Past Year Paper, Mathematics (Set - 3), Delhi, 2015, Class 12, Maths JEE Notes | EduRev

 Page 1


65/1/3 1 [P.T.O. 
 
 
 
 
¸üÖê»Ö ®ÖÓ. 
Roll No. 
 
 
 
ÝÖ×ÞÖŸÖ 
MATHEMATICS  
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100 
Time allowed : 3 hours ] [ Maximum Marks : 100 
 
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö : 
 (i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …  
 (ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
 (iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æîü …  
 (iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æïü …  
 (v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ 
Æïü …  
 (vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …  
 Series : SSO/1/C 
ÛúÖê›ü ®ÖÓ. 
Code No.  
    
65/1/3
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …  
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …  
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …  
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê 
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê       
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …  
• Please check that this question paper contains 8 printed pages.  
• Code number given on the right hand side of the question paper should be written on the 
title page of the answer-book by the candidate. 
• Please check that this question paper contains 26 questions. 
• Please write down the Serial Number of the question before attempting it. 
• 15 minutes time has been allotted to read this question paper. The question paper will be 
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the 
question paper only and will not write any answer on the answer-book during this period. 
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü 
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë … 
Candidates must write the Code on 
the title page of the answer-book. 
 
SET – 3 
Page 2


65/1/3 1 [P.T.O. 
 
 
 
 
¸üÖê»Ö ®ÖÓ. 
Roll No. 
 
 
 
ÝÖ×ÞÖŸÖ 
MATHEMATICS  
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100 
Time allowed : 3 hours ] [ Maximum Marks : 100 
 
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö : 
 (i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …  
 (ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
 (iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æîü …  
 (iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æïü …  
 (v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ 
Æïü …  
 (vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …  
 Series : SSO/1/C 
ÛúÖê›ü ®ÖÓ. 
Code No.  
    
65/1/3
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …  
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …  
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …  
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê 
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê       
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …  
• Please check that this question paper contains 8 printed pages.  
• Code number given on the right hand side of the question paper should be written on the 
title page of the answer-book by the candidate. 
• Please check that this question paper contains 26 questions. 
• Please write down the Serial Number of the question before attempting it. 
• 15 minutes time has been allotted to read this question paper. The question paper will be 
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the 
question paper only and will not write any answer on the answer-book during this period. 
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü 
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë … 
Candidates must write the Code on 
the title page of the answer-book. 
 
SET – 3 
65/1/3 2  
General Instructions :   
 (i) All questions are compulsory. 
 (ii) Please check that this Question Paper contains 26 Questions. 
 (iii) Questions 1 – 6 in Section-A are Very Short Answer Type Questions carrying 1 mark 
each. 
 (iv) Questions 7 – 19 in Section-B are Long Answer I Type Questions carrying 4 marks 
each. 
 (v) Questions 20 – 26 in Section-C are Long Answer II Type Questions carrying               
6 marks each 
 (vi) Please write down the serial number of the Question before attempting it. 
  
   
ÜÖÞ›ü – † 
SECTION – A 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …  
 Question numbers 1 to 6 carry 1 mark each. 
 
1. C ÛúÖ ×¾Ö»ÖÖê¯Ö®Ö Ûú¸üÛêú ¾ÖÛÎúÖë Ûêú Ûãú»Ö xy = C cos x ÛúÖê ×®Öºþ×¯ÖŸÖ Ûú¸ü®Öê ¾ÖÖ»ÖÖ †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ …  1 
 Write the differential equation obtained by eliminating the arbitrary constant C in the 
equation representing the family of curves xy = C cos x. 
 
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7
?
?
?
?
?
? d
2
y
dx
2
3 
Ûúß ÛúÖê×™ü ŸÖ£ÖÖ ‘ÖÖŸÖ ÛúÖ µÖÖêÝÖ ×»Ö×ÜÖ‹ …  1 
 Write the sum of the order and degree of the differential equation 
 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7 
?
?
?
?
?
? d
2
y
dx
2
3 
 
3. ÃÖפü¿Ö 
?
a = 
^
i – 2
^
j Ûúß ×¤ü¿ÖÖ ´Öë ‹Ûú ÃÖפü¿Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛúÖ ¯Ö׸ü´ÖÖÞÖ 7 ‡ÛúÖ‡Ô Æîü …  1 
 Find a Vector in the direction of 
?
a = 
^
i – 2
^
j that has magnitude 7 units. 
 
