Class 12 Exam > Class 12 Notes > Sample Papers for Class 12 Medical and Non-Medical > Class 12 Mathematics: CBSE Marking Scheme (2024-25)
Class 12 Mathematics: CBSE Marking Scheme (2024-25) | Sample Papers for Class 12 Medical and Non-Medical PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
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
Read More
|
159 docs|4 tests
|
FAQs on Class 12 Mathematics: CBSE Marking Scheme (2024-25) - Sample Papers for Class 12 Medical and Non-Medical
| 1. What is the CBSE marking scheme for Class 12 Mathematics in 2024-25? |  |
Ans. The CBSE marking scheme for Class 12 Mathematics includes a total of 100 marks, which are divided into various sections: 20 marks for internal assessment, 80 marks for the theory exam. The theory exam consists of different types of questions, including very short answer, short answer, and long answer questions.
| 2. How are internal assessment marks calculated in Class 12 Mathematics? | |
Ans. Internal assessment marks in Class 12 Mathematics are calculated based on various components such as periodic tests, notebook submission, and practical work. Schools typically evaluate students continuously throughout the year, contributing a total of 20 marks to the final assessment.
| 3. What types of questions can we expect in the Class 12 Mathematics exam? | |
Ans. In the Class 12 Mathematics exam, students can expect a variety of question types, including multiple-choice questions (MCQs), very short answer questions, short answer questions, and long answer questions. The distribution of marks is designed to cover all topics in the syllabus comprehensively.
| 4. How can students prepare effectively for the Class 12 Mathematics exam? | |
Ans. Students can prepare effectively for the Class 12 Mathematics exam by understanding the syllabus thoroughly, practicing previous years' question papers, and solving sample papers. Regular revision and attending doubt-clearing sessions with teachers are also crucial for better understanding and retention of concepts.
| 5. What is the importance of the Class 12 Mathematics exam in a student's academic career? | |
Ans. The Class 12 Mathematics exam is crucial as it contributes significantly to a student's overall academic performance and is often a deciding factor for college admissions, especially for courses in science, engineering, and other mathematics-related fields. Good performance can open up various opportunities for higher education.
Related Exams
About this Document
|
Nov 22, 2024 Last updated |
Document Description: Class 12 Mathematics: CBSE Marking Scheme (2024-25) for Class 12 2024 is part of Sample Papers for Class 12 Medical and Non-Medical preparation.
The notes and questions for Class 12 Mathematics: CBSE Marking Scheme (2024-25) have been prepared according to the Class 12 exam syllabus. Information about Class 12 Mathematics: CBSE Marking Scheme (2024-25) covers topics
like and Class 12 Mathematics: CBSE Marking Scheme (2024-25) Example, for Class 12 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Class 12 Mathematics: CBSE Marking Scheme (2024-25).
Introduction of Class 12 Mathematics: CBSE Marking Scheme (2024-25) in English is available as part of our Sample Papers for Class 12 Medical and Non-Medical
for Class 12 & Class 12 Mathematics: CBSE Marking Scheme (2024-25) in Hindi for Sample Papers for Class 12 Medical and Non-Medical course.
Download more important topics related with notes, lectures and mock test series for Class 12
Exam by signing up for free. Class 12: Class 12 Mathematics: CBSE Marking Scheme (2024-25) | Sample Papers for Class 12 Medical and Non-Medical
Description
Full syllabus notes, lecture & questions for Class 12 Mathematics: CBSE Marking Scheme (2024-25) | Sample Papers for Class 12 Medical and Non-Medical - Class 12 | Plus excerises question with solution to help you revise complete syllabus for Sample Papers for Class 12 Medical and Non-Medical | Best notes, free PDF download
Information about Class 12 Mathematics: CBSE Marking Scheme (2024-25)
In this doc you can find the meaning of Class 12 Mathematics: CBSE Marking Scheme (2024-25) defined & explained in the simplest way possible. Besides explaining types of
Class 12 Mathematics: CBSE Marking Scheme (2024-25) theory, EduRev gives you an ample number of questions to practice Class 12 Mathematics: CBSE Marking Scheme (2024-25) tests, examples and also practice Class 12
tests
|
159 docs|4 tests
|
Download as PDF
|
Explore Courses for Class 12 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Free
,Objective type Questions
,Important questions
,shortcuts and tricks
,Exam
,practice quizzes
,Class 12 Mathematics: CBSE Marking Scheme (2024-25) | Sample Papers for Class 12 Medical and Non-Medical
,Semester Notes
,Previous Year Questions with Solutions
,mock tests for examination
,Class 12 Mathematics: CBSE Marking Scheme (2024-25) | Sample Papers for Class 12 Medical and Non-Medical
,MCQs
,ppt
,Viva Questions
,past year papers
,Sample Paper
,Extra Questions
,Class 12 Mathematics: CBSE Marking Scheme (2024-25) | Sample Papers for Class 12 Medical and Non-Medical
,Summary
,video lectures
,study material
;
Additional Information about Class 12 Mathematics: CBSE Marking Scheme (2024-25) for Class 12 Preparation
Class 12 Mathematics: CBSE Marking Scheme (2024-25) Free PDF Download
The Class 12 Mathematics: CBSE Marking Scheme (2024-25) is an invaluable resource that delves deep into the core of the Class 12 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 Class 12 Mathematics: CBSE Marking Scheme (2024-25) now and kickstart your journey towards success in the Class 12 exam.
Importance of Class 12 Mathematics: CBSE Marking Scheme (2024-25)
The importance of Class 12 Mathematics: CBSE Marking Scheme (2024-25) cannot be overstated, especially for Class 12 aspirants.
This document holds the key to success in the Class 12 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.
Class 12 Mathematics: CBSE Marking Scheme (2024-25) Notes
Class 12 Mathematics: CBSE Marking Scheme (2024-25) Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Class 12 Mathematics: CBSE Marking Scheme (2024-25).
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, Class 12 Mathematics: CBSE Marking Scheme (2024-25) Notes on EduRev are your ultimate resource for success.
Class 12 Mathematics: CBSE Marking Scheme (2024-25) Class 12 Questions
The "Class 12 Mathematics: CBSE Marking Scheme (2024-25) Class 12 Questions" guide is a valuable resource for all aspiring students preparing for the
Class 12 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 Class 12 Mathematics: CBSE Marking Scheme (2024-25) on the App
Students of Class 12 can study Class 12 Mathematics: CBSE Marking Scheme (2024-25) alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Class 12 Mathematics: CBSE Marking Scheme (2024-25),
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 Class 12 Mathematics: CBSE Marking Scheme (2024-25) is prepared as per the latest Class 12 syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup