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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 …
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FAQs on Past Year Paper, Mathematics (Set - 3), Delhi, 2015, Class 12, Maths - Additional Study Material for JEE
| 1. What are the important topics to study in Mathematics for the Class 12 Maths JEE exam? |  |
Ans. Some important topics to study in Mathematics for the Class 12 Maths JEE exam include calculus, algebra, coordinate geometry, vectors and 3D geometry, probability, and complex numbers.
| 2. How should I prepare for the Class 12 Maths JEE exam? | |
Ans. To prepare for the Class 12 Maths JEE exam, it is important to understand the concepts thoroughly. Practice solving a variety of problems from different topics and refer to textbooks and study materials recommended for JEE preparation. Additionally, solving past year papers and taking mock tests can help in evaluating your preparation level.
| 3. Are there any specific tips to solve calculus problems effectively in the Class 12 Maths JEE exam? | |
Ans. Yes, to solve calculus problems effectively in the Class 12 Maths JEE exam, it is important to understand the concepts of limits, derivatives, and integrals. Practice different types of problems and understand the applications of calculus in real-life scenarios. Also, make sure to revise the formulas and properties related to calculus regularly.
| 4. Is it necessary to solve the Class 12 Maths JEE exam using pen and paper, or can it be solved using a calculator? | |
Ans. In the Class 12 Maths JEE exam, you are not allowed to use a calculator. It is important to solve the problems using pen and paper to demonstrate your understanding of concepts and problem-solving skills. Calculators are not provided or allowed during the exam.
| 5. How can I manage my time effectively during the Class 12 Maths JEE exam? | |
Ans. Time management is crucial during the Class 12 Maths JEE exam. Start by allocating a specific amount of time to each section based on the number of questions and the difficulty level. Prioritize solving the questions you find easier first, and then move on to the more challenging ones. Avoid spending too much time on a single question and make sure to leave some time for revising your answers before submitting the exam.
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