4. µÖפü 
?
a ŸÖ£ÖÖ 
?
b ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü, ŸÖÖê 
?
a ŸÖ£ÖÖ 
?
b Ûêú ²Öß“Ö ÛúÖ ÛúÖêÞÖ ŒµÖÖ Æîü, •Ö²Ö×Ûú 2
?
a – 
?
b ‹Ûú ´ÖÖ¡ÖÛú 
ÃÖפü¿Ö Æîü ?    1 
 If 
?
a and 
?
b are unit Vectors, then what is the angle between 
?
a and 
?
b so that 2
?
a – 
?
b 
is a unit vector ? 
Page 3


65/1/3 1 [P.T.O. 
 
 
 
 
¸üÖê»Ö ®ÖÓ. 
Roll No. 
 
 
 
ÝÖ×ÞÖŸÖ 
MATHEMATICS  
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100 
Time allowed : 3 hours ] [ Maximum Marks : 100 
 
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö : 
 (i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …  
 (ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
 (iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æîü …  
 (iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æïü …  
 (v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ 
Æïü …  
 (vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …  
 Series : SSO/1/C 
ÛúÖê›ü ®ÖÓ. 
Code No.  
    
65/1/3
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …  
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …  
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …  
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê 
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê       
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …  
• Please check that this question paper contains 8 printed pages.  
• Code number given on the right hand side of the question paper should be written on the 
title page of the answer-book by the candidate. 
• Please check that this question paper contains 26 questions. 
• Please write down the Serial Number of the question before attempting it. 
• 15 minutes time has been allotted to read this question paper. The question paper will be 
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the 
question paper only and will not write any answer on the answer-book during this period. 
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü 
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë … 
Candidates must write the Code on 
the title page of the answer-book. 
 
SET – 3 
65/1/3 2  
General Instructions :   
 (i) All questions are compulsory. 
 (ii) Please check that this Question Paper contains 26 Questions. 
 (iii) Questions 1 – 6 in Section-A are Very Short Answer Type Questions carrying 1 mark 
each. 
 (iv) Questions 7 – 19 in Section-B are Long Answer I Type Questions carrying 4 marks 
each. 
 (v) Questions 20 – 26 in Section-C are Long Answer II Type Questions carrying               
6 marks each 
 (vi) Please write down the serial number of the Question before attempting it. 
  
   
ÜÖÞ›ü – † 
SECTION – A 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …  
 Question numbers 1 to 6 carry 1 mark each. 
 
1. C ÛúÖ ×¾Ö»ÖÖê¯Ö®Ö Ûú¸üÛêú ¾ÖÛÎúÖë Ûêú Ûãú»Ö xy = C cos x ÛúÖê ×®Öºþ×¯ÖŸÖ Ûú¸ü®Öê ¾ÖÖ»ÖÖ †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ …  1 
 Write the differential equation obtained by eliminating the arbitrary constant C in the 
equation representing the family of curves xy = C cos x. 
 
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7
?
?
?
?
?
? d
2
y
dx
2
3 
Ûúß ÛúÖê×™ü ŸÖ£ÖÖ ‘ÖÖŸÖ ÛúÖ µÖÖêÝÖ ×»Ö×ÜÖ‹ …  1 
 Write the sum of the order and degree of the differential equation 
 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7 
?
?
?
?
?
? d
2
y
dx
2
3 
 
3. ÃÖפü¿Ö 
?
a = 
^
i – 2
^
j Ûúß ×¤ü¿ÖÖ ´Öë ‹Ûú ÃÖפü¿Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛúÖ ¯Ö׸ü´ÖÖÞÖ 7 ‡ÛúÖ‡Ô Æîü …  1 
 Find a Vector in the direction of 
?
a = 
^
i – 2
^
j that has magnitude 7 units. 
 
4. µÖפü 
?
a ŸÖ£ÖÖ 
?
b ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü, ŸÖÖê 
?
a ŸÖ£ÖÖ 
?
b Ûêú ²Öß“Ö ÛúÖ ÛúÖêÞÖ ŒµÖÖ Æîü, •Ö²Ö×Ûú 2
?
a – 
?
b ‹Ûú ´ÖÖ¡ÖÛú 
ÃÖפü¿Ö Æîü ?    1 
 If 
?
a and 
?
b are unit Vectors, then what is the angle between 
?
a and 
?
b so that 2
?
a – 
?
b 
is a unit vector ? 
65/1/3 3 [P.T.O. 
5. µÖפü ‹Ûú ¸êüÜÖÖ ×®Ö¤ìü¿ÖÖÓÛú †õÖÖë Ûúß ¬Ö®ÖÖŸ´ÖÛú פü¿ÖÖ†Öë Ûêú ÃÖÖ£Ö ÛúÖêÞÖ a, ß, ? ²Ö®ÖÖŸÖß Æîü, ŸÖÖê sin
2
a + sin
2
ß + 
sin
2
? ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …  1 
 If a line makes angles a, ß, ? with the positive direction of coordinate axes, then write 
the value of sin
2
a + sin
2
ß + sin
2
?. 
 
6. µÖפü A = 
?
?
?
?
?
?
1 2
3 –1
, B = 
?
?
?
?
?
?
1 3
–1 1
 Æïü, ŸÖÖê AB ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …  1 
 If A = 
?
?
?
?
?
?
1 2
3 –1
 and B = 
?
?
?
?
?
?
1 3
–1 1
, write the value of AB . 
 
ÜÖÞ›ü – ²Ö 
SECTION – B 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …  
 Question numbers 7 to 19 carry 4 marks each. 
 
7. µÖפü y = 
x cos
–1
 x
1 – x
2
 – log 1 – x
2
 Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú 
dy
dx
 = 
cos
–1
 x
(1 – x
2
)
3/2
 4 
 If y = 
x cos
–1
 x
1 – x
2
 – log 1 – x
2
, then prove that 
dy
dx
 = 
cos
–1
 x
(1 – x
2
)
3/2
 
 
8. x Ûêú ÃÖÖ¯ÖêõÖ (sin x)
x
 + sin
–1
 x ÛúÖ †¾ÖÛú»Ö•Ö –ÖÖŸÖ Ûúßו֋ …  4 
 Find the derivative of (sin x)
x
 + sin
–1
 x w.r.t. x 
 
9. µÖפü x = a sec
3
 ?, y = a tan
3
 ? Æîü, ŸÖÖê 
d
2
y
dx
2
 ÛúÖ ´ÖÖ®Ö ? = 
p
4
 ¯Ö¸ü –ÖÖŸÖ Ûúßו֋ …  4 
 If x = a sec
3
 ?, y = a tan
3
 ?, find 
d
2
y
dx
2
 at ? = 
p
4
 
 
10. –ÖÖŸÖ Ûúßו֋ : 
)
?
(
 
(x
2
 + 1)e
x
(x + 1)
2
 dx 4 
 Find 
)
?
(
 
(x
2
 + 1)e
x
(x + 1)
2
 dx 
Page 4


65/1/3 1 [P.T.O. 
 
 
 
 
¸üÖê»Ö ®ÖÓ. 
Roll No. 
 
 
 
ÝÖ×ÞÖŸÖ 
MATHEMATICS  
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100 
Time allowed : 3 hours ] [ Maximum Marks : 100 
 
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö : 
 (i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …  
 (ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
 (iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æîü …  
 (iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æïü …  
 (v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ 
Æïü …  
 (vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …  
 Series : SSO/1/C 
ÛúÖê›ü ®ÖÓ. 
Code No.  
    
65/1/3
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …  
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …  
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …  
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê 
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê       
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …  
• Please check that this question paper contains 8 printed pages.  
• Code number given on the right hand side of the question paper should be written on the 
title page of the answer-book by the candidate. 
• Please check that this question paper contains 26 questions. 
• Please write down the Serial Number of the question before attempting it. 
• 15 minutes time has been allotted to read this question paper. The question paper will be 
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the 
question paper only and will not write any answer on the answer-book during this period. 
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü 
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë … 
Candidates must write the Code on 
the title page of the answer-book. 
 
SET – 3 
65/1/3 2  
General Instructions :   
 (i) All questions are compulsory. 
 (ii) Please check that this Question Paper contains 26 Questions. 
 (iii) Questions 1 – 6 in Section-A are Very Short Answer Type Questions carrying 1 mark 
each. 
 (iv) Questions 7 – 19 in Section-B are Long Answer I Type Questions carrying 4 marks 
each. 
 (v) Questions 20 – 26 in Section-C are Long Answer II Type Questions carrying               
6 marks each 
 (vi) Please write down the serial number of the Question before attempting it. 
  
   
ÜÖÞ›ü – † 
SECTION – A 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …  
 Question numbers 1 to 6 carry 1 mark each. 
 
1. C ÛúÖ ×¾Ö»ÖÖê¯Ö®Ö Ûú¸üÛêú ¾ÖÛÎúÖë Ûêú Ûãú»Ö xy = C cos x ÛúÖê ×®Öºþ×¯ÖŸÖ Ûú¸ü®Öê ¾ÖÖ»ÖÖ †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ …  1 
 Write the differential equation obtained by eliminating the arbitrary constant C in the 
equation representing the family of curves xy = C cos x. 
 
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7
?
?
?
?
?
? d
2
y
dx
2
3 
Ûúß ÛúÖê×™ü ŸÖ£ÖÖ ‘ÖÖŸÖ ÛúÖ µÖÖêÝÖ ×»Ö×ÜÖ‹ …  1 
 Write the sum of the order and degree of the differential equation 
 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7 
?
?
?
?
?
? d
2
y
dx
2
3 
 
3. ÃÖפü¿Ö 
?
a = 
^
i – 2
^
j Ûúß ×¤ü¿ÖÖ ´Öë ‹Ûú ÃÖפü¿Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛúÖ ¯Ö׸ü´ÖÖÞÖ 7 ‡ÛúÖ‡Ô Æîü …  1 
 Find a Vector in the direction of 
?
a = 
^
i – 2
^
j that has magnitude 7 units. 
 
4. µÖפü 
?
a ŸÖ£ÖÖ 
?
b ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü, ŸÖÖê 
?
a ŸÖ£ÖÖ 
?
b Ûêú ²Öß“Ö ÛúÖ ÛúÖêÞÖ ŒµÖÖ Æîü, •Ö²Ö×Ûú 2
?
a – 
?
b ‹Ûú ´ÖÖ¡ÖÛú 
ÃÖפü¿Ö Æîü ?    1 
 If 
?
a and 
?
b are unit Vectors, then what is the angle between 
?
a and 
?
b so that 2
?
a – 
?
b 
is a unit vector ? 
65/1/3 3 [P.T.O. 
5. µÖפü ‹Ûú ¸êüÜÖÖ ×®Ö¤ìü¿ÖÖÓÛú †õÖÖë Ûúß ¬Ö®ÖÖŸ´ÖÛú פü¿ÖÖ†Öë Ûêú ÃÖÖ£Ö ÛúÖêÞÖ a, ß, ? ²Ö®ÖÖŸÖß Æîü, ŸÖÖê sin
2
a + sin
2
ß + 
sin
2
? ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …  1 
 If a line makes angles a, ß, ? with the positive direction of coordinate axes, then write 
the value of sin
2
a + sin
2
ß + sin
2
?. 
 
6. µÖפü A = 
?
?
?
?
?
?
1 2
3 –1
, B = 
?
?
?
?
?
?
1 3
–1 1
 Æïü, ŸÖÖê AB ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …  1 
 If A = 
?
?
?
?
?
?
1 2
3 –1
 and B = 
?
?
?
?
?
?
1 3
–1 1
, write the value of AB . 
 
ÜÖÞ›ü – ²Ö 
SECTION – B 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …  
 Question numbers 7 to 19 carry 4 marks each. 
 
7. µÖפü y = 
x cos
–1
 x
1 – x
2
 – log 1 – x
2
 Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú 
dy
dx
 = 
cos
–1
 x
(1 – x
2
)
3/2
 4 
 If y = 
x cos
–1
 x
1 – x
2
 – log 1 – x
2
, then prove that 
dy
dx
 = 
cos
–1
 x
(1 – x
2
)
3/2
 
 
8. x Ûêú ÃÖÖ¯ÖêõÖ (sin x)
x
 + sin
–1
 x ÛúÖ †¾ÖÛú»Ö•Ö –ÖÖŸÖ Ûúßו֋ …  4 
 Find the derivative of (sin x)
x
 + sin
–1
 x w.r.t. x 
 
9. µÖפü x = a sec
3
 ?, y = a tan
3
 ? Æîü, ŸÖÖê 
d
2
y
dx
2
 ÛúÖ ´ÖÖ®Ö ? = 
p
4
 ¯Ö¸ü –ÖÖŸÖ Ûúßו֋ …  4 
 If x = a sec
3
 ?, y = a tan
3
 ?, find 
d
2
y
dx
2
 at ? = 
p
4
 
 
10. –ÖÖŸÖ Ûúßו֋ : 
)
?
(
 
(x
2
 + 1)e
x
(x + 1)
2
 dx 4 
 Find 
)
?
(
 
(x
2
 + 1)e
x
(x + 1)
2
 dx 
65/1/3 4  
11. ¤üÖê ×¾ÖªÖ»ÖµÖ A ŸÖ£ÖÖ B †¯Ö®Öê ×¾ÖªÖÙ£ÖµÖÖë ÛúÖê ÃÖÆüµÖÖêÝÖ Ûúß ³ÖÖ¾Ö®ÖÖ, ÃÖŸµÖ¾ÖÖפüŸÖÖ ŸÖ£ÖÖ ÃÖÆü®Ö¿Ö߻֟ÖÖ Ûêú ´Ö滵ÖÖë Ûêú 
×»Ö‹ ¯ÖÏ×ŸÖ ×¾ÖªÖ£Öá ÛÎú´Ö¿Ö: ` x, ` y ŸÖ£ÖÖ ` z ÛúÖ ¯Öã¸üÃÛúÖ¸ü ¤êü®ÖÖ “ÖÖÆüŸÖê Æïü … ×¾ÖªÖ»ÖµÖ A †¯Ö®Öê ÛÎú´Ö¿Ö: 3, 1, 2 
×¾ÖªÖÙ£ÖµÖÖë ÛúÖê ‡®Ö ´Ö滵ÖÖë Ûêú ×»Ö‹ Ûãú»Ö ` 1,100 ¯Öã¸üÃÛúÖ¸ü þֺþ¯Ö ¤êü®ÖÖ “ÖÖÆüŸÖÖ Æîü … ×¾ÖªÖ»ÖµÖ B †¯Ö®Öê ÛÎú´Ö¿Ö: 1, 
2, 3 ×¾ÖªÖÙ£ÖµÖÖë ÛúÖê ` 1,400 ¯Öã¸üÃÛúÖ¸ü þֺþ¯Ö ¤êü®ÖÖ “ÖÖÆüŸÖÖ Æîü … µÖפü ‡®Ö ŸÖß®ÖÖë ´Ö滵ÖÖë ¯Ö¸ü פü‹ ÝÖ‹ ‹Ûú-‹Ûú 
¯Öã¸üÃÛúÖ¸ü Ûúß Ûãú»Ö ¸üÖ×¿Ö ` 600 Æîü, ŸÖÖê  
 (i) ˆ¯Ö¸üÖêŒŸÖ ×ãÖ×ŸÖ ÛúÖê ¸îü×ÜÖÛú ÃÖ´ÖßÛú¸üÞÖ ²Ö®ÖÖÛú¸ü ‹Ûú †Ö¾µÖæÆü ÃÖ´ÖßÛú¸üÞÖ Ûêú ºþ¯Ö ´Öë ¾µÖŒŸÖ Ûúßו֋ …  
 (ii) ŒµÖÖ ÃÖ´ÖßÛú¸üÞÖ ×®ÖÛúÖµÖ ÛúÖê †Ö¾µÖæÆüÖë Ûêú ¯ÖϵÖÖêÝÖ ÃÖê Æü»Ö ×ÛúµÖÖ •ÖÖ ÃÖÛúŸÖÖ Æîü ? 
 (iii) †Ö¯Ö ×ÛúÃÖ ´Öæ»µÖ ¯Ö¸ü †×¬ÖÛúŸÖ´Ö ¯Öã¸üÃÛúÖ¸ü ¤êü®ÖÖ “ÖÖÆüŸÖê Æïü †Öî¸ü ŒµÖÖë ?  4 
 Two schools A and B decided to award prizes to their students for three values, team 
spirit, truthfulness and tolerance at the rate of ` x, ` y and ` z per student respectively. 
School A, decided to award a total of ` 1,100 for the three values to 3, 1 and 2 students 
respectively while school B decided to award ` 1,400 for the three values to 1, 2 and 3 
students respectively. If one prize for all the three values together amount to ` 600 
then 
 (i) Represent the above situation by a matrix equation after forming linear 
equations. 
 (ii) Is it possible to solve the system of equations so obtained using matrices ? 
 (iii) Which value you prefer to be rewarded most and why ? 
 
12. µÖפü [2x  3] 
?
?
?
?
?
?
1 2
–3 0
 
?
?
?
?
?
?
x
3
 = O Æîü, ŸÖÖê x –ÖÖŸÖ Ûúßו֋ … 4 
    †£Ö¾ÖÖ  
 †Ö¾µÖæÆü A = 
?
?
?
?
?
?
1 3 3
1 4 3
1 3 4
 ÛúÖ ¾µÖãŸÛÎú´Ö –ÖÖŸÖ Ûúßו֋ …  
 If [2x  3] 
?
?
?
?
?
?
1 2
–3 0
 
?
?
?
?
?
?
x
3
 = O, find x 
         OR 
 Find the inverse of the matrix A = 
?
?
?
?
?
?
1 3 3
1 4 3
1 3 4
 
Page 5


65/1/3 1 [P.T.O. 
 
 
 
 
¸üÖê»Ö ®ÖÓ. 
Roll No. 
 
 
 
ÝÖ×ÞÖŸÖ 
MATHEMATICS  
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100 
Time allowed : 3 hours ] [ Maximum Marks : 100 
 
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö : 
 (i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …  
 (ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
 (iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æîü …  
 (iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ Æïü …  
 (v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ׸üŸÖ 
Æïü …  
 (vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …  
 Series : SSO/1/C 
ÛúÖê›ü ®ÖÓ. 
Code No.  
    
65/1/3
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …  
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …  
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …  
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …  
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê 
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê       
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …  
• Please check that this question paper contains 8 printed pages.  
• Code number given on the right hand side of the question paper should be written on the 
title page of the answer-book by the candidate. 
• Please check that this question paper contains 26 questions. 
• Please write down the Serial Number of the question before attempting it. 
• 15 minutes time has been allotted to read this question paper. The question paper will be 
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the 
question paper only and will not write any answer on the answer-book during this period. 
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü 
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë … 
Candidates must write the Code on 
the title page of the answer-book. 
 
SET – 3 
65/1/3 2  
General Instructions :   
 (i) All questions are compulsory. 
 (ii) Please check that this Question Paper contains 26 Questions. 
 (iii) Questions 1 – 6 in Section-A are Very Short Answer Type Questions carrying 1 mark 
each. 
 (iv) Questions 7 – 19 in Section-B are Long Answer I Type Questions carrying 4 marks 
each. 
 (v) Questions 20 – 26 in Section-C are Long Answer II Type Questions carrying               
6 marks each 
 (vi) Please write down the serial number of the Question before attempting it. 
  
   
ÜÖÞ›ü – † 
SECTION – A 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …  
 Question numbers 1 to 6 carry 1 mark each. 
 
1. C ÛúÖ ×¾Ö»ÖÖê¯Ö®Ö Ûú¸üÛêú ¾ÖÛÎúÖë Ûêú Ûãú»Ö xy = C cos x ÛúÖê ×®Öºþ×¯ÖŸÖ Ûú¸ü®Öê ¾ÖÖ»ÖÖ †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ …  1 
 Write the differential equation obtained by eliminating the arbitrary constant C in the 
equation representing the family of curves xy = C cos x. 
 
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7
?
?
?
?
?
? d
2
y
dx
2
3 
Ûúß ÛúÖê×™ü ŸÖ£ÖÖ ‘ÖÖŸÖ ÛúÖ µÖÖêÝÖ ×»Ö×ÜÖ‹ …  1 
 Write the sum of the order and degree of the differential equation 
 1 + 
?
?
?
?
?
?
dy
dx
4 
= 7 
?
?
?
?
?
? d
2
y
dx
2
3 
 
3. ÃÖפü¿Ö 
?
a = 
^
i – 2
^
j Ûúß ×¤ü¿ÖÖ ´Öë ‹Ûú ÃÖפü¿Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛúÖ ¯Ö׸ü´ÖÖÞÖ 7 ‡ÛúÖ‡Ô Æîü …  1 
 Find a Vector in the direction of 
?
a = 
^
i – 2
^
j that has magnitude 7 units. 
 
4. µÖפü 
?
a ŸÖ£ÖÖ 
?
b ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü, ŸÖÖê 
?
a ŸÖ£ÖÖ 
?
b Ûêú ²Öß“Ö ÛúÖ ÛúÖêÞÖ ŒµÖÖ Æîü, •Ö²Ö×Ûú 2
?
a – 
?
b ‹Ûú ´ÖÖ¡ÖÛú 
ÃÖפü¿Ö Æîü ?    1 
 If 
?
a and 
?
b are unit Vectors, then what is the angle between 
?
a and 
?
b so that 2
?
a – 
?
b 
is a unit vector ? 
65/1/3 3 [P.T.O. 
5. µÖפü ‹Ûú ¸êüÜÖÖ ×®Ö¤ìü¿ÖÖÓÛú †õÖÖë Ûúß ¬Ö®ÖÖŸ´ÖÛú פü¿ÖÖ†Öë Ûêú ÃÖÖ£Ö ÛúÖêÞÖ a, ß, ? ²Ö®ÖÖŸÖß Æîü, ŸÖÖê sin
2
a + sin
2
ß + 
sin
2
? ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …  1 
 If a line makes angles a, ß, ? with the positive direction of coordinate axes, then write 
the value of sin
2
a + sin
2
ß + sin
2
?. 
 
6. µÖפü A = 
?
?
?
?
?
?
1 2
3 –1
, B = 
?
?
?
?
?
?
1 3
–1 1
 Æïü, ŸÖÖê AB ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …  1 
 If A = 
?
?
?
?
?
?
1 2
3 –1
 and B = 
?
?
?
?
?
?
1 3
–1 1
, write the value of AB . 
 
ÜÖÞ›ü – ²Ö 
SECTION – B 
  
 ¯ÖÏ¿®Ö ÃÖÓܵÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …  
 Question numbers 7 to 19 carry 4 marks each. 
 
7. µÖפü y = 
x cos
–1
 x
1 – x
2
 – log 1 – x
2
 Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú 
dy
dx
 = 
cos
–1
 x
(1 – x
2
)
3/2
 4 
 If y = 
x cos
–1
 x
1 – x
2
 – log 1 – x
2
, then prove that 
dy
dx
 = 
cos
–1
 x
(1 – x
2
)
3/2
 
 
8. x Ûêú ÃÖÖ¯ÖêõÖ (sin x)
x
 + sin
–1
 x ÛúÖ †¾ÖÛú»Ö•Ö –ÖÖŸÖ Ûúßו֋ …  4 
 Find the derivative of (sin x)
x
 + sin
–1
 x w.r.t. x 
 
9. µÖפü x = a sec
3
 ?, y = a tan
3
 ? Æîü, ŸÖÖê 
d
2
y
dx
2
 ÛúÖ ´ÖÖ®Ö ? = 
p
4
 ¯Ö¸ü –ÖÖŸÖ Ûúßו֋ …  4 
 If x = a sec
3
 ?, y = a tan
3
 ?, find 
d
2
y
dx
2
 at ? = 
p
4
 
 
10. –ÖÖŸÖ Ûúßו֋ : 
)
?
(
 
(x
2
 + 1)e
x
(x + 1)
2
 dx 4 
 Find 
)
?
(
 
(x
2
 + 1)e
x
(x + 1)
2
 dx 
65/1/3 4  
11. ¤üÖê ×¾ÖªÖ»ÖµÖ A ŸÖ£ÖÖ B †¯Ö®Öê ×¾ÖªÖÙ£ÖµÖÖë ÛúÖê ÃÖÆüµÖÖêÝÖ Ûúß ³ÖÖ¾Ö®ÖÖ, ÃÖŸµÖ¾ÖÖפüŸÖÖ ŸÖ£ÖÖ ÃÖÆü®Ö¿Ö߻֟ÖÖ Ûêú ´Ö滵ÖÖë Ûêú 
×»Ö‹ ¯ÖÏ×ŸÖ ×¾ÖªÖ£Öá ÛÎú´Ö¿Ö: ` x, ` y ŸÖ£ÖÖ ` z ÛúÖ ¯Öã¸üÃÛúÖ¸ü ¤êü®ÖÖ “ÖÖÆüŸÖê Æïü … ×¾ÖªÖ»ÖµÖ A †¯Ö®Öê ÛÎú´Ö¿Ö: 3, 1, 2 
×¾ÖªÖÙ£ÖµÖÖë ÛúÖê ‡®Ö ´Ö滵ÖÖë Ûêú ×»Ö‹ Ûãú»Ö ` 1,100 ¯Öã¸üÃÛúÖ¸ü þֺþ¯Ö ¤êü®ÖÖ “ÖÖÆüŸÖÖ Æîü … ×¾ÖªÖ»ÖµÖ B †¯Ö®Öê ÛÎú´Ö¿Ö: 1, 
2, 3 ×¾ÖªÖÙ£ÖµÖÖë ÛúÖê ` 1,400 ¯Öã¸üÃÛúÖ¸ü þֺþ¯Ö ¤êü®ÖÖ “ÖÖÆüŸÖÖ Æîü … µÖפü ‡®Ö ŸÖß®ÖÖë ´Ö滵ÖÖë ¯Ö¸ü פü‹ ÝÖ‹ ‹Ûú-‹Ûú 
¯Öã¸üÃÛúÖ¸ü Ûúß Ûãú»Ö ¸üÖ×¿Ö ` 600 Æîü, ŸÖÖê  
 (i) ˆ¯Ö¸üÖêŒŸÖ ×ãÖ×ŸÖ ÛúÖê ¸îü×ÜÖÛú ÃÖ´ÖßÛú¸üÞÖ ²Ö®ÖÖÛú¸ü ‹Ûú †Ö¾µÖæÆü ÃÖ´ÖßÛú¸üÞÖ Ûêú ºþ¯Ö ´Öë ¾µÖŒŸÖ Ûúßו֋ …  
 (ii) ŒµÖÖ ÃÖ´ÖßÛú¸üÞÖ ×®ÖÛúÖµÖ ÛúÖê †Ö¾µÖæÆüÖë Ûêú ¯ÖϵÖÖêÝÖ ÃÖê Æü»Ö ×ÛúµÖÖ •ÖÖ ÃÖÛúŸÖÖ Æîü ? 
 (iii) †Ö¯Ö ×ÛúÃÖ ´Öæ»µÖ ¯Ö¸ü †×¬ÖÛúŸÖ´Ö ¯Öã¸üÃÛúÖ¸ü ¤êü®ÖÖ “ÖÖÆüŸÖê Æïü †Öî¸ü ŒµÖÖë ?  4 
 Two schools A and B decided to award prizes to their students for three values, team 
spirit, truthfulness and tolerance at the rate of ` x, ` y and ` z per student respectively. 
School A, decided to award a total of ` 1,100 for the three values to 3, 1 and 2 students 
respectively while school B decided to award ` 1,400 for the three values to 1, 2 and 3 
students respectively. If one prize for all the three values together amount to ` 600 
then 
 (i) Represent the above situation by a matrix equation after forming linear 
equations. 
 (ii) Is it possible to solve the system of equations so obtained using matrices ? 
 (iii) Which value you prefer to be rewarded most and why ? 
 
12. µÖפü [2x  3] 
?
?
?
?
?
?
1 2
–3 0
 
?
?
?
?
?
?
x
3
 = O Æîü, ŸÖÖê x –ÖÖŸÖ Ûúßו֋ … 4 
    †£Ö¾ÖÖ  
 †Ö¾µÖæÆü A = 
?
?
?
?
?
?
1 3 3
1 4 3
1 3 4
 ÛúÖ ¾µÖãŸÛÎú´Ö –ÖÖŸÖ Ûúßו֋ …  
 If [2x  3] 
?
?
?
?
?
?
1 2
–3 0
 
?
?
?
?
?
?
x
3
 = O, find x 
         OR 
 Find the inverse of the matrix A = 
?
?
?
?
?
?
1 3 3
1 4 3
1 3 4
 
65/1/3 5 [P.T.O. 
13. ÃÖÖ¸ü×ÞÖÛúÖë Ûêú ÝÖãÞÖ¬Ö´ÖÖí ÛúÖ ¯ÖϵÖÖêÝÖ Ûú¸üÛêú ×ÃÖ¨ü Ûúßו֋ ×Ûú  4 
 
?
?
?
?
?
?
(a + 1) (a + 2) a + 2 1
(a + 2) (a + 3) a + 3 1
(a + 3) (a + 4) a + 4 1
= –2 
 Using properties of determinants, prove that 
 
?
?
?
?
?
?
(a + 1) (a + 2) a + 2 1
(a + 2) (a + 3) a + 3 1
(a + 3) (a + 4) a + 4 1
= –2 
 
14. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ : 
)
?
(
0
p/2
 
dx
1 + tan x
 4 
 Evaluate : 
)
?
(
0
p/2
 
dx
1 + tan x
 
 
15. –ÖÖŸÖ Ûúßו֋ : 
)
?
(
 
x
(x – 1)
2
 (x + 2)
 dx 4 
   †£Ö¾ÖÖ  
 –ÖÖŸÖ Ûúßו֋ : 
)
?
(
 
x + 2
2x
2
 + 6x + 5
 dx 
 Find : 
)
?
(
 
x
(x – 1)
2
 (x + 2)
 dx 
    OR 
 Find : 
)
?
(
 
x + 2
2x
2
 + 6x + 5
 dx 
 
16. 15 ²Ö»²ÖÖë Ûêú œÌêü¸ü ÃÖê, וÖÃÖ´Öë 5 ²Ö»²Ö ÜÖ¸üÖ²Ö Æïü, 2 ²Ö»²ÖÖë ÛúÖ ‹Ûú ®Ö´Öæ®ÖÖ µÖÖ¥ü“”ûµÖÖ, ײ֮ÖÖ ¯ÖÏןÖãÖÖ¯Ö®ÖÖ Ûêú ×®ÖÛúÖ»ÖÖ 
•ÖÖŸÖÖ Æîü … ÜÖ¸üÖ²Ö ²Ö»²ÖÖë Ûúß ÃÖÓܵÖÖ ÛúÖ ¯ÖÏÖ×µÖÛúŸÖÖ ²ÖÓ™ü®Ö –ÖÖŸÖ Ûúßו֋ …  4 
†£Ö¾ÖÖ 
 ‹Ûú ×¾Ö¿ÖêÂÖ ÃÖ´ÖõÖÖ ÛúÖê A ŸÖ£ÖÖ B «üÖ¸üÖ Ã¾ÖŸÖÓ¡Ö ºþ¯Ö ÃÖê Æü»Ö Ûú¸ü®Öê Ûúß ¯ÖÏÖ×µÖÛúŸÖÖ‹Ñ ÛÎú´Ö¿Ö: 
1
2
 ŸÖ£ÖÖ 
1
3
 Æïü … µÖפü 
¤üÖê®ÖÖë þ֟ÖÓ¡Ö ºþ¯Ö ÃÖê ÃÖ´ÖõÖÖ ÛúÖê Æü»Ö Ûú¸ü®Öê ÛúÖ ¯ÖϵÖÖÃÖ Ûú¸üŸÖê Æïü, ŸÖÖê ¯ÖÏÖ×µÖÛúŸÖÖ –ÖÖŸÖ Ûúßו֋ ×Ûú  
 (i) ÃÖ´ÖõÖÖ Æü»Ö ÆüÖê •ÖÖŸÖß Æîü …  
 (ii) ˆ®Ö´Öë ÃÖê ŸÖ£µÖŸÖÖ ÛúÖê‡Ô ‹Ûú ÃÖ´ÖõÖÖ Æü»Ö Ûú¸ü »ÖêŸÖÖ Æîü …