Tag: Best Notes on Unit 5 Mathematical Reasoning and Aptitude for UGC NET NTA exam Paper 1

UGC NET Paper 1 – Unit 5: Mathematical Reasoning & Aptitude-MCQs-Set-2

Section A – Mathematical Reasoning (Q1–Q25)

Q1. “Either it rains or it snows” is represented by
A) p ∧ q B) p ∨ q C) p → q D) ¬p → q
Answer: B
Solution: “Either…or” means disjunction (∨).

Q2. Negation of “Some students are honest” is
A) No student is honest B) All students are dishonest C) All students are not honest D) All students are dishonest
Answer: A
Solution: Negation of “Some A are B” → “No A are B.”

Q3. Which of the following is not a statement?
A) 5 + 7 = 12 B) Today is Monday C) Read this book D) Moon reflects light
Answer: C
Solution: Imperative sentence ≠ logical statement.

Q4. Truth value of (p ∧ q) when p = T, q = F is
A) T B) F C) Undefined D) Both
Answer: B

Q5. The compound statement “If 3 > 5 then 10 > 8” is
A) True B) False C) Cannot say D) Contradiction
Answer: A
Solution: Antecedent false ⇒ implication true.

Q6. “All cats are mammals” ⇒ Contrapositive is
A) If not mammal then not cat B) If mammal then cat C) If cat then mammal D) None
Answer: A

Q7. Identify the tautology:
A) p ∧ ¬p B) p ∨ ¬p C) ¬(p ∨ q) D) p → ¬p
Answer: B

Q8. Which of the following is a contradiction?
A) p ∨ ¬p B) p ∧ ¬p C) ¬(p ∧ ¬p) D) p → p
Answer: B

Q9. If p = T, q = F then ¬(p ∨ q) = ?
A) T B) F C) Both D) Undefined
Answer: A

Q10. “If it rains then I take umbrella” – converse is
A) If I take umbrella then it rains.
B) If it doesn’t rain then no umbrella.
C) I never use umbrella.
D) None.
Answer: A

Q11. Valid conclusion:
All A are B; Some B are C ⇒ ?
A) Some A are C B) All C are A C) Cannot be concluded D) No A is C
Answer: C

Q12. “Either the train is late or the driver is ill.” – when false?
A) Both true B) Both false C) Only one true D) Never
Answer: B

Q13. Which statement pair forms contradictory propositions?
A) All A are B & No A are B B) Some A are B & Some A are not B C) All A are B & Some A are not B D) No A are B & Some A are not B
Answer: C

Q14. “If p then q” and “¬p or q” are
A) Contradictory B) Equivalent C) Independent D) Inverse
Answer: B

Q15. If all statements are true except one, which cannot be true together?
A) p ∧ ¬p B) p ∨ q C) ¬p ∨ ¬q D) p → q
Answer: A

Q16. A valid argument is also called
A) Tautology B) Contradiction C) Deductive truth D) Inference
Answer: D

Q17. If “No books are pens” is true, then “Some books are pens” is
A) True B) False C) Uncertain D) Independent
Answer: B

Q18. “All students in the class are attentive.” – negation is
A) Some students in the class are not attentive.
Answer: A

Q19. A proposition that is always false is called a
A) Tautology B) Contradiction C) Contingency D) Negation
Answer: B

Q20. If a valid argument has true premises, then its conclusion is
A) True B) False C) Indeterminate D) Independent
Answer: A

Q21. Symbolic form of “If it is not cold then I go out.”
A) ¬p → q B) p → ¬q C) q → p D) ¬q → p
Answer: A

Q22. Inference rule “If p→q and q→r then p→r” is
A) Law of syllogism B) Law of identity C) Law of negation D) Law of equivalence
Answer: A

Q23. When a conclusion is probable but not certain, it is
A) Deduction B) Induction C) Fallacy D) Contradiction
Answer: B

Q24. “Either India wins or the match is drawn.” – if both false ⇒
A) Statement true B) Statement false C) Cannot say D) Partially true
Answer: B

Q25. Identify the fallacy:
If it rains, streets wet. Streets wet ⇒ it rained.
A) Denying antecedent B) Affirming consequent C) Inverse error D) Contrapositive
Answer: B

Section B – Quantitative Aptitude (Q26–Q50)

Q26. 20 % of a number = 60. Find the number.
→ N = 60 × 100 / 20 = 300.
Answer: 300

Q27. Find the mean of 10, 20, 30, 40, 50.
Sum = 150 → Avg = 150/5 = 30.
Answer: 30

Q28. A and B can finish a job in 12 and 18 days. Together?

$T = \frac{12 \times 18}{12 + 18} = 7.2 days$

Answer: 7.2 days

Q29. Compound interest on ₹ 800 at 5 % for 2 years = ?

$800 [(1.05)^{2} – 1] = 800 \times 0.1025 = 82$

Answer: ₹ 82

Q30. Ratio A:B = 4:5. If A = 80, B =?
B = 80 × 5/4 = 100.
Answer: 100

Q31. Two numbers have HCF 6 and LCM 72. If one is 18, the other =?

$6 \times 72 = 18 \times x \Rightarrow x = 24$

Answer: 24

Q32. A car runs 60 km at 30 km/h and 60 km at 90 km/h. Average speed = ?

$2 \times 30 \times 90 / (30 + 90) = 45$

Answer: 45 km/h

Q33. Discount 10 % on ₹ 2000 ⇒ SP = ?
= 2000 – 200 = 1800.
Answer: ₹ 1800

Q34. If SP = ₹ 600, loss = 25 %, CP = ?

$600 \times 100 / 75 = 800$

Answer: ₹ 800

Q35. 15 % of x = 75. Find x.

$x = 75 \times 100 / 15 = 500$

Answer: 500

Q36. Find probability of odd number on dice.
= 3/6 = 1/2.
Answer: 0.5

Q37. If 10 workers finish job in 15 days, work done in 9 days by 10 workers = ?
= 9/15 = 3/5 = 60 %.
Answer: 60 % of work.

Q38. Find simple interest on ₹ 5000 at 12 % for 2 years.

$5000 \times 12 \times 2 / 100 = 1200$

Answer: ₹ 1200

Q39. Average of first n natural numbers = ?

$\frac{n + 1}{2}$

Answer: (n + 1)/2

Q40. Number of factors of 36 = ?
36 = 2²×3² → (2+1)(2+1)=9.
Answer: 9

Q41. Find next term: 5, 11, 23, 47, ?
Pattern ×2 +1 ⇒ (47×2)+1 = 95.
Answer: 95

Q42. 8 men can build a wall in 20 days. How long for 10 men?
8×20 = 10×x ⇒ x = 16.
Answer: 16 days

Q43. Price of item ₹ 5000 increased by 8 %, then decreased by 8 %. Net effect =?
Loss = (8×8)/100 = 0.64 %.
Answer: 0.64 % loss.

Q44. Find median of 3, 7, 9, 11, 13.
= 9.
Answer: 9

Q45. Mean of first 5 even numbers = ?
(2+4+6+8+10)/5 = 6.
Answer: 6

Q46. If 2x + 3 = 11 ⇒ x =?
x = 4.
Answer: 4

Q47. Simplify:

$(1 / 2 + 1 / 3 + 1 / 6) \times 6$

= (1 + 2 + 1) = 4.
Answer: 4

Q48. A train 150 m long crosses a pole in 15 s. Speed = ?
= 150/15 = 10 m/s = 36 km/h.
Answer: 36 km/h

Q49. If sum of digits of a number is divisible by 9, the number is divisible by ?
A) 3 B) 6 C) 9 D) 12
Answer: C

Q50. Find value of √(144 + 25).
= √169 = 13.
Answer: 13

October 22, 2025
UGC NET Paper – 1- UNIT 5 – MATHEMATICAL REASONING & APTITUDE

(50 MCQs with Solutions)

🔹 Section A: Logical & Mathematical Reasoning (Q1–Q20)

Q1. Which of the following is a statement in mathematical logic?
A) “Please open the door.”
B) “What a beautiful day!”
C) “5 is a prime number.”
D) “Close the window.”
Answer: C
Solution: Only declarative sentences that are either true or false are statements. “5 is a prime number” is true; the rest are commands or exclamations.

Q2. The negation of “All teachers are honest” is:
A) “No teacher is honest.”
B) “Some teachers are not honest.”
C) “All teachers are dishonest.”
D) “Some teachers are honest.”
Answer: B
Solution: Negation of “All A are B” = “Some A are not B.”

Q3. If p: “It is raining.” and q: “The ground is wet.”, then “If it is raining, then the ground is wet” is symbolically:
A) p ∧ q
B) p ∨ q
C) p → q
D) q → p
Answer: C
Solution: “If p then q” = p → q.

