Inequality

Q1. Which of the following expression will be true if the expression Z < Y ≥ W = V is definitely true:
a) V > Y
b) Z < W
c) V ≥ Z
d) W ≤ Z
e) None of these

Q2. Which of the following expression will not be true if the expression A = C ≥ B > D is definitely true:
a) B > A
b) D < C
c) A ≥ B
d) D < A
e) All are True

Q3. In which of the following expression will the expression L > M be definitely true:
a) M > N ≥ P > L
b) L > N ≤ M > P
c) M ≤ N = P ≥ L
d) L > N ≥ M < P
e) None of these

Directions (1–4): In these questions, relationship between different elements is shown in the statements. The statements are followed by two conclusions. Study the conclusions based on the given statements and select the appropriate answer.

Give answer:
(a) If only conclusion I follows.
(b) If only conclusion II follows.
(c) If either conclusion I or II follows.
(d) If neither conclusion I nor II follows.
(e) If both conclusion I and II follow.

1. Statements: A > B = C ≥ D < E < F
   Conclusions: I. F > A II. A > D
2. Statements: P < Q < R > T = S > X
   Conclusions: I. Q > P II. X < P
3. Statements: B > C = D < E < P
   Conclusions: I. B > E II. C < P
4. Statements: W ≥ Y = X > Q > S
   Conclusions: I. W > X II. W = X
5. Statements: B < C ≤ D ≥ P ≥ Q > S
   Conclusions: I. B < D II. D > S
6. Statements: X > Y ≥ W = T ≤ R < P
   Conclusions: I. Y > P II. P < W
7. Statements: X > Y ≥ Z, X > P ≥ Q
   Conclusions: I. X > Z II. Q < Z
8. Statements: A < B < C, C < D > E > F
   Conclusions: I. D > A II. A > D
9. Statements: H ≤ I ≤ P ≤ J, J ≥ E = T
   Conclusions: I. H < J II. H = J
10. Statements: P > S ≥ R > T ≥ Q, R < A
    Conclusions: I. P > A II. T < A
11. Statements: T ≥ S ≥ R > V, V < P < X, V > B
    Conclusions: I. X > T II. X > B
12. Statements: P > Q = S > T, T > R = Y
    Conclusions: I. P ≥ T II. P < T
13. Statements: S > A < B > C, C > D > G
    Conclusions: I. B > G II. A < D
14. Statements: I > E > P ≥ H = G, A ≥ B = G
    Conclusions: I. A > E II. P = G
15. Statements: R = T ≥ X ≥ V < W > P
    Conclusions: I. R ≥ V II. V > P
16. Statements: T > S ≥ R > O
    Conclusions: I. S > Y II. S < Y
17. Statements: A > L = T, T < R ≤ H > K
    Conclusions: I. A > R II. T < H
18. Statements: O ≥ S ≥ P = T ≤ Y ≤ W
    Conclusions: I. O ≥ T II. P ≤ W
19. Statements: A ≥ B = C, B < D ≤ E
    Conclusions: I. D > A II. E > C
20. Statements: V ≥ K > M = N, M > S, T < K
    Conclusions: I. T < N II. V = S
21. Statements: O ≥ P = Q, R < P, S < Q
    Conclusions: I. R < S II. O > S
22. Statements: T < P ≤ U, L > U ≤ K, P ≥ R
    Conclusions: I. K ≥ R II. L > R
23. Statements: V ≥ K > M = N, M > S, T < K
    Conclusions: I. T < N II. V = S
24. Statements: Z ≤ Y, X < Z, Y > L, L ≥ W
    Conclusions: I. Y ≥ L II. Y > X
25. Statements: A ≥ P, A < U, U = T, R > U
    Conclusions: I. P = A II. A > P
26. Statements: U ≥ T, V ≤ T, X > V, V = Y > A
    Conclusions: I. U > A II. X > T
27. Statements: A ≥ B > C ≥ F, Z < C ≤ D < E
    Conclusions: I. A > Z II. F < E
28. Statements: D > B, K ≤ N, B < C, N < B
    Conclusions: I. D > N II. C ≥ K
29. Statements: B < C, D > B, C > K, K ≤ N, N > B
    Conclusions: I. C > N II. B > K
30. Statements: P ≥ Q ≥ R, T ≥ U ≤ R, U = A ≥ S ≥ Y
    Conclusions: I. T < P II. T ≥ Y
31. Statements: C ≥ V ≤ R = N ≥ T > Q; Y ≥ N < A
    Conclusions: I. Q > V II. Q < Y
32. Statements: C ≥ V ≤ R = N ≥ T > Q; Y ≥ N < A
    Conclusions: I. C ≥ Y II. A > Q
33. Statements: P ≥ R < U ≤ M < V; T ≤ U; L < M
    Conclusions: I. T < L II. L < V
34. Statements: P ≥ R < U ≤ M < V; T ≤ U; L < M
    Conclusions: I. V > T II. T ≤ P
35. Statements: L > K ≥ X < J = N ≤ Y < D
    Conclusions: I. K > J II. D > J
36. Statements: S ≥ T ≤ A; E ≤ T; R > D = T
    Conclusions: I. R > S II. E ≤ A
37. Statements: R ≤ A > B; C > A; F ≤ B
    Conclusions: I. C > R II. F < A
38. Statements: L > K ≥ X < J = N ≤ Y < D
    Conclusions: I. X ≥ Y II. L > N
39. Statements: A = B ≥ C ≥ D < E
    Conclusions: I. A > E II. D < E
40. Statements: M < O < U ≤ R ≤ T; P ≥ R ≤ T ≤ C < L
    Conclusions: I. L > M II. O ≤ C
41. Statements: C < L = I ≤ N > G; I < M ≥ O > R > T
    Conclusions: I. C < O II. G > T
42. Statements: M < O ≤ U ≤ R ≥ T; P ≥ R ≤ I ≤ C < L
    Conclusions: I. P > T II. P = T
43. Statements: A > X = Z ≥ T < C; T < V = M ≥ E = O
    Conclusions: I. C > E II. E > C
44. Statements: T < V = M ≥ E = O; A > X = Z ≥ T < C
    Conclusions: I. M > C II. V ≥ O
45. Statements: H = D; D < U; B ≥ U
    Conclusions: I. U ≥ H II. D > U
46. Statement: P > Q ≥ R = S < T = U
    Conclusions: I. Q > S II. S = Q
47. Statement: J < D = L ≤ K ≤ Q ≥ R
    Conclusions: I. J < Q II. L ≤ R
48. Statement: O > P = G > B ≥ X = M ≤ H
    Conclusions: I. G > X II. O > M
49. Expression: H ≥ I = J ? K ≤ L < M
    Correct option: d) either < or ≤
50. Which expression makes ‘P > S’ definitely false?
    Correct option: a) P > Q ≥ R = S

c) R = P > Q ≥ S
d) S > Q ≥ R < P
e) S < Q ≤ R < P

52. Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the given expression in such a manner that ‘N < K’ definitely holds true?
    K __ L __ M __ N
    a) ≥, =, >
    b) ≤, <, =
    c) ≥, =, <
    d) >, ≥, <
    e) None of these

Coded Inequality

Directions (1–5): In the following question, symbols @, #, %, $, and © are used with the following meanings:

‘A @ B’ means ‘A is not smaller than B’.
‘A # B’ means ‘A is neither smaller than nor equal to B’.
‘A % B’ means ‘A is not greater than B’.
‘A $ B’ means ‘A is neither greater than nor equal to B’.
‘A © B’ means ‘A is neither smaller than nor greater than B’.

Now in each of the following questions assuming the given statements to be true, find out which of the three conclusions I, II and III given below them is/are definitely true.

1. Statements: J # H, H © T, T $ R, R % F
   Conclusions:
   I. J # R
   II. R # F
   III. J # T
   (a) Only I is true
   (b) Only I & II are true
   (c) Only III is true
   (d) All are true
   (e) None of these
2. Statements: E $ P, P % H, H @ I, I # K
   Conclusions:
   I. P © I
   II. I % E
   III. H % K
   (a) Only I is true
   (b) Only II is true
   (c) Only III is true
   (d) Only I and II are true
   (e) None is true

c) R = P > Q ≥ S
d) S > Q ≥ R < P
e) S < Q ≤ R < P

51. Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the given expression in such a manner that ‘N < K’ definitely holds true?
    K __ L __ M __ N
    a) ≥, =, >
    b) ≤, <, =
    c) ≥, =, <
    d) >, ≥, <
    e) None of these

Directions (1–5): In the following question, symbols @, #, %, $, and © are used with the following meanings:

‘A @ B’ means A is not smaller than B (A ≥ B)
‘A # B’ means A is greater than B (A > B)
‘A % B’ means A is not greater than B (A ≤ B)
‘A $ B’ means A is smaller than B (A < B)
‘A © B’ means A is equal to B (A = B)

Now in each question, assuming the statements are true, find which conclusions I, II and III are definitely true.

