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Page 1
65/1 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.
65/1
• 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 – 1
Page 2
65/1 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.
65/1
• 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 – 1
65/1 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. 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.
2. 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.
3. 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.
4. †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.
Page 3
65/1 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.
65/1
• 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 – 1
65/1 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. 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.
2. 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.
3. 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.
4. †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.
65/1 3 [P.T.O.
5. †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
.
6. †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
.
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. ×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 ?
8. †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
.
Page 4
65/1 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.
65/1
• 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 – 1
65/1 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. 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.
2. 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.
3. 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.
4. †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.
65/1 3 [P.T.O.
5. †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
.
6. †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
.
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. ×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 ?
8. †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
.
65/1 4
9. 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
10. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
11. –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.
12. 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֋ …
Page 5
65/1 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.
65/1
• 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 – 1
65/1 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. 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.
2. 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.
3. 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.
4. †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.
65/1 3 [P.T.O.
5. †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
.
6. †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
.
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. ×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 ?
8. †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
.
65/1 4
9. 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
10. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
11. –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.
12. 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֋ …
65/1 5 [P.T.O.
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.
13. ×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
14. ¸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
15. ×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
16. ? 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.
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FAQs on Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths - Additional Study Material for JEE
| 1. What is the exam pattern for the Class 12 Maths JEE exam? |  |
Ans. The Class 12 Maths JEE exam follows a set pattern that includes multiple-choice questions, numerical value-based questions, and integer answer type questions. The exam also has a negative marking scheme where 1 mark is deducted for every incorrect answer.
| 2. How can I prepare effectively for the Class 12 Maths JEE exam? | |
Ans. To prepare effectively for the Class 12 Maths JEE exam, it is recommended to thoroughly understand the concepts and practice a variety of questions. Solve previous year papers, mock tests, and sample papers to get a good grasp of the exam pattern and time management. Seek guidance from teachers or join coaching institutes if necessary.
| 3. What are the important topics to focus on for the Class 12 Maths JEE exam? | |
Ans. Some important topics to focus on for the Class 12 Maths JEE exam include calculus, algebra, coordinate geometry, probability, and vectors. These topics have a significant weightage in the exam and a strong understanding of these concepts will enhance your chances of scoring well.
| 4. Are calculators allowed in the Class 12 Maths JEE exam? | |
Ans. No, calculators are not allowed in the Class 12 Maths JEE exam. Students are expected to perform calculations manually and should practice solving problems without relying on calculators.
| 5. What is the marking scheme for the Class 12 Maths JEE exam? | |
Ans. The marking scheme for the Class 12 Maths JEE exam varies for different types of questions. Multiple-choice questions carry 4 marks for each correct answer and -1 mark for each incorrect answer. Numerical value-based questions and integer answer type questions carry 4 marks for each correct answer but do not have negative marking.
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,Maths | Additional Study Material for JEE
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Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths Notes
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