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Page 1
65/1/2 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 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.
65/1/2
• 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
65/1/2 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 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.
65/1/2
• 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
65/1/2 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. 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
?.
2. 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 .
3. 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.
4. †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
Page 3
65/1/2 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 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.
65/1/2
• 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
65/1/2 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. 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
?.
2. 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 .
3. 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.
4. †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
65/1/2 3 [P.T.O.
5. 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.
6. 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 ?
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. –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
8. 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 …
Page 4
65/1/2 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 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.
65/1/2
• 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
65/1/2 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. 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
?.
2. 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 .
3. 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.
4. †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
65/1/2 3 [P.T.O.
5. 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.
6. 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 ?
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. –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
8. 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 …
65/1/2 4
From a lot of 15 bulbs which include 5 defectives, a sample of 2 bulbs is drawn at
random (without replacement). Find the probability distribution of the number of
defective bulbs.
OR
Probability of solving specific problem independently by A and B are
1
2
and
1
3
respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem.
9. ? ÛúÖ ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ ŸÖÖ×Ûú “ÖÖ¸ü ز֤ãü A, B, C ŸÖ£ÖÖ D ו֮ÖÛêú ×ãÖ×ŸÖ ÃÖפü¿Ö ÛÎú´Ö¿Ö: 4
^
i + 5
^
j +
^
k,
–
^
j –
^
k, 3
^
i + ?
^
j + 4
^
k ŸÖ£ÖÖ –4
^
i + 4
^
j + 4
^
k Æïü, ÃÖ´ÖŸÖ»ÖßµÖ Æïü … 4
Find the value of ? so that the four points A, B, C and D with position vectors
4
^
i + 5
^
j +
^
k, –
^
j –
^
k, 3
^
i + ?
^
j + 4
^
k and –4
^
i + 4
^
j + 4
^
k respectively are coplanar.
10. ¸êüÜÖÖ†Öë
?
r = (
^
i + 2
^
j +
^
k) + ?(
^
i –
^
j +
^
k) ŸÖ£ÖÖ
?
r = (2
^
i –
^
j –
^
k) + µ(2
^
i +
^
j + 2
^
k) Ûêú ²Öß“Ö Ûúß
®µÖæ®ÖŸÖ´Ö ¤æü¸üß –ÖÖŸÖ Ûúßו֋ … 4
Find the shortest distance between the lines
?
r = (
^
i + 2
^
j +
^
k) + ?(
^
i –
^
j +
^
k) and
?
r = (2
^
i –
^
j –
^
k) + µ(2
^
i +
^
j + 2
^
k)
11. Æü»Ö Ûúßו֋ : tan
–1
2x + tan
–1
3x =
p
4
4
†£Ö¾ÖÖ
×ÃÖ¨ü Ûúßו֋ ×Ûú : tan
–1
63
16
= sin
–1
5
13
+ cos
–1
3
5
Solve : tan
–1
2x + tan
–1
3x =
p
4
OR
Prove that : tan
–1
63
16
= sin
–1
5
13
+ cos
–1
3
5
Page 5
65/1/2 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 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.
65/1/2
• 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
65/1/2 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. 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
?.
2. 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 .
3. 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.
4. †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
65/1/2 3 [P.T.O.
5. 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.
6. 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 ?
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. –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
8. 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 …
65/1/2 4
From a lot of 15 bulbs which include 5 defectives, a sample of 2 bulbs is drawn at
random (without replacement). Find the probability distribution of the number of
defective bulbs.
OR
Probability of solving specific problem independently by A and B are
1
2
and
1
3
respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem.
9. ? ÛúÖ ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ ŸÖÖ×Ûú “ÖÖ¸ü ز֤ãü A, B, C ŸÖ£ÖÖ D ו֮ÖÛêú ×ãÖ×ŸÖ ÃÖפü¿Ö ÛÎú´Ö¿Ö: 4
^
i + 5
^
j +
^
k,
–
^
j –
^
k, 3
^
i + ?
^
j + 4
^
k ŸÖ£ÖÖ –4
^
i + 4
^
j + 4
^
k Æïü, ÃÖ´ÖŸÖ»ÖßµÖ Æïü … 4
Find the value of ? so that the four points A, B, C and D with position vectors
4
^
i + 5
^
j +
^
k, –
^
j –
^
k, 3
^
i + ?
^
j + 4
^
k and –4
^
i + 4
^
j + 4
^
k respectively are coplanar.
10. ¸êüÜÖÖ†Öë
?
r = (
^
i + 2
^
j +
^
k) + ?(
^
i –
^
j +
^
k) ŸÖ£ÖÖ
?
r = (2
^
i –
^
j –
^
k) + µ(2
^
i +
^
j + 2
^
k) Ûêú ²Öß“Ö Ûúß
®µÖæ®ÖŸÖ´Ö ¤æü¸üß –ÖÖŸÖ Ûúßו֋ … 4
Find the shortest distance between the lines
?
r = (
^
i + 2
^
j +
^
k) + ?(
^
i –
^
j +
^
k) and
?
r = (2
^
i –
^
j –
^
k) + µ(2
^
i +
^
j + 2
^
k)
11. Æü»Ö Ûúßו֋ : tan
–1
2x + tan
–1
3x =
p
4
4
†£Ö¾ÖÖ
×ÃÖ¨ü Ûúßו֋ ×Ûú : tan
–1
63
16
= sin
–1
5
13
+ cos
–1
3
5
Solve : tan
–1
2x + tan
–1
3x =
p
4
OR
Prove that : tan
–1
63
16
= sin
–1
5
13
+ cos
–1
3
5
65/1/2 5 [P.T.O.
12. 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
13. 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
14. 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
15. –ÖÖŸÖ Ûúßו֋ :
)
?
(
(x
2
+ 1)e
x
(x + 1)
2
dx 4
Find
)
?
(
(x
2
+ 1)e
x
(x + 1)
2
dx
16. ¤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 ?
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FAQs on Past Year Paper, Mathematics (Set - 2), Delhi, 2015, Class 12, Maths - Additional Study Material for JEE
| 1. What is the exam pattern of JEE Mathematics paper for Class 12? |  |
Ans. The JEE Mathematics paper for Class 12 consists of 30 questions in total. The paper is divided into two sections - Section A and Section B. Section A consists of 20 multiple-choice questions (MCQs) with four options, out of which only one is correct. Section B consists of 10 questions with a single-digit integer answer ranging from 0 to 9.
| 2. How should I prepare for the JEE Mathematics paper for Class 12? | |
Ans. To prepare for the JEE Mathematics paper for Class 12, it is essential to have a thorough understanding of the concepts covered in the syllabus. Start by reviewing the textbooks and class notes, and make sure to practice solving a variety of problems. It is also helpful to solve past year papers and take mock tests to familiarize yourself with the exam pattern and improve time management skills.
| 3. What are some important topics to focus on for the JEE Mathematics paper for Class 12? | |
Ans. Some important topics to focus on for the JEE Mathematics paper for Class 12 include calculus, algebra, coordinate geometry, vectors and 3D geometry, and probability. These topics carry significant weightage in the exam and mastering them will help you score well. However, it is important to have a balanced preparation and not neglect other topics as well.
| 4. Are calculators allowed in the JEE Mathematics paper for Class 12? | |
Ans. No, calculators or any electronic devices are not allowed in the JEE Mathematics paper for Class 12. The exam is designed to test your problem-solving abilities and mathematical skills without the aid of calculators. It is important to practice solving problems manually to build speed and accuracy.
| 5. How can I manage time effectively during the JEE Mathematics paper for Class 12? | |
Ans. Time management is crucial during the JEE Mathematics paper for Class 12. Start by allocating a specific amount of time to each section based on the number of questions. Prioritize solving the questions you are confident about first and then move on to the more challenging ones. If you encounter a difficult question, it is better to skip it and come back to it later if time permits. Regular practice and mock tests will help you improve your speed and efficiency.
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