Class 12 Exam > Class 12 Notes > Toppers Answer Sheets for Class 12 > CBSE Class 12 Mathematics: Question Paper for 2019 (Term-II)
CBSE Class 12 Mathematics: Question Paper for 2019 (Term-II) | Toppers Answer Sheets for Class 12 PDF Download
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Page 1
65/1/2 1 [P.T.O.
Roll No.
????
MATHEMATICS
3 100
Time allowed : 3 hours Maximum Marks : 100
Series : BVM/1
Code No.
65/1/2
? ????? ???? ?? ?? ?? ?? ???-?? ?? ????? ???? 11 ?? ?
? ???-?? ?? ?????? ??? ?? ???? ??? ?? ???? ???? ??? ??? ???-??????? ?? ???-???? ?? ???? ?
? ????? ???? ?? ?? ?? ?? ???-?? ?? 29 ??? ?? ?
? ????? ??? ?? ??? ????? ??? ???? ?? ????, ??? ?? ????? ???? ???? ?
? ?? ???-?? ??? ???? ?? ??? 15 ???? ?? ??? ???? ??? ?? ? ???-?? ?? ????? ?????? ?? 10.15
??? ???? ????? ? 10.15 ??? ?? 10.30 ??? ?? ??? ???? ???-?? ??? ????? ???? ?? ???? ??
?????? ?? ???-??????? ?? ????? ??? ??? ?????? ?
? Please check that this question paper contains 11 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 29 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.
??????? ???? ??? ???-??????? ?? ???-????
?? ???? ???? ?
Candidates must write the Code on
the title page of the answer-book.
SET – 2
Page 2
65/1/2 1 [P.T.O.
Roll No.
????
MATHEMATICS
3 100
Time allowed : 3 hours Maximum Marks : 100
Series : BVM/1
Code No.
65/1/2
? ????? ???? ?? ?? ?? ?? ???-?? ?? ????? ???? 11 ?? ?
? ???-?? ?? ?????? ??? ?? ???? ??? ?? ???? ???? ??? ??? ???-??????? ?? ???-???? ?? ???? ?
? ????? ???? ?? ?? ?? ?? ???-?? ?? 29 ??? ?? ?
? ????? ??? ?? ??? ????? ??? ???? ?? ????, ??? ?? ????? ???? ???? ?
? ?? ???-?? ??? ???? ?? ??? 15 ???? ?? ??? ???? ??? ?? ? ???-?? ?? ????? ?????? ?? 10.15
??? ???? ????? ? 10.15 ??? ?? 10.30 ??? ?? ??? ???? ???-?? ??? ????? ???? ?? ???? ??
?????? ?? ???-??????? ?? ????? ??? ??? ?????? ?
? Please check that this question paper contains 11 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 29 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.
??????? ???? ??? ???-??????? ?? ???-????
?? ???? ???? ?
Candidates must write the Code on
the title page of the answer-book.
SET – 2
65/1/2 2
?????? ????? ?
(i) ??? ??? ??????? ?? ?
(ii) ?? ???-?? ?? 29 ??? ?? ??? ??? ????? ?? ??????? ?? ? ?, ?, ? ??? ? ? ??? ? ?? 4 ??? ??
????? ?? ????? ?? ??? ?? ?? ? ??? ? ?? 8 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ??
11 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ?? 6 ??? ?? ????? ?? ????? ?? ??? ?? ?? ?
(iii) ??? ? ?? ??? ????? ?? ??? ?? ???, ?? ???? ???? ??? ?? ????????????? ??? ?? ???? ?? ?
(iv) ???? ???-?? ?? ????? ??? ?? ? ??? ?? ??? ? ?? 1 ???, ??? ? ?? 3 ????? ??, ??? ? ?? 3
????? ?? ??? ??? ? ?? 3 ????? ?? ??????? ????? ?? ? ???? ??? ????? ?? ?? ?????? ?? ??
????? ?? ???? ?? ?
(v) ????????? ?? ????? ?? ?????? ??? ?? ? ??? ?????? ???, ??? ??? ???????? ???????? ????
???? ?? ?
General Instructions :
(i) All questions are compulsory.
(ii) This question paper contains 29 questions divided into four sections A, B, C and D.
