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Page 1
Class-XII/Sample Paper/2024-25/Mathematics/Page 1 of 9
SAMPLE QUESTION PAPER (2024 - 25)
CLASS- XII
SUBJECT: Mathematics (041)
Time: 3 Hours Maximum Marks: 80
General Instructions:
Read the following instructions very carefully and strictly follow them:
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) This Question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and
20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks
each.
(v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B,
3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E.
(ix) Use of calculators is not allowed.
SECTION-A ? ? 1 20 20 ??
(This section comprises of multiple choice questions (MCQs) of 1 mark each)
Select the correct option (Question 1 - Question 18):
Q.1. If for a square matrix , A ?? . (?????? ?? ) = [
???????? ?? ?? ?? ???????? ?? ?? ?? ????????
], then the value of A adj A ? is
equal to:
(A) 1 (B) 2025 1 ? (C) ? ?
2
2025 45 ? (D) ? ?
2
2025 2025 ?
Q.2. Assume , , , X Y Z W and P are matrices of order 2 , 3 , 2 , 3 n k p n ? ? ? ? and , pk ?
respectively. Then the restriction on , nk
and p so that PY WY ? will be defined are:
(A) 3, k p n ?? (B) is arbitrary, 2 kp ?
(C) is arbitrary, 3 pk ? (D) 2, 3 kp ??
Q.3. The interval in which the function f defined by ?? (?? ) = ?? ??
is strictly increasing, is
(A) [1, 8) (B) ? ? ,0 ?? (C) (-8, 8) (D) (0,8)
Page 2
Class-XII/Sample Paper/2024-25/Mathematics/Page 1 of 9
SAMPLE QUESTION PAPER (2024 - 25)
CLASS- XII
SUBJECT: Mathematics (041)
Time: 3 Hours Maximum Marks: 80
General Instructions:
Read the following instructions very carefully and strictly follow them:
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) This Question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and
20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks
each.
(v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B,
3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E.
(ix) Use of calculators is not allowed.
SECTION-A ? ? 1 20 20 ??
(This section comprises of multiple choice questions (MCQs) of 1 mark each)
Select the correct option (Question 1 - Question 18):
Q.1. If for a square matrix , A ?? . (?????? ?? ) = [
???????? ?? ?? ?? ???????? ?? ?? ?? ????????
], then the value of A adj A ? is
equal to:
(A) 1 (B) 2025 1 ? (C) ? ?
2
2025 45 ? (D) ? ?
2
2025 2025 ?
Q.2. Assume , , , X Y Z W and P are matrices of order 2 , 3 , 2 , 3 n k p n ? ? ? ? and , pk ?
respectively. Then the restriction on , nk
and p so that PY WY ? will be defined are:
(A) 3, k p n ?? (B) is arbitrary, 2 kp ?
(C) is arbitrary, 3 pk ? (D) 2, 3 kp ??
Q.3. The interval in which the function f defined by ?? (?? ) = ?? ??
is strictly increasing, is
(A) [1, 8) (B) ? ? ,0 ?? (C) (-8, 8) (D) (0,8)
Class-XII/Sample Paper/2024-25/Mathematics/Page 2 of 9
Q.4. If and AB are non-singular matrices of same order with ?????? (?? ) = ?? , then ?????? (?? -?? ???? )
?? is equal to
(A)
5
(B)
2
5
(C)
4
5
(D)
5
5
Q.5. The value of '' n , such that the differential equation ?? ?? ????
????
= ?? (???????? - ???????? + ?? );
(?????????? ?? , ?? ? ?? +
) is homogeneous, is
(A) 0 (B)1 (C) 2 (D) 3
Q.6. If the points (?? ?? , ?? ?? ), (?? ?? , ?? ?? ) and (?? ?? + ?? ?? , ?? ?? + ?? ?? )are collinear, then ?? ?? ?? ?? is equal to
(A)
21
xy (B)
11
xy (C)
22
xy (D)
12
xx
Q.7. If
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
is a skew-symmetric matrix then the value of a b c ? ? ?
(A)1 (B) 2 (C) 3 (D) 4
Q.8. For any two events A and B , if
? ?
1
2
PA ? ,
? ?
2
3
PB ? and ? ?
