Unit 3: Mathematical and Logical Reasoning - Free Study Resources | Boundless Maths
🧠 Unit 3: Mathematical and Logical Reasoning
Coding-Decoding · Blood Relations · Syllogism · Odd Man Out
Weightage: 6 Marks in School Exam
🧠
Topics Covered
1. Coding-Decoding
Identify the pattern used to code letters or numbers, then apply it to encode or decode new words
2. Blood Relations
Determine family relationships from given clues using step-by-step logical deduction and family tree diagrams
3. Syllogism
Draw logical conclusions from two or more given statements using "All", "Some", "No" type reasoning
4. Odd Man Out
Identify the one element that does NOT share the common property of the rest of the group
🔑 Coding-Decoding Tips
Always check the first 2–3 letter pairs to identify the rule (e.g. +1, −2, reverse, mirror). Verify it holds for all letters before applying. Common rules: alphabet shift, position coding, reversal, mirror (A↔Z).
👨👩👧 Blood Relations Tips
Draw a family tree diagram immediately. Use boxes for males and circles for females. Mark marriages with "=" and children with downward lines. Trace the path step by step — never guess.
🔗 Syllogism Tips
Key rules: "All A are B" + "All B are C" → "All A are C". But "Some A are B" + "All B are C" → only "Some A are C". Never assume more than the statements give you.
🔍 Odd Man Out Tips
Look for the common property in the group: number type (prime, square, odd), category (animals, cities, planets), pattern in a series. The odd one out breaks that property.
In a certain code, BOOK is written as DQQM. How is DESK coded in the same language?
(a) FGUM
(b) FGUL
(c) EGUM
(d) FHUM
✓ Correct Answer: (a) FGUM
Explanation:
Find the rule: B→D(+2), O→Q(+2), O→Q(+2), K→M(+2). Rule: each letter +2.
Apply to DESK: D+2=F, E+2=G, S+2=U, K+2=M → FGUM
Question 2
In a code language, TIGER is written as QDFBO. What is the rule being applied?
(a) Each letter is replaced by the letter 3 positions before it
(b) Each letter is replaced by the letter 3 positions after it
(c) The word is reversed and each letter moved +1
(d) Each letter is replaced by its mirror in the alphabet
✓ Correct Answer: (a) Each letter is replaced by the letter 3 positions before it
Explanation:
T(20)−3=Q(17) ✓, I(9)−3=F(6) ✓, G(7)−3=D(4) ✓, E(5)−3=B(2) ✓, R(18)−3=O(15) ✓
Rule confirmed: each letter → letter 3 before it (−3 shift).
Question 3
If CAT = 3120 and PEN = 16514, using the same rule (each letter = its alphabetical position), what is the code for LOG?
Pointing to a photograph, Arjun says, "She is the daughter of the only son of my paternal grandfather." How is the girl related to Arjun?
(a) Niece
(b) Cousin
(c) Sister
(d) Daughter
✓ Correct Answer: (c) Sister
Explanation:
"Only son of my paternal grandfather" = Arjun's father.
"Daughter of my father" = Arjun's sister.
The girl is Arjun's sister.
Question 7
A is the brother of B. B is the mother of C. D is the father of B. What is A to D?
(a) Son
(b) Grandson
(c) Father
(d) Brother
✓ Correct Answer: (a) Son
Explanation:
D is the father of B. A is the brother of B.
Since A and B are siblings, they share the same parents. D is B's father → D is also A's father.
Therefore A is D's son.
Question 8
Introducing a woman, a man says, "Her mother is the only daughter of my mother-in-law." How is the woman related to the man?
(a) Daughter
(b) Sister
(c) Niece
(d) Wife
✓ Correct Answer: (a) Daughter
Explanation:
"My mother-in-law" = the man's wife's mother.
"Only daughter of my mother-in-law" = the man's wife (only daughter of his wife's mother).
The woman's mother = the man's wife → the woman is the man's daughter.
Question 9
P is the son of Q. Q is the daughter of R. S is the husband of R. How is S related to P?
(a) Father
(b) Grandfather
(c) Uncle
(d) Brother
✓ Correct Answer: (b) Grandfather
Explanation:
P is Q's son → P is R's grandson (since Q is R's daughter).
S is R's husband → S is R's spouse → S is P's grandfather.
S is P's grandfather.