Section A comprises of 4 questions of one mark each, Section B comprises of 8
questions of two marks each, Section C comprises of 11 questions of four marks
each and Section D comprises of 6 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 1
question of Section A, 3 questions of Section B, 3 questions of Section C and 3
questions of Section D. You have to attempt only one of the alternatives in all such
questions.
(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.
??? – ?
SECTION – A
??? ????? 1 ?? 4 ?? ?? ????? ??? 1 ??? ?? ?? ?
Question numbers 1 to 4 carry 1 mark each.
1. ???? ?????? x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
?? ????? ? ??? ??? ????? ?
Find the order and the degree of the differential equation x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
.
Page 3
65/1/2 1 [P.T.O.
Roll No.
????
MATHEMATICS
3 100
Time allowed : 3 hours Maximum Marks : 100
Series : BVM/1
Code No.
65/1/2
? ????? ???? ?? ?? ?? ?? ???-?? ?? ????? ???? 11 ?? ?
? ???-?? ?? ?????? ??? ?? ???? ??? ?? ???? ???? ??? ??? ???-??????? ?? ???-???? ?? ???? ?
? ????? ???? ?? ?? ?? ?? ???-?? ?? 29 ??? ?? ?
? ????? ??? ?? ??? ????? ??? ???? ?? ????, ??? ?? ????? ???? ???? ?
? ?? ???-?? ??? ???? ?? ??? 15 ???? ?? ??? ???? ??? ?? ? ???-?? ?? ????? ?????? ?? 10.15
??? ???? ????? ? 10.15 ??? ?? 10.30 ??? ?? ??? ???? ???-?? ??? ????? ???? ?? ???? ??
?????? ?? ???-??????? ?? ????? ??? ??? ?????? ?
? Please check that this question paper contains 11 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 29 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.
??????? ???? ??? ???-??????? ?? ???-????
?? ???? ???? ?
Candidates must write the Code on
the title page of the answer-book.
SET – 2
65/1/2 2
?????? ????? ?
(i) ??? ??? ??????? ?? ?
(ii) ?? ???-?? ?? 29 ??? ?? ??? ??? ????? ?? ??????? ?? ? ?, ?, ? ??? ? ? ??? ? ?? 4 ??? ??
????? ?? ????? ?? ??? ?? ?? ? ??? ? ?? 8 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ??
11 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ?? 6 ??? ?? ????? ?? ????? ?? ??? ?? ?? ?
(iii) ??? ? ?? ??? ????? ?? ??? ?? ???, ?? ???? ???? ??? ?? ????????????? ??? ?? ???? ?? ?
(iv) ???? ???-?? ?? ????? ??? ?? ? ??? ?? ??? ? ?? 1 ???, ??? ? ?? 3 ????? ??, ??? ? ?? 3
????? ?? ??? ??? ? ?? 3 ????? ?? ??????? ????? ?? ? ???? ??? ????? ?? ?? ?????? ?? ??
????? ?? ???? ?? ?
(v) ????????? ?? ????? ?? ?????? ??? ?? ? ??? ?????? ???, ??? ??? ???????? ???????? ????
???? ?? ?
General Instructions :
(i) All questions are compulsory.
(ii) This question paper contains 29 questions divided into four sections A, B, C and D.
Section A comprises of 4 questions of one mark each, Section B comprises of 8
questions of two marks each, Section C comprises of 11 questions of four marks
each and Section D comprises of 6 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 1
question of Section A, 3 questions of Section B, 3 questions of Section C and 3
questions of Section D. You have to attempt only one of the alternatives in all such
questions.
(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.
??? – ?
SECTION – A
??? ????? 1 ?? 4 ?? ?? ????? ??? 1 ??? ?? ?? ?
Question numbers 1 to 4 carry 1 mark each.
1. ???? ?????? x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
?? ????? ? ??? ??? ????? ?
Find the order and the degree of the differential equation x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
.
65/1/2 3
[P.T.O.
2. ??? f(x) = x + 7 ???? g(x) = x – 7, x ? R ???, ???
d
dx
(fog) (x) ??? ????? ?
If f(x) = x + 7 and g(x) = x – 7, x ? R, then find
d
dx
(fog)(x).
3. ??? 2
?
?
?
?
?
?
1 3
0 x
+
?
?
?
?
?
?
y 0
1 2
=
?
?
?
?
?
?
5 6
1 8
??, ??? x – y ?? ??? ??? ????? ?
