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# Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Commerce Notes | EduRev

## Commerce : Past Year Paper, Mathematics (Set - 1), Foreign, 2016, Class 12, Maths Commerce Notes | EduRev

``` Page 1

65/2/1/F 1 [P.T.O.

¸üÖê»Ö ®ÖÓ.
Roll No.

ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ×¸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü †×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours   Maximum Marks : 100

ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æîü …
(iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …
(v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ
Æïü …
(vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …
Series : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.

65/2/1/F
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´Öã×¦üŸÖ ¯ÖéÂšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü ×¤ü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯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×ÃŸÖÛúÖ ¯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×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.

SET – 1
Page 2

65/2/1/F 1 [P.T.O.

¸üÖê»Ö ®ÖÓ.
Roll No.

ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ×¸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü †×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours   Maximum Marks : 100

ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æîü …
(iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …
(v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ
Æïü …
(vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …
Series : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.

65/2/1/F
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´Öã×¦üŸÖ ¯ÖéÂšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü ×¤ü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯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×ÃŸÖÛúÖ ¯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×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.

SET – 1
65/2/1/F 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying
6 marks each
(vi) Please write down the serial number of the Question before attempting it.

ÜÖÞ›ü – †
SECTION – A

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …
Question numbers 1 to 6 carry 1 mark each.

1. µÖ×¤ü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.

2. µÖ×¤ü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.

3.  µÖ×¤ü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
Page 3

65/2/1/F 1 [P.T.O.

¸üÖê»Ö ®ÖÓ.
Roll No.

ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ×¸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü †×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours   Maximum Marks : 100

ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æîü …
(iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …
(v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ
Æïü …
(vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …
Series : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.

65/2/1/F
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´Öã×¦üŸÖ ¯ÖéÂšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü ×¤ü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯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×ÃŸÖÛúÖ ¯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×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.

SET – 1
65/2/1/F 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying
6 marks each
(vi) Please write down the serial number of the Question before attempting it.

ÜÖÞ›ü – †
SECTION – A

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …
Question numbers 1 to 6 carry 1 mark each.

1. µÖ×¤ü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.

2. µÖ×¤ü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.

3.  µÖ×¤ü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
65/2/1/F 3 [P.T.O.
4.  µÖ×¤
?
a ,
?
b ŸÖ£ÖÖ
?
c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖ×¤ü¿Ö Æïü ×Ûú
?
a +
?
b +
?
c =
?
0 Æîü, ŸÖÖê
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a ÛúÖ ´ÖÖ®Ö
×»Ö×ÜÖ‹ …
If
?
a ,
?
b,
?
c are unit vectors such that
?
a +
?
b +
?
c =
?
0, then write the value of
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a .

5.  µÖ×¤ü
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 Æîü ŸÖ£ÖÖ
?
a = 5 Æîü, ŸÖÖê
?
b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 and
?
a = 5, then write the value of
?
b .

6.  ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö Ø²Ö¤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö
ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …
Write the equation of a plane which is at a distance of 5 3 units from origin and the
normal to which is equally inclined to coordinate axes.

ÜÖÞ›ü – ²Ö
SECTION – B

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …
Question numbers 7 to 19 carry 4 marks each.

7.  ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú
cot
–1

1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2

†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúß×•Ö‹ :
tan
–1

?
?
?
?
?
?
x – 2
x – 1
+ tan
–1

?
?
?
?
?
?
x + 2
x + 1
=
p
4

Prove that :
cot
–1

1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2

OR
Solve for x :
tan
–1

?
?
?
?
?
?
x – 2
x – 1
+ tan
–1

?
?
?
?
?
?
x + 2
x + 1
=
p
4

Page 4

65/2/1/F 1 [P.T.O.

¸üÖê»Ö ®ÖÓ.
Roll No.

ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ×¸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü †×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours   Maximum Marks : 100

ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æîü …
(iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …
(v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ
Æïü …
(vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …
Series : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.

65/2/1/F
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´Öã×¦üŸÖ ¯ÖéÂšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü ×¤ü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯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×ÃŸÖÛúÖ ¯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×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.

SET – 1
65/2/1/F 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying
6 marks each
(vi) Please write down the serial number of the Question before attempting it.

ÜÖÞ›ü – †
SECTION – A

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …
Question numbers 1 to 6 carry 1 mark each.

1. µÖ×¤ü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.

2. µÖ×¤ü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.

3.  µÖ×¤ü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
65/2/1/F 3 [P.T.O.
4.  µÖ×¤
?
a ,
?
b ŸÖ£ÖÖ
?
c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖ×¤ü¿Ö Æïü ×Ûú
?
a +
?
b +
?
c =
?
0 Æîü, ŸÖÖê
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a ÛúÖ ´ÖÖ®Ö
×»Ö×ÜÖ‹ …
If
?
a ,
?
b,
?
c are unit vectors such that
?
a +
?
b +
?
c =
?
0, then write the value of
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a .

5.  µÖ×¤ü
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 Æîü ŸÖ£ÖÖ
?
a = 5 Æîü, ŸÖÖê
?
b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 and
?
a = 5, then write the value of
?
b .

6.  ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö Ø²Ö¤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö
ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …
Write the equation of a plane which is at a distance of 5 3 units from origin and the
normal to which is equally inclined to coordinate axes.

ÜÖÞ›ü – ²Ö
SECTION – B

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …
Question numbers 7 to 19 carry 4 marks each.

7.  ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú
cot
–1

1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2

†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúß×•Ö‹ :
tan
–1

?
?
?
?
?
?
x – 2
x – 1
+ tan
–1

?
?
?
?
?
?
x + 2
x + 1
=
p
4

Prove that :
cot
–1

1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2

OR
Solve for x :
tan
–1

?
?
?
?
?
?
x – 2
x – 1
+ tan
–1

?
?
?
?
?
?
x + 2
x + 1
=
p
4

65/2/1/F 4
8.  †ÓÝÖÏê•Öß ×¾ÖÂÖµÖ ¯ÖœÌüÖ®Öê ¾ÖÖ»Öß ÛúÖêØ“ÖÝÖ ÃÖÓÃ£ÖÖ ¤üÖê ²Öî“Ö I †Öî¸ü II ´Öë ÛúõÖÖ »ÖêŸÖß Æîü ×•Ö®Ö´Öë †´Öß¸ü ¾Ö ÝÖ¸üß²Ö ²Ö““ÖÖë Ûêú
×»Ö‹ ±úßÃÖ †»ÖÝÖ-†»ÖÝÖ Æîü … ²Öî“Ö I ´Öë 20 ÝÖ¸üß²Ö ŸÖ£ÖÖ 5 †´Öß¸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö
` 9,000 Æîü, •Ö²Ö×Ûú ²Öî“Ö II ´Öë 5 ÝÖ¸üß²Ö †Öî¸ü 25 †´Öß¸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö ` 26,000 Æî …
†Ö¾µÖæÆü ×¾Ö×¬Ö ÃÖê ¤üÖê®ÖÖë ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏŸµÖêÛú ²Ö““Öê «üÖ¸üÖ ¤üß ÝÖ‡Ô ´ÖÖ×ÃÖÛú ±úßÃÖ –ÖÖŸÖ Ûúß×•Ö‹ … ‹êÃÖÖ Ûú¸üÛêú ÛúÖêØ“ÖÝÖ
ÃÖÓÃ£ÖÖ ÃÖ´ÖÖ•Ö ´Öë ŒµÖÖ ´Öæ»µÖ ¸üÜÖ ¸üÆüß Æîü ?
A coaching institute of English (subject) conducts classes in two batches I and II and
fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich
children and total monthly collection is ` 9,000, whereas in batch II, it has 5 poor and
25 rich children and total monthly collection is ` 26,000. Using matrix method, find
monthly fees paid by each child of two types. What values the coaching institute is
inculcating in the society ?

9.  µÖ×¤ü ±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 Ûúß×•Ö‹ …
Find the values of a and b, if the function f defined by
f(x) =
? ?
?
?
?

x
2
+ 3x + a ,   x = 1
bx + 2        ,   x > 1

is differentiable at x = 1.

10.  tan
–1

?
?
?
?
?
?
1 + x
2
– 1
x
ÛúÖ sin
–1

2x
1 + x
2
Ûêú ÃÖÖ¯ÖêõÖ †¾ÖÛú»Ö®Ö Ûúß×•Ö‹, •Ö²Ö×Ûú x ? (–1, 1) Æîü …
†£Ö¾ÖÖ
µÖ×¤ü x = sin t Æîü ŸÖ£ÖÖ y = sin pt Æîü, ŸÖÖê ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0
Differentiate tan
–1

?
?
?
?
?
?
1 + x
2
– 1
x
w.r.t. sin
–1

2x
1 + x
2
, if x ? (–1, 1)
OR
If x = sin t and y = sin pt, prove that (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0.

11.  ¾ÖÛÎúÖë y
2
= 4ax ŸÖ£ÖÖ x
2
= 4by Ûêú ²Öß“Ö ÛúÖ ¯ÖÏ×ŸÖ“”êû¤üß ÛúÖêÞÖ –ÖÖŸÖ Ûúß×•Ö‹ …
Find the angle of intersection of the curves y
2
= 4ax and x
2
= 4by.

12.  ´ÖÖ®Ö –ÖÖŸÖ Ûúß×•Ö‹ :
)
?
(
0
p

x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p

x
1 + sin a sin x
dx
Page 5

65/2/1/F 1 [P.T.O.

¸üÖê»Ö ®ÖÓ.
Roll No.

ÝÖ×ÞÖŸÖ
MATHEMATICS
×®Ö¬ÖÖÔ×¸üŸÖ ÃÖ´ÖµÖ : 3 ‘ÖÞ™êüü †×¬ÖÛúŸÖ´Ö †ÓÛú : 100
Time allowed : 3 hours   Maximum Marks : 100

ÃÖÖ´ÖÖ®µÖ ×®Ö¤ìü¿Ö :
(i) ÃÖ³Öß ¯ÖÏ¿®ÖÖë Ûêú ˆ¢Ö¸ü ×»ÖÜÖ®Öê Æïü …
(ii) Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë 26 ¯ÖÏ¿®Ö Æîü …
(iii) ÜÖÞ›ü-† Ûêú ¯ÖÏ¿®Ö 1–6 ŸÖÛú †×ŸÖ »Ö‘Öã-ˆ¢Ö¸ü ¾ÖÖ»Öê ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 1 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æîü …
(iv) ÜÖÞ›ü-²Ö Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 7–19 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü I ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 4 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ Æïü …
(v) ÜÖÞ›ü-ÃÖ Ûêú ¯ÖÏ¿®Ö ÃÖÓ. 20–26 ŸÖÛú ¤üß‘ÖÔ-ˆ¢Ö¸ü II ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏ¿®Ö Æïü †Öî¸ü ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö Ûêú ×»Ö‹ 6 †ÓÛú ×®Ö¬ÖÖÔ×¸üŸÖ
Æïü …
(vi) ˆ¢Ö¸ü ×»ÖÜÖ®ÖÖ ¯ÖÏÖ¸Óü³Ö Ûú¸ü®Öê ÃÖê ¯ÖÆü»Öê Ûéú¯ÖµÖÖ ¯ÖÏ¿®Ö ÛúÖ ÛÎú´ÖÖÓÛú †¾Ö¿µÖ ×»Ö×ÜÖ‹ …
Series : ONS/2
ÛúÖê›ü ®ÖÓ.
Code No.

