JEE Exam > JEE Notes > Mathematics (Maths) Class 12 > Past Year Paper, Mathematics (Set - 3), Foreign, 2016, Class 12, Maths
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Page 1
65/2/3/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.
65/2/3/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 – 3
Page 2
65/2/3/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.
65/2/3/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 – 3
65/2/3/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. 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 .
2. 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 .
3. 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
65/2/3/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.
65/2/3/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 – 3
65/2/3/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. 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 .
2. 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 .
3. 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.
65/2/3/F 3 [P.T.O.
4. 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.
5. 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.
6. 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.
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. 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
65/2/3/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.
65/2/3/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 – 3
65/2/3/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. 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 .
2. 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 .
3. 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.
65/2/3/F 3 [P.T.O.
4. 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.
5. 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.
6. 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.
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. 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.
65/2/3/F 4
8. 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.
9. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p
x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p
x
1 + sin a sin x
dx
10. –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
11. –ÖÖŸÖ Ûúßו֋ :
)
?
(
x sin
–1
x
1 – x
2
dx
Find :
)
?
(
x sin
–1
x
1 – x
2
dx
12. ×®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
13. ×®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
65/2/3/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.
65/2/3/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 – 3
65/2/3/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. 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 .
2. 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 .
3. 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.
65/2/3/F 3 [P.T.O.
4. 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.
5. 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.
6. 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.
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. 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.
65/2/3/F 4
8. 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.
9. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p
x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p
x
1 + sin a sin x
dx
10. –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
11. –ÖÖŸÖ Ûúßו֋ :
)
?
(
x sin
–1
x
1 – x
2
dx
Find :
)
?
(
x sin
–1
x
1 – x
2
dx
12. ×®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
13. ×®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
65/2/3/F 5 [P.T.O.
14. 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 .
15. ×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).
16. ‹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 - 3), Foreign, 2016, Class 12, Maths - Mathematics (Maths) Class 12 - JEE
| 1. How can I prepare for the Class 12 Mathematics exam? |  |
Ans. To prepare for the Class 12 Mathematics exam, it is essential to have a clear understanding of the syllabus and exam pattern. Start by going through the textbook thoroughly and make note of important concepts and formulas. Practice solving a variety of problems from previous year papers and sample papers to improve your problem-solving skills. Additionally, seek help from your teachers or join a coaching institute for extra guidance and support.
| 2. How can I score well in the Class 12 Mathematics exam? | |
Ans. Scoring well in the Class 12 Mathematics exam requires a systematic approach and consistent practice. Start by understanding the concepts thoroughly and practice solving different types of problems. Focus on improving your problem-solving speed and accuracy. Make a habit of solving at least one or two math problems every day to strengthen your skills. Additionally, revise regularly and attempt mock tests to assess your preparation level and identify areas that need improvement.
| 3. Are there any important topics that I should focus on for the Class 12 Mathematics exam? | |
Ans. Yes, there are a few important topics that you should focus on for the Class 12 Mathematics exam. Some of these include calculus, algebra, coordinate geometry, and probability. These topics carry significant weightage in the exam and mastering them can help you score well. Additionally, make sure to go through the previous year papers and analyze the distribution of marks to identify the topics that are frequently asked.
| 4. What are some effective time management strategies for the Class 12 Mathematics exam? | |
Ans. Time management is crucial for the Class 12 Mathematics exam. Here are some effective strategies to manage your time effectively:
- Create a study schedule and allocate specific time slots for each topic or chapter.
- Break down complex topics into smaller subtopics and allocate time accordingly.
- Practice solving problems within a time limit to improve your speed.
- Prioritize topics based on their weightage in the exam and focus more on those topics.
- Avoid spending too much time on a single problem. If you get stuck, move on and come back to it later.
| 5. How can I overcome exam anxiety for the Class 12 Mathematics exam? | |
Ans. Exam anxiety is common, but there are ways to overcome it. Here are some tips:
- Practice deep breathing exercises and meditation to calm your mind.
- Break down your study sessions into smaller chunks to avoid feeling overwhelmed.
- Stay positive and believe in your abilities. Focus on your preparation rather than worrying about the outcome.
- Get enough sleep and maintain a healthy lifestyle to keep your mind and body in good shape.
- Seek support from friends, family, or teachers for guidance and motivation.
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