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**Instructions**

Choose which of the conclusion(s) follow(s) based on the statements given:

**Question 1:Â **$ E \geq D$; $ A = C$; $ A > B$; $ F = E$; $ B < D$;

Conclusions:

I: $ E > A$

II: $ F \leq C$

a)Â Only conclusion I is true.

b)Â Only conclusion II is true.

c)Â Either conclusion I or conclusion II is true.

d)Â Both the conclusions are true.

e)Â Neither of the conclusions is true.

**Question 2:Â **$ R = S$; $Q \geq P$; $S < T$; $ T > Q$;

Conclusions:

I: $ T \geq P$

II: $ R \leq Q$

a)Â Only conclusion I is true.

b)Â Only conclusion II is true.

c)Â Either conclusion I or conclusion II is true.

d)Â Both the conclusions are true.

e)Â Neither of the conclusions is true.

**Question 3:Â **Statements:

$ M \geq O$; $ L \leq P$; $ N > P$; $ P \leq O$;

Conclusions:

I: $ N > L$;

II: $M \geq L$

a)Â Only conclusion I is true.

b)Â Only conclusion II is true.

c)Â Either conclusion I or conclusion II is true.

d)Â Both the conclusions are true.

e)Â Neither of the conclusions is true.

**Question 4:Â **Statements:

$A>C$; $B \leq D$; $E \geq A$; $D > E$;

Conclusions:

I:$ B \leq A$

II: $ D > C$

a)Â Only conclusion I is true.

b)Â Only conclusion II is true.

c)Â Either conclusion I or conclusion II is true.

d)Â Both the conclusions are true.

e)Â Neither of the conclusions is true.

**Question 5:Â **Statements:

$P \leq Q$; $S > R$; $ R<P$

Conclusions:

I: $R < Q$;

II: $P> S$;

a)Â Only conclusion I is true.

b)Â Only conclusion II is true.

c)Â Either conclusion I or conclusion II is true.

d)Â Both the conclusions are true.

e)Â Neither of the conclusions is true.

**Question 6:Â **For which of the following options, the conclusion $F \leq D$Â will be definitely false?

a)Â $ F= E \leq B \leq A \leq D$

b)Â $ F= E \leq B \geq A \leq D$

c)Â $ F= E \geq B > A \geq D$

d)Â $ F= E > B \leq A\leq D$

e)Â More than one of the above

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**Question 7:Â **For which of the following options, the conclusions $A\geq B$ and $C < D$ will definitely hold true?

a)Â $D > A \geq E = C \geq B$

b)Â $D > A \geq E < C < B$

c)Â $ D \geq A = E < C > B$

d)Â $ D \geq A > E > C = B$

e)Â $ D \leq A > E < C = B$

**Question 8:Â **Select the option that will fill the blanks such that the conclusion $P \leq Q$ is definitely false:

P_____A_____R_____F_____Q

a)Â $ =, \leq , \leq, = $

b)Â $ = , \geq, = , >$

c)Â $ \geq, \leq, = ,>$

d)Â $ >, > , < , =$

e)Â $ \geq, =, <, \geq$

**Question 9:Â **Select the option that will fill the blanks such that the conclusion $D<A$ is definitely true.

A ____ C _____ B____ E_____D

a)Â $= , \geq, = , \leq$

b)Â $=, \geq, >, =$

c)Â $\geq, \geq, \leq, >$

d)Â $\geq , >, \geq, =$

e)Â More than one of the above

**Question 10:Â **In which of the following expressions either F > Z or F = Z is definitely true?

a)Â D = Z < M = T $\geq$ D = F

b)Â Q > F < N = K $\geq$ Z < R

c)Â F $\geq$ H = U $\leq$ Z < M = P

d)Â E > N = E < F $ \leq$ R > Z

e)Â E < F $\geq$ J = Y = C $\geq$ Z

**Question 11:Â **Which of the following expressions will not be definitely true if the expression

â€˜$ E > F = Y \geq Q = J < N \leq D$â€™ is definitely true?

a)Â D > J

b)Â Q < N

c)Â F $\geq$ J

d)Â E < N

e)Â N = D

**Question 12:Â **Which of the following expressions will be definitely true if the expression â€˜$H \leq T = G > U < W$â€™ is definitely true?

a)Â H > W

b)Â U = H

c)Â U < T

d)Â G > W

e)Â H = G

**Instructions**

In the next five questions, the values some elements are compared in the statement. Two conclusions are given after every statement. Assuming the statement to be true, you have to determine which of the conclusions are implied by the statement.

**Question 13:Â **$ A \leq B > C > D; C \leq E = F < G$

Conclusion I: $E < D$

Conclusion II: $ F \geq D$

a)Â Only conclusion I is implied

b)Â Only conclusion II is implied

c)Â Both the conclusions are implied

d)Â Neither conclusion I nor II is implied

e)Â Either I or II is implied

**Question 14:Â **$ A > B < C = D > E; E > F \geq G < H$

Conclusion I: $A < G$

Conclusion II: $ B > F$

a)Â Only conclusion I is implied

b)Â Only conclusion II is implied

c)Â Both the conclusions are implied

d)Â Neither conclusion I nor II is implied

e)Â Either I or II is implied

**Question 15:Â **$G > H \geq I > J; I \leq K < L $

Conclusion I: $G > K $

Conclusion II: $J < L $

a)Â Only conclusion I is implied

b)Â Only conclusion II is implied

c)Â Both the conclusions are implied

d)Â Neither conclusion I nor II is implied

e)Â Either I or II is implied

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**Question 16:Â **$A < B \leq C > D; C \leq E < F $

Conclusion I: $B \leq E $

Conclusion II: $ D < E$

a)Â Only conclusion I is implied

b)Â Only conclusion II is implied

c)Â Both the conclusions are implied

d)Â Neither conclusion I nor II is implied

e)Â Either I or II is implied

**Question 17:Â **$ G > H \leq I = J; J > K < L$

Conclusion I: $G > L$

Conclusion II: $ H < K$

a)Â Only conclusion I is implied

b)Â Only conclusion II is implied

c)Â Both the conclusions are implied

d)Â Neither conclusion I nor II is implied

e)Â Either I or II is implied

**Instructions**

In the next 5 questions, the symbols @, (, !, ), * are used with the following meanings as illustrated below.

P @ Q – â€˜P is either greater than or equal to Qâ€™

P * Q – â€˜P is either less than or equal to Qâ€™

P $ Q – â€˜P is neither greater than nor less than Qâ€™

P ! Q – â€˜P is greater than Qâ€™

P ~ Q – â€˜P is less than Qâ€™

**Question 18:Â **If C ~ Y and K ! N, what should come in the place of ____?

K @ B ! C ___ N ~ Y

a)Â @

b)Â *

c)Â ~

d)Â $

e)Â More than one of the above

**Question 19:Â **If A ~ O is true, which of the following can come in place of ____?

A * B $ T ____ J ~ O

a)Â !

b)Â *

c)Â @

d)Â $

e)Â More than one of the above

**Question 20:Â **Statements:

1. F * G

2. O $ G

3. F @ O

4. F ~ Y

Conclusions:

1. Y ! G

2. F $ G

a)Â Only conclusion 1 is true

b)Â Only conclusion 2 is true

c)Â Either conclusion 1 or conclusion 2 is true

d)Â Neither conclusion 1 nor conclusion 2 is true

e)Â Both conclusion 1 and conclusion 2 are true

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**Answers & Solutions:**

**1)Â AnswerÂ (C)**

The given statements can be combined and expressed as $ C = A > B < D \leq E = F$.

$F \leq C$ can be written as $ A \geq E$.

The 2 conclusions, state that $ A < E $ or $ A \geq C$.

One among these 2 conclusions must definitely follow.

Therefore, either conclusion I or conclusion II follows. Therefore, option C is the right answer.

**2)Â AnswerÂ (E)**

The given statements can be combined as $R = S < T > Q \geq P$.

$T$ is greater than $Q$. Therefore, $T$ is definitely greater than $P$. Conclusion I is false since it states that $T \geq P$.

No relationship can be established between $R$ and $Q$ since there is a change in the sign of the inequality. Therefore, conclusion II does not follow.

Neither of the conclusions follow. Therefore, option E is the right answer.

**3)Â AnswerÂ (D)**

The given statements can be combined as $ M \geq O \geq P \geq L$ and $ N > P \geq L$.

As we can see, $N$ is definitely greater than $L$. Conclusion I definitely follows.

Also, we can infer than $M \geq L$ from the first relationship. Conclusion II definitely follows.

Both the conclusions follow. Therefore, option D is the right answer.

**4)Â AnswerÂ (B)**

The given statements can be combined as $ B \leq D > E \geq A > C$.

As we can see, $ D > C$ and no relationship can be established between $B$ and $A$ due to the change in the inequality. Therefore, only conclusion II follows and hence, option B is the right answer.

**5)Â AnswerÂ (A)**

The given statements can be combined as $S > R < P \leq Q$. As we can see, no relationship can be established between $P$ and $S$ as there is a change in the inequality. Also, we can see that $R$ is less than $Q$.

Only conclusion I follows and hence, option A is the right answer.

**6)Â AnswerÂ (C)**

First, let us eliminate the options in which there is a change in the direction of the inequality between F and D. We can eliminate optionÂ B and option D using this case.

Now, let us check options AÂ and C.

Option A:

$ F= E \leq B \leq A \leq D$

From this option, we can conclude that $ F\leq D$. Therefore, the statement $F \leq D$ is definitely true.Â We can eliminate optionÂ A.

OptionÂ C:

$ F= E \geq B > A \geq D$

We can conclude that $F> D$. Therefore, the conclusion $F \leq D$ is definitely false in this case. Therefore, optionÂ C is the right answer.

**7)Â AnswerÂ (A)**

Any option in which there is a change of sign between A and B or C and D will render the terms unrelatable. Therefore, let us first eliminate such choices.

Except options AÂ and D, all other options involve a change in the inequality. Therefore, we can eliminate options B,Â C and E.

OptionÂ A:

$D > A \geq E = C \geq B$

Both the conclusions $A \geq B$ and $C<D$ hold true in this case.

OptionÂ D:

$ D \geq A > E > C = B$

The conclusion $C<D$ is true but the conclusionÂ $A \geq B$ is false in this option. Therefore, we can eliminate option D.

Therefore, option AÂ is the right answer.

**8)Â AnswerÂ (B)**

P and Q are at the either ends of the inequality. If there is an inconsistency in the sign (like change from > to < in between), no relationship can be established between P and Q. Therefore, we can eliminate options C,Â D and E.

Let us check options AÂ and B.

Option A:

$P = A \leq R \leq F =Q$

We can say that $P \leq Q$ from this relation. Therefore, option A can be eliminated.

OptionÂ B:

$P = A \geq R = F > Q$

We can say with conviction that $ P > Q$.Therefore, the conclusion $P\leqQ$ will be definitely false in this case. Therefore, option B is the right answer.

**9)Â AnswerÂ (E)**

We can easily eliminate the options in which there is a change in the direction of the inequality between AÂ and D, since no relationship can be established in such cases. This eliminates options AÂ and C. Let us check options B and D.

OptionÂ B:

$A= C \geq B> E = D$

In this option, the conclusion $D<A$ is definitely true.

Option D:

$A\geq C > B\geq E = D$

In optionÂ D too, the conclusion $D<A$ is definitely true.

The conclusion $D<A$ is true for both options B and D. Therefore, optionÂ E is the right answer.

**10)Â AnswerÂ (E)**

In E < F $\geq$ J = Y = C $\geq$ Z we can say that,

F $\geq$ C and C $\geq$ Z

Thus, F $\geq$ Z

Thus, option E does not definitely follow.

**11)Â AnswerÂ (D)**

It is given that E > Q and Q < N. No definite relationship between E and N can be found out.

Thus, option D does not definitely follow.

**12)Â AnswerÂ (C)**

It is given that T = G and G > U so T > U.

Thus, option C definitely follow.

**13)Â AnswerÂ (E)**

Given statement is $ A \leq B > C > D; C \leq E = F < G$

E < D is false.

We know that E = F from the given question.

Substituting E = F in Conclusion 2

=>Â $ F \geq D$ –>Â $ E \geq D$. This can be divided and written as E > D or E = D

Since, we cannot find relationship between E and D, it can be written as E < D or E > D or E = D.

Hence, ‘Either I or II’ is implied is correct

**14)Â AnswerÂ (D)**

The given statement is

$ A > B < C = D > E; E > F \geq G < H$

B is less than C and E is also less than C. Hence we cannot compare B and E.

Since only â€˜Eâ€™ is common between the first and the second parts of the inequality, we cannot compare B with any of the terms from the second part. Hence conclusion II is not implied.

Similarly no relation can be established between A and G and thus we cannot comment about which one is greater among K and Q. Thus conclusion I is also not implied. Hence the correct answer is option D.

**15)Â AnswerÂ (B)**

The given statement is

$G > H \geq I > J; I \leq K < L $

G is greater than H and H is also greater than or equal to I . Hence G is definitely greater than I. K is also greater than or equal to I. Hence we cannot compare G and K. Thus, I is not implied.

J is less than I and I is less than or equal to K which in turn is less than L. Hence J is definitely less than L. Thus, conclusion II is implied. Hence, the correct answer is option B

**16)Â AnswerÂ (C)**

The given statement is

$A < B \leq C > D; C \leq E < F $

A is less than B which is less than or equal to C. Now C is less than or equal to E. Hence B is definitely less than E. Thus conclusion I is valid.

D is less than C and C is less than or equal to E. Hence D is definitely less than E. Thus, conclusion II is also valid. Hence option C is the correct answer.

**17)Â AnswerÂ (D)**

The given inequality is

$ G > H \leq I = J; J > K < L$

G is greater than H and I is greater than or equal to H. Hence we cannot compare G and I. Thus, we cannot compare G and H also. Since H is the only term which is common between the first and the second term, we cannot compare G and L. Thus, I is not implied.

H is less than or equal to I which is equal to J. So H is less than or equal to J. Now K is also less than J. Hence we cannot say anything about the relationship between H and K.

Thus, II is also not implied. Hence option D is the correct answer.

**18)Â AnswerÂ (D)**

Replacing the symbols we get:

$K \geq B > C$ __ $N < Y$

Also, $C < Y$ and $K > N$

This is only satisfied if C = N.

**19)Â AnswerÂ (E)**

Replacing the symbols we get:

$A \leq B = T$ __ $J < O$

given, $A < O$

This can be true if T = J or T < J or $T \leq J$.

**20)Â AnswerÂ (E)**

Given,

$F \leq G$, O = G, $F \geq O$ and $F < Y$

This possible only if F = O = G.

Thus, Y > G and F = O