Mathematical Inequalities -Aptitude Questions with Answers – Free Practice!

Ans: D

Explanation: Let’s decode the symbols and rewrite the statements and conclusions:

* A # B: A > B
* A \$ B: A ≤ B
* A & B: A < B
* A ! B: A ≥ B
* A © B: A = B

Statements:
* F ! G: F ≥ G
* H & F: H < F
* I © J: I = J
* J # F: J > F

Conclusions:
* I. I # H: I > H
* II. G & J: G < J
* III. J # H: J > H

Let’s analyze the conclusions based on the statements:

1. I. I # H: I > H
* We know I = J and H < F and J > F. Therefore, I = J > F > H. Thus, I > H. Conclusion I is true.

2. II. G & J: G < J
* We know F ≥ G and J > F. Combining, we get J > F ≥ G. Thus, J > G, or G < J. Conclusion II is true.

3. III. J # H: J > H
* We know H < F and J > F. Therefore, J > F > H. Thus, J > H. Conclusion III is true.

Since all conclusions are true, the correct answer is D.

Ans: C

Explanation:
Let’s decode the symbols:
* A @ B: A = B
* A % B: A ≤ B
* A # B: A > B
* A & B: A ≥ B
* A \$ B: A < B

The statements are: P # R, R @ L, L & T
This translates to: P > R, R = L, L ≥ T
Combining these, we get: P > R = L ≥ T

Now let’s check the conclusions:
I. L \$ P: L < P. Since we know P > L, this conclusion is true.
II. P # T: P > T. Since P > L and L ≥ T, it follows that P > T. This conclusion is true.
Since both conclusions are true, the correct answer is C.

Ans: E

Explanation: Let’s translate the symbols:
* ‘@’ means ‘=’
* ‘%’ means ‘<='
* ‘#’ means ‘>’
* ‘&’ means ‘>=’
* ‘\$’ means ‘<'

The statements are: J > K, K = P, P < R
Combining these, we get: J > K = P < R

Now let’s evaluate the conclusions:
* I. J # R => J > R. From the combined statement, we know J > K and P < R. But we don't have a direct relationship between J and R. So, we can't definitively say J > R.
* II. R \$ J => R < J. Again, we know J > K = P < R, which means J could be greater than R, equal to R or less than R. We can't definitively say R < J.

Since neither conclusion is definitely true, the answer is E.

Ans: E

Explanation:
Let’s analyze the statements and conclusions.
Statements: A > B; E ≥ F < G ≥ B
* We can rewrite the statements involving B as: A > B and G ≥ B
* Conclusion I: G > E. We know E ≥ F and G ≥ B and A > B, but there’s no direct link between G and E that guarantees G > E or any relationship other than the fact that they are connected to B. This conclusion might be true, but not necessarily.
* Conclusion II: E ≥ B. We have E ≥ F and G ≥ B and A > B, but this does not tell us anything about the relationship between E and B. We can deduce, based on the statement G ≥ B, that if G = B, then F ≤ E, but it’s not certain that E ≥ B or other relationships can be deduced from the provided statements.

Therefore, we cannot determine whether G > E or E ≥ B is definitely true based on the given statements.

Ans: C

Explanation:
Let’s analyze the statements and conclusions:
Statements: D > S > L = P ≤ K ≤ N < O

Conclusion I: P = L
From the statement, we have L = P. Therefore, conclusion I is true.

Conclusion II: O > P
From the statement, we have L = P and P ≤ K ≤ N < O. Combining these, it means P < O. Therefore, conclusion II is true.

Since both conclusions I and II are true, the correct option is C.

Ans: B

Explanation:
Let’s analyze the statements and conclusions.
Statements:
1. P > Q ≤ S ≥ T (This implies P > Q and S ≥ T)
2. V = W ≤ T (This implies V ≤ T)

Conclusion I: P > T
From statement 1, we know P > Q and S ≥ T. We can’t directly relate P and T. However, from statement 1 and 2, we know that T ≤ S and V ≤ T. We don’t have enough information to say for certain if P > T.

Conclusion II: S ≥ V
From statement 1, we have S ≥ T. From statement 2, we have V ≤ T. Combining these, S ≥ T and T ≥ V which means S ≥ V.

Therefore, only conclusion II is definitely true.

Ans: A

Explanation:
From the statements, we have the following relationships:
* P < R < D < A < N
* P > F = S

Combining the relationships, we know that A > D > R > P and also P > F. This implies that A is greater than P, and P is greater than or equal to F. Thus, A > P >= F. This means A > F. Therefore, conclusion II is true.