1
,
4
P A B ?? then ?? (
?? ¯
?? ¯
/ ) equals:
(A)
3
8
(B)
8
9
(C)
5
8
(D)
1
4
Q.9. The value of ?? if the angle between ?? ? = 2?? 2
?? ^ - 3?? ?? ^ + ?? ^
and ?? ? = ?? ^ + ?? ^ + ?? ?? ^
is obtuse, is
(A) ?? - [0, 1] (B) (0, 1) (C) [0, 8) (D) [1, 8)
Q.10. If |?? ? | = 3, |?? ? ?
| = 4 and |?? ? + ?? ? ?
| =5, then |?? ? - ?? ? ?
| =
(A) 3 (B) 4 (C) 5 (D) 8
Q.11. For the linear programming problem (LPP), the objective function is 43 Z x y ?? and the feasible
region determined by a set of constraints is shown in the graph:
Page 3
Class-XII/Sample Paper/2024-25/Mathematics/Page 1 of 9
SAMPLE QUESTION PAPER (2024 - 25)
CLASS- XII
SUBJECT: Mathematics (041)
Time: 3 Hours Maximum Marks: 80
General Instructions:
Read the following instructions very carefully and strictly follow them:
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) This Question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and
20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks
each.
(v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B,
3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E.
(ix) Use of calculators is not allowed.
SECTION-A ? ? 1 20 20 ??
(This section comprises of multiple choice questions (MCQs) of 1 mark each)
Select the correct option (Question 1 - Question 18):
Q.1. If for a square matrix , A ?? . (?????? ?? ) = [
???????? ?? ?? ?? ???????? ?? ?? ?? ????????
], then the value of A adj A ? is
equal to:
(A) 1 (B) 2025 1 ? (C) ? ?
2
2025 45 ? (D) ? ?
2
2025 2025 ?
Q.2. Assume , , , X Y Z W and P are matrices of order 2 , 3 , 2 , 3 n k p n ? ? ? ? and , pk ?
respectively. Then the restriction on , nk
and p so that PY WY ? will be defined are:
(A) 3, k p n ?? (B) is arbitrary, 2 kp ?
(C) is arbitrary, 3 pk ? (D) 2, 3 kp ??
Q.3. The interval in which the function f defined by ?? (?? ) = ?? ??
is strictly increasing, is
(A) [1, 8) (B) ? ? ,0 ?? (C) (-8, 8) (D) (0,8)
Class-XII/Sample Paper/2024-25/Mathematics/Page 2 of 9
Q.4. If and AB are non-singular matrices of same order with ?????? (?? ) = ?? , then ?????? (?? -?? ???? )
?? is equal to
(A)
5
(B)
2
5
(C)
4
5
(D)
5
5
Q.5. The value of '' n , such that the differential equation ?? ?? ????
????
= ?? (???????? - ???????? + ?? );
(?????????? ?? , ?? ? ?? +
) is homogeneous, is
(A) 0 (B)1 (C) 2 (D) 3
Q.6. If the points (?? ?? , ?? ?? ), (?? ?? , ?? ?? ) and (?? ?? + ?? ?? , ?? ?? + ?? ?? )are collinear, then ?? ?? ?? ?? is equal to
(A)
21
xy (B)
11
xy (C)
22
xy (D)
12
xx
Q.7. If
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
is a skew-symmetric matrix then the value of a b c ? ? ?
(A)1 (B) 2 (C) 3 (D) 4
Q.8. For any two events A and B , if
? ?
1
2
PA ? ,
? ?
2
3
PB ? and ? ?
1
,
4
P A B ?? then ?? (
?? ¯
?? ¯
/ ) equals:
(A)
3
8
(B)
8
9
(C)
5
8
(D)
1
4
Q.9. The value of ?? if the angle between ?? ? = 2?? 2
?? ^ - 3?? ?? ^ + ?? ^
and ?? ? = ?? ^ + ?? ^ + ?? ?? ^
is obtuse, is
(A) ?? - [0, 1] (B) (0, 1) (C) [0, 8) (D) [1, 8)
Q.10. If |?? ? | = 3, |?? ? ?
| = 4 and |?? ? + ?? ? ?
| =5, then |?? ? - ?? ? ?
| =
(A) 3 (B) 4 (C) 5 (D) 8
Q.11. For the linear programming problem (LPP), the objective function is 43 Z x y ?? and the feasible
region determined by a set of constraints is shown in the graph:
Class-XII/Sample Paper/2024-25/Mathematics/Page 3 of 9
(Note: The figure is not to scale.)
Which of the following statements is true?