Question 10
If X is the brother of Y, Y is the sister of Z, and Z is the son of W, how is X related to W?
(a) Son
(b) Grandson
(c) Nephew
(d) Brother
✓ Correct Answer: (a) Son
Explanation:
Z is son of W. Y is sister of Z → Y is also child of W. X is brother of Y → X is also child of W.
Therefore X is W's son.
Part C — Syllogism
Question 11
Statements: (1) All doctors are educated. (2) All educated people read books. Which conclusion logically follows?
(a) Some doctors do not read books.
(b) All doctors read books.
(c) All book readers are doctors.
(d) No doctor reads books.
✓ Correct Answer: (b) All doctors read books.
Explanation:
Stmt 1: All doctors → educated. Stmt 2: All educated → read books.
Chain: All doctors → educated → read books.
Conclusion: All doctors read books. ✓
Rule: "All A are B" + "All B are C" → "All A are C".
Question 12
Statements: (1) Some fruits are sweet. (2) All mangoes are fruits. Conclusions: (I) Some mangoes are sweet. (II) All fruits are mangoes. Which conclusion(s) follow?
(a) Only (I) follows
(b) Only (II) follows
(c) Both follow
(d) Neither follows
✓ Correct Answer: (d) Neither follows
Explanation:
Let's check each conclusion carefully:
Conclusion (I): "Some mangoes are sweet"
We know: (a) All mangoes are fruits, and (b) Some fruits are sweet.
The key question is: are those "some sweet fruits" definitely mangoes? Not necessarily. The sweet fruits could be apples, grapes or any other fruit — the statements do not say the sweet fruits must include mangoes. So we cannot guarantee that even one mango is sweet. Conclusion (I) does NOT follow.
Conclusion (II): "All fruits are mangoes"
This is the reverse of Statement 2 ("All mangoes are fruits"). Reversing an "All" statement is never valid in syllogism. Clearly does NOT follow.
💡 Common mistake: Many students assume that because mangoes are fruits, and some fruits are sweet, some mangoes must be sweet. But syllogism requires certainty — if it is only possible (not guaranteed), it does not follow.
Therefore: Neither conclusion follows.
Part D — Odd Man Out
Question 13
Find the odd one out: 4, 9, 16, 20, 25
(a) 4
(b) 9
(c) 20
(d) 25
✓ Correct Answer: (c) 20
Explanation:
4=2², 9=3², 16=4², 25=5² — all are perfect squares.
20 is NOT a perfect square (√20 ≈ 4.47).
Therefore 20 is the odd one out.
Question 14
Find the odd one out: Mercury, Venus, Earth, Moon, Mars
(a) Mercury
(b) Venus
(c) Moon
(d) Mars
✓ Correct Answer: (c) Moon
Explanation:
Mercury, Venus, Earth, and Mars are all planets of our solar system.
Moon is a natural satellite of Earth — not a planet.
Therefore Moon is the odd one out.
Part E — Assertion-Reason Questions
📋 Instructions for Assertion-Reason Questions
Statement I is the Assertion (A) and Statement II is the Reason (R). Choose the correct option:
(a) Both A and R are True and R is the correct explanation of A
(b) Both A and R are True but R is not the correct explanation of A
(c) A is True but R is False
(d) A is False but R is True
Assertion-Reason 1
Assertion (A): In a code where each letter is shifted +3, the word CAT is coded as FDW.
Reason (R): In a +3 letter shift, each letter is replaced by the letter that comes 3 positions after it in the alphabet.
(a) Both A and R are True and R is the correct explanation of A
(b) Both A and R are True but R is not the correct explanation of A
(c) A is True but R is False
(d) A is False but R is True
✓ Correct Answer: (a) Both A and R are True and R is the correct explanation of A
Explanation:
R is TRUE — in a +3 shift, each letter moves 3 positions forward in the alphabet.
Verify A: C(3)+3=F(6) ✓, A(1)+3=D(4) ✓, T(20)+3=W(23) ✓ → CAT is indeed coded as FDW. So A is TRUE.
R directly explains why CAT → FDW (by applying the +3 rule to each letter) → answer is (a).
Exam tip: Always verify the assertion by working through the calculation before choosing your option.
Assertion-Reason 2
Assertion (A): If Priya is the daughter of Ravi's father, then Priya is Ravi's sister.
