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Page 1 of 15
MARKING SCHEME
CLASS XII
MATHEMATICS (CODE-041)
SECTION: A (Solution of MCQs of 1 Mark each)
Q no. ANS HINTS/SOLUTION
1. (D)
For a square matrix A of order nn ? , we have ? ? .,
n
A adj A A I ? where
n
I is the identity matrix of
order . nn ?
So, ? ?
3
2025 0 0
. 0 2025 0 2025
0 0 2025
A adj A I
??
??
??
??
??
??
2025 A ?? & ? ?
2 31
2025 adj A A
?
??
? ?
2
2025 2025 . A adj A ? ? ? ?
2. (A)
3. (C)
?? = ?? ?? = >
???? ???? = ?? ??
In the domain (R) of the function,
????
????
> 0 , hence the function is strictly increasing in ( -8, 8)
4. (B)
5, A ?
? ?
2
2
2
1 1 2
5. B AB B A B A
??
? ? ?
5. (B)
A differential equation of the form ? ? ,
dy
f x y
dx
? is said to be homogeneous, if ? ? , f x y is a
homogeneous function of degree 0.
Now, log log
n
ee
dy y
x y e
dx x
??
??
??
??
? ? ? ? log . , ;
e n
dy y y
e f x y Let
dx x x
?? ??
? ? ?
?? ??
?? ??
. ? ? , f x y will be a
homogeneous function of degree 0, if 1. n ?
6. (A) Method 1: ( Short cut)
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
? ? ? ?
? ?
1 2 1 2 1 2 1 2
1 2 2 2 2 1 2 2
1 1 2 1 1 2 22
00
x x y y x x y y
x y x y x y x y
x x x y y y xy
?? ??
? ? ? ? ? ? ? ?
? ? ? ? ??
2 1 1 2
. x y x y ??
Page 2
Page 1 of 15
MARKING SCHEME
CLASS XII
MATHEMATICS (CODE-041)
SECTION: A (Solution of MCQs of 1 Mark each)
Q no. ANS HINTS/SOLUTION
1. (D)
For a square matrix A of order nn ? , we have ? ? .,
n
A adj A A I ? where
n
I is the identity matrix of
order . nn ?
So, ? ?
3
2025 0 0
. 0 2025 0 2025
0 0 2025
A adj A I
??
??
??
??
??
??
2025 A ?? & ? ?
2 31
2025 adj A A
?
??
? ?
2
2025 2025 . A adj A ? ? ? ?
2. (A)
3. (C)
?? = ?? ?? = >
???? ???? = ?? ??
In the domain (R) of the function,
????
????
> 0 , hence the function is strictly increasing in ( -8, 8)
4. (B)
5, A ?
? ?
2
2
2
1 1 2
5. B AB B A B A
??
? ? ?
5. (B)
A differential equation of the form ? ? ,
dy
f x y
dx
? is said to be homogeneous, if ? ? , f x y is a
homogeneous function of degree 0.
Now, log log
n
ee
dy y
x y e
dx x
??
??
??
??
? ? ? ? log . , ;
e n
dy y y
e f x y Let
dx x x
?? ??
? ? ?
?? ??
?? ??
. ? ? , f x y will be a
homogeneous function of degree 0, if 1. n ?
6. (A) Method 1: ( Short cut)
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
? ? ? ?
? ?
1 2 1 2 1 2 1 2
1 2 2 2 2 1 2 2
1 1 2 1 1 2 22
00
x x y y x x y y
x y x y x y x y
x x x y y y xy
?? ??
? ? ? ? ? ? ? ?
? ? ? ? ??
2 1 1 2
. x y x y ??
Page 2 of 15
Method 2:
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
11
22
1 2 1 2
1
10
1
xy
xy
x x y y
?
??
? ? ? ? ? ?
2 1 2 2 1 2 2 2 1 1 1 2 1 1 2 1 1 2 2 1
1. 1 0 x y x y x y x y x y x y x y x y x y x y ? ? ? ? ? ? ? ? ? ? ?
2 1 1 2
. x y x y ??
