Questions & explanations
1. Statements: No A is B. Some B are C. Conclusions: I. Some A are C. II. Some A are not C. Which of the following is correct?
- (a) Only I follows
- (b) Only II follows
- (c) Either I or II follows
- (d) Neither I nor II follows
Answer: (c) Either I or II follows
From the statements, no A is B and some B are C. There is no direct relation between A and C. So neither conclusion I (some A are C) nor conclusion II (some A are not C) individually follows. However, are they a complementary pair? They are I and O types, so they should form a complementary pair. Some B are C, so there is a possibility that A and C have no relation. But the complementary pair rule applies only when there is no definite relation between the terms. Here, we cannot say that either some A are C or some A are not C must be true. It is possible that no A is C (if all C are B, then no A is C, but that would make both conclusions false? Actually if no A is C, then 'some A are C' is false, and 'some A are not C' is true because all A are not C. So in that case II is true. But if some A are C, then I is true. So indeed, one of them must be true. Here, the possibilities are: either some A are C (I true) or no A is C (then II true because some A are not C is true if there is at least one A; but if A is empty? In syllogisms, we assume non-empty sets? Usually UPSC assumes non-empt
2. Statement: 'A is the only daughter of B'. B is the only son of C. C is the father of D. How is D related to A?
- (a) Uncle
- (b) Father
- (c) Grandfather
- (d) Cannot be determined
Answer: (d) Cannot be determined
A is only daughter of B, so B is parent (gender not given yet). B is only son of C, so B is male. C is father of D, so D is child of C. Since B is only son of C, D must be sibling of B (unless D is B himself, but 'only son' implies B is the only male child; D could be sister). Thus D is sibling of B, making D uncle or aunt of A. But gender of D not given, so 'uncle' is appropriate if male; however, option (a) says uncle, which is correct if D is male. But D could be female? D is child of C, so D could be daughter. Then D would be aunt. But the question asks 'How is D related to A?' and options include 'Uncle'. Since D's gender is not given, we cannot be sure. However, the most definite: D is sibling of B, so D is either uncle or aunt. But among options, only uncle is close. C could have daughters. So D could be sister of B, making D aunt. So relationship is not fixed. But the question expects 'Cannot be determined' because gender of D is unknown. C is father of D, so D is child of C. D could be male or female. If D is male, then D is brother of B, but B is only son, so D cannot be ma
3. Rahul walks 10 km towards the east, then turns 45 degrees to his right and walks 10 km. What is the shortest distance between his starting point and final position?
- (a) 10√(2 + √2) km
- (b) 10 km
- (c) 10√2 km
- (d) 20 km
Answer: (a) 10√(2 + √2) km
Rahul walks 10 km towards the east. Let this be the first leg of his journey. Then he turns 45 degrees to his right. This means his new direction of travel is 45 degrees South of East. He then walks another 10 km. The path forms an isosceles triangle where two sides are 10 km each. The angle *inside the triangle* at the turning point is crucial for calculating the shortest distance (displacement). If he turns 45 degrees to his right from an eastward path, the internal angle of the triangle formed by his starting point, turning point, and final point is 180° - 45° = 135°. To find the shortest distance (displacement) between his starting point and final position, we use the Law of Cosines: d² = a² + b² - 2ab cos(C). Here, a = 10 km, b = 10 km, and C = 135°. d² = 10² + 10² - 2 × 10 × 10 × cos(135°)
d² = 100 + 100 - 200 × (-1/√2)
d² = 200 + 200 × (√2 / 2)
d² = 200 + 100√2
Therefore, d = √(200 + 100√2) = 10√(2 + √2) km. Numerically, this is approximately 18.48 km.
4. Using the same two-grid system from the previous question (Grid I row-major, Grid II column-wise), what letter is coded as 42?
- (a) T
- (b) S
- (c) R
- (d) Q
Answer: (d) Q
Code 42 means row 4 from Grid I column 2 from Grid II. Grid I row4: letters at positions 16-20: P Q R S T? Actually row4: positions (4-1)*5+1=16 to 20: 16=P 17=Q 18=R 19=S 20=T. So row4 col? The code gives column from Grid II not from Grid I. So we need the letter that is in Grid I at row4 and also in Grid II at column2. But a letter has a unique position in each grid. So we need to find which letter has Grid I row=4 and Grid II column=2. Grid II column2: B G L Q V. So letters in that column: B G L Q V. Among these which is in Grid I row4? Grid I row4: P Q R S T. The common letter is Q. So code 42 corresponds to Q. But option (a) T (b) S (c) R (d) Q. So (d) Q. But I said (a) T? Let's check: Actually code 42: first digit 4 = Grid I row4 second digit 2 = Grid II column2. Grid I row4: positions 16-20: P(16) Q(17) R(18) S(19) T(20). Grid II column2: B(1,2) G(2,2) L(3,2) Q(4,2) V(5,2). The only common letter is Q. So answer is Q. So correct answer (d).
5. Nine persons A, B, C, D, E, F, G, H, I sit in a row of nine chairs facing north. A does not sit at either end. B sits third to the left of C. D sits to the immediate right of E. F sits at the left end. G sits at the right end. H sits second to the left of I. Who sits at the centre (5th position)?
- (a) A
- (b) B
- (c) C
- (d) Cannot be determined
Answer: (d) Cannot be determined
Chairs 1 to 9. F at 1, G at 9. B third left of C: possible (2,5), (3,6), (4,7), (5,8), (6,9) but 9 is G, so (5,8) and (6,9) not? (6,9) not because 9 is G, so (5,8) works if 8 not taken. So possible: (2,5), (3,6), (4,7), (5,8). D immediate right of E: possible (2,3), (3,4), (4,5), (5,6), (6,7), (7,8), (8,9) but 9 is G, so (7,8) and (8,9) not? (8,9) not because 9 is G, so (7,8) works if 8 not taken. So possible: (2,3), (3,4), (4,5), (5,6), (6,7), (7,8). H second left of I: H at z, I at z+2. Possible: (1,3) no; (2,4); (3,5); (4,6); (5,7); (6,8); (7,9) but 9 is G, so (6,8) and (7,9) not? (7,9) not because 9 is G, so (6,8) works if 8 not taken. So possible: (2,4), (3,5), (4,6), (5,7), (6,8). A not at ends. We need to find who is at 5. Multiple combinations possible. For example, if B,C are (2,5), then position 5 is C. If B,C are (3,6), then position 5 is free. So cannot determine.
6. Assertion (A): All squares are rectangles.
Reason (R): All rectangles have four sides.
Codes:
(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
- (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
Answer: (b) Both A and R are true but R is not the correct explanation of A
Assertion (A) is true: A square is a quadrilateral with four equal sides and four right angles. A rectangle is a quadrilateral with four right angles. Since all squares have four right angles, they fit the definition of a rectangle. Reason (R) is also true: A rectangle is a type of quadrilateral, and all quadrilaterals have four sides. However, Reason (R) does not correctly explain Assertion (A). While squares, as rectangles, do have four sides, the reason why 'all squares are rectangles' is because squares possess the defining properties of a rectangle (i.e., four right angles), not simply because rectangles have four sides. Having four sides is a necessary but not sufficient condition for a shape to be a rectangle, and it doesn't explain the specific relationship between squares and rectangles. Therefore, both A and R are true, but R is not the correct explanation of A.
7. Four persons A, B, C, D sit in a row. Clues: A sits at left end. B sits to the right of A. C sits to the left of D. How many valid arrangements are possible?
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Answer: (c) 3
Fix A at left end. B must be to the right of A, so B can be in positions 2,3,4. But C is left of D, so possible orders: A B C D, A C B D, A C D B? Check: A C D B: C left of D (yes), B right of A (yes), but D left of B? Actually D is left of B? In A C D B, D is left of B, but no condition on D vs B. So all are valid? Let's list systematically: Positions 1:A. For B to be right of A, B can be 2,3,4. For C left of D, possible pairs: (C,D) can be (2,3), (2,4), (3,4). But B also occupies one. So possible arrangements: (1)A, (2)B, (3)C, (4)D -> valid. (1)A, (2)B, (3)D, (4)C -> invalid because C not left of D. (1)A, (2)C, (3)B, (4)D -> valid (C left of D, B right of A). (1)A, (2)C, (3)D, (4)B -> valid? C left of D yes, B right of A yes. (1)A, (2)D, (3)C, (4)B -> invalid because C not left of D. (1)A, (2)D, (3)B, (4)C -> invalid. So valid: A B C D, A C B D, A C D B. That's 3. But.
8. Rohan starts from his home and walks 3 km towards the North. He then turns right and walks 4 km. After that, he turns right again and walks 6 km. What is the shortest distance between his starting point and his final position?
- (a) 3 km
- (b) 5 km
- (c) 7 km
- (d) 13 km
Answer: (b) 5 km
Let Rohan's starting point be the origin (0,0). 1. Walks 3 km towards the North: His position becomes (0, 3). 2. Turns right (East) and walks 4 km: His position becomes (0+4, 3) = (4, 3). 3. Turns right again (South) and walks 6 km: His position becomes (4, 3-6) = (4, -3). So, the starting point is (0,0) and the final position is (4, -3). To find the shortest distance, we use the Pythagorean theorem, considering the net displacement in East-West and North-South directions. Net East displacement = 4 km. Net South displacement = 3 km (since he went 3 km North and then 6 km South, resulting in 3 km net South). Shortest distance = √(Net East displacement² + Net South displacement²)
Shortest distance = √(4² + (-3)²) = √(16 + 9) = √25 = 5 km. Therefore, the shortest distance between his starting point and his final position is 5 km, which corresponds to option (b).
