JEE Exam > JEE Notes > Additional Study Material for JEE > Past Year Paper, Maths (Set - 2),Outside Delhi, 2015, Class 12, Maths
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Page 1
65/2 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.
65/2
• 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 – 2
Page 2
65/2 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.
65/2
• 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 – 2
65/2 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. 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.
2. †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.
3. †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
.
4. †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
.
Page 3
65/2 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.
65/2
• 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 – 2
65/2 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. 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.
2. †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.
3. †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
.
4. †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
.
65/2 3 [P.T.O.
5. 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.
6. 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.
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. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
8. –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.
Page 4
65/2 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.
65/2
• 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 – 2
65/2 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. 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.
2. †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.
3. †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
.
4. †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
.
65/2 3 [P.T.O.
5. 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.
6. 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.
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. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
8. –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.
65/2 4
9. 52 ŸÖÖ¿Ö Ûêú ¯Ö¢ÖÖë Ûúß ³Ö»Öß ³ÖÖÓ×ŸÖ ±ëú™üß ÝÖ‡Ô ÝÖøüß ´Öë ÃÖê 4 ¯Ö¢Öê ˆ¢Ö¸üÖê¢Ö¸ü ¯ÖÏןÖãÖÖ¯Ö®ÖÖ ÃÖ×ÆüŸÖ ×®ÖÛúÖ»Öê •ÖÖŸÖê Æïü …
‡ÃÖÛúß ŒµÖÖ ¯ÖÏÖ×µÖÛúŸÖÖ Æîü ×Ûú 4
(i) ÃÖ³Öß 4 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
(ii) Ûêú¾Ö»Ö 2 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
†£Ö¾ÖÖ
¯ÖÖÃÖÖë Ûêú ‹Ûú •ÖÖê›Ìêü ÛúÖê “ÖÖ¸ü ²ÖÖ¸ü ˆ”ûÖ»Ö®Öê ¯Ö¸ü ׫üÛúÖë Ûúß ÃÖÓܵÖÖ ÛúÖ ¯ÖÏÖ×µÖÛúŸÖÖ ²ÖÓ™ü®Ö –ÖÖŸÖ Ûúßו֋ … ‡ÃÖ ²ÖÓ™ü®Ö ÛúÖ
´ÖÖ¬µÖ ³Öß –ÖÖŸÖ Ûúßו֋ …
Four cards are drawn successively with replacement from a well shuffled deck of 52
cards. What is the probability that
(i) all the four cards are spades ?
(ii) only 2 cards are spades ?
OR
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the
probability distribution of number of successes. Hence find the mean of the
distribution.
10. ×ÃÖ¨ü Ûúßו֋ ×Ûú : [ ]
?
a ,
?
b +
?
c ,
?
d = [ ]
?
a ,
?
b,
?
d + [ ]
?
a ,
?
c ,
?
d 4
Prove that [ ]
?
a ,
?
b +
?
c ,
?
d = [ ]
?
a ,
?
b,
?
d + [ ]
?
a ,
?
c ,
?
d
11. ¸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
12. ×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
Page 5
65/2 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.
65/2
• 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 – 2
65/2 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. 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.
2. †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.
3. †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
.
4. †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
.
65/2 3 [P.T.O.
5. 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.
6. 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.
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. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
8. –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.
65/2 4
9. 52 ŸÖÖ¿Ö Ûêú ¯Ö¢ÖÖë Ûúß ³Ö»Öß ³ÖÖÓ×ŸÖ ±ëú™üß ÝÖ‡Ô ÝÖøüß ´Öë ÃÖê 4 ¯Ö¢Öê ˆ¢Ö¸üÖê¢Ö¸ü ¯ÖÏןÖãÖÖ¯Ö®ÖÖ ÃÖ×ÆüŸÖ ×®ÖÛúÖ»Öê •ÖÖŸÖê Æïü …
‡ÃÖÛúß ŒµÖÖ ¯ÖÏÖ×µÖÛúŸÖÖ Æîü ×Ûú 4
(i) ÃÖ³Öß 4 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
(ii) Ûêú¾Ö»Ö 2 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
†£Ö¾ÖÖ
¯ÖÖÃÖÖë Ûêú ‹Ûú •ÖÖê›Ìêü ÛúÖê “ÖÖ¸ü ²ÖÖ¸ü ˆ”ûÖ»Ö®Öê ¯Ö¸ü ׫üÛúÖë Ûúß ÃÖÓܵÖÖ ÛúÖ ¯ÖÏÖ×µÖÛúŸÖÖ ²ÖÓ™ü®Ö –ÖÖŸÖ Ûúßו֋ … ‡ÃÖ ²ÖÓ™ü®Ö ÛúÖ
´ÖÖ¬µÖ ³Öß –ÖÖŸÖ Ûúßו֋ …
Four cards are drawn successively with replacement from a well shuffled deck of 52
cards. What is the probability that
(i) all the four cards are spades ?
(ii) only 2 cards are spades ?
OR
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the
probability distribution of number of successes. Hence find the mean of the
distribution.
10. ×ÃÖ¨ü Ûúßו֋ ×Ûú : [ ]
?
a ,
?
b +
?
c ,
?
d = [ ]
?
a ,
?
b,
?
d + [ ]
?
a ,
?
c ,
?
d 4
Prove that [ ]
?
a ,
?
b +
?
c ,
?
d = [ ]
?
a ,
?
b,
?
d + [ ]
?
a ,
?
c ,
?
d
11. ¸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
12. ×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
65/2 5 [P.T.O.
13. ? 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.
14. 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
.
15. 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.
16. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
x + 3
5 – 4x – 2x
2
dx 4
Find
)
?
(
.
.
x + 3
5 – 4x – 2x
2
dx.
17. ×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 ?
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FAQs on Past Year Paper, Maths (Set - 2),Outside Delhi, 2015, Class 12, Maths - Additional Study Material for JEE
| 1. What is the difficulty level of the Maths JEE exam for Class 12 students? |  |
Ans. The Maths JEE exam for Class 12 students is considered to be highly challenging. It requires a deep understanding of mathematical concepts and problem-solving skills. Students need to practice extensively and have a strong foundation in mathematics to perform well in this exam.
| 2. How can I prepare effectively for the Maths JEE exam? | |
Ans. Effective preparation for the Maths JEE exam involves several steps. Firstly, students should thoroughly study the prescribed syllabus and understand the concepts. They should then practice solving a variety of problems, including previous year question papers and sample papers. Additionally, joining coaching classes or online programs can provide guidance and additional practice material. Regular revision and time management are also crucial for effective preparation.
| 3. Are calculators allowed in the Maths JEE exam? | |
Ans. No, calculators are not allowed in the Maths JEE exam. Students are expected to solve the mathematical problems manually without any electronic devices. Hence, it is important to practice mental calculations and sharpen problem-solving skills without relying on calculators.
| 4. What are the important topics to focus on for the Maths JEE exam? | |
Ans. The Maths JEE exam covers a wide range of topics from the Class 12 syllabus. Some of the important topics to focus on include calculus, algebra, coordinate geometry, vectors, and three-dimensional geometry. These topics have a significant weightage in the exam and require thorough understanding and practice.
| 5. How should I manage my time during the Maths JEE exam? | |
Ans. Time management is crucial during the Maths JEE exam. It is advisable to allocate specific time slots for each section and question. Start with the easier questions to build confidence and save time for the more challenging ones. It is important to keep track of time and avoid spending too much time on a single question. Regular practice and mock tests can help in developing effective time management skills.
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