7. (A)
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
When the matrix A is skew symmetric then ;
T
ij ji
A A a a ? ? ? ? ?
2; 0 and 3 c a b ? ? ? ? ?
So , 0 3 2 1. a b c ? ? ? ? ? ?
8. (C)
? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ?
1 2 1
;;
2 3 4
11
;
23
1 1 1 7
Wehave,
2 3 4 12
P A P B P A B
P A P B
P A B P A P B P A B
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
? ?
? ?
7
1
1
5
12
.
2
8
3
P A B
P A B P A B
A
P
B
P B P B P B
?
?
?? ? ? ?
? ? ? ? ?
??
??
9. (B) For obtuse angle, cos ?? < 0 => ?? ?. ?? ? < 0
?? ?? ?? - ???? + ?? < ?? => ?? ?? ?? - ???? < ?? => ?? ? ( ?? , ?? )
10. (C)
3, 4, 5 a b a b ? ? ? ?
We have ,
? ?
? ?
2 2 2 2
2 2 9 16 50 5. a b a b a b a b ? ? ? ? ? ? ? ? ? ? ?
11. (B) Corner point Value of the objective function 43 Z x y ??
1. ? ? 0,0 O
0 z ?
2. ? ? 40,0 R
160 z ?
3. ? ? 30,20 Q
120 60 180 z ? ? ?
4. ? ? 0,40 P
120 z ?
Since , the feasible region is bounded so the maximum value of the objective function 180 z ? is at
? ? 30,20 . Q
Page 3
Page 1 of 15
MARKING SCHEME
CLASS XII
MATHEMATICS (CODE-041)
SECTION: A (Solution of MCQs of 1 Mark each)
Q no. ANS HINTS/SOLUTION
1. (D)
For a square matrix A of order nn ? , we have ? ? .,
n
A adj A A I ? where
n
I is the identity matrix of
order . nn ?
So, ? ?
3
2025 0 0
. 0 2025 0 2025
0 0 2025
A adj A I
??
??
??
??
??
??
2025 A ?? & ? ?
2 31
2025 adj A A
?
??
? ?
2
2025 2025 . A adj A ? ? ? ?
2. (A)
3. (C)
?? = ?? ?? = >
???? ???? = ?? ??
In the domain (R) of the function,
????
????
> 0 , hence the function is strictly increasing in ( -8, 8)
4. (B)
5, A ?
? ?
2
2
2
1 1 2
5. B AB B A B A
??
? ? ?
5. (B)
A differential equation of the form ? ? ,
dy
f x y
dx
? is said to be homogeneous, if ? ? , f x y is a
homogeneous function of degree 0.
Now, log log
n
ee
dy y
x y e
dx x
??
??
??
??
? ? ? ? log . , ;
e n
dy y y
e f x y Let
dx x x
?? ??
? ? ?
?? ??
?? ??
. ? ? , f x y will be a
homogeneous function of degree 0, if 1. n ?
6. (A) Method 1: ( Short cut)
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
? ? ? ?
? ?
1 2 1 2 1 2 1 2
1 2 2 2 2 1 2 2
1 1 2 1 1 2 22
00
x x y y x x y y
x y x y x y x y
x x x y y y xy
?? ??
? ? ? ? ? ? ? ?
? ? ? ? ??
2 1 1 2
. x y x y ??
Page 2 of 15
Method 2:
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
11
22
1 2 1 2
1
10
1
xy
xy
x x y y
?
??
? ? ? ? ? ?
2 1 2 2 1 2 2 2 1 1 1 2 1 1 2 1 1 2 2 1
1. 1 0 x y x y x y x y x y x y x y x y x y x y ? ? ? ? ? ? ? ? ? ? ?
2 1 1 2
. x y x y ??
7. (A)
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
When the matrix A is skew symmetric then ;
T
ij ji
A A a a ? ? ? ? ?
2; 0 and 3 c a b ? ? ? ? ?
So , 0 3 2 1. a b c ? ? ? ? ? ?
8. (C)
? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ?
1 2 1
;;
2 3 4
11
;
23
1 1 1 7
Wehave,
2 3 4 12
P A P B P A B
P A P B
P A B P A P B P A B
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
? ?
? ?
7
1
1
5
12
.
2
8
3
P A B
P A B P A B
A
P
B
P B P B P B
?
?
?? ? ? ?
? ? ? ? ?
??
??
9. (B) For obtuse angle, cos ?? < 0 => ?? ?. ?? ? < 0
?? ?? ?? - ???? + ?? < ?? => ?? ?? ?? - ???? < ?? => ?? ? ( ?? , ?? )
10. (C)
3, 4, 5 a b a b ? ? ? ?
We have ,
? ?
? ?
2 2 2 2
2 2 9 16 50 5. a b a b a b a b ? ? ? ? ? ? ? ? ? ? ?
11. (B) Corner point Value of the objective function 43 Z x y ??
1. ? ? 0,0 O
0 z ?
2. ? ? 40,0 R
160 z ?
3. ? ? 30,20 Q
120 60 180 z ? ? ?
4. ? ? 0,40 P
120 z ?
Since , the feasible region is bounded so the maximum value of the objective function 180 z ? is at
? ? 30,20 . Q
Page 3 of 15
Section –B
[This section comprises of solution of very short answer type questions (VSA) of 2 marks each]
12. (A)
?
????
?? 3
( 1 + ?? 4
)
1
2
= ?
????
?? 5
( 1 +
1
?? 4
)
1
2
( Let 1 + ?? -4
= 1 +
1
?? 4
= ?? , ???? = -4?? -5
???? = -
4
?? 5
???? ?
????
?? 5
= -
1
4
???? )
= -
1
4
?
????
?? 1
2
= -
1
4
× 2 × v ?? + ?? , where '' c denotes any arbitrary constant of integration.
= -
1
2
v1 +
1
?? 4
+ ?? = -
1
2?? 2
v1 + ?? 4
+ ??
13. (A)
We know, ? ? ? ? ? ?
2
0
0, if 2
a
f x dx f a x f x ? ? ? ?
?
Let
? ?
7
cos f x ec x ? .
Now,
? ? ? ? ? ?
77
2 cos 2 cos f x ec x ec x f x ?? ? ? ? ? ? ? ?
2
7
0
cos 0; ec x dx
?
??
?
Using the property
? ? ? ? ? ?
2
0
0, if 2 .
a
f x dx f a x f x ? ? ? ?
?
14. (B)
The given differential equation ?? ?? '
= ?? =>
????
????
= log ??
???? = log ?? ???? => ????? = ?log ?? ????
?? = ?? log ?? - ?? + ??
hence the correct option is (B).
15. (B)
The graph represents
1
cos yx
?
? whose domain is ? ? 1,1 ? and range is ? ? 0, . ?
16. (D) Since the inequality 18 10 134 Z x y ? ? ? has no point in common with the feasible region hence
the minimum value of the objective function 18 10 Z x y ?? is 134 at ? ? 3,8 P .
17. (D)
The graph of the function ?? : ?? ? ?? defined by ? ? ? ?; f x x ? ? ? ? ?
where . denotes . . G I F is a straight
line
? ? 2.5 ,2.5 x h h ? ? ? ? , '' h is an infinitesimally small positive quantity. Hence, the function is
continuous and differentiable at 2.5 . x ?
18. (B) The required region is symmetric about the y ? axis.
So, required area (in sq units ) is
4
3
4
2
0
0
64
2 2 4 .
3
3
2
y
ydy
??
??
? ? ?
??
??
??
?
19. (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
20. (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
Page 4
Page 1 of 15
MARKING SCHEME
CLASS XII
MATHEMATICS (CODE-041)
SECTION: A (Solution of MCQs of 1 Mark each)
Q no. ANS HINTS/SOLUTION
1. (D)
For a square matrix A of order nn ? , we have ? ? .,
n
A adj A A I ? where
n
I is the identity matrix of
order . nn ?
So, ? ?
3
2025 0 0
. 0 2025 0 2025
0 0 2025
A adj A I
??
??
??
??
??
??
2025 A ?? & ? ?
2 31
2025 adj A A
?
??
? ?
2
2025 2025 . A adj A ? ? ? ?
