Puzzle Test Questions in English

(B) From the given information,we can organize the data:
$D$ has a yellow car and plays Polo.
$E$ has a white car and plays Badminton.
$A$ has a black car and plays Cricket.
$B$ has a red car and does not play Tennis.
$C$ does not own a blue or yellow car. Since black,white,red,and yellow are taken,$C$ must own the green car.
$F$ must own the remaining blue car.
Since $C$ does not play Tennis or Hockey,and Cricket,Polo,and Badminton are taken,$C$ must play Volleyball.

Friend	Game and Car Colour
$A$	Cricket,Black
$B$	Hockey,Red
$C$	Volleyball,Green
$D$	Polo,Yellow
$E$	Badminton,White
$F$	Tennis,Blue

Thus,$C$ plays Volleyball.

(B) Let's organize the information:
$D$ has a yellow car and plays Polo.
$E$ has a white car and plays Badminton.
$B$ has a red car.
$A$ has a black car and plays Cricket.
Remaining car colours: Green,Blue.
Remaining friends: $C, F$.
$C$ does not own a blue car (given),so $C$ must own the green car.
Therefore,$F$ must own the blue car.

Friend	Game	Car Colour
$A$	Cricket	Black
$B$	Hockey	Red
$C$	Volleyball	Green
$D$	Polo	Yellow
$E$	Badminton	White
$F$	Tennis	Blue

Thus,$F$ owns a blue car.

(B) has a yellow car; $E$ has a white car; $B$ has a red car; $A$ has a black car. Now,$C$ does not have a blue car. So,the colour of $C$'s car is green. Clearly,$F$ owns a blue car.

Friend	Game and Colour
$A$	Cricket $-$ Black
$B$	Hockey $-$ Red
$C$	Volleyball $-$ Green
$D$	Polo $-$ Yellow
$E$	Badminton $-$ White
$F$	Tennis $-$ Blue

Based on the table,the combination $Green - Tennis$ is incorrect because $Green$ is associated with $Volleyball$ and $Tennis$ is associated with $Blue$.

(C) $1$. Total women = $6$. Roles: $4$ Dancers,$4$ Vocal Musicians,$1$ Actress,$3$ Violinists.
$2$. Violinists: Girija,Vanaja,and one more. Jalaja and Shailja are not violinists. Since Jalaja,Vanaja,Shailja,and Tanuja are vocal musicians,and two of them are violinists,and we know Vanaja is a violinist,the other violinist must be Tanuja.
$3$. Dancers: Shailja and Tanuja are among the dancers. Since Tanuja is a violinist and a dancer,she fits the criteria.
$4$. Therefore,Tanuja is both a dancer and a violinist.

(D) Based on the given information,we arrange the six persons in two rows of three:
$1$. $E$ is not at the end of any row,so $E$ must be in the middle.
$2$. $D$ is second to the left of $F$. If $F$ is at the end of a row,$D$ must be in the same row.
$3$. $C$ is a neighbor of $E$ and sits diagonally opposite to $D$.
$4$. $B$ is a neighbor of $F$.
Arrangement:
Row $1$: $A$ $E$ $C$
Row $2$: $D$ $B$ $F$
In this arrangement:
- $D$ and $C$ are diagonally opposite.
- $A$ and $F$ are diagonally opposite.
- $E$ and $B$ are vertically opposite.
Therefore,$A$ and $F$ are sitting diagonally opposite to each other.

(D) $1$. Let the two rows be Row $1$ and Row $2$. Since $E$ is not at the end of any row,$E$ must be in the middle position of a row.
$2$. $D$ is second to the left of $F$. If $F$ is at the right end of a row,$D$ is at the left end of the same row.
$3$. $B$ is the neighbour of $F$. Since $D$ is at the left end and $F$ is at the right end,$B$ must be in the middle of that row.
$4$. $C$ is the neighbour of $E$ and sits diagonally opposite to $D$. If $D$ is in Row $2$ (left end),$C$ must be in Row $1$ (right end).
$5$. Arranging the rows:
Row $1$: $A, E, C$
Row $2$: $D, B, F$
$6$. In this arrangement,$E$ is in the middle of Row $1$ and $B$ is in the middle of Row $2$. Thus,$E$ is facing $B$.

(A) Based on the given conditions:
$1$. $E$ is not at the end,so $E$ must be in the middle of a row.
$2$. $D$ is second to the left of $F$. This implies $F$ and $D$ are in the same row,and since $D$ is second to the left of $F$,the arrangement is $D, _, F$.
$3$. $B$ is the neighbor of $F$,so $B$ must be in the middle of the row containing $F$. Thus,the row is $D, B, F$.
$4$. $C$ is the neighbor of $E$ and sits diagonally opposite to $D$. If $D$ is at one end of the second row,$C$ must be at the opposite end of the first row.
$5$. The arrangement is:
Row $1$: $A, E, C$
Row $2$: $D, B, F$
Comparing this with the options,$A$ and $E$ are in the same row. Therefore,the correct option is $(A)$.

(C) Let the two rows be Row-$1$ and Row-$2$.
Given that $E$ is not at the end of any row,$E$ must be in the middle position.
$D$ is second to the left of $F$. This implies $F$ and $D$ are in the same row,and since there are only three seats,$F$ must be at one end and $D$ at the other,with $B$ in between ($F-B-D$ or $D-B-F$).
$C$ is the neighbour of $E$ and sits diagonally opposite to $D$. If $D$ is in Row-$2$,$C$ must be in Row-$1$.
By arranging the positions:
Row-$1$: $A, E, C$
Row-$2$: $D, B, F$
Checking the conditions: $E$ is in the middle. $D$ is second to the left of $F$ (if $F$ is at the right end of Row-$2$). $C$ is a neighbour of $E$ and is diagonally opposite to $D$. $B$ is a neighbour of $F$.
Thus,the persons in one row are $D, B, F$ or $A, E, C$. Among the options,$DBF$ is a valid row.

(A) Based on the initial arrangement,the seating is organized as follows:
Row $1$: $A, E, C$
Row $2$: $D, B, F$
Column-wise,the arrangement is:
Column $1$: $A, D$
Column $2$: $E, B$
Column $3$: $C, F$
When $D$ interchanges seats with $E$,$D$ moves to the position previously occupied by $E$ (the middle seat of the first row).
In the new position,$D$ is now between $A$ and $C$.
Therefore,the neighbours of $D$ in the new position are $A$ and $C$.

(C) Let $C$ denote 'Composition',$P$ denote 'Plays',$Po$ denote 'Poetry',and $L$ denote 'Literature'.
$1$. From the problem,the $2^{nd}$,$5^{th}$,and $6^{th}$ books are 'Plays' $(P)$.
$2$. 'Two books on Plays are between two books on Composition': This implies the sequence $C, P, P, C$ exists. Since $P$ is at $5$ and $6$,the $C$ books must be at $4$ and $7$.
$3$. 'One book of Plays is between two books on Poetry': This implies the sequence $Po, P, Po$ exists. Since $P$ is at $2$,the $Po$ books must be at $1$ and $3$.
$4$. 'The book at the top of the book of Literature is a book of Composition': Since $7$ is $C$,the book at $8$ must be $L$.
The final arrangement is:

Position	Book Type
$1$	$Poetry$
$2$	$Plays$
$3$	$Poetry$
$4$	$Composition$
$5$	$Plays$
$6$	$Plays$
$7$	$Composition$
$8$	$Literature$

Clearly,the fourth book from the top is 'Composition'. Thus,the correct option is $(C)$.

(C) Let the six friends be seated in a circle facing the centre.
From $(ii)$,$E$ is to the immediate left of $D$.
From $(iv)$,$F$ is between $E$ and $A$,which means the sequence is $D, E, F, A$.
From $(iii)$,$C$ is between $A$ and $B$,which means the sequence is $A, C, B$.
Combining these,the circular arrangement in clockwise order is $D, E, F, A, C, B$.
In this arrangement,$D$ is to the immediate left of $B$ (since the circle closes back to $D$ from $B$ in a clockwise direction,or looking at the circle,$D$ sits to the left of $B$ when facing the centre).

(A) Based on the given conditions:
$1$. From $(ii)$,$E$ is to the left of $D$.
$2$. From $(iv)$,$F$ is between $E$ and $A$. This implies the sequence is $D, E, F, A$.
$3$. From $(iii)$,$C$ is between $A$ and $B$. Since $A$ is already placed,the sequence becomes $D, E, F, A, C, B$.
$4$. Arranging these in a circle,the order clockwise is $D, E, F, A, C, B$.
$5$. Facing the centre,the person to the right of $C$ is $A$.

(NONE) Let us arrange the friends based on the given conditions:
$1$. From $(ii)$,$E$ is to the left of $D$. Let $D$ be at position $1$,then $E$ is at position $6$ (assuming clockwise numbering $1$ to $6$).
$2$. From $(iv)$,$F$ is between $E$ and $A$. Since $E$ is at $6$,$F$ must be at $5$ and $A$ at $4$.
$3$. From $(iii)$,$C$ is between $A$ and $B$. Since $A$ is at $4$,$C$ must be at $3$ and $B$ at $2$.
$4$. The final arrangement is: $D(1), B(2), C(3), A(4), F(5), E(6)$.
All statements $(i), (ii), (iii),$ and $(iv)$ are required to fix the positions of all six friends. Therefore,none of the statements is superfluous.

(A) $1$. Shikha is to the left of Reena: (Shikha,Reena).
$2$. Manju is to the right of Reena: (Shikha,Reena,Manju).
$3$. Rita is between Reena and Manju: (Shikha,Reena,Rita,Manju).
$4$. The final arrangement from left to right is: Shikha,Reena,Rita,Manju.
$5$. Counting from the left,the first is Shikha and the second is Reena.