Reason (R): The daughter of one's father is one's sister (assuming the same father).
(a) Both A and R are True and R is the correct explanation of A
(b) Both A and R are True but R is not the correct explanation of A
(c) A is True but R is False
(d) A is False but R is True
✓ Correct Answer: (a) Both A and R are True and R is the correct explanation of A
Explanation:
A is TRUE — "Priya is the daughter of Ravi's father" means Priya and Ravi share the same father, so Priya is Ravi's sister.
R is TRUE — the logical rule stated in R directly justifies the conclusion in A.
R correctly explains A → answer is (a).
Assertion-Reason 3
Assertion (A): From the statements "All birds can fly" and "A penguin is a bird," the conclusion "A penguin can fly" is logically valid.
Reason (R): In syllogism, if "All A are B" and "C is an A," then "C is a B" is always a logically valid conclusion.
(a) Both A and R are True and R is the correct explanation of A
(b) Both A and R are True but R is not the correct explanation of A
(c) A is True but R is False
(d) A is False but R is True
✓ Correct Answer: (a) Both A and R are True and R is the correct explanation of A
Explanation:
A is TRUE — given the two premises as stated, the conclusion follows logically. Syllogism tests logical validity, not real-world accuracy. Even though penguins cannot actually fly, the conclusion is logically valid from those premises.
R is TRUE — this is a standard valid syllogistic pattern: "All A are B" + "C is A" → "C is B."
R correctly explains why A is logically valid → answer is (a).
Key distinction: In syllogism questions, always judge logical validity, not real-world truth.
Assertion-Reason 4
Assertion (A): In the group {Cow, Horse, Parrot, Dog, Cat}, Parrot is the odd one out.
Reason (R): Cow, Horse, Dog, and Cat are all four-legged animals, whereas Parrot is a bird with two legs.
(a) Both A and R are True and R is the correct explanation of A
(b) Both A and R are True but R is not the correct explanation of A
(c) A is True but R is False
(d) A is False but R is True
✓ Correct Answer: (a) Both A and R are True and R is the correct explanation of A
Explanation:
A is TRUE — Parrot is the odd one out in this group.
R is TRUE — the common property of Cow, Horse, Dog, and Cat is that they are all four-legged mammals. Parrot is a bird (two-legged, feathered), which breaks the common property.
R correctly explains why Parrot is the odd one out → answer is (a).
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Quick-reference rules for all reasoning types — plus formulas for all 8 units — in one organised Class 11 Formula Deck.
✅ Coding-Decoding rule types and step-by-step approach
✅ Blood Relations — family tree method and common relation types
✅ Syllogism — all valid and invalid conclusion patterns
In a code language, each letter is replaced by the letter 3 positions BEFORE it in the alphabet (D→A, E→B, … A→X, B→Y, C→Z). (i) Encode: MATHS (ii) Decode: OBPQBO
Solution:
Rule: each letter → letter 3 before it (−3, wrapping: A−3=X, B−3=Y, C−3=Z)
(i) Encode MATHS: M→J, A→X, T→Q, H→E, S→P → JXQEP
(ii) Decode OBPQBO: apply the reverse rule (+3): O+3=R, B+3=E, P+3=S, Q+3=T, B+3=E, O+3=R → RESTER
(i) MATHS → JXQEP | (ii) OBPQBO → RESTER
Blood Relations
Example 2
Read and answer: "Sunita is the daughter of Ramesh. Ramesh is the brother of Kavita. Kavita's husband is Mohan. Mohan and Kavita have a son Dev." (i) How is Sunita related to Kavita? (ii) How is Mohan related to Sunita? (iii) How is Dev related to Sunita?
Solution:
Family Tree:
Ramesh (brother) ↔ Kavita + Mohan (married)
↓ ↓
Sunita Dev
(i) Kavita is Ramesh's sister. Sunita is Ramesh's daughter. → Kavita is Sunita's father's sister → Kavita is Sunita's aunt.
(ii) Mohan is Kavita's husband. Kavita is Sunita's aunt. → Mohan is Sunita's uncle.
(iii) Dev is Kavita and Mohan's son. Kavita is Sunita's aunt. → Dev is Sunita's cousin.
(i) Kavita is Sunita's aunt | (ii) Mohan is Sunita's uncle | (iii) Dev is Sunita's cousin
Example 3
A woman says to a man, "Your mother's husband's sister is my grandmother." How is the man related to the woman?
Solution:
Trace the path step by step from the man's side:
"Your mother's husband" = the man's father.
"Your father's sister" = the man's paternal aunt.
This paternal aunt of the man is the woman's grandmother.
So: Man's paternal aunt = Woman's grandmother.
This means the man's father and the woman's grandmother are siblings (father = brother of the woman's grandmother, or the woman's grandmother IS the man's father's sister).
If the woman's grandmother is the man's father's sister → the woman's mother/parent is the man's father's niece/nephew → the woman is the man's father's great-niece → the man is the woman's great-uncle's son. More directly: the woman is the man's cousin's child or the man is the woman's grand-uncle. The closest standard answer: the man is the woman's maternal/paternal grand-uncle.
The man is the woman's grand-uncle (the woman is the man's grand-niece).
Syllogism & Odd Man Out
Example 4
(a) Syllogism: Statements: (1) Some students are athletes. (2) All athletes are disciplined. Conclusions: (I) Some students are disciplined. (II) All students are disciplined.
(b) Odd Man Out: Find the odd one out from: 3, 7, 11, 14, 19. Justify.
Solution:
(a) Some students → athletes (from Stmt 1). All athletes → disciplined (Stmt 2).
(I) Those students who are athletes must be disciplined → Some students are disciplined. Follows ✓
(II) We only know SOME students are athletes; other students may not be disciplined → Does NOT follow ✗
(b) 3, 7, 11, 19 are all prime numbers. 14 = 2 × 7 — NOT prime. 14 is the odd one out.
(a) Only Conclusion (I) follows. | (b) 14 — not a prime number.
📐 Keep Every Rule at Your Fingertips
Coding-decoding patterns, blood relation types, syllogism rules — plus formulas for all 8 units — one printable Class 11 Formula Deck.
Read the following family information and answer all four questions: "Rajan and Meena are a married couple with two children — a son Ankit and a daughter Pooja. Ankit is married to Sheetal and they have a daughter Riya. Pooja is married to Vivek. Meena's mother is Kamla." (i) How is Riya related to Pooja? (ii) How is Vivek related to Ankit? (iii) How is Kamla related to Ankit? (iv) How is Sheetal related to Rajan?
Complete Solution:
Family Tree:
Kamla → Meena = Rajan (married)
↓ ↓
Ankit = Sheetal Pooja = Vivek
↓
Riya
(i) Riya is Ankit's daughter. Pooja is Ankit's sister. → Riya is Pooja's niece.
(ii) Vivek is Pooja's husband. Ankit is Pooja's brother. → Vivek is Ankit's brother-in-law.
(iii) Kamla is Meena's mother. Meena is Ankit's mother. → Kamla is Ankit's maternal grandmother.
(iv) Sheetal is Ankit's wife. Rajan is Ankit's father. → Sheetal is Rajan's daughter-in-law.
Six members of a family — A, B, C, D, E, and F — are described as follows: • A and B are a married couple. • C is the son of A. D is the daughter of B. • E is the brother of C. F is the daughter of D. • D is married to G (not in the list). Answer: (i) How is E related to D? (ii) How is F related to C? (iii) How is A related to F? (iv) How is B related to E?
Complete Solution:
Family Tree:
A = B (married)
↓ (children): C, D, E (E is C's brother, so E is also A & B's son)
D = G → F (F is D's daughter)
(i) E is the brother of C. D is the daughter of B. Since A and B are parents of C, D, and E (E is C's brother so same parents) → D and E are siblings → E is D's brother.
(ii) F is D's daughter. C is D's brother (both are children of A and B). → F is C's niece.
(iii) A is the parent of D (D is B's daughter and A is B's spouse → A is also D's parent). F is D's daughter. → A is F's grandparent (grandfather or grandmother).
(iv) B is the mother/father of E (E is C's brother; C is A's son and D is B's daughter — same parents). → B is E's parent (mother or father).
(i) E is D's brother | (ii) F is C's niece | (iii) A is F's grandparent | (iv) B is E's parent
Coding-Decoding
Question 3
In a code language, each letter is replaced by the letter in its MIRROR position in the alphabet (A↔Z, B↔Y, C↔X, D↔W, … M↔N). (i) Encode SMART. (ii) Encode KIND. (iii) Decode OLLV. (iv) If TEACHER is encoded as GVZXSVI, verify whether this follows the same rule.
Complete Solution:
Mirror rule: mirror position = 27 − original. So A(1)↔Z(26), B(2)↔Y(25), ..., M(13)↔N(14).
For each set, state which conclusions logically follow. Justify.
Set A: Statements: (1) All managers are leaders. (2) Some leaders are visionary. Conclusions: (I) Some managers are visionary. (II) All managers are visionary.
Set B: Statements: (1) No student failed the test. (2) Priya is a student. Conclusions: (I) Priya did not fail. (II) Priya passed with distinction.
Complete Solution:
Set A: All managers → leaders (Stmt 1). Some leaders → visionary (Stmt 2).
(I) "Some managers are visionary" — The "some leaders" who are visionary may or may not be managers. We cannot be certain. Does NOT follow ✗.
(II) "All managers are visionary" — definitely does NOT follow ✗.
Neither conclusion follows.
Set B: No student failed (Stmt 1). Priya is a student (Stmt 2).
(I) Priya is a student and no student failed → Priya did not fail. Follows ✓
(II) "Passed with distinction" — not implied; we only know she didn't fail. Does NOT follow ✗.
Set A: Neither conclusion follows. | Set B: Only Conclusion (I) follows.
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Coding patterns, blood relation steps, syllogism rules — and formulas for all 8 Class 11 units — in one PDF.
Context: A company gives a logical reasoning test to all job applicants. The test has three sections. Read and answer the questions below.
Section 1 — Coding: In the company's internal system, employee names are coded using the rule: each letter → next letter in alphabet (+1 shift). A manager's coded name is SBKFTI.
Section 2 — Blood Relation: The HR head, Mrs Sharma, says to applicant Ajay: "My mother's only son's daughter is your interviewer today."
Section 3 — Syllogism: All shortlisted candidates are sent an offer letter. Meera received an offer letter. Raj did not receive an offer letter.
(i) Decode the manager's name: SBKFTI
Solution:
Rule: each letter was shifted +1 to encode. To decode, apply −1 to each letter.
S−1=R, B−1=A, K−1=J, F−1=E, T−1=S, I−1=H → RAJESH
The manager's name is RAJESH.
(ii) Determine how the interviewer is related to Mrs Sharma.
Solution:
"My mother's only son" = Mrs Sharma's brother.
"My mother's only son's daughter" = Mrs Sharma's brother's daughter = Mrs Sharma's niece.
The interviewer is Mrs Sharma's niece.
(iii)(a) Using the syllogism clues, what can we conclude about Meera and Raj?
Solution:
Rule: All shortlisted → receive offer letter. (p → q)
Meera received an offer letter — but this is the conclusion (q), not the hypothesis. We CANNOT conclude Meera is shortlisted (converse does not hold necessarily).
Raj did NOT receive an offer letter (¬q). Contrapositive: ¬q → ¬p = "If no offer letter → not shortlisted." So Raj is NOT shortlisted.
Meera: Cannot confirm if shortlisted from this rule alone. Raj: Definitely NOT shortlisted (contrapositive reasoning).
(iii)(b) OR: Encode the word RESULT using the same +1 coding rule, and also decode BQQMF.
Decode BQQMF (−1): B−1=A, Q−1=P, Q−1=P, M−1=L, F−1=E → APPLE
RESULT → SFTVMU | BQQMF → APPLE
Case Study 2: A Logic Puzzle at a Family Event
Context: At a school quiz, students are given a mixed reasoning puzzle. Read the following and answer:
Family information: Suresh and Lata are married. They have three children: Rohit, Sonia, and Nisha. Rohit is married to Priya. Sonia is married to Anil. Priya's father is Mr Verma. Rohit and Priya have a son Karan.
Syllogism set: Statement 1: All quiz participants are students. Statement 2: All students carry an ID card. Statement 3: Tanya is participating in the quiz.
Odd Man Out: Delhi, Mumbai, Chennai, Kolkata, Amazon
(i) How is Karan related to (a) Lata and (b) Sonia?
Solution:
(a) Karan is Rohit's son. Rohit is Lata's son. → Karan is Lata's grandson.
(b) Karan is Rohit's son. Sonia is Rohit's sister. → Karan is Sonia's nephew.
(a) Grandson | (b) Nephew
(ii) Using the syllogism statements, what can we conclude about Tanya?
Solution:
Tanya is a quiz participant → Tanya is a student (Stmt 1).
Tanya is a student → Tanya carries an ID card (Stmt 2).
Tanya carries an ID card. (Two-step syllogistic chain)
(iii)(a) Identify the odd one out from: Delhi, Mumbai, Chennai, Kolkata, Amazon. Justify.
Solution:
Delhi, Mumbai, Chennai, Kolkata — all major Indian cities.
Amazon — a river, not a city.
Odd one out: Amazon — it is a river; all others are Indian cities.
(iii)(b) OR: How is Mr Verma related to Karan? Show step-by-step reasoning.
Solution:
Karan is the son of Rohit and Priya. Mr Verma is Priya's father. Priya is Karan's mother.
Mr Verma is the father of Karan's mother → Mr Verma is Karan's maternal grandfather.
Mr Verma is Karan's maternal grandfather.
Case Study 3: The Mehta Family Tree
Context: The Mehta family is gathered for a reunion. Study the following information carefully and use a family tree to answer the questions.
• Ramanlal and Savitri are grandparents. They have two sons — Dinesh and Prakash — and one daughter — Hema.
• Dinesh is married to Sushma. They have a son Nikhil and a daughter Priti.
• Prakash is married to Rekha. They have two sons — Aryan and Kartik.
• Hema is married to Suresh. They have one daughter — Diya.
• Nikhil is married to Pooja and they have a son Veer.
(i) How are Nikhil and Aryan related? How are Priti and Diya related?
Solution:
Nikhil is the son of Dinesh. Aryan is the son of Prakash. Dinesh and Prakash are brothers. Therefore Nikhil and Aryan are cousins (sons of two brothers).
Priti is Dinesh's daughter. Diya is Hema's daughter. Dinesh and Hema are siblings (brother and sister). Therefore Priti and Diya are cousins.
Nikhil and Aryan are cousins. | Priti and Diya are cousins.
(ii) How is Veer related to (a) Savitri and (b) Diya?
Solution:
(a) Veer is Nikhil's son. Nikhil is Dinesh's son. Dinesh is Savitri's son. Veer is Savitri's great-grandson (son → grandson → great-grandson). → Veer is Savitri's great-grandson.
(b) Veer is Nikhil's son. Diya is Hema's daughter. Nikhil's father Dinesh and Hema are siblings. So Nikhil and Diya are cousins → Veer (Nikhil's son) is Diya's cousin's son, and Diya is Veer's cousin's mother — the closest standard relation: Diya is Veer's first cousin once removed (or in common terms, Veer is Diya's nephew by extension).
(a) Veer is Savitri's great-grandson. | (b) Diya is Veer's first cousin once removed (Diya's father's brother's son = Nikhil; Nikhil's son = Veer).
(iii)(a) How many grandsons does Ramanlal have in total? List them.
Solution:
Ramanlal's children: Dinesh, Prakash (sons), Hema (daughter).
Grandsons (male grandchildren):
— Dinesh's son: Nikhil
— Prakash's sons: Aryan and Kartik
— Hema has a daughter Diya (not a grandson)
Ramanlal has 3 grandsons: Nikhil, Aryan, and Kartik.
(iii)(b) OR: How is Pooja related to (a) Sushma and (b) Prakash?
Solution:
(a) Pooja is Nikhil's wife. Sushma is Nikhil's mother. → Pooja is Sushma's daughter-in-law.
(b) Prakash is Nikhil's uncle (brother of Nikhil's father Dinesh). Pooja is Nikhil's wife. → Pooja is Prakash's nephew's wife (in common terms, Pooja is Prakash's niece-in-law).
(a) Pooja is Sushma's daughter-in-law. | (b) Pooja is Prakash's nephew's wife (niece-in-law).
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Unit 3 is 6 reliable marks — with practised patterns for coding, a clear family tree habit for blood relations, and a few syllogism rules, most students can score full marks here.
✅ Coding-Decoding — common rule types and step-by-step approach
✅ Blood Relations — family tree method with common relation types
✅ Syllogism — all valid conclusion patterns with examples
✅ Odd Man Out — quick identification strategies
✅ Formulas for all 8 Class 11 Applied Maths units — chapter-wise
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