Q4. The contrapositive of p → q is:
A) q → p
B) ¬p → ¬q
C) ¬q → ¬p
D) p ↔ q
Answer: C
Solution: Contrapositive = reverse order + negation → ¬q → ¬p.

Q5. The converse of “If you study, you will pass” is:
A) If you do not study, you will not pass.
B) If you pass, you studied.
C) If you pass, you studied hard.
D) If you pass, you have not studied.
Answer: B
Solution: Converse reverses positions: p → q ⇒ q → p.

Q6. In a truth table for p ∨ q, when is it false?
A) When both p and q are true
B) When p true, q false
C) When both p and q are false
D) Never
Answer: C
Solution: Disjunction (∨) is false only when both statements are false.

Q7. For p = True, q = False, what is truth value of p → q?
A) True
B) False
C) Cannot say
D) Both
Answer: B
Solution: In implication, p→q is false only if p is true and q is false.

Q8. Identify the tautology.
A) p ∨ ¬p
B) p ∧ ¬p
C) ¬(p ∨ q)
D) ¬p ∨ ¬q
Answer: A
Solution: p ∨ ¬p is always true, hence a tautology.

Q9. “Some birds can swim.” is logically equivalent to:
A) All birds can swim.
B) There exists at least one bird that can swim.
C) No bird can swim.
D) Some birds cannot swim.
Answer: B
Solution: “Some” = “there exists at least one.”

Q10. Deductive reasoning example:
A) Observing several black crows and concluding all crows are black.
B) If all A are B, and X is A, then X is B.
C) Predicting future based on experience.
D) Assuming probability.
Answer: B

Q11. The conclusion follows logically from:
Premise 1: All roses are flowers.
Premise 2: All flowers are plants.
∴ ?
A) All roses are plants.
B) All plants are roses.
C) All flowers are roses.
D) No roses are plants.
Answer: A

Q12. In logic, Fallacy means:
A) Correct reasoning
B) False reasoning
C) True argument
D) Valid deduction
Answer: B

Q13. Inductive reasoning leads to:
A) Definite truth
B) Probable conclusion
C) Mathematical proof
D) Contradiction
Answer: B

Q14. “Either the meeting is today or tomorrow.” is a:
A) Conjunction
B) Disjunction
C) Implication
D) Negation
Answer: B

Q15. Identify invalid argument:
A) If p→q, q→r ⇒ p→r
B) If p→q and p true ⇒ q true
C) If p→q and q true ⇒ p true
D) If ¬q true ⇒ ¬p true (contrapositive)
Answer: C
Solution: q true does not guarantee p true (fallacy of affirming the consequent).

Q16. A valid argument’s conclusion:
A) Must be true
B) Must follow logically
C) Must be false
D) May be uncertain
Answer: B

Q17. In syllogism:
All dogs are animals.
Some animals are cats.
Conclusion: ?
A) Some cats are dogs.
B) All dogs are cats.
C) No cats are dogs.
D) Cannot be concluded.
Answer: D
Solution: No direct relationship between dogs and cats.

Q18. If “All men are mortal” is true, “Some mortals are men” is:
A) True
B) False
C) Indeterminate
D) Contradictory
Answer: A

Q19. If p and q are both false, then ¬(p ∨ q) is:
A) True
B) False
C) Indeterminate
D) Same as q
Answer: A
Solution: p∨q = False ⇒ ¬(False) = True.

Q20. “If 7 is even, then 2+2=5.” — what is truth value?
A) True
B) False
C) Unknown
D) Conditional
Answer: A
Solution: Antecedent (7 is even) is false → conditional p→q is automatically true.

🔹 Section B: Quantitative Aptitude (Q21–Q50)

Q21. Find average of 25, 35, 45, 55, 65.
Solution:
Sum = 225 → Average = 225/5 = 45
Answer: 45

Q22. If cost price = ₹80, selling price = ₹100, profit % = ?

$\frac{100 - 80}{80} \times 100 = 25 %$

Answer: 25%

Q23. Find simple interest on ₹2000 at 5% for 3 years.

$S I = \frac{2000 \times 5 \times 3}{100} = 300$

Answer: ₹300

Q24. Compound Interest on ₹1000 at 10% for 2 years

$C I = 1000 [(1 + 0.10)^{2} - 1] = 1000 (1.21 - 1) = 210$

Answer: ₹210

Q25. A and B can complete work in 10 and 15 days respectively. Time together = ?