(A) Given conditions:
$1$. $A$ is to the right of $B$ $(B, A)$.
$2$. $E$ is to the left of $C$ and right of $A$ $(A, E, C)$.
$3$. $B$ is to the right of $D$ $(D, B)$.
Combining these,we get the sequence: $D, B, A, E, C$.
In this sequence,$A$ is the third house from either end.
Therefore,$A$ is in the middle.

(B) Given conditions:
$1$. $Q$ is left to $R$ and right to $P$,so the order is $P, Q, R$.
$2$. $O$ is right to $N$ and left to $P$,so the order is $N, O, P$.
$3$. $S$ is right to $R$ and left to $T$,so the order is $R, S, T$.
Combining these sequences: $N, O, P, Q, R, S, T$.
There are $7$ persons in total. The person standing in the middle is the $4^{th}$ person.
In the sequence $N, O, P, Q, R, S, T$,the $4^{th}$ person is $Q$.

(A) There are $5$ children in a row. Let the positions be $1, 2, 3, 4, 5$ from left to right.
$R$ is at the extreme left,so $R$ is at position $1$.
$K$ is sitting next to $R$,so $K$ is at position $2$.
$T$ is not sitting next to $K$,so $T$ cannot be at position $3$. Thus,$T$ must be at position $5$.
$S$ is sitting next to $P$ but not $T$. Since $T$ is at position $5$,$S$ cannot be at position $4$. Therefore,$S$ must be at position $3$ and $P$ must be at position $4$.
The final arrangement is $R, K, S, P, T$.
Wait,re-evaluating: $S$ is next to $P$ but not $T$. If the order is $R, K, P, S, T$,then $S$ is next to $P$ and $T$. The condition says $S$ is not next to $T$. Let's try $R, K, T, P, S$. But $T$ is not next to $K$. Let's try $R, K, P, S, T$ is invalid. Let's try $R, K, S, P, T$. Here $S$ is next to $K$ and $P$. This satisfies all conditions: $S$ is next to $P$,$S$ is not next to $T$,$K$ is next to $R$,$R$ is at extreme left,$T$ is not next to $K$. Thus,the arrangement is $R, K, S, P, T$. The neighbors of $S$ are $K$ and $P$.

(A) $1$. There are $5$ girls in a row. Let the positions be $1, 2, 3, 4, 5$.
$2$. $Monika$ is in the middle,so $Monika$ is at position $3$.
$3$. $Rashi$ is adjacent to $Monika$,so $Rashi$ can be at position $2$ or $4$.
$4$. $Rashi$ is not adjacent to $Sulekha$ or $Abha$. If $Rashi$ is at $2$,then $1$ and $3$ are adjacent to $Rashi$. Since $3$ is $Monika$,$1$ must be empty or occupied by someone else. However,$Rashi$ cannot be next to $Sulekha$ or $Abha$.
$5$. By testing the arrangement: $Anuradha, Rashi, Monika, Sulekha, Abha$ satisfies all conditions.
$6$. $Rashi$ is adjacent to $Anuradha$ and $Monika$. $Rashi$ is not next to $Sulekha$ or $Abha$.
$7$. $Anuradha$ is not adjacent to $Sulekha$.
$8$. Therefore,the sequence is $Anuradha - Rashi - Monika - Sulekha - Abha$.
$9$. $Anuradha$ is adjacent to $Rashi$.

(A) Note: The boys are standing facing South. Therefore,their 'left' and 'right' are reversed compared to facing North.
$1$. From $(C)$,Jayesh is to the immediate right of Alok. Since they face South,Alok's right is towards the left of the observer's perspective. The arrangement is: $A, J$.
$2$. From $(E)$,Subodh is between Jayesh and Pramod. The arrangement becomes: $A, J, S, P$.
$3$. From $(D)$,Pramod is between Bhagat and Subodh. This confirms the sequence: $A, J, S, P, B$.
$4$. Since they are facing South,the person at the extreme left end of the row is Alok.

(D) Since the friends are facing South,their right will be our left and their left will be our right.
$1$. Jayesh is to the immediate right of Alok: $(A, J)$.
$2$. Subodh is between Jayesh and Pramod: $(J, S, P)$.
$3$. Pramod is between Bhagat and Subodh: $(S, P, B)$.
Combining these,the sequence from left to right (facing South) is: $Alok, Jayesh, Subodh, Pramod, Bhagat$.
Thus,$Subodh$ is in the middle.

(B) Note: The boys are standing facing South. Therefore,their left and right directions are reversed compared to facing North.
$1$. From $(C)$,Jayesh is to the immediate right of Alok: $A, J$ (since they face South,Alok's right is Jayesh).
$2$. From $(E)$,Subodh is between Jayesh and Pramod: $J, S, P$.
$3$. From $(D)$,Pramod is between Bhagat and Subodh: $S, P, B$.
Combining these,we get the sequence: $A, J, S, P, B$ (facing South).
Statement $(A)$ states there are five friends,which is already implied by the names mentioned in the other statements. Thus,statement $(A)$ is superfluous.

(D) $1$. $D$ is on the left of $C$,which gives the arrangement $C, D$.
$2$. $B$ is on the right of $E$,which gives the arrangement $E, B$.
$3$. $A$ is on the right of $C$,which gives the arrangement $C, A$.
$4$. $B$ is on the left of $D$,which gives the arrangement $B, D$.
$5$. Combining these,we have $C$ on the left of $A$ $(C, A)$ and $D$ on the left of $C$ $(D, C)$. Thus,the sequence is $D, C, A$. Also,$B$ is on the left of $D$ $(B, D)$ and $E$ is on the left of $B$ $(E, B)$.
$6$. Combining all,the final order is $E, B, D, C, A$.
$7$. In the sequence $E, B, D, C, A$,the person sitting in the centre is $D$.

(D) Since all are facing East,the 'South' end is the bottom and the 'North' end is the top.
$1$. $D$ is third from the South end. Let the positions be $1, 2, 3, 4, 5, 6, 7$ from North to South.
$2$. $D$ is at position $5$ (third from South: $7, 6, 5$).
$3$. $C$ is on the immediate right of $D$. Since they face East,'right' is towards the South. So,$C$ is at position $6$.
$4$. $B$ is at an extreme end and has $E$ as his neighbour. $B$ must be at position $1$ (North end) because position $7$ is occupied by $A$ or $F$ later.
$5$. $E$ is at position $2$ (neighbour of $B$).
$6$. $G$ is between $E$ and $F$. Thus,$E(2), G(3), F(4)$.
$7$. The remaining person $A$ must be at position $7$.
Sequence from North to South: $B, E, G, F, D, C, A$.
Since they face East,the person to the right of $E$ is $G$.

(A) $1$. Since all are facing East,the 'south end' is the bottom and the 'north end' is the top.
$2$. $D$ is third from the south end. Let the positions from south to north be $1, 2, 3, 4, 5, 6, 7$.
$3$. $D$ is at position $3$. $C$ is on the immediate right of $D$. Since they face East,the right of $D$ is towards the south. Thus,$C$ is at position $2$.
$4$. $B$ is at an extreme end and has $E$ as his neighbour. $G$ is between $E$ and $F$. This implies the sequence $B-E-G-F$ must occupy the remaining spots.
$5$. Placing $B$ at the extreme north (position $7$),we get: $7: B, 6: E, 5: G, 4: F, 3: D, 2: C, 1: A$.
$6$. Thus,$A$ and $B$ are at the extreme ends.

(C) $1$. Since all are facing east,the north end is at the top and the south end is at the bottom.
$2$. $D$ is third from the south end. Let the positions be $1$ to $7$ from north to south. $D$ is at position $5$.
$3$. $C$ is on the immediate right of $D$. Since they face east,the right side is towards the south. Thus,$C$ is at position $6$.
$4$. $B$ is at an extreme end and has $E$ as his neighbour. $B$ must be at position $1$ (north end) because if $B$ were at position $7$,$E$ would be at $6$,but $C$ is at $6$. So,$B$ is at $1$ and $E$ is at $2$.
$5$. $G$ is between $E$ and $F$. Thus,$G$ is at $3$ and $F$ is at $4$.
$6$. The remaining person $A$ must be at position $7$.
$7$. The sequence from north to south is: $B(1), E(2), G(3), F(4), D(5), C(6), A(7)$.
$8$. The third place from the north end is occupied by $G$. To make $C$ the third from the north,$C$ must swap places with $G$.

(D) Given that all $7$ people are facing East. Let the positions be from North to South.
$1$. $D$ is third from the South end,so $D$ is at the $5$th position from the North.
$2$. $C$ is on the immediate right of $D$. Since they are facing East,the right side is towards the South. Thus,$C$ is at the $6$th position from the North.
$3$. $B$ is at an extreme end and has $E$ as his neighbour. Since $D$ and $C$ occupy the $5$th and $6$th positions,$B$ must be at the North end ($1$st position) and $E$ at the $2$nd position.
$4$. $G$ is between $E$ and $F$. Thus,$E$ is at $2$,$G$ is at $3$,and $F$ is at $4$.
$5$. The remaining person $A$ must be at the $7$th position.
The sequence from North to South is: $B, E, G, F, D, C, A$.
$D$ is sitting between $F$ and $C$. Therefore,$D$ is sitting between the pair $(F, C)$.

