Mathematical and Logical Reasoning

Given sentence: "8 is less than 6."

Concept: A sentence is a mathematically acceptable statement if it is either true or false (but not both).

Working: The sentence "8 is less than 6" is false because 8 > 6. Since it has a definite truth value (false), it qualifies as a statement.

Conclusion: Yes, it is a statement (a false statement).

Given sentence: "Every set is a finite set."

Concept: A sentence is a statement if it is either true or false.

Working: The sentence is false because there exist sets that are not finite (e.g., the set of natural numbers $\mathbb{N}$ is infinite). Since it has a definite truth value (false), it qualifies as a statement.

Conclusion: Yes, it is a statement (a false statement).

Given sentence: "The sun is a star."

Concept: A sentence is a statement if it is either true or false.

Working: It is a scientifically established fact that the Sun is a star. Therefore, this sentence is always true and has a definite truth value.

Conclusion: Yes, it is a statement (a true statement).

Given sentence: "Mathematics is fun."

Concept: A sentence is a statement only if it is either definitely true or definitely false.

Working: This sentence is subjective — for those who enjoy mathematics it may be true, but for others it may not be. It does not have a definite, universal truth value.

Conclusion: No, it is NOT a statement because its truth value varies from person to person.

Given sentence: "There is no rain without clouds."

Concept: A sentence is a statement if it is either true or false.

Working: This is a scientifically true fact — rain requires clouds. The sentence has a definite truth value (true).

Conclusion: Yes, it is a statement (a true statement).

Given sentence: "How far is Chennai from here?"

Concept: Questions, exclamations, and commands are not statements because they cannot be assigned a truth value of true or false.

Working: This is an interrogative sentence (a question). It does not assert anything and cannot be said to be true or false.

Conclusion: No, it is NOT a statement because it is a question and does not have a truth value.

Given statement: "The school is closed if it is a holiday or a Sunday."

Concept:
- Inclusive 'Or': At least one of the two conditions is true (both can be true simultaneously).
- Exclusive 'Or': Exactly one of the two conditions is true (both cannot be true simultaneously).

Working: A day can be both a holiday AND a Sunday at the same time (e.g., a national holiday falling on a Sunday). In either case or both cases, the school remains closed. So both conditions can hold simultaneously.

Conclusion: The "Or" used here is Inclusive Or, because the school is closed when it is a holiday, or a Sunday, or both.

Given statement: "Two lines intersect at a point or are parallel."

Concept: Exclusive 'Or' means exactly one of the two alternatives holds; both cannot be true simultaneously.

Working: Two lines in a plane cannot intersect at a point AND be parallel at the same time. These are mutually exclusive situations — if two lines intersect, they are not parallel, and if they are parallel, they do not intersect.

Conclusion: The "Or" used here is Exclusive Or, because both conditions cannot be true simultaneously.

Given statement: "Students can take French or Sanskrit as their third language."

Concept: Exclusive 'Or' means exactly one of the two alternatives holds.

Working: A student can choose only one language as their third language — either French or Sanskrit, but not both at the same time. The two choices are mutually exclusive.

Conclusion: The "Or" used here is Exclusive Or, because a student cannot take both French and Sanskrit simultaneously as their third language.

Given Statements:
- Some actors are singers. (partial overlap between actors and singers)
- All singers are dancers. (singers ⊂ dancers)

Analysis of Conclusion (1): "Some actors are dancers."
Since some actors are singers, and all singers are dancers, those actors who are singers must also be dancers. Therefore, some actors are dancers. ✓ Conclusion (1) follows.

Analysis of Conclusion (2): "No singer is actor."
Statement I says "Some actors are singers," which directly means some singers ARE actors. So the claim "No singer is actor" is contradicted by Statement I. ✗ Conclusion (2) does not follow.

Answer: (A) — Only conclusion (1) follows.

Given Statements:
- All harmoniums are instruments. (harmoniums ⊂ instruments)
- All instruments are flutes. (instruments ⊂ flutes)

Analysis of Conclusion (1): "All the flutes are instruments."
Statement II says all instruments are flutes, i.e., instruments ⊂ flutes. This does NOT mean all flutes are instruments (the converse is not necessarily true). ✗ Conclusion (1) does not follow.

Analysis of Conclusion (2): "All the harmoniums are flutes."
Since all harmoniums are instruments (Statement I) and all instruments are flutes (Statement II), by transitivity: all harmoniums are flutes. ✓ Conclusion (2) follows.

Answer: (B) — Only conclusion (2) follows.

Given Statements:
- Some mangoes are yellow.
- Some fruits are mangoes.

Analysis of Conclusion (1): "Some mangoes are green."
The statements only tell us that some mangoes are yellow. Nothing is said about mangoes being green. This cannot be concluded from the given statements. ✗ Conclusion (1) does not follow.

Analysis of Conclusion (2): "Fruits are yellow."
Statement II says only SOME fruits are mangoes, and Statement I says only SOME mangoes are yellow. We cannot conclude that all fruits are yellow. ✗ Conclusion (2) does not follow.

Answer: (D) — Neither conclusion (1) nor conclusion (2) follows.

Given Statements:
- Some ants are parrots. (partial overlap: ants ∩ parrots ≠ ∅)
- All parrots are apples. (parrots ⊂ apples)

Analysis of Conclusion (1): "All the apples are parrots."
Statement II says all parrots are apples, i.e., parrots ⊂ apples. The converse (all apples are parrots) does not necessarily follow. ✗ Conclusion (1) does not follow.

Analysis of Conclusion (2): "Some ants are apples."
Since some ants are parrots (Statement I) and all parrots are apples (Statement II), those ants that are parrots must also be apples. Therefore, some ants are apples. ✓ Conclusion (2) follows.

Answer: (B) — Only conclusion (2) follows.

Given: CAMEL → DBNFM

Finding the pattern:
$$C \xrightarrow{+1} D, \quad A \xrightarrow{+1} B, \quad M \xrightarrow{+1} N, \quad E \xrightarrow{+1} F, \quad L \xrightarrow{+1} M$$

Each letter is replaced by the next letter in the alphabet (shifted by +1).

Applying the same rule to ROOM:
$$R \xrightarrow{+1} S, \quad O \xrightarrow{+1} P, \quad O \xrightarrow{+1} P, \quad M \xrightarrow{+1} N$$

So ROOM is coded as SPPN.

Answer: (b) SPPN

Given:
- BLEPIN = 9 8 7 4 1 6
- MATPIN = 1 2 3 4 1 6

Decoding individual letters:
$$B=9,\ L=8,\ E=7,\ P=4,\ I=1,\ N=6$$
$$M=1,\ A=2,\ T=3,\ P=4,\ I=1,\ N=6$$

So: $T=3,\ A=2,\ B=9,\ L=8,\ E=7$

TABLE = $3\ 2\ 9\ 8\ 7$ = 32987

Answer: (a) 32987

Given codes:
- 'hi mi si' = 'Air is life'
- 'si ni zi' = 'Balloon of Air'
- 'ci zi mi' = 'Life of PI'

Step 1: Find the code for 'Air'.
Comparing statement 1 and statement 2: common word is 'Air' and common code is 'si'. So si = Air.

Step 2: Find the code for 'life'.
Comparing statement 1 and statement 3: common word is 'life' and common code is 'mi'. So mi = life.

Step 3: From statement 1 (hi mi si = Air is life), with si = Air and mi = life, the remaining word 'is' = hi.

Step 4: Find the code for 'of'.
Comparing statement 2 and statement 3: common word is 'of' and common code is 'zi'. So zi = of.

Step 5: From statement 2 (si ni zi = Balloon of Air), with si = Air and zi = of, the remaining word 'Balloon' = ni.

Step 6: From statement 3 (ci zi mi = Life of PI), with zi = of and mi = life, the remaining word 'PI' = ci.

