JEE Exam > JEE Notes > Mathematics (Maths) Class 12 > Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths
Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths | Mathematics (Maths) Class 12 - JEE PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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).
Read More
|
204 videos|290 docs|139 tests
|
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.
About this Document
|
4.91/5 Rating |
|
Nov 30, 2024 Last updated |
Document Description: Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths for JEE 2024 is part of Mathematics (Maths) Class 12 preparation.
The notes and questions for Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths have been prepared according to the JEE exam syllabus. Information about Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths covers topics
like and Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths.
Introduction of Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths in English is available as part of our Mathematics (Maths) Class 12
for JEE & Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths in Hindi for Mathematics (Maths) Class 12 course.
Download more important topics related with notes, lectures and mock test series for JEE
Exam by signing up for free. JEE: Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths | Mathematics (Maths) Class 12 - JEE
Description
Full syllabus notes, lecture & questions for Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths | Mathematics (Maths) Class 12 - JEE - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) Class 12 | Best notes, free PDF download
Information about Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths
In this doc you can find the meaning of Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths defined & explained in the simplest way possible. Besides explaining types of
Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths theory, EduRev gives you an ample number of questions to practice Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths tests, examples and also practice JEE
tests
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Foreign
,Class 12
,2016
,Class 12
,Foreign
,Semester Notes
,shortcuts and tricks
,2016
,Mathematics (Set - 1)
,Maths | Mathematics (Maths) Class 12 - JEE
,video lectures
,Mathematics (Set - 1)
,Important questions
,Foreign
,Exam
,2016
,Maths | Mathematics (Maths) Class 12 - JEE
,Maths | Mathematics (Maths) Class 12 - JEE
,MCQs
,Past Year Paper
,practice quizzes
,Objective type Questions
,past year papers
,Past Year Paper
,Free
,Previous Year Questions with Solutions
,Sample Paper
,Class 12
,Extra Questions
,mock tests for examination
,ppt
,Past Year Paper
,Summary
,Mathematics (Set - 1)
,Viva Questions
,study material
;
Additional Information about Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths for JEE Preparation
Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Free PDF Download
The Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths now and kickstart your journey towards success in the JEE exam.
Importance of Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths
The importance of Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths cannot be overstated, especially for JEE aspirants.
This document holds the key to success in the JEE exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Notes
Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Notes on EduRev are your ultimate resource for success.
Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths JEE Questions
The "Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths on the App
Students of JEE can study Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup