JEE Exam > JEE Notes > Additional Study Material for JEE > Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths
Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths | Additional Study Material for JEE PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
65/1 1 [P.T.O.
¸üÖê»Ö ®ÖÓ.
Roll No.
ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours ] [ Maximum Marks : 100
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/1
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …
• Please check that this question paper contains 8 printed pages.
• Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
• Please check that this question paper contains 26 questions.
• Please write down the Serial Number of the question before attempting it.
• 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.
SET – 1
Page 2
65/1 1 [P.T.O.
¸üÖê»Ö ®ÖÓ.
Roll No.
ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours ] [ Maximum Marks : 100
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/1
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …
• Please check that this question paper contains 8 printed pages.
• Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
• Please check that this question paper contains 26 questions.
• Please write down the Serial Number of the question before attempting it.
• 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.
SET – 1
65/1 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. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
2. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
3. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
4. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
Page 3
65/1 1 [P.T.O.
¸üÖê»Ö ®ÖÓ.
Roll No.
ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours ] [ Maximum Marks : 100
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/1
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …
• Please check that this question paper contains 8 printed pages.
• Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
• Please check that this question paper contains 26 questions.
• Please write down the Serial Number of the question before attempting it.
• 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.
SET – 1
65/1 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. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
2. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
3. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
4. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
65/1 3 [P.T.O.
5. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
6. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 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. ×ÛúÃÖß ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö Ûêú ¯ÖÖÃÖ ` 35,000 ÛúÖ ÛúÖêÂÖ Æîü וÖÃÖê ¤üÖê ׳֮®Ö-׳֮®Ö ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üüÖë ´Öë ×®Ö¾Öê×¿ÖŸÖ Ûú¸ü®ÖÖ
Æîü … ¯ÖÏ£Ö´Ö ²ÖÖÑ›ü ¯Ö¸ü 8% ¾ÖÖÙÂÖÛú ²µÖÖ•Ö Æîü, וÖÃÖê ‹Ûú †®ÖÖ£ÖÖ»ÖµÖ ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü ŸÖ£ÖÖ ×«üŸÖßµÖ ²ÖÖÑ›ü ¯Ö¸ü 10%
²µÖÖ•Ö Æîü וÖÃÖê ‹Ûú ‹®Ö.•Öß.†Öê. (ÛïúÃÖ¸ü ‹ò›ü ÃÖÖêÃÖÖ‡™üß) ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü … †Ö¾µÖæÆü ÝÖãÞÖ®Ö Ûêú ¯ÖϵÖÖêÝÖ ÃÖê µÖÆü
×®Ö¬ÖÖÔ׸üŸÖ Ûúßו֋ ×Ûú ` 35,000 Ûêú ÛúÖêÂÖ ÛúÖê ¤üÖê ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üÖë ´Öë ×®Ö¾Öê¿Ö Ûú¸ü®Öê Ûêú ×»Ö‹ ×ÛúÃÖ ¯ÖÏÛúÖ¸ü ²ÖÖÑ™ëü
וÖÃÖÃÖê ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö ÛúÖê ¯ÖÏÖ¯ŸÖ Ûãú»Ö ²µÖÖ•Ö ` 3,200 ÆüÖê ?
‡ÃÖ ¯ÖÏ¿®Ö ÃÖê ŒµÖÖ ´Öæ»µÖ •Ö×®ÖŸÖ ÆüÖêŸÖê Æïü ? 4
A trust fund has ` 35,000 is to be invested in two different types of bonds. The first
bond pays 8% interest per annum which will be given to orphanage and second bond
pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society).
Using matrix multiplication, determine how to divide ` 35,000 among two types of
bonds if the trust fund obtains an annual total interest of ` 3,200. What are the values
reflected in this question ?
8. †Ö¾µÖæÆü A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
ÛúÖê ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü ŸÖ£ÖÖ ‹Ûú ×¾ÖÂÖ´Ö-ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Ûêú µÖÖêÝÖ Ûêú ºþ¯Ö ´Öë
¾µÖŒŸÖ Ûúßו֋ … 4
Express the matrix A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
as the sum of a symmetric and skew
symmetric matrix.
†£Ö¾ÖÖ/OR
µÖפü A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
Æîü, ŸÖÖê ÃÖŸµÖÖ×¯ÖŸÖ Ûúßו֋ ×Ûú (AB)
–1
= B
–1
A
–1
If A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
, verify that (AB)
–1
= B
–1
A
–1
.
Page 4
65/1 1 [P.T.O.
¸üÖê»Ö ®ÖÓ.
Roll No.
ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours ] [ Maximum Marks : 100
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/1
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …
• Please check that this question paper contains 8 printed pages.
• Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
• Please check that this question paper contains 26 questions.
• Please write down the Serial Number of the question before attempting it.
• 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.
SET – 1
65/1 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. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
2. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
3. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
4. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
65/1 3 [P.T.O.
5. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
6. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 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. ×ÛúÃÖß ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö Ûêú ¯ÖÖÃÖ ` 35,000 ÛúÖ ÛúÖêÂÖ Æîü וÖÃÖê ¤üÖê ׳֮®Ö-׳֮®Ö ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üüÖë ´Öë ×®Ö¾Öê×¿ÖŸÖ Ûú¸ü®ÖÖ
Æîü … ¯ÖÏ£Ö´Ö ²ÖÖÑ›ü ¯Ö¸ü 8% ¾ÖÖÙÂÖÛú ²µÖÖ•Ö Æîü, וÖÃÖê ‹Ûú †®ÖÖ£ÖÖ»ÖµÖ ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü ŸÖ£ÖÖ ×«üŸÖßµÖ ²ÖÖÑ›ü ¯Ö¸ü 10%
²µÖÖ•Ö Æîü וÖÃÖê ‹Ûú ‹®Ö.•Öß.†Öê. (ÛïúÃÖ¸ü ‹ò›ü ÃÖÖêÃÖÖ‡™üß) ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü … †Ö¾µÖæÆü ÝÖãÞÖ®Ö Ûêú ¯ÖϵÖÖêÝÖ ÃÖê µÖÆü
×®Ö¬ÖÖÔ׸üŸÖ Ûúßו֋ ×Ûú ` 35,000 Ûêú ÛúÖêÂÖ ÛúÖê ¤üÖê ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üÖë ´Öë ×®Ö¾Öê¿Ö Ûú¸ü®Öê Ûêú ×»Ö‹ ×ÛúÃÖ ¯ÖÏÛúÖ¸ü ²ÖÖÑ™ëü
וÖÃÖÃÖê ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö ÛúÖê ¯ÖÏÖ¯ŸÖ Ûãú»Ö ²µÖÖ•Ö ` 3,200 ÆüÖê ?
‡ÃÖ ¯ÖÏ¿®Ö ÃÖê ŒµÖÖ ´Öæ»µÖ •Ö×®ÖŸÖ ÆüÖêŸÖê Æïü ? 4
A trust fund has ` 35,000 is to be invested in two different types of bonds. The first
bond pays 8% interest per annum which will be given to orphanage and second bond
pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society).
Using matrix multiplication, determine how to divide ` 35,000 among two types of
bonds if the trust fund obtains an annual total interest of ` 3,200. What are the values
reflected in this question ?
8. †Ö¾µÖæÆü A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
ÛúÖê ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü ŸÖ£ÖÖ ‹Ûú ×¾ÖÂÖ´Ö-ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Ûêú µÖÖêÝÖ Ûêú ºþ¯Ö ´Öë
¾µÖŒŸÖ Ûúßו֋ … 4
Express the matrix A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
as the sum of a symmetric and skew
symmetric matrix.
†£Ö¾ÖÖ/OR
µÖפü A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
Æîü, ŸÖÖê ÃÖŸµÖÖ×¯ÖŸÖ Ûúßו֋ ×Ûú (AB)
–1
= B
–1
A
–1
If A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
, verify that (AB)
–1
= B
–1
A
–1
.
65/1 4
9. ÃÖÖ¸ü×ÞÖÛúÖë Ûêú ÝÖãÞÖ¬Ö´ÖÖí Ûêú ¯ÖϵÖÖêÝÖ ÃÖê ×®Ö´®Ö ÛúÖê x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : 4
?
?
?
?
?
?
?
?
a + x a – x a – x
a – x a + x a – x
a – x a – x a + x
= 0
Using properties of determinants, solve for x :
?
?
?
?
?
?
?
?
a + x a – x a – x
a – x a + x a – x
a – x a – x a + x
= 0
10. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
11. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
x
(x
2
+ 1) (x – 1)
dx 4
†£Ö¾ÖÖ
´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
1
2
.
.
sin
–1
x
(1 – x
2
)
3/2
dx
Find
)
?
(
.
.
x
(x
2
+ 1) (x – 1)
dx.
OR
Find
)
?
(
0
1
2
.
.
sin
–1
x
(1 – x
2
)
3/2
dx.
12. 52 ŸÖÖ¿Ö Ûêú ¯Ö¢ÖÖë Ûúß ³Ö»Öß ³ÖÖÓ×ŸÖ ±ëú™üß ÝÖ‡Ô ÝÖøüß ´Öë ÃÖê 4 ¯Ö¢Öê ˆ¢Ö¸üÖê¢Ö¸ü ¯ÖÏןÖãÖÖ¯Ö®ÖÖ ÃÖ×ÆüŸÖ ×®ÖÛúÖ»Öê •ÖÖŸÖê Æïü …
‡ÃÖÛúß ŒµÖÖ ¯ÖÏÖ×µÖÛúŸÖÖ Æîü ×Ûú 4
(i) ÃÖ³Öß 4 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
(ii) Ûêú¾Ö»Ö 2 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
†£Ö¾ÖÖ
¯ÖÖÃÖÖë Ûêú ‹Ûú •ÖÖê›Ìêü ÛúÖê “ÖÖ¸ü ²ÖÖ¸ü ˆ”ûÖ»Ö®Öê ¯Ö¸ü ׫üÛúÖë Ûúß ÃÖÓܵÖÖ ÛúÖ ¯ÖÏÖ×µÖÛúŸÖÖ ²ÖÓ™ü®Ö –ÖÖŸÖ Ûúßו֋ … ‡ÃÖ ²ÖÓ™ü®Ö ÛúÖ
´ÖÖ¬µÖ ³Öß –ÖÖŸÖ Ûúßו֋ …
Page 5
65/1 1 [P.T.O.
¸üÖê»Ö ®ÖÓ.
Roll No.
ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ׸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü] [†×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours ] [ Maximum Marks : 100
ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®Ö †×®Ö¾ÖÖµÖÔ Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 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 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 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 ×»ÖÜÖë …
Series : SSO/C
ÛúÖê›ü ®ÖÓ.
Code No.
65/1
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´ÖãצüŸÖ ¯Öéšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü פü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü ¯Ö¸ü ×»ÖÜÖë …
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æïü …
• Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¿Öãºþ Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê, ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»ÖÜÖë …
• ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌü®Öê Ûêú ×»Ö‹ 15 ×´Ö®Ö™ü ÛúÖ ÃÖ´ÖµÖ ×¤üµÖÖ ÝÖµÖÖ Æîü … ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖ ×¾ÖŸÖ¸üÞÖ ¯Öæ¾ÖÖÔÆËü®Ö ´Öë 10.15 ²Ö•Öê
×ÛúµÖÖ •ÖÖµÖêÝÖÖ … 10.15 ²Ö•Öê ÃÖê 10.30 ²Ö•Öê ŸÖÛú ”ûÖ¡Ö Ûêú¾Ö»Ö ¯ÖÏ¿®Ö-¯Ö¡Ö ÛúÖê ¯ÖœÌëüÝÖê †Öî¸ü ‡ÃÖ †¾Ö×¬Ö Ûêú ¤üÖî¸üÖ®Ö ¾Öê
ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ ¯Ö¸ü ÛúÖê‡Ô ˆ¢Ö¸ü ®ÖÆüà ×»ÖÜÖëÝÖê …
• Please check that this question paper contains 8 printed pages.
• Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
• Please check that this question paper contains 26 questions.
• Please write down the Serial Number of the question before attempting it.
• 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
¯Ö¸üßõÖÖ£Öá ÛúÖê›ü ÛúÖê ˆ¢Ö¸ü-¯Öã×ßÖÛúÖ Ûêú ´ÖãÜÖ-¯Öéšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.
SET – 1
65/1 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. ÃÖפü¿Ö 3
?
a + 2
?
b Ûêú פüÛËú †®Öã¯ÖÖŸÖ ×»Ö×ÜÖ‹ •ÖÆüÖÑ
?
a =
^
i +
^
j – 2
^
k ŸÖ£ÖÖ
?
b = 2
^
i – 4
^
j + 5
^
k Æïü … 1
Write the direction ratio’s of the vector 3
?
a + 2
?
b where
?
a =
^
i +
^
j – 2
^
k and
?
b = 2
^
i – 4
^
j + 5
^
k.
2. ÃÖפü¿Ö
?
a = 2
^
i + 3
^
j + 2
^
k ÛúÖ ÃÖפü¿Ö
?
b = 2
^
i + 2
^
j +
^
k ¯Ö¸ü ¯ÖÏõÖê¯Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the projection of the vector
?
a = 2
^
i + 3
^
j + 2
^
k on the vector
?
b = 2
^
i + 2
^
j +
^
k.
3. ز֤ãü (1, 2, 3) ÃÖê ÆüÖêÛú¸ü •ÖÖ®Öê ¾ÖÖ»Öß ˆÃÖ ¸êüÜÖÖ ÛúÖ ÃÖפü¿Ö ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ÃÖ´ÖŸÖ»Ö
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0 ¯Ö¸ü »ÖÓ²Ö Æîü … 1
Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the
plane
?
r · ( )
^
i + 2
^
j – 5
^
k + 9 = 0.
4. †ÓŸÖ¸üÖ»Ö p/2 < x < p ´Öë x ÛúÖ ¾ÖÆü ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ וÖÃÖÛêú ×»Ö‹ †Ö¾µÖæÆü
?
?
?
?
?
?
2 sin x 3
1 2 sin x
†¾µÖãŸÛÎú´ÖÞÖßµÖ Æîü … 1
In the interval p/2 < x < p, find the value of x for which the matrix
?
?
?
?
?
?
2 sin x 3
1 2 sin x
is singular.
65/1 3 [P.T.O.
5. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ
dy
dx
= x
3
e
–2y
ÛúÖ Æü»Ö –ÖÖŸÖ Ûúßו֋ … 1
Find the solution of the differential equation
dy
dx
= x
3
e
–2y
.
6. †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ x
dy
dx
+ y = e
–2 x
ÛúÖ ÃÖ´ÖÖÛú»Ö®Ö ÝÖãÞÖÛú –ÖÖŸÖ Ûúßו֋ … 1
Write the integrating factor of the differential equation
x
dy
dx
+ y = e
–2 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. ×ÛúÃÖß ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö Ûêú ¯ÖÖÃÖ ` 35,000 ÛúÖ ÛúÖêÂÖ Æîü וÖÃÖê ¤üÖê ׳֮®Ö-׳֮®Ö ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üüÖë ´Öë ×®Ö¾Öê×¿ÖŸÖ Ûú¸ü®ÖÖ
Æîü … ¯ÖÏ£Ö´Ö ²ÖÖÑ›ü ¯Ö¸ü 8% ¾ÖÖÙÂÖÛú ²µÖÖ•Ö Æîü, וÖÃÖê ‹Ûú †®ÖÖ£ÖÖ»ÖµÖ ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü ŸÖ£ÖÖ ×«üŸÖßµÖ ²ÖÖÑ›ü ¯Ö¸ü 10%
²µÖÖ•Ö Æîü וÖÃÖê ‹Ûú ‹®Ö.•Öß.†Öê. (ÛïúÃÖ¸ü ‹ò›ü ÃÖÖêÃÖÖ‡™üß) ÛúÖê ¤êü פüµÖÖ •ÖÖ®ÖÖ Æîü … †Ö¾µÖæÆü ÝÖãÞÖ®Ö Ûêú ¯ÖϵÖÖêÝÖ ÃÖê µÖÆü
×®Ö¬ÖÖÔ׸üŸÖ Ûúßו֋ ×Ûú ` 35,000 Ûêú ÛúÖêÂÖ ÛúÖê ¤üÖê ¯ÖÏÛúÖ¸ü Ûêú ²ÖÖÑ›üÖë ´Öë ×®Ö¾Öê¿Ö Ûú¸ü®Öê Ûêú ×»Ö‹ ×ÛúÃÖ ¯ÖÏÛúÖ¸ü ²ÖÖÑ™ëü
וÖÃÖÃÖê ¾µÖÖ¯ÖÖ¸ü ÃÖÓ‘Ö ÛúÖê ¯ÖÏÖ¯ŸÖ Ûãú»Ö ²µÖÖ•Ö ` 3,200 ÆüÖê ?
‡ÃÖ ¯ÖÏ¿®Ö ÃÖê ŒµÖÖ ´Öæ»µÖ •Ö×®ÖŸÖ ÆüÖêŸÖê Æïü ? 4
A trust fund has ` 35,000 is to be invested in two different types of bonds. The first
bond pays 8% interest per annum which will be given to orphanage and second bond
pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society).
Using matrix multiplication, determine how to divide ` 35,000 among two types of
bonds if the trust fund obtains an annual total interest of ` 3,200. What are the values
reflected in this question ?
8. †Ö¾µÖæÆü A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
ÛúÖê ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü ŸÖ£ÖÖ ‹Ûú ×¾ÖÂÖ´Ö-ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Ûêú µÖÖêÝÖ Ûêú ºþ¯Ö ´Öë
¾µÖŒŸÖ Ûúßו֋ … 4
Express the matrix A =
?
?
?
?
?
?
?
?
2 4 – 6
7 3 5
1 – 2 4
as the sum of a symmetric and skew
symmetric matrix.
†£Ö¾ÖÖ/OR
µÖפü A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
Æîü, ŸÖÖê ÃÖŸµÖÖ×¯ÖŸÖ Ûúßו֋ ×Ûú (AB)
–1
= B
–1
A
–1
If A =
?
?
?
?
?
?
2 3
1 – 4
, B =
?
?
?
?
?
?
1 –2
–1 3
, verify that (AB)
–1
= B
–1
A
–1
.
65/1 4
9. ÃÖÖ¸ü×ÞÖÛúÖë Ûêú ÝÖãÞÖ¬Ö´ÖÖí Ûêú ¯ÖϵÖÖêÝÖ ÃÖê ×®Ö´®Ö ÛúÖê x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : 4
?
?
?
?
?
?
?
?
a + x a – x a – x
a – x a + x a – x
a – x a – x a + x
= 0
Using properties of determinants, solve for x :
?
?
?
?
?
?
?
?
a + x a – x a – x
a – x a + x a – x
a – x a – x a + x
= 0
10. ´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
p/4
.
.
log (1 + tan x) dx 4
Evaluate
)
?
(
0
p/4
.
.
log (1 + tan x) dx.
11. –ÖÖŸÖ Ûúßו֋ :
)
?
(
.
.
x
(x
2
+ 1) (x – 1)
dx 4
†£Ö¾ÖÖ
´ÖÖ®Ö –ÖÖŸÖ Ûúßו֋ :
)
?
(
0
1
2
.
.
sin
–1
x
(1 – x
2
)
3/2
dx
Find
)
?
(
.
.
x
(x
2
+ 1) (x – 1)
dx.
OR
Find
)
?
(
0
1
2
.
.
sin
–1
x
(1 – x
2
)
3/2
dx.
12. 52 ŸÖÖ¿Ö Ûêú ¯Ö¢ÖÖë Ûúß ³Ö»Öß ³ÖÖÓ×ŸÖ ±ëú™üß ÝÖ‡Ô ÝÖøüß ´Öë ÃÖê 4 ¯Ö¢Öê ˆ¢Ö¸üÖê¢Ö¸ü ¯ÖÏןÖãÖÖ¯Ö®ÖÖ ÃÖ×ÆüŸÖ ×®ÖÛúÖ»Öê •ÖÖŸÖê Æïü …
‡ÃÖÛúß ŒµÖÖ ¯ÖÏÖ×µÖÛúŸÖÖ Æîü ×Ûú 4
(i) ÃÖ³Öß 4 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
(ii) Ûêú¾Ö»Ö 2 ¯Ö¢Öê ÆãüÛãú´Ö Ûêú Æïü ?
†£Ö¾ÖÖ
¯ÖÖÃÖÖë Ûêú ‹Ûú •ÖÖê›Ìêü ÛúÖê “ÖÖ¸ü ²ÖÖ¸ü ˆ”ûÖ»Ö®Öê ¯Ö¸ü ׫üÛúÖë Ûúß ÃÖÓܵÖÖ ÛúÖ ¯ÖÏÖ×µÖÛúŸÖÖ ²ÖÓ™ü®Ö –ÖÖŸÖ Ûúßו֋ … ‡ÃÖ ²ÖÓ™ü®Ö ÛúÖ
´ÖÖ¬µÖ ³Öß –ÖÖŸÖ Ûúßו֋ …
65/1 5 [P.T.O.
Four cards are drawn successively with replacement from a well shuffled deck of 52
cards. What is the probability that
(i) all the four cards are spades ?
(ii) only 2 cards are spades ?
OR
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the
probability distribution of number of successes. Hence find the mean of the
distribution.
13. ×ÃÖ¨ü Ûúßו֋ ×Ûú : [ ]
?
a ,
?
b +
?
c ,
?
d = [ ]
?
a ,
?
b,
?
d + [ ]
?
a ,
?
c ,
?
d 4
Prove that [ ]
?
a ,
?
b +
?
c ,
?
d = [ ]
?
a ,
?
b,
?
d + [ ]
?
a ,
?
c ,
?
d
14. ¸êüÜÖÖ†Öë
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k ŸÖ£ÖÖ
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k Ûêú ²Öß“Ö
®µÖæ®ÖŸÖ´Ö ¤æü¸üß –ÖÖŸÖ Ûúßו֋ … 4
Find the shortest distance between the following lines :
?
r = 2
^
i – 5
^
j +
^
k + ? ( ) 3
^
i + 2
^
j + 6
^
k and
?
r = 7
^
i – 6
^
k + µ ( )
^
i + 2
^
j + 2
^
k
15. ×ÃÖ¨ü Ûúßו֋ ×Ûú 2tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
4
†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúßו֋ : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
Prove that 2 tan
–1
?
?
?
?
?
?
1
2
+ tan
–1
?
?
?
?
?
?
1
7
= sin
–1
?
?
?
?
?
?
31
25 2
OR
Solve for x : tan
–1
?
?
?
?
?
?
1 – x
1 + x
=
1
2
tan
–1
x, x > 0
16. ? Ûêú ×ÛúÃÖ ´ÖÖ®Ö Ûêú ×»Ö‹ ±ú»Ö®Ö f(x) =
?
?
?
?
?
?(x
2
+ 2), µÖפü x = 0
4x + 6 , µÖפü x > 0
x = 0 ¯Ö¸ü ÃÖÓŸÖŸÖ Æîü … †ŸÖ: x = 0 ¯Ö¸ü
±ú»Ö®Ö Ûúß †¾ÖÛú»Ö®ÖßµÖŸÖÖ Ûúß •ÖÖÑ“Ö Ûúßו֋ … 4
For what value of ? the function defined by f(x) =
? ?
?
?
?
?(x
2
+ 2), if x = 0
4x + 6 , if x > 0
is continuous at
x = 0 ? Hence check the differentiability of f(x) at x = 0.
Read More
|
22 videos|162 docs|17 tests
|
FAQs on Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths - Additional Study Material for JEE
| 1. What is the exam pattern for the Class 12 Maths JEE exam? |  |
Ans. The Class 12 Maths JEE exam follows a set pattern that includes multiple-choice questions, numerical value-based questions, and integer answer type questions. The exam also has a negative marking scheme where 1 mark is deducted for every incorrect answer.
| 2. How can I prepare effectively for the Class 12 Maths JEE exam? | |
Ans. To prepare effectively for the Class 12 Maths JEE exam, it is recommended to thoroughly understand the concepts and practice a variety of questions. Solve previous year papers, mock tests, and sample papers to get a good grasp of the exam pattern and time management. Seek guidance from teachers or join coaching institutes if necessary.
| 3. What are the important topics to focus on for the Class 12 Maths JEE exam? | |
Ans. Some important topics to focus on for the Class 12 Maths JEE exam include calculus, algebra, coordinate geometry, probability, and vectors. These topics have a significant weightage in the exam and a strong understanding of these concepts will enhance your chances of scoring well.
| 4. Are calculators allowed in the Class 12 Maths JEE exam? | |
Ans. No, calculators are not allowed in the Class 12 Maths JEE exam. Students are expected to perform calculations manually and should practice solving problems without relying on calculators.
| 5. What is the marking scheme for the Class 12 Maths JEE exam? | |
Ans. The marking scheme for the Class 12 Maths JEE exam varies for different types of questions. Multiple-choice questions carry 4 marks for each correct answer and -1 mark for each incorrect answer. Numerical value-based questions and integer answer type questions carry 4 marks for each correct answer but do not have negative marking.
Related Exams
About this Document
|
4.90/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths for JEE 2024 is part of Additional Study Material for JEE preparation.
The notes and questions for Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths have been prepared according to the JEE exam syllabus. Information about Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths covers topics
like and Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths.
Introduction of Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths in English is available as part of our Additional Study Material for JEE
for JEE & Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths in Hindi for Additional Study Material for JEE 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, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths | Additional Study Material for JEE
Description
Full syllabus notes, lecture & questions for Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths | Additional Study Material for JEE - JEE | Plus excerises question with solution to help you revise complete syllabus for Additional Study Material for JEE | Best notes, free PDF download
Information about Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths
In this doc you can find the meaning of Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths defined & explained in the simplest way possible. Besides explaining types of
Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths theory, EduRev gives you an ample number of questions to practice Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, 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
Maths (Set - 1)
,Important questions
,2015
,Maths | Additional Study Material for JEE
,Maths (Set - 1)
,Class 12
,practice quizzes
,Maths | Additional Study Material for JEE
,mock tests for examination
,ppt
,Outside Delhi
,Summary
,Viva Questions
,MCQs
,Outside Delhi
,Outside Delhi
,Objective type Questions
,Maths | Additional Study Material for JEE
,video lectures
,Previous Year Questions with Solutions
,Free
,2015
,Class 12
,Class 12
,Past Year Paper
,study material
,shortcuts and tricks
,past year papers
,Sample Paper
,Past Year Paper
,Exam
,Maths (Set - 1)
,Past Year Paper
,Extra Questions
,2015
,Semester Notes
;
Additional Information about Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths for JEE Preparation
Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths Free PDF Download
The Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths now and kickstart your journey towards success in the JEE exam.
Importance of Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths
The importance of Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths Notes
Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths Notes on EduRev are your ultimate resource for success.
Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths JEE Questions
The "Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths on the App
Students of JEE can study Past Year Paper, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, 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, Maths (Set - 1),Outside Delhi, 2015, Class 12, Maths is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup