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In an art exhibition held during the 70th Independence Day celebrations in Delhi, five portraits of five wellknown freedom fighters namely Mahatma Gandhi, Bhagat Singh, Lal Bahadur Shastri, Lala Lajpat Rai and Jawaharlal Nehru were displayed. All these portraits were of different colors: red, orange, blue, violet, pink(not necessarily in the same order). All these five portraits were displayed with a different number of story books, with each portrait having at least one story book.

Some additional information is given below:

• The number of story books displayed at the violet colored portrait was equal to two-fifths of the absolute difference between the number of story books displayed at the portraits of Lala Lajpat Rai and Jawaharlal Nehru.


• Portraits of Lala Lajpat Rai and Jawaharlal Nehru were not of violet color.


• The number of story books displayed at the blue colored portrait was one-third of the sum of the number of story books displayed at the portraits of Bhagat Singh and Lal Bahadur Shastri.


• Portraits of Bhagat Singh and Lal Bahadur Shastri were not of blue color.


• The number of story books displayed at the Jawaharlal Nehru portrait was half the number of story books displayed at the pink colored portrait.


• Mahatma Gandhi portrait was neither red colored nor violet colored.


• The number of story books displayed at the Jawaharlal Nehru portrait was equal to the sum of the number of story books displayed at the red and blue colored portraits.


• Jawaharlal Nehru portrait was neither red colored nor blue colored

1. Color(s) of how many portraits can be determined uniquely?




2. If Lal Bahadur Shastri portrait was of Pink color and the absolute difference between the number of story books displayed at Lala Lajpat Rai portrait and Jawaharlal Nehru portrait was 10, find the number of story books displayed at Mahatma Gandhi portrait.




3. Which of the following statement(s) is/are necessarily TRUE?
   
   (i) The violet colored portrait could not be of Lala Lajpat Rai, Jawaharlal Nehru or Mahatma Gandhi.
   
   (ii) The blue colored portrait could be of Lal Bahadur Shastri.
   
   (iii) The red colored portrait could not be of Lala Lajpat Rai.




4. Which of the following statements are required, to uniquely determine colours of all the portraits?
   
   (i) Lala Lajpat Rai portrait is red coloured
   
   (ii) Portrait of Bhagat Singh is not pink coloured
   
   (iii) Portrait of Mahatma Gandhi is not orange coloured

Color(s) of how many portraits can be determined uniquely?

Nehru is orange, I believe that's all we have. We know that from (2) Nehru is not violet, from (5) is not pink, and from (8) is neither red nor blue. Therefore he must be orange. You know that Rai can't be violet, Singh and Shastri can't be blue, and Gandhi can't be red or violet but you can't determine from the 8 bullet points any more information from this.

If Lal Bahadur Shastri portrait was of Pink color and the absolute difference between the number of story books displayed at Lala Lajpat Rai portrait and Jawaharlal Nehru portrait was 10, find the number of story books displayed at Mahatma Gandhi portrait.

If Shastri is pink, Gandhi must be blue, since his is not red or violet. Then Rai is red because his is not violet, and so Singh is violet. Then blue = Nehru - red => Gandhi = Nehru - Rai = 10. Therefore 10 story books at the Gandhi portrait.

Which of the following statement(s) is/are necessarily TRUE?
(i) The violet colored portrait could not be of Lala Lajpat Rai, Jawaharlal Nehru or Mahatma Gandhi.

TRUE: (2) says Rai and Nehru aren't violet; (6) says Gandhi is not either.

(ii) The blue colored portrait could be of Lal Bahadur Shastri.

FALSE: Shastri is not blue, because of (4).

(iii) The red colored portrait could not be of Lala Lajpat Rai.

FALSE: The red coloured portrait could be of Rai (Based on the second question asked, if Shastri is pink, then Rai is red).

Which of the following statements are required, to uniquely determine colours of all the portraits?
(i) Lala Lajpat Rai portrait is red coloured

Not unique. Singh and Shastri could be either violet or pink.

(ii) Portrait of Bhagat Singh is not pink coloured

Not unique either. Gandhi could be pink, then Rai would be blue, and Singh could be red and Shastri violet. Or Gandhi pink, Rai blue, Singh violet, and Shastri red. More than one possibility, therefore not unique.

(iii) Portrait of Mahatma Gandhi is not orange coloured

NOT REQUIRED (We know Nehru is Orange already).

Answer:

You'd need the combination of (i) and (ii) to determine the colours uniquely. Why? Nehru is orange. Then (i) says that Rai is red. Then Singh and Shastri both can't be blue, so Gandhi must be Blue. Then (ii) says that Singh is not pink, so Shastri must be pink. Then Singh must be violet.