2. (A)
3. (C)
?? = ?? ?? = >
???? ???? = ?? ??
In the domain (R) of the function,
????
????
> 0 , hence the function is strictly increasing in ( -8, 8)
4. (B)
5, A ?
? ?
2
2
2
1 1 2
5. B AB B A B A
??
? ? ?
5. (B)
A differential equation of the form ? ? ,
dy
f x y
dx
? is said to be homogeneous, if ? ? , f x y is a
homogeneous function of degree 0.
Now, log log
n
ee
dy y
x y e
dx x
??
??
??
??
? ? ? ? log . , ;
e n
dy y y
e f x y Let
dx x x
?? ??
? ? ?
?? ??
?? ??
. ? ? , f x y will be a
homogeneous function of degree 0, if 1. n ?
6. (A) Method 1: ( Short cut)
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
? ? ? ?
? ?
1 2 1 2 1 2 1 2
1 2 2 2 2 1 2 2
1 1 2 1 1 2 22
00
x x y y x x y y
x y x y x y x y
x x x y y y xy
?? ??
? ? ? ? ? ? ? ?
? ? ? ? ??
2 1 1 2
. x y x y ??
Page 2 of 15
Method 2:
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
11
22
1 2 1 2
1
10
1
xy
xy
x x y y
?
??
? ? ? ? ? ?
2 1 2 2 1 2 2 2 1 1 1 2 1 1 2 1 1 2 2 1
1. 1 0 x y x y x y x y x y x y x y x y x y x y ? ? ? ? ? ? ? ? ? ? ?
2 1 1 2
. x y x y ??
7. (A)
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
When the matrix A is skew symmetric then ;
T
ij ji
A A a a ? ? ? ? ?
2; 0 and 3 c a b ? ? ? ? ?
So , 0 3 2 1. a b c ? ? ? ? ? ?
8. (C)
? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ?
1 2 1
;;
2 3 4
11
;
23
1 1 1 7
Wehave,
2 3 4 12
P A P B P A B
P A P B
P A B P A P B P A B
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
? ?
? ?
7
1
1
5
12
.
2
8
3
P A B
P A B P A B
A
P
B
P B P B P B
?
?
?? ? ? ?
? ? ? ? ?
??
??
9. (B) For obtuse angle, cos ?? < 0 => ?? ?. ?? ? < 0
?? ?? ?? - ???? + ?? < ?? => ?? ?? ?? - ???? < ?? => ?? ? ( ?? , ?? )
10. (C)
3, 4, 5 a b a b ? ? ? ?
We have ,
? ?
? ?
2 2 2 2
2 2 9 16 50 5. a b a b a b a b ? ? ? ? ? ? ? ? ? ? ?
11. (B) Corner point Value of the objective function 43 Z x y ??
1. ? ? 0,0 O
0 z ?
2. ? ? 40,0 R
160 z ?
3. ? ? 30,20 Q
120 60 180 z ? ? ?
4. ? ? 0,40 P
120 z ?
Since , the feasible region is bounded so the maximum value of the objective function 180 z ? is at
? ? 30,20 . Q
Page 3 of 15
Section –B
[This section comprises of solution of very short answer type questions (VSA) of 2 marks each]
12. (A)
?
????
?? 3
( 1 + ?? 4
)
1
2
= ?
????
?? 5
( 1 +
1
?? 4
)
1
2
( Let 1 + ?? -4
= 1 +
1
?? 4
= ?? , ???? = -4?? -5
???? = -
4
?? 5
???? ?
????
?? 5
= -
1
4
???? )
= -
1
4
?
????
?? 1
2
= -
1
4
× 2 × v ?? + ?? , where '' c denotes any arbitrary constant of integration.
= -
1
2
v1 +
1
?? 4
+ ?? = -
1
2?? 2
v1 + ?? 4
+ ??
13. (A)
We know, ? ? ? ? ? ?
2
0
0, if 2
a
f x dx f a x f x ? ? ? ?
?
Let
? ?
7
cos f x ec x ? .
Now,
? ? ? ? ? ?
