JEE Exam > JEE Notes > Mathematics (Maths) Class 12 > Past Year Paper, Mathematics (Set - 2), Foreign, 2016, Class 12, Maths
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Page 1
65/2/2/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.
65/2/2/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 – 2
Page 2
65/2/2/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.
65/2/2/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 – 2
65/2/2/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 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.
2. 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 .
3. 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 .
4. 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.
Page 3
65/2/2/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.
65/2/2/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 – 2
65/2/2/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 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.
2. 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 .
3. 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 .
4. 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.
65/2/2/F 3 [P.T.O.
5. 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.
6. 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.
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. 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.
8. 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.
Page 4
65/2/2/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.
65/2/2/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 – 2
65/2/2/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 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.
2. 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 .
3. 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 .
4. 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.
65/2/2/F 3 [P.T.O.
5. 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.
6. 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.
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. 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.
8. 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.
65/2/2/F 4
9. 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.
10. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p
x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p
x
1 + sin a sin x
dx
11. –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
12. –ÖÖŸÖ Ûúßו֋ :
)
?
(
x sin
–1
x
1 – x
2
dx
Find :
)
?
(
x sin
–1
x
1 – x
2
dx
13. ×®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
14. ×®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
Page 5
65/2/2/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.
65/2/2/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 – 2
65/2/2/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 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.
2. 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 .
3. 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 .
4. 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.
65/2/2/F 3 [P.T.O.
5. 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.
6. 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.
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. 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.
8. 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.
65/2/2/F 4
9. 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.
10. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p
x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p
x
1 + sin a sin x
dx
11. –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
12. –ÖÖŸÖ Ûúßו֋ :
)
?
(
x sin
–1
x
1 – x
2
dx
Find :
)
?
(
x sin
–1
x
1 – x
2
dx
13. ×®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
14. ×®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
65/2/2/F 5 [P.T.O.
15. 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 .
16. ×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).
17. ‹Ûú ×›ü²²Öê ´Öë 20 ¯Öê®Ö Æîü וÖÃÖ´Öë ÃÖê 2 ÜÖ¸üÖ²Ö Æïü … µÖפü ˆ¢Ö¸üÖê¢Ö¸ü ¯ÖÏןÖãÖÖ¯Ö®ÖÖ Ûêú ÃÖÖ£Ö 5 ¯Öê®Ö ×®ÖÛúÖ»Öê •ÖÖ‹Ñ, ŸÖÖê
¯ÖÏÖ×µÖÛúŸÖÖ –ÖÖŸÖ Ûúßו֋ ×Ûú †×¬ÖÛúŸÖ´Ö 2 ¯Öê®Ö ÜÖ¸üÖ²Ö ÆüÖëÝÖë …
†£Ö¾ÖÖ
´ÖÖ®ÖÖ ×Ûú X, ÛúÖò»Öê•ÖÖë Ûúß ÃÖÓܵÖÖ ÃÖæ×“ÖŸÖ Ûú¸üŸÖÖ Æîü •ÖÆüÖÑ ¯Ö¸ü †Ö¯Ö †¯Ö®ÖÖ ¯Ö׸üÞÖÖ´Ö †Ö®Öê Ûêú ²ÖÖ¤ü †Ö¾Öê¤ü®Ö Ûú¸ëüÝÖë †Öî¸ü
P (X = x) ¯ÖÏÖ×µÖÛúŸÖÖ ÃÖæ×“ÖŸÖ Ûú¸üŸÖÖ Æîü •Ö²Ö×Ûú †Ö¯ÖÛúÖê x ÛúÖò»Öê•Ö ´Öë ¯ÖϾÖê¿Ö ×´Ö»Ö ÃÖÛúŸÖÖ Æîü … פüµÖÖ ÝÖµÖÖ Æîü ×Ûú
P(X = x) =
?
?
?
?
?
kx , µÖפü x = 0 µÖÖ 1
2 kx , µÖפü x = 2
k(5 – x) , µÖפü x = 3 µÖÖ 4
0 , µÖפü x > 4
,
•Ö²Ö×Ûú k ‹Ûú ¬Ö®ÖÖŸ´ÖÛú †“Ö¸ü Æîü …
k ÛúÖ ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ … µÖÆü ¯ÖÏÖ×µÖÛúŸÖÖ ³Öß –ÖÖŸÖ Ûúßו֋ ×Ûú †Ö¯ÖÛúÖê (i) ‹Ûú †Öî¸ü Ûêú¾Ö»Ö ‹Ûú ÛúÖò»Öê•Ö ´Öë ¯ÖϾÖê¿Ö
×´Ö»ÖêÝÖÖ (ii) †×¬ÖÛú ÃÖê †×¬ÖÛú ¤üÖê ÛúÖò»Öê•ÖÖë ´Öë ¯ÖϾÖê¿Ö ×´Ö»ÖêÝÖÖ (iii) Ûú´Ö ÃÖê Ûú´Ö ¤üÖê ÛúÖò»Öê•ÖÖë ´Öë ¯ÖϾÖê¿Ö ×´Ö»ÖêÝÖÖ …
A box has 20 pens of which 2 are defective. Calculate the probability that out of
5 pens drawn one by one with replacement, at most 2 are defective.
OR
Let, X denote the number of colleges where you will apply after your results and
P (X = x) denotes your probability of getting admission in x number of colleges. It is
given that
P(X = x) =
?
?
?
kx , if x = 0 or 1
2 kx , if x = 2
k(5 – x) , if x = 3 or 4
0 , if x > 4
,
where k is a positive constant. Find the value of k. Also find the probability that you
will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2
colleges.
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FAQs on Past Year Paper, Mathematics (Set - 2), Foreign, 2016, Class 12, Maths - Mathematics (Maths) Class 12 - JEE
| 1. What are the important topics to study in the Mathematics (Set - 2) exam for Class 12? |  |
Ans. Some important topics to study for the Mathematics (Set - 2) exam for Class 12 include algebra, trigonometry, calculus, coordinate geometry, and probability. It is crucial to have a strong understanding of these topics to perform well in the exam.
| 2. How should I prepare for the Mathematics (Set - 2) exam for Class 12? | |
Ans. To prepare for the Mathematics (Set - 2) exam for Class 12, it is essential to thoroughly understand the concepts and practice solving different types of problems. Students can refer to the textbook, solve previous year papers, and take mock tests to assess their knowledge and identify areas that require improvement.
| 3. Are there any important tips to solve mathematical problems quickly during the exam? | |
Ans. Yes, there are a few tips to solve mathematical problems quickly during the exam. Firstly, practice mental calculations to speed up calculations. Secondly, read the question carefully and identify the given information and what needs to be found. Break down complex problems into smaller steps to make them more manageable. Lastly, manage your time effectively by allocating specific time slots for different sections or types of problems.
| 4. How can I improve my problem-solving skills in Mathematics? | |
Ans. Improving problem-solving skills in Mathematics requires consistent practice. Start by understanding the concepts and principles behind different topics. Then, solve a variety of problems from textbooks, reference books, and previous year papers. As you solve more problems, try to analyze the different approaches and strategies used. Seek help from teachers or peers if you face difficulties in solving certain types of problems.
| 5. What are some common mistakes to avoid in the Mathematics (Set - 2) exam for Class 12? | |
Ans. Some common mistakes to avoid in the Mathematics (Set - 2) exam for Class 12 include not reading the question properly, skipping steps in calculations, not showing complete working, and not revising the answers before submitting the exam. It is important to stay focused, pay attention to details, and double-check your answers to minimize errors.
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