Question 1

Given: X ≥ Y, Y > Z, Z ≤ W, W < V, V = X. Which of the following is INDETERMINATE (cannot be determined)?

Accepted Answer

From the given: X ≥ Y > Z ≤ W < V and V = X. Since V = X, we have X = V. From V > W, we get X > W. From W ≥ Z, we get W ≥ Z. From Y > Z and Z ≤ W, we cannot determine if Y > W or Y < W or Y = W because Z is a lower bound for both but their relative positions are not fixed.

Question 2

Given: P ≤ Q, Q < R, R ≥ S, S > T, T ≤ U. Which conclusion is DEFINITELY FALSE?

Accepted Answer

From the given: P ≤ Q < R ≥ S > T ≤ U. We can establish: P < R (P ≤ Q < R), S > T (given), R ≥ S (given). However, we cannot determine if P > T because P and T are in separate chains with no direct connection. The relationship between U and other variables is also indeterminate except that U ≥ T.

Question 3

Given: M > N > O, O ≥ P, P > Q, Q ≤ R. Which of the following is DEFINITELY FALSE?

Accepted Answer

Chain: M > N > O ≥ P > Q ≤ R. From this: M > O (definite), O > Q (definite), M > Q (definite). But R vs Q: Q ≤ R means R ≥ Q, so R could equal Q or be greater. R vs M: no direct path. R vs O: no direct path. Let's check each option: (a) M > Q is TRUE. (b) R ≥ Q is TRUE (given Q ≤ R). (c) O > Q is TRUE (O ≥ P > Q). (d) R > M cannot be determined, but let's check if it's definitely false. R ≤ Q is definitely false because Q ≤ R means R ≥ Q, so R ≤ Q is impossible.

Question 4

Given: P ≤ Q, Q < R, R ≥ S, S > T, T ≤ U. Which conclusion is INDETERMINATE (cannot be proven true or false)?

Accepted Answer

Chain: P ≤ Q < R ≥ S > T ≤ U. We can determine: P < R (definite), S > T (given), R ≥ S (given). But P vs U: P ≤ Q < R ≥ S > T ≤ U. We know P ≤ Q < R and R ≥ S > T ≤ U. There is no direct path from P to U that guarantees a definite relationship. P could be less than, equal to, or greater than U.

Question 5

Study the following statements and answer the question: A > B ≥ C D < C E ≤ B F > A Which of the following conclusions is definitely TRUE? (I) F > C (II) D < A (III) E < F

Accepted Answer

Let us analyze each conclusion systematically: Given statements: - A > B ≥ C (A is greater than B, and B is greater than or equal to C) - D < C (D is less than C) - E ≤ B (E is less than or equal to B) - F > A (F is greater than A) Conclusion (I): F > C From F > A and A > B ≥ C, we can chain: F > A > B ≥ C Therefore, F > C is DEFINITELY TRUE. ✓ Conclusion (II): D < A We know D < C and A > B ≥ C. From D < C and C ≤ B < A, we get D < A is DEFINITELY TRUE. ✓ Conclusion (III): E < F We know E ≤ B and F > A > B. Since E ≤ B and F > A > B, we have E ≤ B < A < F. Therefore, E < F is DEFINITELY TRUE. ✓ Since all three conclusions (I), (II), and (III) are definitely true, the correct answer is the option stating that all conclusions are true.

Question 6

Given: M < N, N ≤ O, O > P, P ≥ Q, Q = R. Which conclusion is definitely TRUE?

Accepted Answer

From the chain: M < N ≤ O > P ≥ Q = R. We can deduce M < N ≤ O (so M < O), O > P ≥ Q (so O > Q), and Q = R (so any property of Q applies to R). However, the relationship between N and P is indeterminate because N ≤ O > P does not establish N > P or N ≤ P. Similarly, M and P have no direct relationship. But M < O is definite, and O > Q is definite.

Question 7

Given: X < Y, Y ≤ Z, Z > W, W ≥ V, V = T. Which conclusion is definitely TRUE?

