Past Year Paper - Solutions, Maths (Set - 1, 2 & 3),Outside Delhi, 2015, Class 12, Maths | Additional Study Material for JEE PDF Download

 Page 1


1
Strictly Confidential — (For Internal and Restricted Use Only)
Senior School Certificate Examination
July — 2015 (Comptt.)
Marking Scheme — Mathematics (Outside Delhi) 65/1, 65/2, 65/3
General Instructions:
1. The Marking Scheme provides general guidelines to reduce subjectivity in the marking.
The answers given in the Marking Scheme are suggested answers. The content is thus
indicative. If a student has given any other answer which is different from the one given
in the Marking Scheme, but conveys the meaning, such answers should be given full
weightage.
2. Evaluation is to be done as per instructions provided in the marking scheme. It should
not be done according to one’s own interpretation or any other consideration — Marking
Scheme should be strictly adhered to and religiously followed.
3. Alternative methods are accepted. Proportional marks are to be awarded.
4. In question (s) on differential equations, constant of integration has to be written.
5. If a candidate has attempted an extra question, marks obtained in the question attempted
first should be retained and the other answer should be scored out.
6. A full scale of marks - 0 to 100 has to be used. Please do not hesitate to award full marks
if the answer deserves it.
7. Separate Marking Scheme for all the three sets has been given.
8. As per orders of the Hon’ble Supreme Court. The candidates would now be permitted
to obtain photocopy of the Answer book on request on payment of the prescribed fee.
All examiners/Head Examiners are once again reminded that they must ensure that
evaluation is carried out strictly as per value points for each answer as given in the
Marking Scheme.
Page 2


1
Strictly Confidential — (For Internal and Restricted Use Only)
Senior School Certificate Examination
July — 2015 (Comptt.)
Marking Scheme — Mathematics (Outside Delhi) 65/1, 65/2, 65/3
General Instructions:
1. The Marking Scheme provides general guidelines to reduce subjectivity in the marking.
The answers given in the Marking Scheme are suggested answers. The content is thus
indicative. If a student has given any other answer which is different from the one given
in the Marking Scheme, but conveys the meaning, such answers should be given full
weightage.
2. Evaluation is to be done as per instructions provided in the marking scheme. It should
not be done according to one’s own interpretation or any other consideration — Marking
Scheme should be strictly adhered to and religiously followed.
3. Alternative methods are accepted. Proportional marks are to be awarded.
4. In question (s) on differential equations, constant of integration has to be written.
5. If a candidate has attempted an extra question, marks obtained in the question attempted
first should be retained and the other answer should be scored out.
6. A full scale of marks - 0 to 100 has to be used. Please do not hesitate to award full marks
if the answer deserves it.
7. Separate Marking Scheme for all the three sets has been given.
8. As per orders of the Hon’ble Supreme Court. The candidates would now be permitted
to obtain photocopy of the Answer book on request on payment of the prescribed fee.
All examiners/Head Examiners are once again reminded that they must ensure that
evaluation is carried out strictly as per value points for each answer as given in the
Marking Scheme.
2
QUESTION PAPER CODE 65/1
EXPECTED ANSWER V ALE POINTS
SECTION A
Marks
1.
3a 2b +
r
r
 = 
7i 5 j 4k - +
r r r
½
? D.R’s are 7, –5, 4 ½
2.
(2i 3 j 2k) (2i 2 j k) + + · + +
r r r r r r
 = 12 ½
p = 
12 12
4
3 | b |
= = r
½
3.
r
r
 = 
(i 2 j 3k) (i 2 j 5k) + + + ? + -
r r r r r r
1
4. For singular matrix
     4 sin
2
 x – 3 = 0 ½
sin x = 
3
2
± ? x = 
2
3
p
½
5.
2y
e dy
?
 = 
3
x dx
?
½
? 
2y
e
2
 = 
4
x
c
4
+ ½
6. I.F. = 
1
dx
x
e
? ½
= 
2 x
e
½
SECTION B
7. Let investment in first type of bonds be Rs x.
? Investment in 2nd type = Rs (35000 – x) ½
Page 3


1
Strictly Confidential — (For Internal and Restricted Use Only)
Senior School Certificate Examination
July — 2015 (Comptt.)
Marking Scheme — Mathematics (Outside Delhi) 65/1, 65/2, 65/3
General Instructions:
1. The Marking Scheme provides general guidelines to reduce subjectivity in the marking.
The answers given in the Marking Scheme are suggested answers. The content is thus
indicative. If a student has given any other answer which is different from the one given
in the Marking Scheme, but conveys the meaning, such answers should be given full
weightage.
2. Evaluation is to be done as per instructions provided in the marking scheme. It should
not be done according to one’s own interpretation or any other consideration — Marking
Scheme should be strictly adhered to and religiously followed.
3. Alternative methods are accepted. Proportional marks are to be awarded.
4. In question (s) on differential equations, constant of integration has to be written.
5. If a candidate has attempted an extra question, marks obtained in the question attempted
first should be retained and the other answer should be scored out.
6. A full scale of marks - 0 to 100 has to be used. Please do not hesitate to award full marks
if the answer deserves it.
7. Separate Marking Scheme for all the three sets has been given.
8. As per orders of the Hon’ble Supreme Court. The candidates would now be permitted
to obtain photocopy of the Answer book on request on payment of the prescribed fee.
All examiners/Head Examiners are once again reminded that they must ensure that
evaluation is carried out strictly as per value points for each answer as given in the
Marking Scheme.
2
QUESTION PAPER CODE 65/1
EXPECTED ANSWER V ALE POINTS
SECTION A
Marks
1.
3a 2b +
r
r
 = 
7i 5 j 4k - +
r r r
½
? D.R’s are 7, –5, 4 ½
2.
(2i 3 j 2k) (2i 2 j k) + + · + +
r r r r r r
 = 12 ½
p = 
12 12
4
3 | b |
= = r
½
3.
r
r
 = 
(i 2 j 3k) (i 2 j 5k) + + + ? + -
r r r r r r
1
4. For singular matrix
     4 sin
2
 x – 3 = 0 ½
sin x = 
3
2
± ? x = 
2
3
p
½
5.
2y
e dy
?
 = 
3
x dx
?
½
? 
2y
e
2
 = 
4
x
c
4
+ ½
6. I.F. = 
1
dx
x
e
? ½
= 
2 x
e
½
SECTION B
7. Let investment in first type of bonds be Rs x.
? Investment in 2nd type = Rs (35000 – x) ½
3
8
x
100
35000 x 10
100
? ?
? ?
? ?
? ?
? ?
-
? ?? ?
? ?
? ?
 = (3200) 1½
?
8 10
x (35000 x)
100 100
+ -
= 3200
1
? x = Rs 15000
? Investment in first = Rs 15000
1
and in 2nd = Rs 20000
8. Getting A' = 
2 7 1
4 3 2
6 5 4
? ?
? ?
-
? ?
? ?
-
? ?
1
Let P = 
4 11 5 2 11/ 2 5/ 2
1 1
(A A ) 11 6 3 11/ 2 3 3/ 2
2 2
5 3 8 5/ 2 3/ 2 4
- -
? ? ? ?
? ? ? ?
+ ' = =
? ? ? ?
? ? ? ?
- -
? ? ? ?
1
Since P' = P ? P is a symmetric matrix
Let Q = 
1
(A A )
2
- ' =
0 3 7 0 3/ 2 7 / 2
1
3 0 7 3/ 2 0 7 / 2
2
7 7 0 7 / 2 7 / 2 0
- - - -
? ? ? ?
? ? ? ?
=
? ? ? ?
? ? ? ?
- -
? ? ? ?
1
Since Q' = –Q ? Q is skew symmetric
Also
P + Q = 
2 11/ 2 5/ 2 0 3/ 2 7 / 2
11/ 2 3 3/ 2 3/ 2 0 7 / 2
5/ 2 3/ 2 4 7 / 2 7 / 2 0
- - -
? ? ? ?
? ? ? ?
+
? ? ? ?
? ? ? ?
- -
? ? ? ?
 = A A 1
OR
AB = 
2 3 1 2 1 5
1 4 1 3 5 14
- -
? ?? ? ? ?
=
? ?? ? ? ?
- - -
? ?? ? ? ?
1
?
?
?
?
?
?
Page 4


1
Strictly Confidential — (For Internal and Restricted Use Only)
Senior School Certificate Examination
July — 2015 (Comptt.)
Marking Scheme — Mathematics (Outside Delhi) 65/1, 65/2, 65/3
General Instructions:
1. The Marking Scheme provides general guidelines to reduce subjectivity in the marking.
The answers given in the Marking Scheme are suggested answers. The content is thus
indicative. If a student has given any other answer which is different from the one given
in the Marking Scheme, but conveys the meaning, such answers should be given full
weightage.
2. Evaluation is to be done as per instructions provided in the marking scheme. It should
not be done according to one’s own interpretation or any other consideration — Marking
Scheme should be strictly adhered to and religiously followed.
3. Alternative methods are accepted. Proportional marks are to be awarded.
4. In question (s) on differential equations, constant of integration has to be written.
5. If a candidate has attempted an extra question, marks obtained in the question attempted
first should be retained and the other answer should be scored out.
6. A full scale of marks - 0 to 100 has to be used. Please do not hesitate to award full marks
if the answer deserves it.
7. Separate Marking Scheme for all the three sets has been given.
8. As per orders of the Hon’ble Supreme Court. The candidates would now be permitted
to obtain photocopy of the Answer book on request on payment of the prescribed fee.
All examiners/Head Examiners are once again reminded that they must ensure that
evaluation is carried out strictly as per value points for each answer as given in the
Marking Scheme.
2
QUESTION PAPER CODE 65/1
EXPECTED ANSWER V ALE POINTS
SECTION A
Marks
1.
3a 2b +
r
r
 = 
7i 5 j 4k - +
r r r
½
? D.R’s are 7, –5, 4 ½
2.
(2i 3 j 2k) (2i 2 j k) + + · + +
r r r r r r
 = 12 ½
p = 
12 12
4
3 | b |
= = r
½
3.
r
r
 = 
(i 2 j 3k) (i 2 j 5k) + + + ? + -
r r r r r r
1
4. For singular matrix
     4 sin
2
 x – 3 = 0 ½
sin x = 
3
2
± ? x = 
2
3
p
½
5.
2y
e dy
?
 = 
3
x dx
?
½
? 
2y
e
2
 = 
4
x
c
4
+ ½
6. I.F. = 
1
dx
x
e
? ½
= 
2 x
e
½
SECTION B
7. Let investment in first type of bonds be Rs x.
? Investment in 2nd type = Rs (35000 – x) ½
3
8
x
100
35000 x 10
100
? ?
? ?
? ?
? ?
? ?
-
? ?? ?
? ?
? ?
 = (3200) 1½
?
8 10
x (35000 x)
100 100
+ -
= 3200
1
? x = Rs 15000
? Investment in first = Rs 15000
1
and in 2nd = Rs 20000
8. Getting A' = 
2 7 1
4 3 2
6 5 4
? ?
? ?
-
? ?
? ?
-
? ?
1
Let P = 
4 11 5 2 11/ 2 5/ 2
1 1
(A A ) 11 6 3 11/ 2 3 3/ 2
2 2
5 3 8 5/ 2 3/ 2 4
- -
? ? ? ?
? ? ? ?
+ ' = =
? ? ? ?
? ? ? ?
- -
? ? ? ?
1
Since P' = P ? P is a symmetric matrix
Let Q = 
1
(A A )
2
- ' =
0 3 7 0 3/ 2 7 / 2
1
3 0 7 3/ 2 0 7 / 2
2
7 7 0 7 / 2 7 / 2 0
- - - -
? ? ? ?
? ? ? ?
=
? ? ? ?
? ? ? ?
- -
? ? ? ?
1
Since Q' = –Q ? Q is skew symmetric
Also
P + Q = 
2 11/ 2 5/ 2 0 3/ 2 7 / 2
11/ 2 3 3/ 2 3/ 2 0 7 / 2
5/ 2 3/ 2 4 7 / 2 7 / 2 0
- - -
? ? ? ?
? ? ? ?
+
? ? ? ?
? ? ? ?
- -
? ? ? ?
 = A A 1
OR
AB = 
2 3 1 2 1 5
1 4 1 3 5 14
- -
? ?? ? ? ?
=
? ?? ? ? ?
- - -
? ?? ? ? ?
1
?
?
?
?
?
?
4
LHS = (AB)
–1
 = 
14 5 14 5
1 1
or
5 1 5 1 11 11
- -
? ? ? ?
-
? ? ? ?
- -
? ? ? ?
1
RHS = B
–1
A
–1
 = 
3 2 4 3 14 5
1 1
1
1 1 1 2 5 1 11 11
- -
? ? ? ? ? ? -
=
? ? ? ? ? ?
-
? ? ? ? ? ?
1+1
? LHS = RHS
9.
a x a x a x
a x a x a x
a x a x a x
+ - -
- + -
- - +
 = 0
R
1
 ? R
1
 + R
2
 + R
3
,
? 
3a x 3a x 3a x
a x a x a x
a x a x a x
- - -
- + -
- - +
 = 0 1
C
2
 ? C
2
 – C
1
, C
3
 ? C
3
 – C
1
? 
3a x 0 0
a x 2x 0
a x 0 2x
-
-
-
 = 0 1+1
? 4x
2
(3a – x) = 0
? x = 0, x = 3a 1
10. I = 
/ 4
0
log(1 tan x)dx
p
+
?
...(i)
= 
/ 4 / 4
0 0
1 tan x
log 1 tan x dx log 1 dx
4 1 tan x
p p
p - ? ?
? ? ? ?
+ - = +
? ?
? ? ? ?
+
? ? ? ? ? ?
? ?
1 + ½
= 
/ 4
0
[log 2 log(1 tan x)]dx
p
- +
?
...(ii) 1
Page 5


1
Strictly Confidential — (For Internal and Restricted Use Only)
Senior School Certificate Examination
July — 2015 (Comptt.)
Marking Scheme — Mathematics (Outside Delhi) 65/1, 65/2, 65/3
General Instructions:
1. The Marking Scheme provides general guidelines to reduce subjectivity in the marking.
The answers given in the Marking Scheme are suggested answers. The content is thus
indicative. If a student has given any other answer which is different from the one given
in the Marking Scheme, but conveys the meaning, such answers should be given full
weightage.
2. Evaluation is to be done as per instructions provided in the marking scheme. It should
not be done according to one’s own interpretation or any other consideration — Marking
Scheme should be strictly adhered to and religiously followed.
3. Alternative methods are accepted. Proportional marks are to be awarded.
4. In question (s) on differential equations, constant of integration has to be written.
5. If a candidate has attempted an extra question, marks obtained in the question attempted
first should be retained and the other answer should be scored out.
6. A full scale of marks - 0 to 100 has to be used. Please do not hesitate to award full marks
if the answer deserves it.
7. Separate Marking Scheme for all the three sets has been given.
8. As per orders of the Hon’ble Supreme Court. The candidates would now be permitted
to obtain photocopy of the Answer book on request on payment of the prescribed fee.
All examiners/Head Examiners are once again reminded that they must ensure that
evaluation is carried out strictly as per value points for each answer as given in the
Marking Scheme.
2
QUESTION PAPER CODE 65/1
EXPECTED ANSWER V ALE POINTS
SECTION A
Marks
1.
3a 2b +
r
r
 = 
7i 5 j 4k - +
r r r
½
? D.R’s are 7, –5, 4 ½
2.
(2i 3 j 2k) (2i 2 j k) + + · + +
r r r r r r
 = 12 ½
p = 
12 12
4
3 | b |
= = r
½
3.
r
r
 = 
(i 2 j 3k) (i 2 j 5k) + + + ? + -
r r r r r r
1
4. For singular matrix
     4 sin
2
 x – 3 = 0 ½
sin x = 
3
2
± ? x = 
2
3
p
½
5.
2y
e dy
?
 = 
3
x dx
?
½
? 
2y
e
2
 = 
4
x
c
4
+ ½
6. I.F. = 
1
dx
x
e
? ½
= 
2 x
e
½
SECTION B
7. Let investment in first type of bonds be Rs x.
? Investment in 2nd type = Rs (35000 – x) ½
3
8
x
100
35000 x 10
100
? ?
? ?
? ?
? ?
? ?
-
? ?? ?
? ?
? ?
 = (3200) 1½
?
8 10
x (35000 x)
100 100
+ -
= 3200
1
? x = Rs 15000
? Investment in first = Rs 15000
1
and in 2nd = Rs 20000
8. Getting A' = 
2 7 1
4 3 2
6 5 4
? ?
? ?
-
? ?
? ?
-
? ?
1
Let P = 
4 11 5 2 11/ 2 5/ 2
1 1
(A A ) 11 6 3 11/ 2 3 3/ 2
2 2
5 3 8 5/ 2 3/ 2 4
- -
? ? ? ?
? ? ? ?
+ ' = =
? ? ? ?
? ? ? ?
- -
? ? ? ?
1
Since P' = P ? P is a symmetric matrix
Let Q = 
1
(A A )
2
- ' =
0 3 7 0 3/ 2 7 / 2
1
3 0 7 3/ 2 0 7 / 2
2
7 7 0 7 / 2 7 / 2 0
- - - -
? ? ? ?
? ? ? ?
=
? ? ? ?
? ? ? ?
- -
? ? ? ?
1
Since Q' = –Q ? Q is skew symmetric
Also
P + Q = 
2 11/ 2 5/ 2 0 3/ 2 7 / 2
11/ 2 3 3/ 2 3/ 2 0 7 / 2
5/ 2 3/ 2 4 7 / 2 7 / 2 0
- - -
? ? ? ?
? ? ? ?
+
? ? ? ?
? ? ? ?
- -
? ? ? ?
 = A A 1
OR
AB = 
2 3 1 2 1 5
1 4 1 3 5 14
- -
? ?? ? ? ?
=
? ?? ? ? ?
- - -
? ?? ? ? ?
1
?
?
?
?
?
?
4
LHS = (AB)
–1
 = 
14 5 14 5
1 1
or
5 1 5 1 11 11
- -
? ? ? ?
-
? ? ? ?
- -
? ? ? ?
1
RHS = B
–1
A
–1
 = 
3 2 4 3 14 5
1 1
1
1 1 1 2 5 1 11 11
- -
? ? ? ? ? ? -
=
? ? ? ? ? ?
-
? ? ? ? ? ?
1+1
? LHS = RHS
9.
a x a x a x
a x a x a x
a x a x a x
+ - -
- + -
- - +
 = 0
R
1
 ? R
1
 + R
2
 + R
3
,
? 
3a x 3a x 3a x
a x a x a x
a x a x a x
- - -
- + -
- - +
 = 0 1
C
2
 ? C
2
 – C
1
, C
3
 ? C
3
 – C
1
? 
3a x 0 0
a x 2x 0
a x 0 2x
-
-
-
 = 0 1+1
? 4x
2
(3a – x) = 0
? x = 0, x = 3a 1
10. I = 
/ 4
0
log(1 tan x)dx
p
+
?
...(i)
= 
/ 4 / 4
0 0
1 tan x
log 1 tan x dx log 1 dx
4 1 tan x
p p
p - ? ?
? ? ? ?
+ - = +
? ?
? ? ? ?
+
? ? ? ? ? ?
? ?
1 + ½
= 
/ 4
0
[log 2 log(1 tan x)]dx
p
- +
?
...(ii) 1
5
adding (i) and (ii) to get
2I = 
/ 4
0
log 2 1 dx log 2
4
p
p
· =
? 1
? I = 
log 2
8
p
½
11. Writing I = 
2 2
x A Bx C
dx dx
x 1
(x 1)(x 1) x 1
+
? ?
= +
? ?
-
+ - + ? ?
? ?
1
= 
2
1 1
x
1/ 2
2 2
dx dx
x 1
x 1
- +
+
-
+
? ?
1½
= 
2 1
1 1 1
log | x 1| log(x 1) tan x C
2 4 2
-
- - + + +
1½
OR
I = 
1/ 2
1
2 3/ 2
0
sin x
dx
(1 x )
-
-
?
Putting x = sin ?, ? dx = cos ? d? and x = 0 then ? = 0
x ? 
1
then
4 2
p
? =
1
I = 
/ 4 / 4
2
3
0 0
cos
d sec d
cos
p p
?
?· ? = ?· ? ?
?
? ?
1
= [ ]
/ 4
0
tan log | sec |
p
? ? - ? 1
= 
1
log2
4 2
p
-
1
12. (i)  P (all four spades) = 
4 0
4
4
13 39 1
C
52 52 256
? ? ? ?
=
? ? ? ?
? ? ? ?
2
?
?
?
?
?
?
?