(A) Maximum value of Z is at ? ? 40,0 R .
(B) Maximum value of Z is at ? ? 30,20 Q .
(C) Value of Z at ? ? 40,0 R is less than the value at ? ? 0,40 P .
(D) The value of Z at ? ? 30,20 Q is less than the value at ? ? 40,0 R .
Q.12. ?
????
?? ?? (?? +?? ?? )
?? ??
equals
(A) -
1
2?? 2
v1 + ?? 4
+ ?? (B)
1
2?? v1 + ?? 4
+ ??
(C) -
1
4?? v1 + ?? 4
+ ??
(D)
1
4?? 2
v1 + ?? 4
+ ??
Q.13. ? ?????????? ?? ?? ????
?? ???? =
(A) 0 (B)1 (C) 4 (D) 2 ?
Q.14. What is the general solution of the differential equation e
y
'
= x?
(A)?? = ?????????? + ?? (B) ?? = ?????????? - ?? + ?? (C) ?? = ?????????? + ?? + ?? (D) ?? = ?? + ??
Q.15. The graph drawn below depicts
(A) y = ?????? -1
?? (B) y = ?????? -1
?? (C) y = ?????? ?? ?? -1
?? (D) y = ?????? -1
??
Q.16. A linear programming problem (LPP) along with the graph of its constraints is shown below.
Page 4
Class-XII/Sample Paper/2024-25/Mathematics/Page 1 of 9
SAMPLE QUESTION PAPER (2024 - 25)
CLASS- XII
SUBJECT: Mathematics (041)
Time: 3 Hours Maximum Marks: 80
General Instructions:
Read the following instructions very carefully and strictly follow them:
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) This Question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and
20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks
each.
(v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B,
3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E.
(ix) Use of calculators is not allowed.
SECTION-A ? ? 1 20 20 ??
(This section comprises of multiple choice questions (MCQs) of 1 mark each)
Select the correct option (Question 1 - Question 18):
Q.1. If for a square matrix , A ?? . (?????? ?? ) = [
???????? ?? ?? ?? ???????? ?? ?? ?? ????????
], then the value of A adj A ? is
equal to:
(A) 1 (B) 2025 1 ? (C) ? ?
2
2025 45 ? (D) ? ?
2
2025 2025 ?
Q.2. Assume , , , X Y Z W and P are matrices of order 2 , 3 , 2 , 3 n k p n ? ? ? ? and , pk ?
respectively. Then the restriction on , nk
and p so that PY WY ? will be defined are:
(A) 3, k p n ?? (B) is arbitrary, 2 kp ?
(C) is arbitrary, 3 pk ? (D) 2, 3 kp ??
Q.3. The interval in which the function f defined by ?? (?? ) = ?? ??
is strictly increasing, is
(A) [1, 8) (B) ? ? ,0 ?? (C) (-8, 8) (D) (0,8)
Class-XII/Sample Paper/2024-25/Mathematics/Page 2 of 9
Q.4. If and AB are non-singular matrices of same order with ?????? (?? ) = ?? , then ?????? (?? -?? ???? )
?? is equal to
(A)
5
(B)
2
5
(C)
4
5
(D)
5
5
Q.5. The value of '' n , such that the differential equation ?? ?? ????
????
= ?? (???????? - ???????? + ?? );
(?????????? ?? , ?? ? ?? +
) is homogeneous, is
(A) 0 (B)1 (C) 2 (D) 3
Q.6. If the points (?? ?? , ?? ?? ), (?? ?? , ?? ?? ) and (?? ?? + ?? ?? , ?? ?? + ?? ?? )are collinear, then ?? ?? ?? ?? is equal to
(A)
21
xy (B)
11
xy (C)
22
xy (D)
12
xx
Q.7. If
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
is a skew-symmetric matrix then the value of a b c ? ? ?
(A)1 (B) 2 (C) 3 (D) 4
Q.8. For any two events A and B , if
? ?
1
2
PA ? ,
? ?
2
3
PB ? and ? ?