$T = \frac{10 \times 15}{10 + 15} = \frac{150}{25} = 6$

Answer: 6 days

Q26. Train covers 120 km in 3 hours. Find speed in km/hr.

$S p e e d = 120 / 3 = 40$

Answer: 40 km/h

Q27. Convert 54 km/hr into m/s.

$= 54 \times \frac{5}{18} = 15 m/s$

Answer: 15 m/s

Q28. Ratio of 2 quantities is 3:5 and their sum = 80. Find quantities.
Sum ratio parts = 8 → each part = 10.
Quantities = 30, 50.
Answer: 30 and 50

Q29. If 25% of a number is 20, find number.

$N = 20 \times \frac{100}{25} = 80$

Answer: 80

Q30. Find HCF of 24 and 36.
Factors: 24 (2³×3), 36 (2²×3²) → Common = 2²×3 = 12
Answer: 12

Q31. Find LCM of 15 and 20.
LCM = (15×20)/HCF(5) = 300/5 = 60
Answer: 60

Q32. A sells an article at 10% loss. If SP = ₹450, find CP.

$C P = \frac{S P \times 100}{100 - 10} = 450 \times \frac{100}{90} = 500$

Answer: ₹500

Q33. A’s salary increases from ₹20,000 to ₹22,000. % increase = ?

$\frac{2000}{20000} \times 100 = 10 %$

Answer: 10%

Q34. Average of first 5 odd numbers = ?
(1+3+5+7+9)/5 = 25/5 = 5
Answer: 5

Q35. If 12 workers can finish a job in 10 days, 8 workers will take?

$Man-days = 12 \times 10 = 120, Time = 120 / 8 = 15$

Answer: 15 days

Q36. A sum doubles in 8 years under simple interest. Rate = ?

$S I = P, T = 8 \Rightarrow P \times R \times 8 / 100 = P \Rightarrow R = 12.5 %$

Answer: 12.5%

Q37. Probability of getting head in a coin toss = ?

$1 / 2$

Answer: 0.5

Q38. In how many ways can letters of word “MATH” be arranged?

$4! = 24$

Answer: 24

Q39. From 5 persons, choose 2 for a committee.

$^{5} C_{2} = 10$

Answer: 10

Q40. Find next term: 2, 6, 12, 20, ?
Differences: 4, 6, 8 → next difference = 10 → next = 30
Answer: 30

Q41. In a class of 60, 40% are girls. How many boys?
Girls = 24 ⇒ Boys = 60 – 24 = 36
Answer: 36

Q42. Speed ratio of A and B = 2:3. If A takes 45 minutes, B takes ?
Time ∝ 1/Speed ⇒ ratio 3:2 ⇒ B = 45×(2/3)=30 min
Answer: 30 minutes

Q43. If selling price = ₹500, profit = 25%, then CP = ?

$C P = 500 \times 100 / 125 = 400$

Answer: ₹400

Q44. A shopkeeper allows 10% discount on ₹800. Selling price = ?
SP = 800 – 10% of 800 = 720
Answer: ₹720

Q45. Average of 8 numbers is 15. If one number 9 removed, new average = ?
Sum = 120 – 9 =111; Average =111/7=15.857≈15.86
Answer: 15.86

Q46. If A’s income is 20% more than B’s, then B’s income is how much less than A’s?

$\frac{20}{120} \times 100 = 16.67 %$

Answer: 16.67% less

Q47. If CP of 12 pens = SP of 8 pens, find profit %.
Profit on 8 pens = cost of 12 – cost of 8 = 4/8 = 50%
Answer: 50%

Q48. A travels 60 km at 30 km/h and 60 km at 60 km/h. Find average speed.

$= \frac{2 \times 30 \times 60}{30 + 60} = 40$

Answer: 40 km/h

Q49. The next term of series 1, 4, 9, 16, ?
Square numbers → 25
Answer: 25

Q50. If 4 workers complete job in 6 days, how many workers needed to finish in 3 days?
Work ∝ (Workers × Time) → 4×6= W×3 → W=8
Answer: 8 workers

October 22, 2025

UGC – NET – UNIT V: MATHEMATICAL REASONING AND APTITUDE

(For NTA UGC NET Paper-I)

🧠 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.