65/2/1/F
• Ûéú¯ÖµÖÖ •ÖÖÑ“Ö Ûú¸ü »Öë ×Ûú ‡ÃÖ ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ´Öã×¦üŸÖ ¯ÖéÂšü 8 Æïü …
• ¯ÖÏ¿®Ö-¯Ö¡Ö ´Öë ¤üÖ×Æü®Öê ÆüÖ£Ö Ûúß †Öê¸ü ×¤ü‹ ÝÖ‹ ÛúÖê›ü ®Ö´²Ö¸ü ÛúÖê ”ûÖ¡Ö ˆ¢Ö¸ü-¯Öã×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü ¯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×ÃŸÖÛúÖ ¯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×ÃŸÖÛúÖ Ûêú ´ÖãÜÖ-¯ÖéÂšü
¯Ö¸ü †¾Ö¿µÖ ×»ÖÜÖë …
Candidates must write the Code on
the title page of the answer-book.

SET – 1
65/2/1/F 2
General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one
mark each.
(iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks
each.
(v) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying
6 marks each
(vi) Please write down the serial number of the Question before attempting it.

ÜÖÞ›ü – †
SECTION – A

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 1 ÃÖê 6 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 1 †ÓÛú ÛúÖ Æîü …
Question numbers 1 to 6 carry 1 mark each.

1. µÖ×¤ü (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A Æîü, ŸÖÖê †Ö¾µÖæÆü A Ûúß ÛúÖê×™ü ×»Ö×ÜÖ‹ …
If (2 1 3)
?
?
?
?
?
?
–1 0 –1
–1 1 0
0 1 1

?
?
?
?
?
?
1
0
–1
= A, then write the order of matrix A.

2. µÖ×¤ü
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8 Æîü, ŸÖÖê x ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
?
?
?
?
?
x sin ? cos ?
–sin ? –x 1
cos ? 1 x
= 8, write the value of x.

3.  µÖ×¤ü A =
?
?
?
?
?
?
3 5
7 9
ÛúÖê A = P + Q Ûêú ºþ¯Ö ´Öë ×»ÖÜÖÖ •ÖÖŸÖÖ Æîü •ÖÆüÖÑ P ‹Ûú ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü ŸÖ£ÖÖ Q ‹Ûú
×¾ÖÂÖ´Ö ÃÖ´Ö×´ÖŸÖ †Ö¾µÖæÆü Æîü, ŸÖÖê †Ö¾µÖæÆü P ×»Ö×ÜÖ‹ …
If A =
?
?
?
?
?
?
3 5
7 9
is written as A = P + Q, where P is a symmetric matrix and Q is skew
symmetric matrix, then write the matrix P.
65/2/1/F 3 [P.T.O.
4.  µÖ×¤
?
a ,
?
b ŸÖ£ÖÖ
?
c ‹êÃÖê ´ÖÖ¡ÖÛú ÃÖ×¤ü¿Ö Æïü ×Ûú
?
a +
?
b +
?
c =
?
0 Æîü, ŸÖÖê
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a ÛúÖ ´ÖÖ®Ö
×»Ö×ÜÖ‹ …
If
?
a ,
?
b,
?
c are unit vectors such that
?
a +
?
b +
?
c =
?
0, then write the value of
?
a ·
?
b +
?
b ·
?
c +
?
c ·
?
a .

5.  µÖ×¤ü
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 Æîü ŸÖ£ÖÖ
?
a = 5 Æîü, ŸÖÖê
?
b ÛúÖ ´ÖÖ®Ö ×»Ö×ÜÖ‹ …
If
?
a ×
?
b
2
+
?
a ·
?
b
2
= 400 and
?
a = 5, then write the value of
?
b .

6.  ˆÃÖ ÃÖ´ÖŸÖ»Ö ÛúÖ ÃÖ´ÖßÛú¸üÞÖ ×»Ö×ÜÖ‹ •ÖÖê ´Öæ»Ö Ø²Ö¤ãü ÃÖê 5 3 Ûúß ¤æü¸üß ¯Ö¸ü Æîü ŸÖ£ÖÖ ×•ÖÃÖÛúÖ †×³Ö»ÖÓ²Ö †õÖÖë ¯Ö¸ü ÃÖ´ÖÖ®Ö
ºþ¯Ö ÃÖê —ÖãÛúÖ Æîü …
Write the equation of a plane which is at a distance of 5 3 units from origin and the
normal to which is equally inclined to coordinate axes.