1
,
4
P A B ?? then ?? (
?? ¯
?? ¯
/ ) equals:
(A)
3
8
(B)
8
9
(C)
5
8
(D)
1
4
Q.9. The value of ?? if the angle between ?? ? = 2?? 2
?? ^ - 3?? ?? ^ + ?? ^
and ?? ? = ?? ^ + ?? ^ + ?? ?? ^
is obtuse, is
(A) ?? - [0, 1] (B) (0, 1) (C) [0, 8) (D) [1, 8)
Q.10. If |?? ? | = 3, |?? ? ?
| = 4 and |?? ? + ?? ? ?
| =5, then |?? ? - ?? ? ?
| =
(A) 3 (B) 4 (C) 5 (D) 8
Q.11. For the linear programming problem (LPP), the objective function is 43 Z x y ?? and the feasible
region determined by a set of constraints is shown in the graph:
Class-XII/Sample Paper/2024-25/Mathematics/Page 3 of 9
(Note: The figure is not to scale.)
Which of the following statements is true?
(A) Maximum value of Z is at ? ? 40,0 R .
(B) Maximum value of Z is at ? ? 30,20 Q .
(C) Value of Z at ? ? 40,0 R is less than the value at ? ? 0,40 P .
(D) The value of Z at ? ? 30,20 Q is less than the value at ? ? 40,0 R .
Q.12. ?
????
?? ?? (?? +?? ?? )
?? ??
equals
(A) -
1
2?? 2
v1 + ?? 4
+ ?? (B)
1
2?? v1 + ?? 4
+ ??
(C) -
1
4?? v1 + ?? 4
+ ??
(D)
1
4?? 2
v1 + ?? 4
+ ??
Q.13. ? ?????????? ?? ?? ????
?? ???? =
(A) 0 (B)1 (C) 4 (D) 2 ?
Q.14. What is the general solution of the differential equation e
y
'
= x?
(A)?? = ?????????? + ?? (B) ?? = ?????????? - ?? + ?? (C) ?? = ?????????? + ?? + ?? (D) ?? = ?? + ??
Q.15. The graph drawn below depicts
(A) y = ?????? -1
?? (B) y = ?????? -1
?? (C) y = ?????? ?? ?? -1
?? (D) y = ?????? -1
??
Q.16. A linear programming problem (LPP) along with the graph of its constraints is shown below.
Class-XII/Sample Paper/2024-25/Mathematics/Page 4 of 9
The corresponding objective function is: 18 10 Z x y ?? , which has to be minimized. The smallest value of
the objective function Z is 134 and is obtained at the corner point ? ? 3,8 ,
(Note: The figure is not to scale.)
The optimal solution of the above linear programming problem __________.
(A) does not exist as the feasible region is unbounded.
(B) does not exist as the inequality 18 10 134 xy ??
does not have any point in common with the feasible
region.
(C) exists as the inequality 18 10 134 xy ??
has infinitely many points in common with the feasible region.
(D) exists as the inequality 18 10 134 xy ??
does not have any point in common with the feasible region.
Q.17. The function ?? : ?? ? ?? defined by ? ? ? ?; f x x ? where
? ? . denotes the greatest integer function, is
(A) Continuous at 2.5 x ?
but not differentiable at 2.5 x ?
(B) Not Continuous at 2.5 x ?
but differentiable at 2.5 x ?
(C) Not Continuous at 2.5 x ? and not differentiable at 2.5 x ?
(D) Continuous as well as differentiable at 2.5 x ?
Q.18. A student observes an open-air Honeybee nest on the branch of a tree, whose plane figure is parabolic
shape given by
2
4 xy ? . Then the area (in sq units) of the region bounded by parabola
2
4 xy ? and the line
4 y ? is
(A)
32
3
(B)
64
3
(C)
128
3
(D)
256
3
ASSERTION-REASON BASED QUESTIONS
(Question numbers 19 and 20 are Assertion-Reason based questions carrying 1 mark each. Two
statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct
answer from the options (A), (B), (C) and (D) as given below.)
Page 5
Class-XII/Sample Paper/2024-25/Mathematics/Page 1 of 9
SAMPLE QUESTION PAPER (2024 - 25)
CLASS- XII
SUBJECT: Mathematics (041)
Time: 3 Hours Maximum Marks: 80
General Instructions:
Read the following instructions very carefully and strictly follow them:
(i) This Question paper contains 38 questions. All questions are compulsory.
(ii) This Question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and Questions no. 19 and
20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2 marks
each.
(v) In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B,
3 questions in Section C, 2 questions in Section D and one subpart each in 2 questions of Section E.
(ix) Use of calculators is not allowed.
SECTION-A ? ? 1 20 20 ??