9. If 'A + B' means A is the mother of B, 'A – B' means A is the father of B, 'A × B' means A is the sister of B, and 'A ÷ B' means A is the brother of B, then which of the following means that M is the maternal uncle of N?
- (a) M ÷ N + O
- (b) M × O + N
- (c) M + O ÷ N
- (d) M ÷ O + N
Answer: (d) M ÷ O + N
The codes are: A+B means A is the mother of B. A-B means A is the father of B. A x B means A is the sister of B. A divided by B means A is the brother of B. M is the maternal uncle of N if M is the brother of N's mother. Option (d) M divided by O + N works: M divided by O means M is the brother of O, and O + N means O is the mother of N. So M is the brother of N's mother, which makes M the maternal uncle of N. Why the others are wrong: (a) M divided by N + O means M is the brother of N and N is the mother of O, so M is just N's brother, not a maternal uncle. (b) M x O + N means M is the sister of O and O is the mother of N, so M is the maternal aunt, not uncle. (c) M + O divided by N means M is the mother of O and O is the brother of N, so M turns out to be the mother of N, not a maternal uncle.
10. Given: Some A are B. All B are C. Which of the following is a valid possibility?
- (a) All A are C
- (b) Some A are not C
- (c) Both (a) and (b)
- (d) Neither (a) nor (b)
Answer: (c) Both (a) and (b)
From the statements 'Some A are B' and 'All B are C', we can definitively conclude that 'Some A are C'. This means 'No A are C' is not a valid possibility. Let's evaluate the given options:
(a) All A are C: This is a valid possibility. For instance, if all A happen to be B (e.g., A=B={x,y} and C={x,y,z}), then 'Some A are B' (which is true if all A are B) and 'All B are C' would lead to 'All A are C'. (b) Some A are not C: This is also a valid possibility. For example, if only a part of A is B (e.g., A={x,y,p}, B={x,y}, C={x,y,q}), then 'Some A are B' (x,y are B) and 'All B are C' (x,y are C). In this scenario, 'Some A are not C' (p is A but not C) is true. Since both (a) and (b) are logically possible scenarios given the initial statements, the correct answer is (c) Both (a) and (b).
11. Six persons P, Q, R, S, T, and U are sitting in a circle facing the center. P is second to the left of T, and Q is opposite to R. If S is not adjacent to P, who is sitting to the immediate right of U?
- (a) P
- (b) Q
- (c) R
- (d) S
Answer: (a) P
Six people sit in a circle facing the centre. When you face the centre, your left hand points one way round the circle and your right hand points the other way. Put T in a seat. P is second to T's left, so P sits two seats away from T on T's left side. Q sits opposite R, and with T and P already placed, the only opposite pair of empty seats left for Q and R is the one across from each other, fixing Q and R. That leaves two seats for S and U. S must not sit next to P, and one of the two leftover seats is right beside P, so S takes the far seat and U takes the seat next to P. Now check U's immediate right: the person on U's right turns out to be P. So the answer is P. (Q and R sit opposite each other and are not on U's right. S is the one placed away from P, not beside U.)
12. Seven persons sit in a row. Clues: (i) P sits at the center. (ii) Q sits two places to the left of R. (iii) S sits to the immediate right of T. (iv) U sits somewhere to the left of V. (v) W sits at one of the ends. How many valid arrangements exist?
- (a) 1
- (b) 2
- (c) 4
- (d) More than 4
Answer: (d) More than 4
Let positions 1-7. P at 4 (center). W at an end: 1 or 7. Q two left of R: possible pairs (Q,R) = (1,3), (2,4), (3,5), (4,6), (5,7). But P at 4, so (2,4) invalid because R would be at 4 but P is there. Similarly (4,6) invalid. So possible: (1,3), (3,5), (5,7). S immediate right of T: possible pairs (T,S) = (1,2), (2,3), (3,4), (4,5), (5,6), (6,7). But P at 4, so (3,4) invalid, (4,5) invalid. So possible: (1,2), (2,3), (5,6), (6,7). U left of V: many possibilities. W at end: 1 or 7. Now combine these, there are many combinations. For example, if W=1, then Q,R cannot be (1,3) because 1 taken. So Q,R can be (3,5) or (5,7). For each, S,T can be placed in remaining positions. U left of V can be satisfied in multiple ways. So total arrangements >4. Hence answer (d).