ÜÖÞ›ü – ²Ö
SECTION – B

¯ÖÏ¿®Ö ÃÖÓÜµÖÖ 7 ÃÖê 19 ŸÖÛú ¯ÖÏŸµÖêÛú ¯ÖÏ¿®Ö 4 †ÓÛú ÛúÖ Æîü …
Question numbers 7 to 19 carry 4 marks each.

7.  ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú
cot
–1

1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2

†£Ö¾ÖÖ
x Ûêú ×»Ö‹ Æü»Ö Ûúß×•Ö‹ :
tan
–1

?
?
?
?
?
?
x – 2
x – 1
+ tan
–1

?
?
?
?
?
?
x + 2
x + 1
=
p
4

Prove that :
cot
–1

1 + sin x + 1 – sin x
1 + sin x – 1 – sin x
=
x
2
, 0 < x <
p
2

OR
Solve for x :
tan
–1

?
?
?
?
?
?
x – 2
x – 1
+ tan
–1

?
?
?
?
?
?
x + 2
x + 1
=
p
4

65/2/1/F 4
8.  †ÓÝÖÏê•Öß ×¾ÖÂÖµÖ ¯ÖœÌüÖ®Öê ¾ÖÖ»Öß ÛúÖêØ“ÖÝÖ ÃÖÓÃ£ÖÖ ¤üÖê ²Öî“Ö I †Öî¸ü II ´Öë ÛúõÖÖ »ÖêŸÖß Æîü ×•Ö®Ö´Öë †´Öß¸ü ¾Ö ÝÖ¸üß²Ö ²Ö““ÖÖë Ûêú
×»Ö‹ ±úßÃÖ †»ÖÝÖ-†»ÖÝÖ Æîü … ²Öî“Ö I ´Öë 20 ÝÖ¸üß²Ö ŸÖ£ÖÖ 5 †´Öß¸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö
` 9,000 Æîü, •Ö²Ö×Ûú ²Öî“Ö II ´Öë 5 ÝÖ¸üß²Ö †Öî¸ü 25 †´Öß¸ü ²Ö““Öê Æïü †Öî¸ü Ûãú»Ö ´ÖÖ×ÃÖÛú ÃÖÓ“ÖµÖ ¸üÖ×¿Ö ` 26,000 Æî …
†Ö¾µÖæÆü ×¾Ö×¬Ö ÃÖê ¤üÖê®ÖÖë ¯ÖÏÛúÖ¸ü Ûêú ¯ÖÏŸµÖêÛú ²Ö““Öê «üÖ¸üÖ ¤üß ÝÖ‡Ô ´ÖÖ×ÃÖÛú ±úßÃÖ –ÖÖŸÖ Ûúß×•Ö‹ … ‹êÃÖÖ Ûú¸üÛêú ÛúÖêØ“ÖÝÖ
ÃÖÓÃ£ÖÖ ÃÖ´ÖÖ•Ö ´Öë ŒµÖÖ ´Öæ»µÖ ¸üÜÖ ¸üÆüß Æîü ?
A coaching institute of English (subject) conducts classes in two batches I and II and
fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich
children and total monthly collection is ` 9,000, whereas in batch II, it has 5 poor and
25 rich children and total monthly collection is ` 26,000. Using matrix method, find
monthly fees paid by each child of two types. What values the coaching institute is
inculcating in the society ?

9.  µÖ×¤ü ±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 Ûúß×•Ö‹ …
Find the values of a and b, if the function f defined by
f(x) =
? ?
?
?
?

x
2
+ 3x + a ,   x = 1
bx + 2        ,   x > 1

is differentiable at x = 1.

10.  tan
–1

?
?
?
?
?
?
1 + x
2
– 1
x
ÛúÖ sin
–1

2x
1 + x
2
Ûêú ÃÖÖ¯ÖêõÖ †¾ÖÛú»Ö®Ö Ûúß×•Ö‹, •Ö²Ö×Ûú x ? (–1, 1) Æîü …
†£Ö¾ÖÖ
µÖ×¤ü x = sin t Æîü ŸÖ£ÖÖ y = sin pt Æîü, ŸÖÖê ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0
Differentiate tan
–1

?
?
?
?
?
?
1 + x
2
– 1
x
w.r.t. sin
–1

2x
1 + x
2
, if x ? (–1, 1)
OR
If x = sin t and y = sin pt, prove that (1 – x
2
)
d
2
y
dx
2
– x
dy
dx
+ p
2
y = 0.

11.  ¾ÖÛÎúÖë y
2
= 4ax ŸÖ£ÖÖ x
2
= 4by Ûêú ²Öß“Ö ÛúÖ ¯ÖÏ×ŸÖ“”êû¤üß ÛúÖêÞÖ –ÖÖŸÖ Ûúß×•Ö‹ …
Find the angle of intersection of the curves y
2
= 4ax and x
2
= 4by.

12.  ´ÖÖ®Ö –ÖÖŸÖ Ûúß×•Ö‹ :
)
?
(
0
p

x
1 + sin a sin x
dx
Evaluate :
)
?
(
0
p

x
1 + sin a sin x
dx
65/2/1/F 5 [P.T.O.
13.  –ÖÖŸÖ Ûúß×•Ö‹ :
)
?
(
.
.
(2x + 5) 10 – 4x – 3x
2
dx
†£Ö¾ÖÖ
–ÖÖŸÖ Ûúß×•Ö‹ :
)
?
(

(x
2
+ 1) (x
2
+ 4)
(x
2
+ 3) (x
2
– 5)
dx
Find :
)
?
(
.
.
(2x + 5) 10 – 4x – 3x
2
dx
OR
Find :
)
?
(

(x
2
+ 1) (x
2
+ 4)
(x
2
+ 3) (x
2
– 5)
dx

14.  –ÖÖŸÖ Ûúß×•Ö‹ :
)
?
(

x sin
–1
x
1 – x
2
dx
Find :
)
?
(

x sin
–1
x
1 – x
2
dx

15.  ×®Ö´®Ö †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ÛúÖê Æü»Ö Ûúß×•Ö‹ :
y
2
dx + (x
2
– xy + y
2
)dy = 0
Solve the following differential equation :
y
2
dx + (x
2
– xy + y
2
)dy = 0

16.  ×®Ö´®Ö †¾ÖÛú»Ö ÃÖ´ÖßÛú¸üÞÖ ÛúÖê Æü»Ö Ûúß×•Ö‹ :
(cot
–1
y + x) dy = (1 + y
2
) dx
Solve the following differential equation :
(cot
–1
y + x) dy = (1 + y
2
) dx

17.  µÖ×¤ü
?
a ×
?
b =
?
c ×
?
d †Öî¸ü
?
a ×
?
c =
?
b ×
?
d Æîü, ŸÖÖê ¤ü¿ÖÖÔ‡‹ ×Ûú
?
a –
?
d,
?
b –
?
c Ûêú ÃÖ´ÖÖÓŸÖ¸ü Æîü, •Ö²Ö×Ûú
?
a ?
?
d
†Öî¸ü
?
b ?
?
c Æîü …
If
?
a ×
?
b =
?
c ×
?
d and
?
a ×
?
c =
?
b ×
?
d, show that
?
a –
?
d is parallel to
?
b –
?
c , where
?
a ?
?
d
and
?
b ?
?
c .

18.  ×ÃÖ¨ü Ûúß×•Ö‹ ×Ûú Ø²Ö¤ãü†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 ¯ÖÏ×ŸÖ“”êû¤ü Ûú¸üŸÖß Æîü …
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through
C(3, 9, 4) and D(–4, 4, 4).
```
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## Mathematics (Maths) Class 12

209 videos|231 docs|124 tests

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