Answer: (c) ci

Given codes:
- 321 = 'Glass of Tea'
- 426 = 'Tea is Brown'
- 796 = 'Trunks are Brown'

Step 1: Find the code for 'Tea'.
Comparing statement 1 and statement 2: common word is 'Tea' and common digit is 2 (wait — let us check carefully).
- Statement 1 digits: 3, 2, 1
- Statement 2 digits: 4, 2, 6
Common digit: 2 → common word: 'Tea'. So 2 = Tea.

Step 2: Find the code for 'Brown'.
Comparing statement 2 and statement 3: common word is 'Brown' and common digit:
- Statement 2 digits: 4, 2, 6
- Statement 3 digits: 7, 9, 6
Common digit: 6 → common word: 'Brown'. So 6 = Brown.

Step 3: From statement 2 (426 = Tea is Brown), with 2 = Tea and 6 = Brown, the remaining digit for 'is' = 4.

Answer: (c) 4

Given: The man says, "The lady in the picture is my nephew's maternal grandmother."

Step 1: The man's nephew is the son of the man's sister (since the sister has no other sister, the nephew must be the sister's son).

Step 2: The nephew's maternal grandmother is the mother of the nephew's mother, i.e., the mother of the man's sister.

Step 3: The mother of the man's sister is also the mother of the man (they share the same family).

Step 4: Therefore, the lady in the picture is the mother of the man's sister.

Answer: (c) Mother

Given information:
- Meenakshi is Kirti's sister → Meenakshi and Kirti are siblings.
- Kaavya is Kirti's mother → Kaavya is also Meenakshi's mother.
- Dipesh is Kaavya's father → Dipesh is the grandfather of Kirti and Meenakshi.
- Esha is Dipesh's mother (additional info, not needed here).

Relationship chain:
$$\text{Dipesh} \xrightarrow{\text{father of}} \text{Kaavya} \xrightarrow{\text{mother of}} \text{Meenakshi}$$

So Dipesh is Meenakshi's maternal grandfather, and Meenakshi is Dipesh's granddaughter.

Answer: (d) Granddaughter

Given operations:
- $+$ : P+Q → P is the brother of Q
- $-$ : R−S → R is the father of S
- $/$ : S/T → S is the sister of T
- $\times$ : T×U → T is the mother of U

We need to find the expression that means "O is the mother of N."

Checking option (d): $M+L/O \times N$
- $M+L$: M is the brother of L
- $L/O$: L is the sister of O → O and L are sisters
- $O \times N$: O is the mother of N ✓

This directly gives us O is the mother of N.

Let us verify the other options do not give this:
- Option (a): $L+M/N-O$ → L is brother of M; M is sister of N; N is father of O. This gives N as father of O, not O as mother of N. ✗
- Option (b): $L-M \times O/P$ → L is father of M; M is mother of O; O is sister of P. Does not give O as mother of N. ✗
- Option (c): $N/M \times L/O$ → N is sister of M; M is mother of L; L is sister of O. Does not give O as mother of N. ✗

Answer: (d) $M+L/O \times N$

Given options: Insurance, Provident Fund, Salary, Shares.

Analysis:
- Insurance: a form of financial security/investment.
- Provident Fund: a form of savings/investment for the future.
- Salary: income earned from employment (not an investment or saving instrument).
- Shares: a form of investment.

Insurance, Provident Fund, and Shares are all forms of investment or financial security instruments. Salary is the odd one out as it is earned income, not an investment.

Answer: (c) Salary

Given numbers: 3, 5, 11, 14, 17, 21.

Analysis:
- 3 → odd number
- 5 → odd number
- 11 → odd number
- 14 → even number
- 17 → odd number
- 21 → odd number

All numbers except 14 are odd numbers. 14 is the only even number in the list.

Answer: (c) 14

Note: The actual figures (a, b, c, d) cannot be seen from the OCR. However, based on the official answer key provided in the textbook, the answer is option (c).

General approach for such questions: Examine each figure for a common property (e.g., number of sides, shading pattern, symmetry, rotation). The figure that does not share the common property of the other three is the odd one out.

Answer: (c) — as per the official solution provided in the textbook.