Accepted Answer

From the given: X < Y ≤ Z > W ≥ V = T. Let's check: (a) X < Z: X < Y ≤ Z gives X < Z (TRUE by transitivity). (b) Y > W: Y ≤ Z and Z > W doesn't directly connect Y to W. Y could be 5, Z = 10, W = 8, giving Y < W. Or Y = 9, Z = 10, W = 8, giving Y > W. INDETERMINATE. (c) W ≥ T: W ≥ V = T gives W ≥ T (TRUE). (d) X > V: No chain connects X to V directly.

Question 8

Given: M ≥ N, N > O, O ≤ P, P < Q, Q = R. Which of the following is INDETERMINATE (cannot be determined)?

Accepted Answer

From the given: M ≥ N > O ≤ P < Q = R. Let's evaluate: (a) M > O: M ≥ N > O gives M > O (TRUE). (b) N > P: N > O and O ≤ P doesn't establish N vs P. If O = 5, P = 5, N = 6, then N > P. But if O = 5, P = 10, N = 6, then N < P. INDETERMINATE. (c) P < Q: Given directly (TRUE). (d) Q = R: Given directly (TRUE).

Question 9

Given: A ≥ B, B > C, C ≥ D, D < E, E ≤ F. Which of the following conclusions is DEFINITELY FALSE?

Accepted Answer

From the chain: A ≥ B > C ≥ D < E ≤ F. We can deduce A > C (transitive through B), A ≥ D (transitive through B and C), B > D (transitive through C), and E ≤ F. However, the relationship between C and E is indeterminate because C ≥ D < E does not establish C > E or C ≤ E. The relationship between A and E is also indeterminate. But we can definitively say A > C, so 'A ≤ C' is false.

Question 10

Given: A > B, B ≥ C, C > D, D = E, E < F. Which of the following conclusions is definitely TRUE?

Accepted Answer

From the given inequalities: A > B ≥ C > D = E < F. We need to find what must be true. A > B is given. B ≥ C is given. C > D is given. D = E is given. Therefore A > B ≥ C > D = E. This means A > C is definitely true (transitive property: A > B and B ≥ C implies A > C).

Question 11

Given: A > B, B ≥ C, C > D, D = E, E < F. Which of the following conclusions is definitely TRUE?

Accepted Answer

From the chain: A > B ≥ C > D = E < F. We can definitively conclude A > D (since A > B > C > D). However, we cannot determine the relationship between A and F (F could be less than, equal to, or greater than A). We also cannot conclude B > E because B ≥ C > D = E, so B > E is true. Let me verify: A > B ≥ C > D = E, so A > E is definite. Also B ≥ C > D = E means B > E is definite.

Question 12

Given: P ≤ Q, Q < R, R ≥ S, S > T, T ≥ U. Which conclusion is DEFINITELY FALSE?

Accepted Answer

From the chain: P ≤ Q < R ≥ S > T ≥ U. We know P ≤ Q < R (so P < R), R ≥ S > T (so R > T), and S > T ≥ U (so S > U). However, the relationship between P and T is indeterminate because P ≤ Q < R ≥ S > T does not establish a definite order—R ≥ S creates a branch where S could be much smaller than R. Similarly, Q and S have no direct relationship. But U ≤ T < S ≤ R, and P ≤ Q < R, so we cannot say P > R (since P ≤ Q < R means P < R).

Question 13

Given: X ≥ Y, Y > Z, Z ≤ W, W < V, V ≥ X. Which of the following is INDETERMINATE?

Accepted Answer

From the chain: X ≥ Y > Z ≤ W < V ≥ X. We can deduce X ≥ Y > Z (so X > Z), and W < V ≥ X (so W < V and V ≥ X, but no direct link between W and X). The relationship between Y and W is indeterminate because Y > Z ≤ W does not establish Y > W or Y ≤ W—Z could be equal to W, or W could be much larger than Y. Similarly, Z and V have no direct relationship. However, X > Z is definite, Y > Z is definite, and V ≥ X is definite.