77
2 cos 2 cos f x ec x ec x f x ?? ? ? ? ? ? ? ?
2
7
0
cos 0; ec x dx
?
??
?
Using the property
? ? ? ? ? ?
2
0
0, if 2 .
a
f x dx f a x f x ? ? ? ?
?
14. (B)
The given differential equation ?? ?? '
= ?? =>
????
????
= log ??
???? = log ?? ???? => ????? = ?log ?? ????
?? = ?? log ?? - ?? + ??
hence the correct option is (B).
15. (B)
The graph represents
1
cos yx
?
? whose domain is ? ? 1,1 ? and range is ? ? 0, . ?
16. (D) Since the inequality 18 10 134 Z x y ? ? ? has no point in common with the feasible region hence
the minimum value of the objective function 18 10 Z x y ?? is 134 at ? ? 3,8 P .
17. (D)
The graph of the function ?? : ?? ? ?? defined by ? ? ? ?; f x x ? ? ? ? ?
where . denotes . . G I F is a straight
line
? ? 2.5 ,2.5 x h h ? ? ? ? , '' h is an infinitesimally small positive quantity. Hence, the function is
continuous and differentiable at 2.5 . x ?
18. (B) The required region is symmetric about the y ? axis.
So, required area (in sq units ) is
4
3
4
2
0
0
64
2 2 4 .
3
3
2
y
ydy
??
??
? ? ?
??
??
??
?
19. (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
20. (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
Page 4 of 15
21
cot
-1
( 3?? + 5)>
?? 4
= cot
-1
1
=>3x + 5 < 1 ( as cot
-1
?? is strictly decreasing function in its domain)
=> 3x < – 4
=> ?? < -
4
3
? ?? ? ( -8, -
4
3
)
1
2
1
2
1
22.
The marginal cost function is ? ?
2
' 0.00039 0.004 5 C x x x ? ? ? .
? ? ' 150 C ? ? 14.375 .
1
1
23.(a)
1
tan yx
?
?
and log
e
zx ?
.
Then
2
1
1
dy
dx x
?
?
and
1 dz
dx x
?
So,
2
2
1
1
.
1
1
dy
dy
dx
dz
dz
dx
x
x
x
x
?
?
??
?
1
2
1
2
1
2
1
2
OR
23.(b)
Let
x
yx (cos ) . ? Then,
log cosx x
e
ye ?
On differentiating both sides with respect to
, x
we get
log cos
( log cos )
xx
e
e
dy d
e x x
dx dx
?
(cos ) log cos ( ) (log cos )
x
ee
dy d d
x x x x x
dx dx dx
??
? ? ?
??
??
1
(cos ) log cos . ( sin )
cos
x
e
dy
x x x x
dx x
??
? ? ? ?
??
??
(cos ) (log cos tan ) .
x
e
dy
x x x x
dx
? ? ?
1
2
1
2
1
24.(a)
We have b
? ?
+ ?c ? = ( -1 + 3?) i^ + ( 2 + ? ) j^ + k
^
( b
? ?
+ ?c ?) . a ? ? = 0 => 2( -1 + 3? )+ 2 ( 2 + ? )+ 3 = 0
? = -
5
8
1
2
1
1
2
OR
24.(b)
????
? ?? ? ??
= ????
???? ??
- ????
??????
= ( 4?? ^ + 3?? ^
)- ?? ^
= 4?? ^ + 2?? ^
1
2
Page 5
Page 1 of 15
MARKING SCHEME
CLASS XII
MATHEMATICS (CODE-041)
SECTION: A (Solution of MCQs of 1 Mark each)
Q no. ANS HINTS/SOLUTION
1. (D)
For a square matrix A of order nn ? , we have ? ? .,
n
A adj A A I ? where
n
I is the identity matrix of
order . nn ?
So, ? ?
3
2025 0 0
. 0 2025 0 2025
0 0 2025
A adj A I
??
??
??
??
??
??
2025 A ?? & ? ?
2 31
2025 adj A A
?
??
? ?
2
2025 2025 . A adj A ? ? ? ?
2. (A)
3. (C)
?? = ?? ?? = >
???? ???? = ?? ??