🎯 2. Key Components of the Unit

The unit has two parts:

Mathematical Reasoning → Deals with logic, arguments, statements, validity, and reasoning patterns.
Mathematical Aptitude → Deals with numbers, arithmetic operations, ratios, percentages, averages, profit-loss, interest, time-work, etc.

⚙️ 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)

🔹 3.2. Logical Connectives

Symbol	Connective	Meaning	Example	Truth Condition
¬p	Negation	“Not p”	If p = It rains → ¬p = It does not rain	Opposite truth value
p ∧ q	Conjunction	“p and q”	It rains and it’s cold	True only if both true
p ∨ q	Disjunction	“p or q” (inclusive)	It rains or it’s cloudy	True if at least one is true
p → q	Implication	“If p then q”	If it rains, the ground gets wet	False only if p is true and q is false
p ↔ q	Bi-conditional	“p if and only if q”	Two statements true together	True when both have same truth value

🔹 3.3. Truth Tables

p	q	p ∧ q	p ∨ q	p → q	p ↔ q
T	T	T	T	T	T
T	F	F	T	F	F
F	T	F	T	T	F
F	F	F	F	T	T

🔹 3.4. Types of Reasoning

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.
Inductive Reasoning:
- From specific to general.
- Based on observation and probability.
- Example:
  The sun rose every day so far → it will rise tomorrow. (probable)
Analogical Reasoning:
- Drawing comparison between two situations.
- Example:
  “Hand is to glove as foot is to sock.”
Abductive Reasoning:
- Inferring the most likely explanation.
- Example:
  Wet roads → it probably rained.

🔹 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)

🔹 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.

🔹 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 ❌

🧮 4. MATHEMATICAL APTITUDE

Focus: Numerical ability, calculation, and quantitative logic.

🔹 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

🔹 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.

🔹 4.3. Percentage

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

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

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

🔹 4.4. Profit and Loss

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

🔹 4.5. Simple and Compound Interest

Simple Interest (SI):

$S I = \frac{P \times R \times T}{100}$
Compound Interest (CI):

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

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

🔹 4.6. Average

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

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

🔹 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:

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

🔹 4.8. Speed, Distance, and Time

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

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

🔹 4.9. Permutation and Combination (Basic)

Permutation: Arrangement → $n P r = \frac{n!}{(n - r)!}$
Combination: Selection → $n C r = \frac{n!}{r! (n - r)!}$

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

🔹 4.10. Probability (Elementary)

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

P (E) = Total outcomes Favorable outcomes

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

📚 5. TIPS TO SOLVE QUESTIONS IN EXAM

Read carefully – watch for “not”, “all”, “some”.
Simplify logically – reduce long statements using truth rules.
Make short tables for ratio, percentage, etc.
Avoid lengthy calculations – use mental math and common fractions.
Check for units (e.g., km/hr vs m/s).
Eliminate impossible options in MCQs.
Practise previous-year patterns – NET repeats logic types (syllogism, implication, numerical pattern).

🧩 6. SHORTCUT MEMORY CHART

Concept	Formula / Logic	Shortcut
Profit %	(SP–CP)/CP ×100	If gain 20%, SP = 1.2 × CP
Compound Interest	$P (1 + R / 100)^{T} – P$	Add interest each year successively
Work (together)	$\frac{x y}{x + y}$	Multiply divide rule
% to Fraction	25% = 1/4, 20% = 1/5	Memorize key ones
Ratio sum	a:b = x:y → total = x+y	To divide ₹120 in 2:3 → (2/5, 3/5) × 120
Logical negation	“All A are B” → “Some A are not B”	Use opposite quantifier
If p→q	Contrapositive: ¬q → ¬p	Always logically equivalent

🎓 7. EXAM STRATEGY

Task	Time	Goal
Revise formulas daily	10–15 mins	Retain logic & arithmetic
Practise 10 reasoning questions	15 mins	Improve accuracy
Solve 10 aptitude questions	15 mins	Improve speed
Mock tests	Weekly	Identify weak areas

💡 8. COMMON QUESTION AREAS IN UGC NET

Statement–Conclusion reasoning (2–3 questions)
Truth table evaluation (1–2 questions)
Number/letter series or coding (1–2 questions)
Basic arithmetic: ratio, percentage, average, profit–loss (2–3 questions)
Time, work, distance (1 question)
Simple interest/compound interest (1 question)

✅ 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.

October 22, 2025