JEE Exam > JEE Notes > Mathematics (Maths) Class 12 > Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths
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Page 1
65/2/1/F 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¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(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 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 : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.
65/2/1/F
• 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 minute 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
65/2/1/F 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¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(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 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 : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.
65/2/1/F
• 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 minute 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
65/2/1/F 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. µÖפü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.
2. µÖפü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.
3. µÖפü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
Page 3
65/2/1/F 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¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(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 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 : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.
65/2/1/F
• 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 minute 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
65/2/1/F 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. µÖפü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.
2. µÖפü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.
3. µÖפü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
65/2/1/F 3 [P.T.O.
4. µÖפ
?
a ,
?
b ŸÖ£ÖÖ
?
c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü ×Ûú
?
a +
?
b +
?
c =
?
0 Æîü, ŸÖÖê
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a ÛúÖ ´ÖÖ®Ö
×»Ö×ÜÖ‹ …
If
?
a ,
?
b,
?
c are unit vectors such that
?
a +
?
b +
?
c =
?
0, then write the value of
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a .
5. µÖפü
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 Æîü ŸÖ£ÖÖ
?
a = 5 Æîü, ŸÖÖê
?
b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 and
?
a = 5, then write the value of
?
b .
6. ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö ز֤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö
ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …
Write the equation of a plane which is at a distance of 5 3 units from origin and the
normal to which is equally inclined to coordinate axes.
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. ×ÃÖ¨ü Ûúßו֋ ×Ûú
cot
–1
1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ :
tan
–1
?
?
?
?
?
?
x – 2
x – 1
+ tan
–1
?
?
?
?
?
?
x + 2
x + 1
=
p
4
Prove that :
cot
–1
1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2
OR
Solve for x :
tan
–1
?
?
?
?
?
?
x – 2
x – 1
+ tan
–1
?
?
?
?
?
?
x + 2
x + 1
=
p
4
Page 4
65/2/1/F 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¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(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 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 : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.
65/2/1/F
• 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 minute 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
65/2/1/F 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. µÖפü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.
2. µÖפü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.
3. µÖפü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
65/2/1/F 3 [P.T.O.
4. µÖפ
?
a ,
?
b ŸÖ£ÖÖ
?
c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü ×Ûú
?
a +
?
b +
?
c =
?
0 Æîü, ŸÖÖê
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a ÛúÖ ´ÖÖ®Ö
×»Ö×ÜÖ‹ …
If
?
a ,
?
b,
?
c are unit vectors such that
?
a +
?
b +
?
c =
?
0, then write the value of
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a .
5. µÖפü
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 Æîü ŸÖ£ÖÖ
?
a = 5 Æîü, ŸÖÖê
?
b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 and
?
a = 5, then write the value of
?
b .
6. ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö ز֤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö
ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …
Write the equation of a plane which is at a distance of 5 3 units from origin and the
normal to which is equally inclined to coordinate axes.
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. ×ÃÖ¨ü Ûúßו֋ ×Ûú
cot
–1
1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ :
tan
–1
?
?
?
?
?
?
x – 2
x – 1
+ tan
–1
?
?
?
?
?
?
x + 2
x + 1
=
p
4
Prove that :
cot
–1
1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2
OR
Solve for x :
tan
–1
?
?
?
?
?
?
x – 2
x – 1
+ tan
–1
?
?
?
?
?
?
