguments, validity, connectives, truth tables.

• Aptitude: Arithmetic operations, percentages, averages, time–work, etc.

• Approach: Concept clarity + speed practice + short formula recall.

• Nature of questions: Conceptual, logical, application-based — not lengthy mathematics.