In the domain (R) of the function,
????
????
> 0 , hence the function is strictly increasing in ( -8, 8)
4. (B)
5, A ?
? ?
2
2
2
1 1 2
5. B AB B A B A
??
? ? ?
5. (B)
A differential equation of the form ? ? ,
dy
f x y
dx
? is said to be homogeneous, if ? ? , f x y is a
homogeneous function of degree 0.
Now, log log
n
ee
dy y
x y e
dx x
??
??
??
??
? ? ? ? log . , ;
e n
dy y y
e f x y Let
dx x x
?? ??
? ? ?
?? ??
?? ??
. ? ? , f x y will be a
homogeneous function of degree 0, if 1. n ?
6. (A) Method 1: ( Short cut)
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
? ? ? ?
? ?
1 2 1 2 1 2 1 2
1 2 2 2 2 1 2 2
1 1 2 1 1 2 22
00
x x y y x x y y
x y x y x y x y
x x x y y y xy
?? ??
? ? ? ? ? ? ? ?
? ? ? ? ??
2 1 1 2
. x y x y ??
Page 2 of 15
Method 2:
When the points ? ? ? ?
1 1 2 2
, , , x y x y and ? ?
1 2 1 2
, x x y y ?? are collinear in the Cartesian plane then
11
22
1 2 1 2
1
10
1
xy
xy
x x y y
?
??
? ? ? ? ? ?
2 1 2 2 1 2 2 2 1 1 1 2 1 1 2 1 1 2 2 1
1. 1 0 x y x y x y x y x y x y x y x y x y x y ? ? ? ? ? ? ? ? ? ? ?
2 1 1 2
. x y x y ??
7. (A)
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
When the matrix A is skew symmetric then ;
T
ij ji
A A a a ? ? ? ? ?
2; 0 and 3 c a b ? ? ? ? ?
So , 0 3 2 1. a b c ? ? ? ? ? ?
8. (C)
? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ?
1 2 1
;;
2 3 4
11
;
23
1 1 1 7
Wehave,
2 3 4 12
P A P B P A B
P A P B
P A B P A P B P A B
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ?
? ?
? ?
? ?
? ?
? ?
? ?
7
1
1
5
12
.
2
8
3
P A B
P A B P A B
A
P
B
P B P B P B
?
?
?? ? ? ?
? ? ? ? ?
??
??
9. (B) For obtuse angle, cos ?? < 0 => ?? ?. ?? ? < 0
?? ?? ?? - ???? + ?? < ?? => ?? ?? ?? - ???? < ?? => ?? ? ( ?? , ?? )
10. (C)
3, 4, 5 a b a b ? ? ? ?
We have ,
? ?
? ?
2 2 2 2
2 2 9 16 50 5. a b a b a b a b ? ? ? ? ? ? ? ? ? ? ?
11. (B) Corner point Value of the objective function 43 Z x y ??
1. ? ? 0,0 O
0 z ?
2. ? ? 40,0 R
160 z ?
3. ? ? 30,20 Q
120 60 180 z ? ? ?
4. ? ? 0,40 P
120 z ?
Since , the feasible region is bounded so the maximum value of the objective function 180 z ? is at
? ? 30,20 . Q
Page 3 of 15
Section –B
[This section comprises of solution of very short answer type questions (VSA) of 2 marks each]
12. (A)
?
????
?? 3
( 1 + ?? 4
)
1
2
= ?
????
?? 5
( 1 +
1
?? 4
)
1
2
( Let 1 + ?? -4
= 1 +
1
?? 4
= ?? , ???? = -4?? -5
???? = -
4
?? 5
???? ?
????
?? 5
= -
1
4
???? )
= -
1
4
?
????
?? 1
2
= -
1
4
× 2 × v ?? + ?? , where '' c denotes any arbitrary constant of integration.
= -
1
2
v1 +
1
?? 4
+ ?? = -
1
2?? 2
v1 + ?? 4
+ ??
13. (A)
We know, ? ? ? ? ? ?
2
0
0, if 2
a
f x dx f a x f x ? ? ? ?
?
Let
? ?