(This section comprises of multiple choice questions (MCQs) of 1 mark each)
Select the correct option (Question 1 - Question 18):
Q.1. If for a square matrix , A ?? . (?????? ?? ) = [
???????? ?? ?? ?? ???????? ?? ?? ?? ????????
], then the value of A adj A ? is
equal to:
(A) 1 (B) 2025 1 ? (C) ? ?
2
2025 45 ? (D) ? ?
2
2025 2025 ?
Q.2. Assume , , , X Y Z W and P are matrices of order 2 , 3 , 2 , 3 n k p n ? ? ? ? and , pk ?
respectively. Then the restriction on , nk
and p so that PY WY ? will be defined are:
(A) 3, k p n ?? (B) is arbitrary, 2 kp ?
(C) is arbitrary, 3 pk ? (D) 2, 3 kp ??
Q.3. The interval in which the function f defined by ?? (?? ) = ?? ??
is strictly increasing, is
(A) [1, 8) (B) ? ? ,0 ?? (C) (-8, 8) (D) (0,8)
Class-XII/Sample Paper/2024-25/Mathematics/Page 2 of 9
Q.4. If and AB are non-singular matrices of same order with ?????? (?? ) = ?? , then ?????? (?? -?? ???? )
?? is equal to
(A)
5
(B)
2
5
(C)
4
5
(D)
5
5
Q.5. The value of '' n , such that the differential equation ?? ?? ????
????
= ?? (???????? - ???????? + ?? );
(?????????? ?? , ?? ? ?? +
) is homogeneous, is
(A) 0 (B)1 (C) 2 (D) 3
Q.6. If the points (?? ?? , ?? ?? ), (?? ?? , ?? ?? ) and (?? ?? + ?? ?? , ?? ?? + ?? ?? )are collinear, then ?? ?? ?? ?? is equal to
(A)
21
xy (B)
11
xy (C)
22
xy (D)
12
xx
Q.7. If
01
1
2 3 0
c
A a b
??
??
? ? ?
??
??
??
is a skew-symmetric matrix then the value of a b c ? ? ?
(A)1 (B) 2 (C) 3 (D) 4
Q.8. For any two events A and B , if
? ?
1
2
PA ? ,
? ?
2
3
PB ? and ? ?
1
,
4
P A B ?? then ?? (
?? ¯
?? ¯
/ ) equals:
(A)
3
8
(B)
8
9
(C)
5
8
(D)
1
4
Q.9. The value of ?? if the angle between ?? ? = 2?? 2
?? ^ - 3?? ?? ^ + ?? ^
and ?? ? = ?? ^ + ?? ^ + ?? ?? ^
is obtuse, is
(A) ?? - [0, 1] (B) (0, 1) (C) [0, 8) (D) [1, 8)
Q.10. If |?? ? | = 3, |?? ? ?
| = 4 and |?? ? + ?? ? ?
| =5, then |?? ? - ?? ? ?
| =
(A) 3 (B) 4 (C) 5 (D) 8
Q.11. For the linear programming problem (LPP), the objective function is 43 Z x y ?? and the feasible
region determined by a set of constraints is shown in the graph:
Class-XII/Sample Paper/2024-25/Mathematics/Page 3 of 9
(Note: The figure is not to scale.)
Which of the following statements is true?
(A) Maximum value of Z is at ? ? 40,0 R .
(B) Maximum value of Z is at ? ? 30,20 Q .
(C) Value of Z at ? ? 40,0 R is less than the value at ? ? 0,40 P .
(D) The value of Z at ? ? 30,20 Q is less than the value at ? ? 40,0 R .
Q.12. ?
????
?? ?? (?? +?? ?? )
?? ??
equals
(A) -
1
2?? 2
v1 + ?? 4
+ ?? (B)
1
2?? v1 + ?? 4
+ ??
(C) -
1
4?? v1 + ?? 4
+ ??
(D)
1
4?? 2
v1 + ?? 4
+ ??
Q.13. ? ?????????? ?? ?? ????
?? ???? =
(A) 0 (B)1 (C) 4 (D) 2 ?
Q.14. What is the general solution of the differential equation e
y
'
= x?
(A)?? = ?????????? + ?? (B) ?? = ?????????? - ?? + ?? (C) ?? = ?????????? + ?? + ?? (D) ?? = ?? + ??
Q.15. The graph drawn below depicts
(A) y = ?????? -1
?? (B) y = ?????? -1
?? (C) y = ?????? ?? ?? -1
?? (D) y = ?????? -1
??