Find the value of x – y, if
2
?
?
?
?
?
?
1 3
0 x
+
?
?
?
?
?
?
y 0
1 2
=
?
?
?
?
?
?
5 6
1 8
.
4. ??? ?? ???? x-??, y-?? ??? z-?? ?? ???: 90 ?, 135?, 45? ?? ???? ????? ?? ? ?? ???? ??
????-??????? ??? ????? ?
????
?? ???? ?? ???? ?????? ??? ????? ??? ????? (3, 4, 5) ?? ?????? ?? ??? ???? 2
^
i + 2
^
j – 3
^
k ??
?????? ?? ?
If a line makes angles 90 ?, 135 ?, 45 ? with the x, y and z axes respectively, find its
direction cosines.
OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is
parallel to the vector 2
^
i + 2
^
j – 3
^
k.
??? – ?
SECTION – B
??? ????? 5 ?? 12 ?? ?? ????? ??? 2 ??? ?? ?? ?
Question numbers 5 to 12 carry 2 marks each.
5. ???? ????? ?? ??? ?????? * ??? R ?? a * b = ab + 1 ???? ???????? ?? (i) ??-?????? ??????
????? ?? ??? (ii) ??? ?? ??-?????? ??, ??? ??? ?? ?????? ????? ?? ??? ?
Examine whether the operation * defined on R by a * b = ab + 1 is (i) a binary or not.
(ii) if a binary operation, is it associative or not ?
Page 4
65/1/2 1 [P.T.O.
Roll No.
????
MATHEMATICS
3 100
Time allowed : 3 hours Maximum Marks : 100
Series : BVM/1
Code No.
65/1/2
? ????? ???? ?? ?? ?? ?? ???-?? ?? ????? ???? 11 ?? ?
? ???-?? ?? ?????? ??? ?? ???? ??? ?? ???? ???? ??? ??? ???-??????? ?? ???-???? ?? ???? ?
? ????? ???? ?? ?? ?? ?? ???-?? ?? 29 ??? ?? ?
? ????? ??? ?? ??? ????? ??? ???? ?? ????, ??? ?? ????? ???? ???? ?
? ?? ???-?? ??? ???? ?? ??? 15 ???? ?? ??? ???? ??? ?? ? ???-?? ?? ????? ?????? ?? 10.15
??? ???? ????? ? 10.15 ??? ?? 10.30 ??? ?? ??? ???? ???-?? ??? ????? ???? ?? ???? ??
?????? ?? ???-??????? ?? ????? ??? ??? ?????? ?
? Please check that this question paper contains 11 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 29 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.
??????? ???? ??? ???-??????? ?? ???-????
?? ???? ???? ?
Candidates must write the Code on
the title page of the answer-book.
SET – 2
65/1/2 2
?????? ????? ?
(i) ??? ??? ??????? ?? ?
(ii) ?? ???-?? ?? 29 ??? ?? ??? ??? ????? ?? ??????? ?? ? ?, ?, ? ??? ? ? ??? ? ?? 4 ??? ??
????? ?? ????? ?? ??? ?? ?? ? ??? ? ?? 8 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ??
11 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ?? 6 ??? ?? ????? ?? ????? ?? ??? ?? ?? ?
(iii) ??? ? ?? ??? ????? ?? ??? ?? ???, ?? ???? ???? ??? ?? ????????????? ??? ?? ???? ?? ?
(iv) ???? ???-?? ?? ????? ??? ?? ? ??? ?? ??? ? ?? 1 ???, ??? ? ?? 3 ????? ??, ??? ? ?? 3
????? ?? ??? ??? ? ?? 3 ????? ?? ??????? ????? ?? ? ???? ??? ????? ?? ?? ?????? ?? ??
????? ?? ???? ?? ?
(v) ????????? ?? ????? ?? ?????? ??? ?? ? ??? ?????? ???, ??? ??? ???????? ???????? ????
???? ?? ?
General Instructions :
(i) All questions are compulsory.
(ii) This question paper contains 29 questions divided into four sections A, B, C and D.
Section A comprises of 4 questions of one mark each, Section B comprises of 8
questions of two marks each, Section C comprises of 11 questions of four marks
each and Section D comprises of 6 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 1
question of Section A, 3 questions of Section B, 3 questions of Section C and 3
questions of Section D. You have to attempt only one of the alternatives in all such
questions.
(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.
??? – ?
SECTION – A
??? ????? 1 ?? 4 ?? ?? ????? ??? 1 ??? ?? ?? ?
Question numbers 1 to 4 carry 1 mark each.
1. ???? ?????? x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
?? ????? ? ??? ??? ????? ?
Find the order and the degree of the differential equation x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
.
65/1/2 3
[P.T.O.
2. ??? f(x) = x + 7 ???? g(x) = x – 7, x ? R ???, ???
d
dx
(fog) (x) ??? ????? ?
If f(x) = x + 7 and g(x) = x – 7, x ? R, then find
d
dx
(fog)(x).
3. ??? 2
?
?
?
?
?
?
1 3
0 x
+
?
?
?
?
?
?
y 0
1 2
=
?
?
?
?
?
?
5 6
1 8
??, ??? x – y ?? ??? ??? ????? ?
Find the value of x – y, if
2
?
?
?
?
?
?
1 3
0 x
+
?
?
?
?
?
?
y 0
1 2
=
?
?
?
?
?
?
5 6
1 8
.
4. ??? ?? ???? x-??, y-?? ??? z-?? ?? ???: 90 ?, 135?, 45? ?? ???? ????? ?? ? ?? ???? ??
????-??????? ??? ????? ?
????
?? ???? ?? ???? ?????? ??? ????? ??? ????? (3, 4, 5) ?? ?????? ?? ??? ???? 2
^
i + 2
^
j – 3
^
k ??
?????? ?? ?
If a line makes angles 90 ?, 135 ?, 45 ? with the x, y and z axes respectively, find its
direction cosines.
OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is
parallel to the vector 2
^
i + 2
^
j – 3
^
k.
??? – ?
SECTION – B
??? ????? 5 ?? 12 ?? ?? ????? ??? 2 ??? ?? ?? ?
Question numbers 5 to 12 carry 2 marks each.
5. ???? ????? ?? ??? ?????? * ??? R ?? a * b = ab + 1 ???? ???????? ?? (i) ??-?????? ??????
????? ?? ??? (ii) ??? ?? ??-?????? ??, ??? ??? ?? ?????? ????? ?? ??? ?
Examine whether the operation * defined on R by a * b = ab + 1 is (i) a binary or not.
(ii) if a binary operation, is it associative or not ?
65/1/2 4
6. ??? A =
?
?
?
?
?
?
?
?
2 0 1
2 1 3
1 –1 0
??, ??? (A
2
– 5A) ??? ????? ?
If A =
?
?
?
?
?
?
?
?
2 0 1
2 1 3
1 –1 0
, then find (A
2
– 5A).
7. ??? ????? :
?
?
.
.
1 – sin 2x dx,
?
4
< x <
?
2
????
??? ????? :
?
?
.
.
sin
–1
(2x) dx.
Find :
?
?
.
.
1 – sin 2x dx,
?
4
< x <
?
2
OR
Find :
?
?
.
.
sin
–1
(2x) dx.
8. ???? ?? ??? y = e
2x
(a + bx), ????? a, b ????? ??? ??, ??? ?????? ???? ???? ???? ??????
??? ????? ?
Form the differential equation representing the family of curves y = e
2x
(a + bx), where
‘a’ and ‘b’ are arbitrary constants.
9. ?? ???? ??? ?: ??? ????? ???? ?? ? ??? “???? ?? ???? ????? ??? ?????” ?? ????? ??, ???
(i) 5 ??????? (ii) ?????? 5 ???????, ?? ????????? ???-??? ????? ?
????
?? ??????? ?? X ?? ??????? ???? P(X) ???? ???? ?? ??, ???? ‘k’ ????? ????? ?? :
P(X = x) =
?
?
?
?
?
k , ??? x = 0
2k, ??? x = 1
3k, ??? x = 2
0, ?????
‘k’ ?? ??? ??? ????? ?
Page 5
65/1/2 1 [P.T.O.
Roll No.
????
MATHEMATICS
3 100
Time allowed : 3 hours Maximum Marks : 100
Series : BVM/1
Code No.
65/1/2
? ????? ???? ?? ?? ?? ?? ???-?? ?? ????? ???? 11 ?? ?
? ???-?? ?? ?????? ??? ?? ???? ??? ?? ???? ???? ??? ??? ???-??????? ?? ???-???? ?? ???? ?
? ????? ???? ?? ?? ?? ?? ???-?? ?? 29 ??? ?? ?
? ????? ??? ?? ??? ????? ??? ???? ?? ????, ??? ?? ????? ???? ???? ?
? ?? ???-?? ??? ???? ?? ??? 15 ???? ?? ??? ???? ??? ?? ? ???-?? ?? ????? ?????? ?? 10.15
??? ???? ????? ? 10.15 ??? ?? 10.30 ??? ?? ??? ???? ???-?? ??? ????? ???? ?? ???? ??
?????? ?? ???-??????? ?? ????? ??? ??? ?????? ?
? Please check that this question paper contains 11 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 29 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.
??????? ???? ??? ???-??????? ?? ???-????
?? ???? ???? ?
Candidates must write the Code on
the title page of the answer-book.
SET – 2
65/1/2 2
?????? ????? ?
(i) ??? ??? ??????? ?? ?
(ii) ?? ???-?? ?? 29 ??? ?? ??? ??? ????? ?? ??????? ?? ? ?, ?, ? ??? ? ? ??? ? ?? 4 ??? ??
????? ?? ????? ?? ??? ?? ?? ? ??? ? ?? 8 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ??
11 ??? ?? ????? ?? ????? ??? ??? ?? ?? ? ??? ? ?? 6 ??? ?? ????? ?? ????? ?? ??? ?? ?? ?
(iii) ??? ? ?? ??? ????? ?? ??? ?? ???, ?? ???? ???? ??? ?? ????????????? ??? ?? ???? ?? ?
(iv) ???? ???-?? ?? ????? ??? ?? ? ??? ?? ??? ? ?? 1 ???, ??? ? ?? 3 ????? ??, ??? ? ?? 3
????? ?? ??? ??? ? ?? 3 ????? ?? ??????? ????? ?? ? ???? ??? ????? ?? ?? ?????? ?? ??
????? ?? ???? ?? ?
(v) ????????? ?? ????? ?? ?????? ??? ?? ? ??? ?????? ???, ??? ??? ???????? ???????? ????
???? ?? ?
General Instructions :
(i) All questions are compulsory.
(ii) This question paper contains 29 questions divided into four sections A, B, C and D.
Section A comprises of 4 questions of one mark each, Section B comprises of 8
questions of two marks each, Section C comprises of 11 questions of four marks
each and Section D comprises of 6 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 1
question of Section A, 3 questions of Section B, 3 questions of Section C and 3
questions of Section D. You have to attempt only one of the alternatives in all such
questions.
(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.
??? – ?
SECTION – A
??? ????? 1 ?? 4 ?? ?? ????? ??? 1 ??? ?? ?? ?
Question numbers 1 to 4 carry 1 mark each.
1. ???? ?????? x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
?? ????? ? ??? ??? ????? ?
Find the order and the degree of the differential equation x
2
d
2
y
dx
2
=
?
?
?
?
?
?
?
?
?
?
1 +
?
?
?
?
?
?
dy
dx
2 4
.
65/1/2 3
[P.T.O.
2. ??? f(x) = x + 7 ???? g(x) = x – 7, x ? R ???, ???
d
dx
(fog) (x) ??? ????? ?
If f(x) = x + 7 and g(x) = x – 7, x ? R, then find
d
dx
(fog)(x).
3. ??? 2
?
?
?
?
?
?
1 3
0 x
+
?
?
?
?
?
?
y 0
1 2
=
?
?
?
?
?
?
5 6
1 8
??, ??? x – y ?? ??? ??? ????? ?
Find the value of x – y, if
2
?
?
?
?
?
?
1 3
0 x
+
?
?
?
?
?
?
y 0
1 2
=
?
?
?
?
?
?
5 6
1 8
.
4. ??? ?? ???? x-??, y-?? ??? z-?? ?? ???: 90 ?, 135?, 45? ?? ???? ????? ?? ? ?? ???? ??
????-??????? ??? ????? ?
????
?? ???? ?? ???? ?????? ??? ????? ??? ????? (3, 4, 5) ?? ?????? ?? ??? ???? 2
^
i + 2
^
j – 3
^
k ??
?????? ?? ?
If a line makes angles 90 ?, 135 ?, 45 ? with the x, y and z axes respectively, find its
direction cosines.
OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is
parallel to the vector 2
^
i + 2
^
j – 3
^
k.
??? – ?
SECTION – B
??? ????? 5 ?? 12 ?? ?? ????? ??? 2 ??? ?? ?? ?
Question numbers 5 to 12 carry 2 marks each.
5. ???? ????? ?? ??? ?????? * ??? R ?? a * b = ab + 1 ???? ???????? ?? (i) ??-?????? ??????
????? ?? ??? (ii) ??? ?? ??-?????? ??, ??? ??? ?? ?????? ????? ?? ??? ?
Examine whether the operation * defined on R by a * b = ab + 1 is (i) a binary or not.
(ii) if a binary operation, is it associative or not ?
65/1/2 4
6. ??? A =
?
?
?
?
?
?
?
?
2 0 1
2 1 3
1 –1 0
??, ??? (A
2
– 5A) ??? ????? ?
If A =
?
?
?
?
?
?
?
?
2 0 1
2 1 3
1 –1 0
, then find (A
2
– 5A).
7. ??? ????? :
?
?
.
.
1 – sin 2x dx,
?
4
< x <
?
2
????
??? ????? :
?
?
.
.
sin
–1
(2x) dx.
Find :
?
?
.
.
1 – sin 2x dx,
?
4
< x <
?
2
OR
Find :
?
?
.
.
sin
–1
(2x) dx.
8. ???? ?? ??? y = e
2x
(a + bx), ????? a, b ????? ??? ??, ??? ?????? ???? ???? ???? ??????
??? ????? ?
Form the differential equation representing the family of curves y = e
2x
(a + bx), where
‘a’ and ‘b’ are arbitrary constants.
9. ?? ???? ??? ?: ??? ????? ???? ?? ? ??? “???? ?? ???? ????? ??? ?????” ?? ????? ??, ???
(i) 5 ??????? (ii) ?????? 5 ???????, ?? ????????? ???-??? ????? ?
????
?? ??????? ?? X ?? ??????? ???? P(X) ???? ???? ?? ??, ???? ‘k’ ????? ????? ?? :
P(X = x) =
?
?
?
?
?
k , ??? x = 0
2k, ??? x = 1
3k, ??? x = 2
0, ?????
‘k’ ?? ??? ??? ????? ?
65/1/2 5
[P.T.O.
A die is thrown 6 times. If “getting an odd number” is a “success”, what is the
probability of (i) 5 successes ? (ii) atmost 5 successes ?
OR
The random variable X has a probability distribution P(X) of the following form,
where ‘k’ is some number.
P(X = x) =
?
?
?
k , if x = 0
2k, if x = 1
3k, if x = 2
0, otherwise
Determine the value of ‘k’.
10. ?? ????? ??? ?? 1, 2, 3 ??? ??? ?? ??? 4, 5, 6 ??? ??? ?? ???? ??? ??, ??? ????? ???? ?? ?
“????? ?? ?????” ?? ???? ??? A ?? ? “????? ??? ??? ?? ???? ??” ?? ???? B ?? ???????? ?? ? ???
????? ?? ??? ?? ??? ?????? A ??? B ????? ?? ?? ??? ?
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event “number is
even” and B be the event “number is marked red”. Find whether the events A and B
are independent or not.
11. ??? ??? ???? ?????? ?? ???? ?? ???? ???? ???, ??? ??? ????? ?? ?? ??? ?????? ?? ???? ??
?????? 3 ????? ?
????
???
?
a = 2
^
i + 3
^
j +
^
k,
?
b =
^
i – 2
^
j +
^
k ???
?
c = – 3
^
i +
^
j + 2
^
k ??, ??? [ ]
?
a
?
b
?
c ??? ????? ?
If the sum of two unit vectors is a unit vector, prove that the magnitude of their
difference is 3.
OR
If
?
a = 2
^
i + 3
^
j +
^
k,
?
b =
^
i – 2
^
j +
^
k and
?
c = – 3
^
i +
^
j + 2
^
k, find [ ]
?
a
?
b
?
c .
12. ??? ????? :
?
?
.
.
tan
2
x sec
2
x
1 – tan
6
x
dx.
Find :
?
?
.
.
tan
2
x sec
2
x
1 – tan
6
x
dx.
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