x + 2
x + 1
=
p
4
65/2/1/F 4
8. †ÓÝÖÏê•Öß ×¾ÖÂÖµÖ ¯ÖœÌüÖ®Öê ¾ÖÖ»Öß ÛúÖêØ“ÖÝÖ ÃÖÓãÖÖ ¤üÖê ²Öî“Ö I †Öî¸ü II ´Öë ÛúõÖÖ »ÖêŸÖß Æîü וִ֮Öë †´Ö߸ü ¾Ö ÝÖ¸üß²Ö ²Ö““ÖÖë Ûêú
×»Ö‹ ±úßÃÖ †»ÖÝÖ-†»ÖÝÖ Æîü … ²Öî“Ö I ´Öë 20 ÝÖ¸üß²Ö ŸÖ£ÖÖ 5 †´Ö߸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ׿Ö
` 9,000 Æîü, •Ö²Ö×Ûú ²Öî“Ö II ´Öë 5 ÝÖ¸üß²Ö †Öî¸ü 25 †´Ö߸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö ` 26,000 Æî …
†Ö¾µÖæÆü ×¾Ö×¬Ö ÃÖê ¤üÖê®ÖÖë ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏŸµÖêÛú ²Ö““Öê «üÖ¸üÖ ¤üß ÝÖ‡Ô ´ÖÖ×ÃÖÛú ±úßÃÖ –ÖÖŸÖ Ûúßו֋ … ‹êÃÖÖ Ûú¸üÛêú ÛúÖêØ“ÖÝÖ
ÃÖÓãÖÖ ÃÖ´ÖÖ•Ö ´Öë ŒµÖÖ ´Öæ»µÖ ¸üÜÖ ¸üÆüß Æîü ?
A coaching institute of English (subject) conducts classes in two batches I and II and
fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich
children and total monthly collection is ` 9,000, whereas in batch II, it has 5 poor and
25 rich children and total monthly collection is ` 26,000. Using matrix method, find
monthly fees paid by each child of two types. What values the coaching institute is
inculcating in the society ?
9. µÖפü ±ú»Ö®Ö f(x) =
? ?
?
?
?
x
2
+ 3x + a , x = 1
bx + 2 , x > 1
x = 1 ¯Ö¸ü †¾ÖÛú»Ö®ÖßµÖ Æîü, ŸÖÖê a ŸÖ£ÖÖ b Ûêú ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ …
Find the values of a and b, if the function f defined by
f(x) =
? ?
?
?
?
x
2
+ 3x + a , x = 1
bx + 2 , x > 1
is differentiable at x = 1.
10. tan
–1
?
?
?
?
?
?
1 + x
2
– 1
x
ÛúÖ sin
–1
2x
1 + x
2
Ûêú ÃÖÖ¯ÖêõÖ †¾ÖÛú»Ö®Ö Ûúßו֋, •Ö²Ö×Ûú x ? (–1, 1) Æîü …
†£Ö¾ÖÖ
µÖפü x = sin t Æîü ŸÖ£ÖÖ y = sin pt Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0
Differentiate tan
–1
?
?
?
?
?
?
1 + x
2
– 1
x
w.r.t. sin
–1
2x
1 + x
2
, if x ? (–1, 1)
OR
If x = sin t and y = sin pt, prove that (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0.
11. ¾ÖÛÎúÖë y
2
= 4ax ŸÖ£ÖÖ x
2
= 4by Ûêú ²Öß“Ö ÛúÖ ¯ÖÏן֓”êû¤üß ÛúÖêÞÖ –ÖÖŸÖ Ûúßו֋ …
Find the angle of intersection of the curves y
2
= 4ax and x
2
= 4by.
12. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p
x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p
x
1 + sin a sin x
dx
Page 5
65/2/1/F 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¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(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 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 : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.
65/2/1/F
• 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 minute 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
65/2/1/F 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. µÖפü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1
?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.
2. µÖפü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.
3. µÖפü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
65/2/1/F 3 [P.T.O.
4. µÖפ
?
a ,
?
b ŸÖ£ÖÖ
?
c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖפü¿Ö Æïü ×Ûú
?
a +
?
b +
?
c =
?
0 Æîü, ŸÖÖê
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a ÛúÖ ´ÖÖ®Ö
×»Ö×ÜÖ‹ …
If
?
a ,
?
b,
?
c are unit vectors such that
?
a +
?
b +
?
c =
?
0, then write the value of
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a .
5. µÖפü
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 Æîü ŸÖ£ÖÖ
?
a = 5 Æîü, ŸÖÖê
?
b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 and
?
a = 5, then write the value of
?
b .
6. ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö ز֤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö
ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …
Write the equation of a plane which is at a distance of 5 3 units from origin and the
normal to which is equally inclined to coordinate axes.
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. ×ÃÖ¨ü Ûúßו֋ ×Ûú
cot
–1
1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ :
tan
–1
?
?
?
?
?
?
x – 2
x – 1
+ tan
–1
?
?
?
?
?
?
x + 2
x + 1
=
p
4
Prove that :
cot
–1
1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2
OR
Solve for x :
tan
–1
?
?
?
?
?
?
x – 2
x – 1
+ tan
–1
?
?
?
?
?
?
x + 2
x + 1
=
p
4
65/2/1/F 4
8. †ÓÝÖÏê•Öß ×¾ÖÂÖµÖ ¯ÖœÌüÖ®Öê ¾ÖÖ»Öß ÛúÖêØ“ÖÝÖ ÃÖÓãÖÖ ¤üÖê ²Öî“Ö I †Öî¸ü II ´Öë ÛúõÖÖ »ÖêŸÖß Æîü וִ֮Öë †´Ö߸ü ¾Ö ÝÖ¸üß²Ö ²Ö““ÖÖë Ûêú
×»Ö‹ ±úßÃÖ †»ÖÝÖ-†»ÖÝÖ Æîü … ²Öî“Ö I ´Öë 20 ÝÖ¸üß²Ö ŸÖ£ÖÖ 5 †´Ö߸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ׿Ö
` 9,000 Æîü, •Ö²Ö×Ûú ²Öî“Ö II ´Öë 5 ÝÖ¸üß²Ö †Öî¸ü 25 †´Ö߸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö ` 26,000 Æî …
†Ö¾µÖæÆü ×¾Ö×¬Ö ÃÖê ¤üÖê®ÖÖë ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏŸµÖêÛú ²Ö““Öê «üÖ¸üÖ ¤üß ÝÖ‡Ô ´ÖÖ×ÃÖÛú ±úßÃÖ –ÖÖŸÖ Ûúßו֋ … ‹êÃÖÖ Ûú¸üÛêú ÛúÖêØ“ÖÝÖ
ÃÖÓãÖÖ ÃÖ´ÖÖ•Ö ´Öë ŒµÖÖ ´Öæ»µÖ ¸üÜÖ ¸üÆüß Æîü ?
A coaching institute of English (subject) conducts classes in two batches I and II and
fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich
children and total monthly collection is ` 9,000, whereas in batch II, it has 5 poor and
25 rich children and total monthly collection is ` 26,000. Using matrix method, find
monthly fees paid by each child of two types. What values the coaching institute is
inculcating in the society ?
9. µÖפü ±ú»Ö®Ö f(x) =
? ?
?
?
?
x
2
+ 3x + a , x = 1
bx + 2 , x > 1
x = 1 ¯Ö¸ü †¾ÖÛú»Ö®ÖßµÖ Æîü, ŸÖÖê a ŸÖ£ÖÖ b Ûêú ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ …
Find the values of a and b, if the function f defined by
f(x) =
? ?
?
?
?
x
2
+ 3x + a , x = 1
bx + 2 , x > 1
is differentiable at x = 1.
10. tan
–1
?
?
?
?
?
?
1 + x
2
– 1
x
ÛúÖ sin
–1
2x
1 + x
2
Ûêú ÃÖÖ¯ÖêõÖ †¾ÖÛú»Ö®Ö Ûúßו֋, •Ö²Ö×Ûú x ? (–1, 1) Æîü …
†£Ö¾ÖÖ
µÖפü x = sin t Æîü ŸÖ£ÖÖ y = sin pt Æîü, ŸÖÖê ×ÃÖ¨ü Ûúßו֋ ×Ûú (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0
Differentiate tan
–1
?
?
?
?
?
?
1 + x
2
– 1
x
w.r.t. sin
–1
2x
1 + x
2
, if x ? (–1, 1)
OR
If x = sin t and y = sin pt, prove that (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0.
11. ¾ÖÛÎúÖë y
2
= 4ax ŸÖ£ÖÖ x
2
= 4by Ûêú ²Öß“Ö ÛúÖ ¯ÖÏן֓”êû¤üß ÛúÖêÞÖ –ÖÖŸÖ Ûúßו֋ …
Find the angle of intersection of the curves y
2
= 4ax and x
2
= 4by.
12. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p
x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p
x
1 + sin a sin x
dx
65/2/1/F 5 [P.T.O.
13. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
(2x + 5) 10 – 4x – 3x
2
dx
†£Ö¾ÖÖ
–ÖÖŸÖ Ûúßו֋ :
)
?
(
(x
2
+ 1) (x
2
+ 4)
(x
2
+ 3) (x
2
– 5)
dx
Find :
)
?
(
.
.
(2x + 5) 10 – 4x – 3x
2
dx
OR
Find :
)
?
(
(x
2
+ 1) (x
2
+ 4)
(x
2
+ 3) (x
2
– 5)
dx
14. –ÖÖŸÖ Ûúßו֋ :
)
?
(
x sin
–1
x
1 – x
2
dx
Find :
)
?
(
x sin
–1
x
1 – x
2
dx
15. ×®Ö´®Ö †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ÛúÖê Æü»Ö Ûúßו֋ :
y
2
dx + (x
2
– xy + y
2
)dy = 0
Solve the following differential equation :
y
2
dx + (x
2
– xy + y
2
)dy = 0
16. ×®Ö´®Ö †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ÛúÖê Æü»Ö Ûúßו֋ :
(cot
–1
y + x) dy = (1 + y
2
) dx
Solve the following differential equation :
(cot
–1
y + x) dy = (1 + y
2
) dx
17. µÖפü
?
a ×
?
b =
?
c ×
?
d †Öî¸ü
?
a ×
?
c =
?
b ×
?
d Æîü, ŸÖÖê ¤ü¿ÖÖÔ‡‹ ×Ûú
?
a –
?
d,
?
b –
?
c Ûêú ÃÖ´ÖÖÓŸÖ¸ü Æîü, •Ö²Ö×Ûú
?
a ?
?
d
†Öî¸ü
?
b ?
?
c Æîü …
If
?
a ×
?
b =
?
c ×
?
d and
?
a ×
?
c =
?
b ×
?
d, show that
?
a –
?
d is parallel to
?
b –
?
c , where
?
a ?
?
d
and
?
b ?
?
c .
18. ×ÃÖ¨ü Ûúßו֋ ×Ûú ز֤ãü†Öë A(0, –1, –1) ŸÖ£ÖÖ B(4, 5, 1) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ¸êüÜÖÖ Ø²Ö¤ãü†Öë C(3, 9, 4) ŸÖ£ÖÖ
D(–4, 4, 4) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ¸êüÜÖÖ ÛúÖê ¯ÖÏן֓”êû¤ü Ûú¸üŸÖß Æîü …
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through
C(3, 9, 4) and D(–4, 4, 4).
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FAQs on Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths - Mathematics (Maths) Class 12 - JEE
| 1. What are the important topics covered in the Class 12 Mathematics exam? |  |
Ans. The important topics covered in the Class 12 Mathematics exam include calculus, algebra, coordinate geometry, probability, vectors, and linear programming.
| 2. How should I prepare for the Class 12 Mathematics exam? | |
Ans. To prepare for the Class 12 Mathematics exam, it is important to have a clear understanding of the concepts and practice solving a variety of problems. Make sure to study from the textbook, solve previous year papers, and take mock tests to assess your preparation.
| 3. Are there any tips to score well in the Class 12 Mathematics exam? | |
Ans. Yes, here are a few tips to score well in the Class 12 Mathematics exam:
- Understand the concepts thoroughly and practice solving problems regularly.
- Focus on solving previous year papers to get familiar with the exam pattern and types of questions asked.
- Manage your time effectively during the exam and attempt the questions you are confident about first.
- Revise the important formulas and theorems regularly to have them at your fingertips during the exam.
| 4. Can I use a calculator during the Class 12 Mathematics exam? | |
Ans. Generally, a calculator is not allowed during the Class 12 Mathematics exam. However, it is advisable to check the specific guidelines provided by your examination board to confirm whether a calculator is permitted or not.
| 5. How can I improve my problem-solving skills for the Class 12 Mathematics exam? | |
Ans. To improve your problem-solving skills for the Class 12 Mathematics exam, you can follow these steps:
- Start by understanding the problem statement and identifying the given information.
- Break down the problem into smaller steps and try to relate it to the concepts you have learned.
- Use logical reasoning and critical thinking to devise a plan to solve the problem.
- Execute the plan and solve the problem step by step, making sure to show all the necessary calculations.
- Finally, review your solution and check if it matches the expected answer. If not, retrace your steps and identify any errors.
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