Q.16. A linear programming problem (LPP) along with the graph of its constraints is shown below.
Class-XII/Sample Paper/2024-25/Mathematics/Page 4 of 9
The corresponding objective function is: 18 10 Z x y ?? , which has to be minimized. The smallest value of
the objective function Z is 134 and is obtained at the corner point ? ? 3,8 ,
(Note: The figure is not to scale.)
The optimal solution of the above linear programming problem __________.
(A) does not exist as the feasible region is unbounded.
(B) does not exist as the inequality 18 10 134 xy ??
does not have any point in common with the feasible
region.
(C) exists as the inequality 18 10 134 xy ??
has infinitely many points in common with the feasible region.
(D) exists as the inequality 18 10 134 xy ??
does not have any point in common with the feasible region.
Q.17. The function ?? : ?? ? ?? defined by ? ? ? ?; f x x ? where
? ? . denotes the greatest integer function, is
(A) Continuous at 2.5 x ?
but not differentiable at 2.5 x ?
(B) Not Continuous at 2.5 x ?
but differentiable at 2.5 x ?
(C) Not Continuous at 2.5 x ? and not differentiable at 2.5 x ?
(D) Continuous as well as differentiable at 2.5 x ?
Q.18. A student observes an open-air Honeybee nest on the branch of a tree, whose plane figure is parabolic
shape given by
2
4 xy ? . Then the area (in sq units) of the region bounded by parabola
2
4 xy ? and the line
4 y ? is
(A)
32
3
(B)
64
3
(C)
128
3
(D)
256
3
ASSERTION-REASON BASED QUESTIONS
(Question numbers 19 and 20 are Assertion-Reason based questions carrying 1 mark each. Two
statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct
answer from the options (A), (B), (C) and (D) as given below.)
Class-XII/Sample Paper/2024-25/Mathematics/Page 5 of 9
(A) Both (A) and (R) are true and (R) is the correct explanation of (A).
(B) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(C) (A) is true but (R) is false.
(D) (A) is false but (R) is true.
Q.19. Assertion (A): Consider the function defined as ?? (?? ) = |?? | + |?? - 1|, ?? ? ?? . Then ? ? fx
is not differentiable at 0 and 1 xx ?? .
Reason (R): Suppose f be defined and continuous on ? ? , ab and ? ? , c a b ? , then ? ? fx is not
differentiable at ?? = ?? if lim
h?0
-
?? (?? +h)-?? (?? )
h
? lim
h?0
+
?? (?? +h)-?? (?? )
h
.
Q.20. Assertion (A): The function ?? : ?? - {(2?? + 1)
?? 2
: ?? ? ?? } ? (-8, -1] ? [1, 8) defined by
? ? sec f x x ? is not one-one function in its domain.
Reason (R): The line 2 y ?
meets the graph of the function at more than one point.
SECTION B ? ? 2 5 10 ??
(This section comprises of 5 very short answer (VSA) type questions of 2 marks each.)
Q.21. If ?????? -1
(3?? + 5) >
?? 4
,
then find the range of the values of . x
Q.22. The cost (in rupees) of producing x
items in factory, each day is given by
?? (?? ) = 0.00013 ?? 3
+ 0.002 ?? 2
+ 5?? + 2200
Find the marginal cost when 150
items are produced.
Q.23. (a) Find the derivative of
1
tan
?
x with respect to log ; x (where ? ? 1, x?? ).
OR
Q.23. (b) Differentiate the following function with respect to x : (?????? ?? )
?? ; (where?? ? (0,
?? 2
)).
Q.24. (a) If vectors ?? ? = 2i ^ + 2?^ + 3k
^ , ?? ? ?
= - i ^ + 2?^ + k
^ and ?? ? = 3i ^ + ?^ are such that ?? ? ?
+ ??? ? is perpendicular
to ?? ? , then find the value of ?.
OR
Q.24. (b) A person standing at O ? ? 000 ,, is watching an aeroplane which is at the coordinate point
? ? A , , 4 0 3 . At the same time he saw a bird at the coordinate point ? ? B , , . 0 0 1 Find the angles which
????
? ? ? ???
makes with the x,y and z axes.
Q.25. The two co-initial adjacent sides of a parallelogram are 2i ^ - 4?^ - 5k
^ and 2i ^ + 2?^ + 3k
^ . Find its
diagonals and use them to find the area of the parallelogram.
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