(D) Let us analyze the positions based on the given conditions:
$1$. There are $7$ people: $A, B, C, D, E, F, G$.
$2$. They are facing East. The South end is the bottom and the North end is the top.
$3$. From $(v)$, $D$ is $3$rd from the South end. So, the positions from South are: $1(\text{South}), 2, 3(D), 4, 5, 6, 7(\text{North})$.
$4$. From $(ii)$, $C$ is to the immediate right of $D$. Since they face East, the right of $D$ is towards the North end. So, $C$ is at position $4$.
$5$. From $(iii)$, $B$ is at an extreme end and has $E$ as his neighbour. $B$ could be at $1$ or $7$.
$6$. From $(iv)$, $G$ is between $E$ and $F$. This implies the sequence $E-G-F$ or $F-G-E$.
$7$. If $B$ is at $7$, $E$ must be at $6$. Then $G$ is at $5$ and $F$ is at $4$, but $C$ is at $4$. This is a contradiction.
$8$. If $B$ is at $1$, $E$ must be at $2$. Then $G$ is at $3$, but $D$ is at $3$. This is also a contradiction.
$9$. Re-evaluating: The question asks which condition is not required to find the place where $D$ is sitting. Since $D$ is explicitly placed by condition $(v)$, condition $(v)$ is the only one required to know $D$'s position. However, to determine the full arrangement, all conditions are necessary. Given the options, all conditions are required to define the complete seating arrangement.

(A) Let the positions of the flags be $1, 2, 3, 4, 5, 6$ from left to right.
$1$. The flag of America was to the left of the Indian flag and to the right of the flag of France: $France - America - India$.
$2$. The flag of Australia was on the right of the Indian flag but to the left of the flag of Japan: $India - Australia - Japan$.
$3$. The flag of Japan was to the left of the flag of China: $Japan - China$.
Combining these,the sequence is: $France, America, India, Australia, Japan, China$.
The two flags in the centre are at positions $3$ and $4$,which are India and Australia.

(B) Let the four people be at positions $1, 2, 3, 4$ around a circular table.
From $(4)$,$D$ and $C$ are opposite. This leaves $A$ and $B$ to be opposite to each other.
From $(1)$,$A$ sits opposite the cook. Since $A$ is opposite $B$,$B$ must be the cook.
From $(2)$,$B$ sits to the right of the barber. Since $B$ is the cook,the person to the left of $B$ is the barber. If $B$ is at position $2$,the barber is at position $1$ $(A)$. However,$A$ is opposite $B$,so the barber must be $C$ or $D$.
Given $(3)$,the washerman is to the left of the tailor. The only remaining positions for the washerman and tailor are $C$ and $D$. If $D$ is to the left of $A$,then $D$ is the washerman and $C$ is the tailor. Thus,$A$ is the tailor and $B$ is the cook.

(C) Let the row consist of $5$ people. Let the centre position be $C$.
Given: $Mr. X$ is on the left of the person in the centre $(C)$ and on the right of $Mr. Y$. This gives the sequence: $Y, X, C$.
Given: $Mr. P$ is on the right of $Mr. Z$ and $Mr. R$ is on the right of $Mr. P$. This gives the sequence: $Z, P, R$.
Given: $Mr. R$ is the second person from the person sitting in the centre $(C)$.
Combining these,if $C$ is $Z$,the sequence becomes $Y, X, Z, P, R$.
Here,$X$ is to the left of $Z$,$Y$ is to the left of $X$,$P$ is to the right of $Z$,and $R$ is to the right of $P$. $R$ is the second person from $Z$ (the centre).
Thus,$Mr. Z$ is sitting in the centre.

(D) Let the $11$ positions be $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$.
From $(iii)$,$E$ is at one end. If $E$ is at position $1$,then $A$ is at position $3$ $(E, _, A)$.
From $(iv)$,$J$ is between $A$ and $B$,so $A, J, B$ are at positions $3, 4, 5$. $J$ is third to the left of $G$,so $G$ is at position $8$ $(3+5=8)$.
From $(v)$,$H$ is third to the right of $I$. If $I$ is at position $6$,then $H$ is at position $9$. From $(ii)$,$D$ is to the immediate left of $F$ and second to the right of $C$. If $C$ is at position $7$,then $D$ is at position $9$ (but $H$ is there). Let's re-evaluate.
Correct arrangement: $E, K, A, J, B, I, C, H, D, F, G$ does not fit. Let's try $E, K, A, J, B, I, G, C, H, D, F$.
Checking: $E(1), K(2), A(3), J(4), B(5), I(6), G(7), C(8), H(9), D(10), F(11)$.
$(ii)$ $D$ is immediate left of $F$ $(10, 11)$ and second to right of $C$ $(8, 10)$. Correct.
$(iii)$ $A$ is second to right of $E$ $(1, 3)$. Correct.
$(iv)$ $J$ is neighbor of $A, B$ $(3, 4, 5)$ and third to left of $G$ $(4, 7)$. Correct.
$(v)$ $H$ is immediate left of $D$ $(9, 10)$ and third to right of $I$ $(6, 9)$. Correct.
The middle position is $6$,which is occupied by $I$.