Conclusion I states that A = F, which is definitely not true as we established A > F.

Ans: E

Explanation: Let’s analyze the given statement: E < F ≤ G > H ≥ K.

* Conclusion I: F > H
From the statement, we know F ≤ G and G > H. This means F can be greater than H, or equal to H, but not less than H (because G can be > H and F is less than or equal to G). Therefore, we cannot definitively say F > H.

* Conclusion II: G < E
From the statement, we know E < F ≤ G. This means E is definitely less than or equal to G. Therefore, it's impossible for G to be less than E.

Therefore, neither conclusion is definitely true.

Ans: E

Explanation:
Let’s analyze the conclusions based on the given statement:

Conclusion I: U > I
From the statement, we have: Q > I < N = K ≤ T ≤ E ≤ U. Since U is greater than or equal to E, and E is greater than or equal to T, and so on till I, we can conclude that U > I.

Conclusion II: Y > J
From the statement, we have: Q > I < N = K ≤ T ≤ E ≤ U ≥ Y > R = P > O = D > J. Since Y > R and R > P and P > O and O = D and D > J, we can conclude that Y > J.

Therefore, both conclusions I and II are true.

Ans: D

Explanation:
The given statements are: M ≥ Q ≥ G = R ≥ N = B.
Let’s analyze the conclusions:

I. B > M: From the statements, we have B ≤ N ≤ R ≤ G ≤ Q ≤ M. This means B is less than or equal to M. Therefore, B > M is definitely false.

II. M = B: From the statements, we have B ≤ N ≤ R ≤ G ≤ Q ≤ M. This means B is less than or equal to M. It does not necessarily mean that M = B. For example, if M = 5, Q = 4, G = 3, R = 3, N = 2, B = 2, then B ≠ M. Therefore, M = B is definitely false.

Since both conclusions are false individually, and we don’t have the conditions required for either-or cases (like B < M or B = M, and B ≤ M), neither of them follows.

Ans: B

Explanation:
Let’s analyze the statement and the conclusions.
Statement: Z < X < C ≤ V = B > O ≥ K

Conclusion I: Z > O
From the statement, we have Z < X < C ≤ V = B > O. This means Z is less than X, X is less than C, C is less than or equal to V, V is equal to B, and B is greater than O. Combining this information, we can see that Z is less than O. So, Z > O is false.

Conclusion II: V > X
From the statement, we have X < C ≤ V. This implies V > X, as V is greater than or equal to C, and C is greater than X. So, V > X is true.

Therefore, only conclusion II follows.

Ans: A

Explanation: Let’s analyze the given statement and the conclusions.
Statement: A > L = T < R ≤ H > K

Conclusion I: H > L
From the statement, L = T < R ≤ H. This implies L < H, which means H > L. So, conclusion I is true.

Conclusion II: K > T
From the statement, T < R ≤ H > K. There is no direct relationship between T and K because the inequality signs change direction. Therefore, we cannot definitively say if K > T or not. So, conclusion II is false.

Since only conclusion I is true, the correct answer is A.

Ans: C

Explanation: To determine if H < J is definitely true, we need to trace the relationship between H and J in each option.

* A. G < H = I = J: This implies G < H and H = I = J. Thus, H = J, and since G < H, we cannot definitively say H < J.

* B. H > G > I = J: This implies H > G, G > I, and I = J. Therefore, H > G > I = J, which means H > J. This does not make H < J definitely true.

* C. J = I ≥ G > H: This implies J = I, I ≥ G, and G > H. Therefore, J = I ≥ G > H, so J > H or J = H. This implies J > H, which is equivalent to H < J, making H < J definitely true.

* D. H ≥ G > I < J: This implies H ≥ G, G > I, and I < J. We cannot establish a direct relationship between H and J. We could have H > J, H = J, or H < J.

Ans: A

Explanation:
We need to analyze the given statement and conclusions based on the rules of inequalities.

Statement: C < R < I ≤ T ≤ S = Y > P ≥ A < L < D < F ≥ G ≥ H

Conclusion I: Y > C
From the statement, C < R < I ≤ T ≤ S = Y. This means C < Y. Hence, Y > C is true.

Conclusion II: Y > D
From the statement, S = Y > P ≥ A < L < D. Since Y > S, and D is greater than L, and L is greater than A, the relationship between Y and D is not straightforward, also Y is to the left of D, so Y can never be greater than D. Thus, Y > D is false.

Conclusion III: I < A
From the statement, I ≤ T ≤ S = Y > P ≥ A. This means I ≤ T ≤ S = Y > P ≥ A. Therefore, I cannot be directly compared with A, as the inequality signs change. I is possibly less than or equal to A, not definitely less than A. Thus, I < A is false.

