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Page 1
65/3 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 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/3
• 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
65/3 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 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/3
• 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
65/3 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 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. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 x
.
3. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
4. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
5. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
Page 3
65/3 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 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/3
• 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
65/3 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 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. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 x
.
3. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
4. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
5. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
65/3 3 [P.T.O.
6. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
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. ¸êüÜÖÖ†Öë
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k ŸÖ£ÖÖ
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k Ûêú ²Öß“Ö
®µÖæ®ÖŸÖ´Ö ¤æü¸üß –ÖÖŸÖ Ûúßו֋ … 4
Find the shortest distance between the following lines :
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k and
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k
8. ×ÃÖ¨ü Ûúßו֋ ×Ûú 2tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
4
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
Prove that 2 tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
OR
Solve for x : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
9. ? Ûêú ×ÛúÃÖ ´ÖÖ®Ö Ûêú ×»Ö‹ ±ú»Ö®Ö f(x) =
?
?
?
?
?
?(x
2
+ 2), µÖפü x = 0
4x + 6 , µÖפü x > 0
x = 0 ¯Ö¸ü ÃÖÓŸÖŸÖ Æîü … †ŸÖ: x = 0 ¯Ö¸ü
±ú»Ö®Ö Ûúß †¾ÖÛú»Ö®ÖßµÖŸÖÖ Ûúß •ÖÖÑ“Ö Ûúßו֋ … 4
For what value of ? the function defined by f(x) =
? ?
?
?
?
?(x
2
+ 2), if x = 0
4x + 6 , if x > 0
is continuous at
x = 0 ? Hence check the differentiability of f(x) at x = 0.
Page 4
65/3 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 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/3
• 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
65/3 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 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. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 x
.
3. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
4. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
5. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
65/3 3 [P.T.O.
6. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
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. ¸êüÜÖÖ†Öë
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k ŸÖ£ÖÖ
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k Ûêú ²Öß“Ö
®µÖæ®ÖŸÖ´Ö ¤æü¸üß –ÖÖŸÖ Ûúßו֋ … 4
Find the shortest distance between the following lines :
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k and
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k
8. ×ÃÖ¨ü Ûúßו֋ ×Ûú 2tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
4
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
Prove that 2 tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
OR
Solve for x : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
9. ? Ûêú ×ÛúÃÖ ´ÖÖ®Ö Ûêú ×»Ö‹ ±ú»Ö®Ö f(x) =
?
?
?
?
?
?(x
2
+ 2), µÖפü x = 0
4x + 6 , µÖפü x > 0
x = 0 ¯Ö¸ü ÃÖÓŸÖŸÖ Æîü … †ŸÖ: x = 0 ¯Ö¸ü
±ú»Ö®Ö Ûúß †¾ÖÛú»Ö®ÖßµÖŸÖÖ Ûúß •ÖÖÑ“Ö Ûúßו֋ … 4
For what value of ? the function defined by f(x) =
? ?
?
?
?
?(x
2
+ 2), if x = 0
4x + 6 , if x > 0
is continuous at
x = 0 ? Hence check the differentiability of f(x) at x = 0.
65/3 4
10. µÖפü x = ae
t
(sin t + cos t) ŸÖ£ÖÖ y = ae
t
(sin t – cos t) Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú
dy
dx
=
x + y
x – y
. 4
If x = ae
t
(sin t + cos t) and y = ae
t
(sin t – cos t), prove that
dy
dx
=
x + y
x – y
.
11. µÖפü y = Ae
mx
+ Be
nx
Æîü, ŸÖÖê ¤ü¿ÖÖÔ‡‹ ×Ûú
d
2
y
dx
2
– (m + n)
dy
dx
+ mny = 0 4
If y = Ae
mx
+ Be
nx
, show that
d
2
y
dx
2
– (m + n)
dy
dx
+ mny = 0.
12. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
x + 3
5 – 4x – 2x
2
dx 4
Find
)
?
(
.
.
x + 3
5 – 4x – 2x
2
dx.
13. ×ÛúÃÖß ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö Ûêú ¯ÖÖÃÖ ` 35,000 ÛúÖ ÛúÖêÂÖ Æîü וÖÃÖê ¤üÖê ׳֮®Ö-׳֮®Ö ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üüÖë ´Öë ×®Ö¾Öê×¿ÖŸÖ Ûú¸ü®ÖÖ
Æîü … ¯ÖÏ£Ö´Ö ²ÖÖÑ›ü ¯Ö¸ü 8% ¾ÖÖÙÂÖÛú ²µÖÖ•Ö Æîü, וÖÃÖê ‹Ûú †®ÖÖ£ÖÖ»ÖµÖ ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü ŸÖ£ÖÖ ×«üŸÖßµÖ ²ÖÖÑ›ü ¯Ö¸ü 10%
²µÖÖ•Ö Æîü וÖÃÖê ‹Ûú ‹®Ö.•Öß.†Öê. (ÛïúÃÖ¸ü ‹ò›ü ÃÖÖêÃÖÖ‡™üß) ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü … †Ö¾µÖæÆü ÝÖãÞÖ®Ö Ûêú ¯ÖϵÖÖêÝÖ ÃÖê µÖÆü
×®Ö¬ÖÖÔ׸üŸÖ Ûúßו֋ ×Ûú ` 35,000 Ûêú ÛúÖêÂÖ ÛúÖê ¤üÖê ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üÖë ´Öë ×®Ö¾Öê¿Ö Ûú¸ü®Öê Ûêú ×»Ö‹ ×ÛúÃÖ ¯ÖÏÛúÖ¸ü ²ÖÖÑ™ëü
וÖÃÖÃÖê ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö ÛúÖê ¯ÖÏÖ¯ŸÖ Ûãú»Ö ²µÖÖ•Ö ` 3,200 ÆüÖê ?
‡ÃÖ ¯ÖÏ¿®Ö ÃÖê ŒµÖÖ ´Öæ»µÖ •Ö×®ÖŸÖ ÆüÖêŸÖê Æïü ? 4
A trust fund has ` 35,000 is to be invested in two different types of bonds. The first
bond pays 8% interest per annum which will be given to orphanage and second bond
pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society).
Using matrix multiplication, determine how to divide ` 35,000 among two types of
bonds if the trust fund obtains an annual total interest of ` 3,200. What are the values
reflected in this question ?
Page 5
65/3 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 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/3
• 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
65/3 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 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. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
2. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 x
.
3. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
4. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
5. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
65/3 3 [P.T.O.
6. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
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. ¸êüÜÖÖ†Öë
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k ŸÖ£ÖÖ
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k Ûêú ²Öß“Ö
®µÖæ®ÖŸÖ´Ö ¤æü¸üß –ÖÖŸÖ Ûúßו֋ … 4
Find the shortest distance between the following lines :
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k and
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k
8. ×ÃÖ¨ü Ûúßו֋ ×Ûú 2tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
4
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
Prove that 2 tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
OR
Solve for x : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
9. ? Ûêú ×ÛúÃÖ ´ÖÖ®Ö Ûêú ×»Ö‹ ±ú»Ö®Ö f(x) =
?
?
?
?
?
?(x
2
+ 2), µÖפü x = 0
4x + 6 , µÖפü x > 0
x = 0 ¯Ö¸ü ÃÖÓŸÖŸÖ Æîü … †ŸÖ: x = 0 ¯Ö¸ü
±ú»Ö®Ö Ûúß †¾ÖÛú»Ö®ÖßµÖŸÖÖ Ûúß •ÖÖÑ“Ö Ûúßו֋ … 4
For what value of ? the function defined by f(x) =
? ?
?
?
?
?(x
2
+ 2), if x = 0
4x + 6 , if x > 0
is continuous at
x = 0 ? Hence check the differentiability of f(x) at x = 0.
65/3 4
10. µÖפü x = ae
t
(sin t + cos t) ŸÖ£ÖÖ y = ae
t
(sin t – cos t) Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú
dy
dx
=
x + y
x – y
. 4
If x = ae
t
(sin t + cos t) and y = ae
t
(sin t – cos t), prove that
dy
dx
=
x + y
x – y
.
11. µÖפü y = Ae
mx
+ Be
nx
Æîü, ŸÖÖê ¤ü¿ÖÖÔ‡‹ ×Ûú
d
2
y
dx
2
– (m + n)
dy
dx
+ mny = 0 4
If y = Ae
mx
+ Be
nx
, show that
d
2
y
dx
2
– (m + n)
dy
dx
+ mny = 0.
12. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
x + 3
5 – 4x – 2x
2
dx 4
Find
)
?
(
.
.
x + 3
5 – 4x – 2x
2
dx.
13. ×ÛúÃÖß ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö Ûêú ¯ÖÖÃÖ ` 35,000 ÛúÖ ÛúÖêÂÖ Æîü וÖÃÖê ¤üÖê ׳֮®Ö-׳֮®Ö ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üüÖë ´Öë ×®Ö¾Öê×¿ÖŸÖ Ûú¸ü®ÖÖ
Æîü … ¯ÖÏ£Ö´Ö ²ÖÖÑ›ü ¯Ö¸ü 8% ¾ÖÖÙÂÖÛú ²µÖÖ•Ö Æîü, וÖÃÖê ‹Ûú †®ÖÖ£ÖÖ»ÖµÖ ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü ŸÖ£ÖÖ ×«üŸÖßµÖ ²ÖÖÑ›ü ¯Ö¸ü 10%
²µÖÖ•Ö Æîü וÖÃÖê ‹Ûú ‹®Ö.•Öß.†Öê. (ÛïúÃÖ¸ü ‹ò›ü ÃÖÖêÃÖÖ‡™üß) ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü … †Ö¾µÖæÆü ÝÖãÞÖ®Ö Ûêú ¯ÖϵÖÖêÝÖ ÃÖê µÖÆü
×®Ö¬ÖÖÔ׸üŸÖ Ûúßו֋ ×Ûú ` 35,000 Ûêú ÛúÖêÂÖ ÛúÖê ¤üÖê ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üÖë ´Öë ×®Ö¾Öê¿Ö Ûú¸ü®Öê Ûêú ×»Ö‹ ×ÛúÃÖ ¯ÖÏÛúÖ¸ü ²ÖÖÑ™ëü
וÖÃÖÃÖê ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö ÛúÖê ¯ÖÏÖ¯ŸÖ Ûãú»Ö ²µÖÖ•Ö ` 3,200 ÆüÖê ?
‡ÃÖ ¯ÖÏ¿®Ö ÃÖê ŒµÖÖ ´Öæ»µÖ •Ö×®ÖŸÖ ÆüÖêŸÖê Æïü ? 4
A trust fund has ` 35,000 is to be invested in two different types of bonds. The first
bond pays 8% interest per annum which will be given to orphanage and second bond
pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society).
Using matrix multiplication, determine how to divide ` 35,000 among two types of
bonds if the trust fund obtains an annual total interest of ` 3,200. What are the values
reflected in this question ?
65/3 5 [P.T.O.
14. †Ö¾µÖæÆü A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
ÛúÖê ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü ŸÖ£ÖÖ ‹Ûú ×¾ÖÂÖ´Ö-ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Ûêú µÖÖêÝÖ Ûêú ºþ¯Ö ´Öë
¾µÖŒŸÖ Ûúßו֋ … 4
Express the matrix A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
as the sum of a symmetric and skew
symmetric matrix.
†£Ö¾ÖÖ/OR
µÖפü A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
Æîü, ŸÖÖê ÃÖŸµÖÖ×¯ÖŸÖ Ûúßו֋ ×Ûú (AB)
–1
= B
–1
A
–1
If A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
, verify that (AB)
–1
= B
–1
A
–1
.
15. ÃÖÖ¸ü×ÞÖÛúÖë Ûêú ÝÖãÞÖ¬Ö´ÖÖí Ûêú ¯ÖϵÖÖêÝÖ ÃÖê ×®Ö´®Ö ÛúÖê x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : 4
?
?
?
?
?
?
?
?
a + x a – x a – x
a – x a + x a – x
a – x a – x a + x
= 0
Using properties of determinants, solve for x :
?
?
?
?
?
?
?
?
a + x a – x a – x
a – x a + x a – x
a – x a – x a + x
= 0
16. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
17. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
x
(x
2
+ 1) (x – 1)
dx 4
†£Ö¾ÖÖ
´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
1
2
.
.
sin
–1
x
(1 – x
2
)
3/2
dx
Find
)
?
(
.
.
x
(x
2
+ 1) (x – 1)
dx.
OR
Find
)
?
(
0
1
2
.
.
sin
–1
x
(1 – x
2
)
3/2
dx.
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FAQs on Past Year Paper, Maths (Set - 3),Outside Delhi, 2015, Class 12, Maths - Additional Study Material for JEE
| 1. What are the important topics covered in the Maths JEE exam for Class 12? |  |
Ans. The important topics covered in the Maths JEE exam for Class 12 include calculus, algebra, coordinate geometry, vectors and 3D geometry, differential equations, probability, and statistics.
| 2. How should I prepare for the Maths JEE exam for Class 12? | |
Ans. To prepare for the Maths JEE exam for Class 12, it is important to thoroughly understand the concepts and practice solving a variety of problems. It is recommended to refer to the NCERT textbooks, solve previous year papers, and take mock tests to assess your preparation level.
| 3. What is the marking scheme for the Maths JEE exam for Class 12? | |
Ans. The marking scheme for the Maths JEE exam for Class 12 may vary, but generally, each question carries a certain number of marks. There may be multiple-choice questions, short answer type questions, and long answer type questions. It is important to read the instructions carefully and allocate time accordingly to each section.
| 4. Are there any specific tips to improve speed and accuracy in the Maths JEE exam for Class 12? | |
Ans. Yes, there are some tips to improve speed and accuracy in the Maths JEE exam for Class 12. It is essential to practice time management, solve previous year papers within the given time limit, and develop shortcut techniques for calculations. Regular practice and self-assessment will help in improving speed and accuracy.
| 5. How can I analyze my performance in the Maths JEE exam for Class 12? | |
Ans. To analyze your performance in the Maths JEE exam for Class 12, you can compare your answers with the official answer key provided by the exam conducting authority. Calculate your score and identify the areas where you made mistakes or struggled. This analysis will help you understand your strengths and weaknesses, allowing you to focus on improving specific topics before the final exam.
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