7
cos f x ec x ? .
Now,
? ? ? ? ? ?
77
2 cos 2 cos f x ec x ec x f x ?? ? ? ? ? ? ? ?
2
7
0
cos 0; ec x dx
?
??
?
Using the property
? ? ? ? ? ?
2
0
0, if 2 .
a
f x dx f a x f x ? ? ? ?
?
14. (B)
The given differential equation ?? ?? '
= ?? =>
????
????
= log ??
???? = log ?? ???? => ????? = ?log ?? ????
?? = ?? log ?? - ?? + ??
hence the correct option is (B).
15. (B)
The graph represents
1
cos yx
?
? whose domain is ? ? 1,1 ? and range is ? ? 0, . ?
16. (D) Since the inequality 18 10 134 Z x y ? ? ? has no point in common with the feasible region hence
the minimum value of the objective function 18 10 Z x y ?? is 134 at ? ? 3,8 P .
17. (D)
The graph of the function ?? : ?? ? ?? defined by ? ? ? ?; f x x ? ? ? ? ?
where . denotes . . G I F is a straight
line
? ? 2.5 ,2.5 x h h ? ? ? ? , '' h is an infinitesimally small positive quantity. Hence, the function is
continuous and differentiable at 2.5 . x ?
18. (B) The required region is symmetric about the y ? axis.
So, required area (in sq units ) is
4
3
4
2
0
0
64
2 2 4 .
3
3
2
y
ydy
??
??
? ? ?
??
??
??
?
19. (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
20. (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
Page 4 of 15
21
cot
-1
( 3?? + 5)>
?? 4
= cot
-1
1
=>3x + 5 < 1 ( as cot
-1
?? is strictly decreasing function in its domain)
=> 3x < – 4
=> ?? < -
4
3
? ?? ? ( -8, -
4
3
)
1
2
1
2
1
22.
The marginal cost function is ? ?
2
' 0.00039 0.004 5 C x x x ? ? ? .
? ? ' 150 C ? ? 14.375 .
1
1
23.(a)
1
tan yx
?
?
and log
e
zx ?
.
Then
2
1
1
dy
dx x
?
?
and
1 dz
dx x
?
So,
2
2
1
1
.
1
1
dy
dy
dx
dz
dz
dx
x
x
x
x
?
?
??
?
1
2
1
2
1
2
1
2
OR
23.(b)
Let
x
yx (cos ) . ? Then,
log cosx x
e
ye ?
On differentiating both sides with respect to
, x
we get
log cos
( log cos )
xx
e
e
dy d
e x x
dx dx
?
(cos ) log cos ( ) (log cos )
x
ee
dy d d
x x x x x
dx dx dx
??
? ? ?
??
??
1
(cos ) log cos . ( sin )
cos
x
e
dy
x x x x
dx x
??
? ? ? ?
??
??
(cos ) (log cos tan ) .
x
e
dy
x x x x
dx
? ? ?
1
2
1
2
1
24.(a)
We have b
? ?
+ ?c ? = ( -1 + 3?) i^ + ( 2 + ? ) j^ + k
^
( b
? ?
+ ?c ?) . a ? ? = 0 => 2( -1 + 3? )+ 2 ( 2 + ? )+ 3 = 0
? = -
5
8
1
2
1
1
2
OR
24.(b)
????
? ?? ? ??
= ????
???? ??
- ????
??????
= ( 4?? ^ + 3?? ^
)- ?? ^
= 4?? ^ + 2?? ^
1
2
Page 5 of 15
????
^
=
4
2v5
?? ^ +
2
2v5
?? ^
=
2
v5
?? ^ +
1
v5
?? ^
So, the angles made by the vector ????
? ?? ? ??
with the , xy and the z axes are respectively
??????
-1
(
2
v 5
),
?? 2
, ??????
-1
(
1
v 5
) .
1
2
1
25.
?? 1
? ? ? ??
= ?? ? + ?? ? ?
= 4?? ^ - 2?? ^ - 2?? ^
, ?? 2
? ? ? ??
= ?? ? - ?? ? ?
