**Q11:-Statements:- M ≥T< V; U<V< X; Z≥ Y ≥X**

**Conclusions:-**

**I. T<Z**

**II. U<Y**

**III.M>X**

**IV.M ≥ X**

**Q12:-Statements:-W≤A≤K=N;D≤A≤U=L;G>L**

**Conclusions:-**

**I. W<G**

**II. U≥W**

**III.D≥N**

**IV.A≤N**

**Q13:Statements:N<T≤O;F>N≥E=I>R;R≥Y**

**Conclusions:-**

**I. Y<O**

**II. T<F**

**III.T≥F**

**IV.R<O**

**Q14:Statements: I≥N≥O=X;D≥J≥I=E**

**Conclusions:-**

**I. O<D**

**II. X=D**

**III.D≥N**

**IV.E≥X**

**Q15:Statements: X≥T,Z<K,K<H,F=Q,T<Z,F>H**

**Conclusions:-**

**I. T<F**

**II. Q > K**

**III.Z < F**

Let's analyze each of these questions step by step.

### **Q11:**
**Statements:**
- \( M \geq T < V \)
- \( U < V < X \)
- \( Z \geq Y \geq X \)

**Conclusions:**
1. \( T < Z \)
2. \( U < Y \)
3. \( M > X \)
4. \( M \geq X \)

**Analysis:**
- From the statements \( M \geq T < V \) and \( Z \geq Y \geq X > V > T \), it follows that \( T < Z \), so conclusion I is true.
- \( U < V < X \) and \( Z \geq Y \geq X > U \) imply that \( U < Y \), so conclusion II is true.
- \( M \geq T \) and \( X > V > T \), but there is no direct relationship between \( M \) and \( X \) that allows us to conclude \( M > X \), so conclusion III does **not** follow.
- \( M \geq T \) and \( T < X \), \( M \geq X \) is possible but not certain, so conclusion IV does **not** follow.

**Answer:** 2) **Only conclusion I and II follow.**

### **Q12:**
**Statements:**
- \( W \leq A \leq K = N \)
- \( D \leq A \leq U = L \)
- \( G > L \)

**Conclusions:**
1. \( W < G \)
2. \( U \geq W \)
3. \( D \geq N \)
4. \( A \leq N \)

**Analysis:**
- \( G > L = U \geq A \geq W \), so \( W < G \), thus conclusion I follows.
- \( U = L \geq A \geq W \), so \( U \geq W \), thus conclusion II follows.
- \( N = K \geq A \geq D \), so \( D \geq N \) does **not** follow.
- \( A \leq K = N \), so \( A \leq N \) follows.

**Answer:** 5) **Only conclusions I, II, and IV follow.**

### **Q13:**
**Statements:**
- \( N < T \leq O \)
- \( F > N \geq E = I > R \)
- \( R \geq Y \)

**Conclusions:**
1. \( Y < O \)
2. \( T < F \)
3. \( T \geq F \)
4. \( R < O \)

**Analysis:**
- \( N < T \leq O \) and \( R \geq Y \), \( F > N \geq E = I > R \geq Y \), so \( Y < O \) follows.
- \( T \leq O > N \), and \( F > N \), but no direct comparison between \( T \) and \( F \), so \( T < F \) does **not** follow, and \( T \geq F \) does **not** follow either.
- \( R < O \) could be true since \( R \geq Y < O \), so conclusion IV follows.

**Answer:** 1) **Only conclusions I and IV follow.**

### **Q14:**
**Statements:**
- \( I \geq N \geq O = X \)
- \( D \geq J \geq I = E \)

**Conclusions:**
1. \( O < D \)
2. \( X = D \)
3. \( D \geq N \)
4. \( E \geq X \)

**Analysis:**
- \( D \geq J \geq I \geq N \geq O = X \), so \( O < D \) follows.
- \( X \) is equal to \( O \), but not directly equal to \( D \), so \( X = D \) does **not** follow.
- \( D \geq J \geq I \geq N \), so \( D \geq N \) follows.
- \( E = I \geq N \geq X \), so \( E \geq X \) follows.

**Answer:** 3) **Conclusions I, III, and IV follow.**

### **Q15:**
**Statements:**
- \( X \geq T \)
- \( Z < K < H \)
- \( F = Q \)
- \( T < Z \)
- \( F > H \)

**Conclusions:**
1. \( T < F \)
2. \( Q > K \)
3. \( Z < F \)

**Analysis:**
- \( F > H > K > Z > T \), so \( T < F \) follows.
- \( F = Q \) and \( Q > H > K \), so \( Q > K \) follows.
- \( F > H > K > Z \), so \( Z < F \) follows.

**Answer:** 5) **All conclusions I, II, and III follow.**

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Do you have any questions or would you like further clarification on any of the answers?

**Related Questions:**
1. How does combining inequalities from different statements help in drawing conclusions?
2. What is the significance of equality in comparison to inequalities in logical reasoning?
3. Can multiple conclusions be true simultaneously in such logical reasoning problems?
4. How to approach problems where multiple elements are compared indirectly?
5. Why is it important to check if one conclusion contradicts another?

**Tip:** In questions involving multiple comparisons, always ensure that all relationships are carefully analyzed and consider whether any conclusions might be indirectly related.

Explore logical reasoning problems involving inequalities. Learn how to analyze statements and draw conclusions based on given inequalities. This problem set covers advanced concepts in logical reasoning and is suitable for advanced students or enthusiasts of logical puzzles.

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