1. Statements: J # H, H © T, T $ R, R % F
   Conclusions:
   I. J # R
   II. R # F
   III. J # T
   (a) Only I is true
   (b) Only I & II are true
   (c) Only III is true
   (d) All are true
   (e) None of these
2. Statements: E $ P, P % H, H @ I, I # K
   Conclusions:
   I. P © I
   II. I % E
   III. H % K
   (a) Only I is true
   (b) Only II is true
   (c) Only III is true
   (d) Only I and II are true
   (e) None is true
3. Statements: L @ K, K # R, R $ H, H % N
   Conclusions:
   I. L # R
   II. N # R
   III. L © N
   (a) Only I & II are true
   (b) Only either I or II is true
   (c) Only II & III are true
   (d) Only III is true
   (e) None of these
4. Statements: T $ V, V % K, K © L, L % H
   Conclusions:
   I. V % H
   II. T $ L
   III. T $ H
   (a) Only I & II are true
   (b) Only II & III are true
   (c) Only I & III are true
   (d) All are true
   (e) None of these
5. Statements: V @ W, W # D, D $ M, M % F
   Conclusions:
   I. V # D
   II. F # D
   III. V © F
   (a) Only I is true
   (b) Only II is true
   (c) Only either I or II is true
   (d) Both I and II are true
   (e) None of these

Directions (6–10): In the following question, symbols #, $, %, *, and @ are used with the following meanings:

‘A # B’ means A is not smaller than B (A ≥ B)
‘A $ B’ means A is neither greater than nor equal to B (A < B)
‘A % B’ means A is neither greater nor smaller than B (A = B)
‘A * B’ means A is neither smaller than nor equal to B (A > B)
‘A @ B’ means A is not greater than B (A ≤ B)

Now in each question, assuming the given statements to be true, find out which of the two conclusions I and II given below them is/are definitely true.
(a) Only conclusion I is true
(b) Only conclusion II is true
(c) Either conclusion I or II is true
(d) Neither conclusion I nor II is true
(e) Both conclusion I and II are true

6. Statements: A @ B, B % C, C $ D
7. Statements: A @ B, B % C, C $ D
   Conclusions:
   I. C # A
   II. A @ D
8. Statements: P * Q, Q # S, S % R
   Conclusions:
   I. P # R
   II. R % Q
9. Statements: W $ X, X @ Y, Y * Z
   Conclusions:
   I. W $ Z
   II. W * Z
10. Statements: G # H, H * J, J % K
    Conclusions:
    I. K $ G
    II. J $ G
11. Statements: N # M, M * P, P @ T
    Conclusions:
    I. T @ N
    II. P $ N

Directions (11–15): In the following question, symbols @, ©, $, %, and * are used with the following meanings:

‘P $ Q’ means P ≤ Q
‘P @ Q’ means P ≥ Q
‘P * Q’ means P > Q
‘P % Q’ means P < Q
‘P © Q’ means P = Q

Now in each question, assuming the given statements are true, find which of the two conclusions I and II are definitely true.
(a) Only I is true
(b) Only II is true
(c) Either I or II is true
(d) Neither I nor II is true
(e) Both I and II are true

11. Statements: K % V, V $ N, N @ F
    Conclusions:
    I. F % V
    II. K % N
12. Statements: H $ W, W © M, M % B
    Conclusions:
    I. B * H
    II. M @ H
13. Statements: D @ B, B * T, T © M
    Conclusions:
    I. T $ D
    II. M $ D
14. Statements: M * T, T % K, K $ N
    Conclusions:
    I. N * T
    II. N * M
15. Statements: R © J, J @ D, D * F
    Conclusions:
    I. D © R
    II. D % R

Directions (16–20): In the following question, symbols @, #, $, *, and % are used with the following meanings:

‘A # B’ means A ≥ B
‘A @ B’ means A > B
‘A $ B’ means A = B
‘A % B’ means A ≤ B
‘A * B’ means A < B

16. Statements: L % P, P * V, V @ D
Conclusions:
I. L % V
II. L $ D

17. Statements: V % W, W $ H, H # I
Conclusions:
I. V % I
II. I % W

18. Statements: N # W, W @ H, H * T
Conclusions:
I. H * N
II. T @ W

19. Statements: F @ R, H * R, L % H
Conclusions:
I. F @ L
II. R # L

20. Statements: J # K, K * M, M @ T
Conclusions:
I. K * T
II. K @ T

Directions (21–25): In the following question, symbols @, #, $, *, and © are used with the following meanings:

‘A $ B’ means A is smaller than B (A < B)
‘A # B’ means A is greater than or equal to B (A ≥ B)
‘A © B’ means A is smaller than or equal to B (A ≤ B)
‘A @ B’ means A is greater than B (A > B)
‘A * B’ means A is equal to B (A = B)

Now in each question, assuming the given statements are true, find out which of the two conclusions I and II given below them is/are definitely true.

(a) Only conclusion I is true
(b) Only conclusion II is true
(c) Either conclusion I or II is true
(d) Neither conclusion I nor II is true
(e) Both conclusion I and II are true

21. Statements: M $ L, L # K, K @ R
    Conclusions:
    I. M * R
    II. M $ R
22. Statements: J * N, N @ W, W $ V
    Conclusions:
    I. J # V
    II. J $ W
23. Statements: H © Q, Q $ F, L @ F
    Conclusions:
    I. L @ H
    II. H © F
24. Statements: U © D, D $ R, R # T
    Conclusions:
    I. U $ R
    II. T @ D
25. Statements: J @ T, T $ V, V © M
    Conclusions:
    I. T © M
    II. J # M

Directions (26–30): In the following question, symbols @, ©, $, %, and & are used with the following meanings:

‘M % N’ means M ≥ N
‘M © N’ means M < N
‘M $ N’ means M = N
‘M @ N’ means M ≤ N
‘M & N’ means M > N

Now in each question, assuming the given statements are true, find out which of the two conclusions I and II given below them is/are definitely true.

(a) Only conclusion I is true
(b) Only conclusion II is true
(c) Either conclusion I or II is true
(d) Neither I nor II is true
(e) Both I and II are true

26. Statements: D @ Z, Z % N, N $ F
    Conclusions:
    I. F © Z
    II. N @ D
    Answer: (d) Neither I nor II is true

27. Statements: K © M, M % A, A $ T
    Conclusions:
    I. K © A
    II. T © M
    Answer: (d) Neither I nor II is true

28. Statements: M & R, R @ T, T & N
    Conclusions:
    I. N $ M
    II. T $ M
    Answer: (d) Neither I nor II is true

29. Statements: R & A, A % M, M © N
    Conclusions:
    I. M © R
    II. M & R
    Answer: (d) Neither I nor II is true

30. Statements: B % H, H @ W, W $ M
    Conclusions:
    I. B $ W
    II. M © H
    Answer: (d) Neither I nor II is true
31. Which of the following does not make A # C and D * F definitely not true?
    (a) A % B # C @ D $ E $ F
    (b) A @ B % C @ D $ E $ F
    (c) A % B % C @ D $ E * F
    Answer: (c)

32. Which of the following makes C $ E or B % E definitely true?
    (a) A * B $ C @ D % E # F
    (b) A # B * C $ D @ E # F
    (c) A # B # C % D % E * F
    (d) A @ B % C @ D * E * F
    (e) None of these
    Answer: (c)

33. If "A % B % C * D @ E $ F" is true then which of the following is definitely not true?
    (i) A # D
    (ii) C * F
    (iii) B # D
    (iv) D # F
    (a) Only (i)
    (b) Only (i) and (iv)
    (c) Only (ii) and (iii)
    (d) Only (i), (iii) and (iv)
    (e) All are true
    Answer: (d)

34. Which of the following makes A * C and E # B definitely true?
    (a) A * B $ C @ D % E # F
    (b) A * D $ B * C @ E # F
    (c) A * B # C % D * E # F
    (d) A @ B * D @ C % E @ F
    Answer: (b)

35. What will come in place of blank in following expression below such that both A % D and C # F are definitely true?
    A % B % C _ D # E % F
    (i) @ (ii) * (iii) % (iv) $
    Answer: (a)

36. What will come in place of blank in following expression below such that both B * E and F # B are definitely true?
    A @ B $ C _ D @ E * F $ G
    Answer: (b)

37. Which of the following makes F # D $ B definitely true?
    (a) A @ B $ C * D # E * F
    (b) A * B @ C % D # E * F
    (c) A % B * C $ D @ E * F
    (d) A # B % C @ D * E * F
    (e) None of these
    Answer: (b)

38. What will come in place of blank in following expression below such that both B * E and F # B are definitely true?
    A # B @ C $ D _ E $ F @ G
    Answer: (b)

39. Which of the following is definitely true if A # B @ C # D @ E $ F is true?
    (i) C # F
    (ii) F % D
    (iii) B % E
    (iv) E # A
    Answer: (d)

40. Which of the following does not make A % B and D * F definitely not true?
    (a) A % B % C * D $ E * F
    (b) A % B % C @ D * E * F
    (c) A # B % C @ D $ E $ F
    (d) A % B @ C $ D @ E * F
    (e) None of these
    Answer: (b)