= -6?? ^ - 8?? ^
Area of the parallelogram =
1
2
|?? 1
? ? ? ??
× ?? 2
? ? ? ??
| =
1
2
||
?? ^ ?? ^ ?? ^
4 -2 -2
0 -6 -8
|| = 2|?? ^ + 8?? ^ - 6?? ^
|
Area of the parallelogram = 2v101 sq. units.
1
2
1
1
2
Section –C
[This section comprises of solution short answer type questions (SA) of 3 marks each]
26.
?? 2
+ 3
2
= ?? 2
?? h???? ?? = 5 ?? h???? ?? = 4, ?????? 2??
????
????
= 2??
????
????
4 ( 200 )= 5
????
????
=>
????
????
= 160 cm/s
1
2
1
2
1
1
27.
?? =
1
3
v ?? ?
????
????
=
1
6
?? -
1
2
=
1
6v ?? ; ??? ? ( 5,18)
????
????
=
1
6v ?? ?
?? 2
?? ?? ?? 2
= -
1
12?? v ??
So,
?? 2
?? ?? ?? 2
< 0, ??? ? ( 5,18)
This means that the rate of change of the ability to understand spatial concepts decreases
(slows down) with age.
1
1
1
2
1
2
28(a)
(i) ?? = ?????? -?? (
?? ?? ? ? ??
.?? ?? ? ? ??
|?? ?? ? ? ??
|.|?? ?? ? ? ??
|
) = ?????? -?? (
( i^ -2?^ +3k
^
) .( 3i^ -2?^ + ?? ^
)
|( i^ -2?^ +3k
^
) || ( 3i^ -2?^ + ?? ^
) |
)
= ?????? -?? (
?? +?? +?? v ?? +?? +?? v ?? +?? +?? )= ?????? -?? (
????
????
)= ?????? -?? (
?? ?? ) .
(ii) Scalar projection of ?? ?? ? ? ??
on ?? ?? ? ? ??
=
?? ?? ? ? ??
.?? ?? ? ? ??
|?? ?? ? ? ??
|
=
( i^ -2?^ +3k
^
) .( 3i^ -2?^ + ?? ^
)
| ( 3i^ -2?^ + ?? ^
) |
=
3+4+3
v 9+4+1
=
10
v 14
.
1
1
2
1
1
2
3
3
y
x
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FAQs on Class 12 Mathematics: CBSE Marking Scheme (2024-25) - Mathematics (Maths) Class 12 - JEE
| 1. What is the marking scheme for Class 12 Mathematics in CBSE 2024-25? |  |
Ans. The marking scheme for Class 12 Mathematics in CBSE for the academic year 2024-25 typically includes a total of 100 marks. This is divided into 70 marks for the theory examination and 30 marks for internal assessment, which may include projects, assignments, and periodic tests.
| 2. How is the internal assessment structured in the Class 12 Mathematics exam? | |
Ans. The internal assessment for Class 12 Mathematics consists of 30 marks, which is usually divided among various components such as class tests (10 marks), assignments and projects (10 marks), and periodic assessments (10 marks). This helps in evaluating the students' continuous performance throughout the year.
| 3. What types of questions can be expected in the Class 12 Mathematics exam? | |
Ans. The Class 12 Mathematics exam typically includes a variety of question types such as very short answer questions, short answer questions, and long answer questions. These can cover topics like calculus, algebra, statistics, and geometry, with different marks assigned to each type based on their complexity.
| 4. Are there any changes in the syllabus for Class 12 Mathematics in CBSE 2024-25? | |
Ans. Yes, the syllabus for Class 12 Mathematics may undergo changes as per the CBSE guidelines. Students are advised to check the official CBSE website or the latest syllabus notification to stay updated on any new topics or modifications in the existing curriculum for the academic year 2024-25.
| 5. How can students prepare effectively for the Class 12 Mathematics exam? | |
Ans. Students can prepare effectively for the Class 12 Mathematics exam by following a structured study plan. This includes understanding the concepts thoroughly, practicing previous years' question papers, solving sample papers, and regularly revising the topics. Time management and seeking help from teachers or peers for difficult topics can also enhance preparation.
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