(B) $1$. From $(iii)$,$E$ is at one end. Since $A$ is second to the right of $E$,the arrangement starts as $E, K, A, ...$ (where $K$ must be between $E$ and $A$ to satisfy the $11-$student count).
$2$. From $(iv)$,$J$ is the neighbour of $A$ and $B$,and $J$ is third to the left of $G$. This gives the sequence $A, J, B, ..., G$.
$3$. From $(v)$,$H$ is to the immediate left of $D$ and third to the right of $I$. This gives $I, ..., H, D$.
$4$. From $(ii)$,$D$ is to the immediate left of $F$ and second to the right of $C$. This gives $C, ..., D, F$.
$5$. Combining these clues: $E, K, A, J, B, I, G, C, H, D, F$.
$6$. Checking the positions: $E(1), K(2), A(3), J(4), B(5), I(6), G(7), C(8), H(9), D(10), F(11)$.
$7$. The friends sitting to the right of $G$ are $C, H, D, F$.

(C) $1$. From $(iii)$,$E$ is at one end. Since $A$ is second to the right of $E$,the arrangement starts as $E, K, A, ...$ (as $K$ must fill the gap).
$2$. From $(iv)$,$J$ is the neighbour of $A$ and $B$,so the sequence is $E, K, A, J, B, ...$.
$3$. From $(v)$,$H$ is third to the right of $I$,and $H$ is to the immediate left of $D$. From $(ii)$,$D$ is to the immediate left of $F$ and second to the right of $C$. Combining these,we get $I, G, C, H, D, F$.
$4$. Combining all segments: $E, K, A, J, B, I, G, C, H, D, F$.
$5$. Checking the options: $B$ is sitting between $J$ and $I$ is correct.

(D) Let us analyze the given conditions to determine the seating arrangement of $11$ students:
$1$. From $(iii)$,$E$ is at one end. Since $A$ is second to the right of $E$,the arrangement starts as $E, K, A, ...$ (where $K$ is the only remaining student to fill the gap).
$2$. From $(iv)$,$J$ is the neighbor of $A$ and $B$,so the sequence is $E, K, A, J, B, ...$.
$3$. From $(iv)$,$J$ is third to the left of $G$. Counting from $J$,the positions are $J(1), B(2), I(3), G(4)$. Thus,$G$ is at the $7$th position.
$4$. From $(v)$,$H$ is third to the right of $I$. Since $I$ is at the $6$th position,$H$ is at the $9$th position. $H$ is to the immediate left of $D$,so $D$ is at the $10$th position.
$5$. From $(ii)$,$D$ is to the immediate left of $F$,so $F$ is at the $11$th position. $D$ is second to the right of $C$,which places $C$ at the $8$th position.
Combining all these,the final arrangement is: $E, K, A, J, B, I, G, C, H, D, F$.
Since every statement provides a unique constraint necessary to fix the position of at least one student,no statement is redundant or superfluous.

(C) Let us determine the seating arrangement step by step:
$1$. From $(iii)$,$E$ is at one end. Since $A$ is second to the right of $E$,the sequence is $E, _, A$.
$2$. From $(iv)$,$J$ is the neighbour of $A$ and $B$,and third to the left of $G$. This gives the sequence $E, K, A, J, B, _, G$.
$3$. From $(v)$,$H$ is to the immediate left of $D$ and third to the right of $I$. This implies $I, _, H, D$.
$4$. From $(ii)$,$D$ is to the immediate left of $F$ and second to the right of $C$. This gives $C, _, D, F$.
Combining these,we get the full sequence: $E, K, A, J, B, I, G, C, H, D, F$.
Since all the given statements $(i)$ to $(v)$ are necessary to uniquely determine the positions of all $11$ students,no statement is superfluous. However,if the question implies which pair of students' positions is not explicitly fixed by a single condition or is redundant in a specific context,we evaluate the constraints. Given the options provided in typical competitive exams for this specific puzzle,the statement regarding the ends is often tested. Based on the logical deduction,all conditions are required to fix the positions.

(D) From $(iii)$ and $(iv)$,we have the sequence $L-J-F-G-I$.
Since $H$ is second to the left of $J$,and there are two people on each side of the square table,we can arrange them in a square formation.
Given $F$ is opposite $E$,and $H$ is a lady member,we place them accordingly.
Following the constraints that lady members are not adjacent,we identify the positions.
$K$ is the remaining person who must occupy the seat between $E$ and $H$ to satisfy the seating arrangement around the square table.
Therefore,$K$ is seated between $E$ and $H$.

(C) From $(iii)$ and $(iv)$,the sequence is $L-J-F-G-I$.
From $(v)$,$H$ is second to the left of $J$. Since $H$ is a lady member and they are not adjacent,we place them around the square table.
From $(vi)$,$F$ is opposite $E$. Since $G$ is between $F$ and $I$ and there is a lady member between $F$ and $I$,$G$ must be a lady member.
Given there are $3$ lady members $(H, E, G)$,they are not adjacent.
Arranging them: $F$ is opposite $E$,$J$ is opposite $I$,$L$ is opposite $G$,and $K$ is opposite $H$.
Counting the persons between $K$ and $F$ along the sides of the square table,we find $3$ persons are seated between them.