Question 14

Given: X ≥ Y, Y > Z, Z ≤ W, W < V, V ≥ X. Which conclusion is INDETERMINATE?

Accepted Answer

Chain: X ≥ Y > Z ≤ W < V ≥ X. This creates a cycle: X ≥ Y > Z ≤ W < V ≥ X. Let's trace: X ≥ Y > Z and Z ≤ W < V ≥ X. So X ≥ Y > Z and V ≥ X. We have V ≥ X ≥ Y > Z. Also W < V and Z ≤ W. So Z ≤ W < V. Can we determine Z vs X? X ≥ Y > Z, so X > Z (definite). Can we determine W vs Y? W < V ≥ X ≥ Y, so W could be less than, equal to, or greater than Y. This is indeterminate.

Question 15

Study the following statements and determine which conclusion logically follows:

Statements:
1. A > B
2. B ≥ C
3. C > D
4. D = E
5. E < F

Which of the following conclusions is definitely TRUE?

I. A > D
II. F > B
III. C ≥ E

Accepted Answer

Let us trace through the relationships step by step: From Statement 1: A > B From Statement 2: B ≥ C From Statement 3: C > D From Statement 4: D = E From Statement 5: E < F Now checking each conclusion: **Conclusion I: A > D** - A > B (given) - B ≥ C (given) - C > D (given) - Combining: A > B ≥ C > D - Therefore: A > D ✓ DEFINITELY TRUE **Conclusion II: F > B** - We know E < F, so F > E - We know D = E, so F > D - We know C > D, but we don't know the relationship between C and E beyond C > D = E, meaning C > E - However, we don't know if F > C or F < C. For example: if B = 50, C = 50, D = 40, E = 40, F = 45, then F < B - Therefore: F > B is NOT necessarily true ✗ **Conclusion III: C ≥ E** - From C > D and D = E, we get C > E - C > E means C ≥ E is true ✓ DEFINITELY TRUE Both Conclusions I and III are definitely true. However, examining the logical chain: Since A > B ≥ C > D = E < F, both I and III are valid. The question asks which conclusion is definitely TRUE (singular). Both I and III independently are true, making this question require selecting the most direct conclusion. Conclusion I (A > D) follows from the direct transitive chain and is the most straightforward deduction. Conclusion III is also valid but requires the intermediate step of recognizing D = E.

Question 16

Given: A ≤ B, B < C, C = D, D > E, E ≥ F. Which of the following is DEFINITELY FALSE?

Accepted Answer

From the given: A ≤ B < C = D > E ≥ F. We can establish: A < C (A ≤ B < C), A < D (A ≤ B < C = D), D > E (given), C > E (C = D and D > E), E ≥ F (given). The statement B > D is false because B < C and C = D, which means B < D, not B > D.

Question 17

Given: M < N, N ≤ O, O > P, P ≥ Q, Q = R. Which conclusion is DEFINITELY TRUE?

Accepted Answer

From the given: M < N ≤ O > P ≥ Q = R. We can establish: M < O (M < N ≤ O), O > Q (O > P ≥ Q), O > R (O > P ≥ Q = R), P ≥ R (P ≥ Q = R). However, we cannot determine if M > P or M < P because M and P are in separate chains.

Question 18

Consider the following:
P ≥ Q > R
S < R
T ≤ Q
U > P

Which conclusion is DEFINITELY FALSE?

Accepted Answer

This question requires identifying which conclusion cannot possibly be true given the constraints, testing negation logic and inequality reversal.

Question 19

Given:
J < K ≤ L
M ≥ L > N
O ≤ N < P
K = O

Which of the following conclusions is DEFINITELY FALSE?

Accepted Answer

This question tests the ability to use equality constraints to collapse chains and identify contradictions.

Question 20

Consider the statements:
P ≤ Q < R
S > R ≥ T
U = T < V

Which conclusion is DEFINITELY FALSE?

Accepted Answer

This question requires identifying which relationship cannot possibly be true given the constraints.

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