Therefore, only Conclusion I is true.

Ans: A

Explanation:
From the statements A > B > C and C ≥ D, we can deduce the following:
* Since A > B and B > C, then A > C.
* Since C ≥ D, it means C can be greater than or equal to D.
* If C > D, then A > C > D, which means A > D.
* If C = D, then A > C = D, which means A > D.

Therefore, A will always be greater than D. This means Conclusion I (A > D) is definitely true, but Conclusion II (A = D) is definitely false. So only I is true.

Ans: B

Explanation: Let’s analyze the conclusions based on the given statements.
* I) K > X: We have X > Z and Q ≥ V. Also, P > K > S = Q. Combining these, we cannot directly compare X and K because there’s no direct relationship.
* II) Y < P: We have Y ≤ V and Q ≥ V and P > K. Since Q ≥ V, then P > V (as P > K and K = Q). We also have Y ≤ V, so, Y < V or Y = V. As P > V and Y ≤ V. So, we can conclude Y < P is true.

Ans: A

Explanation:
Let’s analyze the statements and conclusions:
Statements:
1. G < M ≥ N (This implies M ≥ G and M ≥ N)
2. B ≤ U ≤ S < G (This implies B ≤ U, U ≤ S, S < G)

Conclusions:
I. B > N
We need to check if B > N is always true.
From the statements, there’s no direct comparison between B and N. B is related to G through U and S, and N is related to G through M. We know G < M, but we don't know if G or M is greater than/equal to B. Thus, we cannot definitively say B > N.

II. U < M
We need to check if U < M is always true.
From statement 1, we have G < M.
From statement 2, we have U ≤ S < G. Combining these, we have U ≤ S < G < M. This means U < M.

Therefore, only conclusion II is definitely true.

Ans: B

Explanation: Let’s analyze the given statements and conclusions.

Statements:
1. Y > Z
2. K = D
3. A > J ≥ Y
4. D > Y

Conclusions:
I. K ≥ J
We know K = D (from statement 2) and D > Y (from statement 4). Also, A > J ≥ Y (from statement 3). Since D > Y and J ≤ Y, we *cannot* definitively say K ≥ J. It’s possible J < Y < D = K, but it's also possible that J is close enough to Y that J is close to or equal to K. Since the relationship between J and K isn't precisely defined, conclusion I is uncertain.
II. K > Z
We know K = D (from statement 2), D > Y (from statement 4), and Y > Z (from statement 1). Therefore, K = D > Y > Z, and thus, K > Z.
III. A > D
We have A > J ≥ Y (from statement 3) and D > Y (from statement 4). Since we only have the relationship between Y, J, D and A, it is not possible to derive a relationship of A > D.

Thus only conclusion II is definitely true.

Ans: C

Explanation: We need to find the symbols to fill the blanks in the expression X _ N _ O ≥ S _ P such that S < X and N ≥ P are definitely true. Let's analyze the given options:

A. X > N = O ≥ S > P
* S < O (from O ≥ S) and O = N (from N = O) implies S < N.
* Also, S < X (from X > N and N = O, and O ≥ S).
* And N > P (from X > N, and O ≥ S, and S > P) But we need N >= P
* This option doesn’t ensure N >= P.

B. (Not a valid option, it contains “nan”, we will not consider this)

C. X > N ≥ O ≥ S = P
* S < X (from X > N, N ≥ O, and O ≥ S).
* Also, N ≥ P (from N ≥ O ≥ S = P).
* This option satisfies both conditions.

D. X > N > O ≥ S ≥ P
* S < X (from X > N > O, and O ≥ S).
* N > P, we require N ≥ P, thus this option is not valid.

Therefore, the correct option is C.

Ans: E

Explanation:
Let’s analyze the statements and conclusions.

Statements:
1. A < B ≤ C > D
2. C > E ≥ F
3. E > B

Combining the statements, we get:
A < B < E < C > D and E ≥ F

Now let’s analyze the conclusions:

i) A < F
From the combined statements, we have A < B and B < E and E ≥ F. Combining these, we get A < B < E ≥ F. We cannot definitively say if A < F, since F could be equal to E, or smaller than E but still greater than A. This conclusion is *not* definitively true.

ii) D < B
From the combined statements, we have B < E < C and also C > D. However, the relationship between B and D cannot be definitively determined as C is greater than both B and D. But in this combination, the information we have only is that C > D and B ≤ C. This conclusion is *not* definitively true.

Since there is no direct relation to conclude conclusion i) and ii), Both conclusions does not follow.