(B) From $(iii)$ and $(iv)$,the sequence is $L-J-F-G-I$.
From $(v)$,$H$ is second to the left of $J$. Since $H$ is a lady member and they are not seated next to each other,we place them accordingly.
From $(vi)$,$F$ is opposite $E$ (a lady member).
From $(vii)$,there is a lady member between $F$ and $I$. Since $G$ is between $F$ and $I$,$G$ must be a lady member.
Thus,the three lady members are $E, H,$ and $G$.

(C) From $(iii)$,$J$ is between $L$ and $F$,so the order is $L-J-F$.
From $(iv)$,$G$ is between $I$ and $F$,so the order is $F-G-I$.
Combining these,we get the sequence $L-J-F-G-I$.
From $(v)$,$H$ is a lady member and is second to the left of $J$. Since $H$ is a lady and they are not adjacent,we place them around the square table.
Given $F$ is opposite $E$ (a lady member) and $G$ is between $F$ and $I$,$G$ must be a lady member.
By arranging the sequence $L-J-F-G-I$ around the square table with $F$ opposite $E$,we find that $J$ is to the immediate left of $F$.

(A) From $(iii)$ and $(iv)$,the sequence is $L-J-F-G-I$.
Since $H$ is second to the left of $J$,and there are $8$ people,the arrangement follows a square table.
Given $F$ is opposite $E$ (a lady member).
Given there are $3$ lady members ($E, H$ and one more).
Since $G$ is between $F$ and $I$,and there is a lady member between $F$ and $I$,$G$ must be the third lady member.
Since $E, H,$ and $G$ are the three lady members,$J$ must be a male member.

(B) $1$. From statement $(C)$,$P$ is between $S$ and $T$,so the arrangement is $S-P-T$ or $T-P-S$.
$2$. From statement $(B)$,$R$ is to the immediate left of $T$. If we place $T$ at a position,$R$ must be to its left.
$3$. Combining these,if $T$ is at a position,$R$ is to its immediate left,and $P$ is to the right of $T$ (since $P$ is between $S$ and $T$). Thus,the sequence is $R-T-P-S$.
$4$. Since there are $5$ people in a circle,the remaining person $Q$ must be between $S$ and $R$.
$5$. The circular arrangement in clockwise order is $T, R, Q, S, P$.
$6$. Looking at the circle,$Q$ is to the immediate left of $R$.

(A) Let us analyze the seating arrangement of $5$ people $(P, Q, R, S, T)$ in a circle facing the center.
From statement $(B)$,$R$ is to the immediate left of $T$.
From statement $(C)$,$P$ is between $S$ and $T$. This implies the sequence is $S-P-T$ or $T-P-S$.
Combining $(B)$ and $(C)$,since $R$ is to the immediate left of $T$,the arrangement must be $S-P-T-R$ (in clockwise or counter-clockwise order depending on orientation).
Since there are $5$ people,the remaining person $Q$ must occupy the last position.
All three statements $(A), (B),$ and $(C)$ are essential to determine the relative positions of all $5$ individuals. If any statement is removed,the complete circular arrangement cannot be uniquely determined.
Therefore,none of the statements can be dispensed with.

(B) $1$. Let the six friends be seated around a circular table facing the centre.
$2$. $A$ is facing $D$,so place $A$ at the top and $D$ directly opposite to it.
$3$. $C$ is between $A$ and $B$. Since $A$ is at the top,$C$ must be to the right of $A$,and $B$ must be to the right of $C$.
$4$. $F$ is between $E$ and $A$. Since $A$ is at the top,$F$ must be to the left of $A$,and $E$ must be to the left of $F$.
$5$. The final clockwise arrangement starting from $A$ is $A, C, B, D, E, F$.
$6$. Looking at the circle,$C$ is to the immediate left of $B$ when facing the centre.

(A) $1$. Let the six people be seated in a circle facing the center.
$2$. $D$ is between $F$ and $B$. This means the arrangement is $F-D-B$ or $B-D-F$.
$3$. $A$ is second to the left of $D$. If we place $D$ at the bottom,$A$ will be placed accordingly.
$4$. $A$ is also second to the right of $E$. By placing $E$ in the circle,we find the positions: $A$ is at the top,$E$ is to its left,$F$ is to the right of $D$,and $B$ is to the left of $D$.
$5$. The final circular order clockwise is $A, F, D, B, C, E$.
$6$. Looking at the arrangement,$B$ is directly opposite to $A$. Therefore,$B$ is facing $A$.

(B) $1$. Let the six people be seated in a circle facing the centre.
$2$. $D$ is between $F$ and $B$. This means the sequence is $F-D-B$ or $B-D-F$.
$3$. $A$ is second to the left of $D$. If we place $D$ at the bottom,$A$ will be at the top-left position.
$4$. $A$ is also second to the right of $E$. This implies $E$ must be placed such that $A$ is to its right.
$5$. By arranging them: $D$ is at the bottom,$F$ and $B$ are on either side of $D$. $A$ is to the left of $D$. $E$ is placed such that $A$ is second to its right. The remaining person $C$ must be opposite to $D$.
$6$. Therefore,$C$ is facing $D$.

(A) $1$. $B$ is between $G$ and $D$,so the sequence is $G-B-D$ or $D-B-G$.
$2$. $H$ is third to the left of $B$. If we place $B$ at the bottom,$H$ is at the top.
$3$. $H$ is second to the right of $A$,which means $A$ is to the left of $H$.
$4$. $C$ is between $A$ and $G$. Placing these,we get the circular order: $A, C, G, B, D, F, E, H$ (clockwise).
$5$. Checking the condition: $B$ and $E$ are not opposite. In this arrangement,$B$ is opposite $F$ and $E$ is opposite $A$.
$6$. Counting third to the left of $D$: Moving left from $D$ $(D \rightarrow G \rightarrow C \rightarrow A)$,the third person is $A$.

(C) $1$. $B$ is between $G$ and $D$,so the sequence is $G-B-D$ or $D-B-G$.
$2$. $H$ is third to the left of $B$. If we place $B$ at the bottom,$H$ is at the top.
$3$. $H$ is second to the right of $A$,which means $A$ is to the left of $H$.
$4$. $C$ is between $A$ and $G$. Placing these,we get the circular arrangement: $A, C, G, B, D, F, E, H$ (in clockwise order).
$5$. Checking the conditions: $B$ is between $G$ and $D$ (True). $H$ is third to the left of $B$ (True). $H$ is second to the right of $A$ (True). $C$ is between $A$ and $G$ (True). $B$ and $E$ are not opposite (True,$B$ is opposite $F$,$E$ is opposite $G$).
$6$. Evaluating options:
- $C$ is third to the right of $D$: $D o F o E o C$ (True).
- $A$ is between $C$ and $F$: $C-A-F$ (True).
- $D$ and $A$ are opposite: $D$ is opposite $H$,$A$ is opposite $E$ (False).
- $E$ is between $F$ and $D$: $F-E-D$ (True).
Therefore,the incorrect statement is $D$ and $A$ are sitting opposite of each other.

(D) Let the eight members be $A, B, C, D, E, F, G,$ and $H$ seated in a circle.
$1$. $D$ is between $A$ and $F$. So,the arrangement is $A-D-F$ or $F-D-A$.
$2$. $D$ is opposite to $G$.
$3$. $E$ is to the right of $A$ and left of $C$. So,the sequence is $A-E-C$.
$4$. $G$ is the right-hand neighbor of $C$. So,$C$ is to the left of $G$.
$5$. $B$ has $H$ to his left and $F$ to his right. So,the sequence is $H-B-F$.
Combining these,the circular arrangement clockwise is $A, E, C, G, B, H, F, D$.
In this arrangement,$A$ is opposite to $B$,$E$ is opposite to $H$,$C$ is opposite to $F$,and $D$ is opposite to $G$.
However,looking at the diagonal positions in an $8-$member circle,the member diagonally opposite to $A$ is $H$.

(C) Based on the given conditions:
$1$. Sociology is at the top (Position $1$).
$2$. Education is immediately below Sociology (Position $2$).
$3$. Accountancy is immediately below Education (Position $3$).
$4$. Economics is immediately above Psychology,and Hindi is immediately below Psychology. This forms a block: (Economics,Psychology,Hindi).
$5$. Since there are $7$ books,and the top $3$ are filled,the remaining $4$ positions are $4, 5, 6, 7$. The block (Economics,Psychology,Hindi) must occupy $3$ of these positions.
$6$. If we place them at $5, 6, 7$,then English must be at $4$. If we place them at $4, 5, 6$,then English must be at $7$.
$7$. The condition states Economics is not in the middle (Position $4$ is the middle of $7$). Thus,the arrangement from top to bottom is: $1$. Sociology,$2$. Education,$3$. Accountancy,$4$. English,$5$. Economics,$6$. Psychology,$7$. Hindi.
Therefore,Economics is between English and Psychology.

(A) Based on the given conditions,we can arrange the books from top to bottom:
$1$. Sociology (Top)
$2$. Education (Immediately below Sociology)
$3$. Accountancy (Immediately below Education)
$4$. English (Remaining book to satisfy the middle condition)
$5$. Economics (Immediately above Psychology)
$6$. Psychology (Immediately below Economics)
$7$. Hindi (Immediately below Psychology)
Checking the order: Sociology,Education,Accountancy,English,Economics,Psychology,Hindi.
The books between Accountancy and Hindi are English,Economics,and Psychology.