Number of students who passed half-yearly exams in the school

  = (Number of students passed in half-yearly but failed in annual exams)

      + (Number of students passed in both exams)

  = (6 + 17 + 9 + 15) + (64 + 55 + 46 + 76)

  = 288.

Also, Number of students who passed annual exams in the school

  = (Number of students failed in half-yearly but passed in annual exams)

      + (Number of students passed in both exams)

  = (14 + 12 + 8 + 13) + (64 + 55 + 46 + 76)

  = 288.

Since, the number of students passed in half-yearly = the number of students passed in annual exams. Therefore, it can be inferred that both the examinations had almost the same difficulty level.

Thus Statements (a), (b) and (d) are false and Statement (c) is true.

Since the classification of the students on the basis of their results and sections form independent groups, so the total number of students in the class:

= (28 + 23 + 17 + 27 + 14 + 12 + 8 + 13 + 6 + 17 + 9 + 15 + 64 + 55 + 46 + 76)

= 430.

Pass percentages in at least one of the two examinations for different sections are:

For Section A		(14 + 6 + 64)	x 100		% =		84	x 100		% = 75%.
		(28 + 14 + 6 + 64)					112

For Section B		(12 + 17 + 55)	x 100		% =		84	x 100		% = 78.5%.
		(23 + 12 + 17 + 55)					107

For Section C		(8 + 9 + 46)	x 100		% =		63	x 100		% = 78.75%.
		(17 + 8 + 9 + 46)					80

For Section D		(13 + 15 + 76)	x 100		% =		104	x 100		% = 79.39%.
		(27 + 13 + 15 + 76)					131

Clearly, the pass percentage is maximum for Section D.

Total number of students passed in annual exams in a section

= [ (No. of students failed in half-yearly but passed in annual exams)

        + (No. of students passed in both exams)

   ] in that section

Success rate in annual exams in Section A

=		No. of students of Section A passed in annual exams	x 100		%
		Total number of students in Section A

=		(14 + 64)	x 100		%
		(28 + 14 + 6 + 64)

=		78	x 100		%
		112

    = 69.64%.

Similarly, success rate in annual exams in:

Section B		(12 + 55)	x 100		% =		67	x 100		% = 62.62%.
		(23 + 12 + 17 + 55)					107

Section C		(8 + 46)	x 100		% =		54	x 100		% = 67.5%.
		(17 + 8 + 9 + 46)					80

Section D		(13 + 76)	x 100		% =		89	x 100		% = 67.94%.
		(27 + 13 + 15 + 76)					131

Clearly, the success rate in annual examination is maximum for Section A.

Total number of failures in half-yearly exams in a section

= [ (Number of students failed in both exams)

        + (Number of students failed in half-yearly but passed in Annual exams)

   ] in that section

Failure rate in half-yearly exams in Section A

=		Number of students of Section A failed in half-yearly	x 100		%
		Total number of students in Section A

=		(28 + 14)	x 100		%
		(28 + 14 + 6 + 64)

=		42	x 100		%
		112

    = 37.5%.

Similarly, failure rate in half-yearly exams in:

Section B		(23 + 12)	x 100		% =		35	x 100		% = 32.71%.
		(23 + 12 + 17 + 55)					107

Section C		(17 + 8)	x 100		% =		25	x 100		% = 31.25%.
		(17 + 8 + 9 + 46)					80

Section D		(27 + 13)	x 100		% =		40	x 100		% = 30.53%.
		(27 + 13 + 15 + 76)					131

Clearly, the failure rate is minimum for Section D.