Solution of Maths Class XII Board Paper 2025
Series: YXW2Z
Set – 1 Q.P. Code: 65/2/1
SECTION-A
1 MARKS QUESTIONS
1.The projection
vector of vector 
on vector 
is
a) 
b) 
c)
d)
Solution
: Formula
2. The function 
is increasing in the
interval
a) (0, 2) b) 
c)
d) 
Solution:
f(x) is to be increasing 
3. If 
then 
is
a) 
b) 
c)
d)
Solution:
Consider 
=
Put 2a-x=t,
hence dx=-dt
If x=a then
t=a,
If x=2a then
t=0
(used properties 
4. If 
is a symmetric matrix,
then 
is
a) -8 b) 0 c)
6 d) 8
Solution : Since A is symmetric
matrix , hence 
2x+y=4+4=8
5. If 
then the range of y is
a) 
b)
c)
d)
Solution:
Since 
6. If line makes
angles of
and
with the positive directions of x, y z-axis
respectively, then is
a) 
only b)
c) 
d) 
Solution: If line makes angles of
with the positive directions of x, y z-axis
respectively then 
7. If E and F are
two events such that P(E) > 0 P (F) 
then 
is
a) 
b) 
c) 
d) 
Solution:
8. Which of the
following can be both a symmetric and skew-symmetric matrix?
a) Unit Matrix b)
Diagonal Matrix c) Null Matrix d) Row Matrix
Solution: 
9. The equation of
a line parallel to the vector 
and posing through
the point (4,-3, 7) is
a) 
b) x = 3t + 4 y = t + 3 x = 21
c) x = 3t + 4 =t-3, z = 2t + 7
d) x = 3t + 4 =-t+3 x = 21 + 7
Solution: 
10. Four friends
Abhay, Bina, Chhaya and Devesh were asked to simplify 4AB + 3(AB + BA) - 4BA
where
A and B are both matrices of order 2 x 2.
It is known that 
and 
Their answers
are given as:
Abhay: 6 AB
Bina: 7AB-BA
Chhaya: 8 AB
Devesh :7 BA-AB
Who answered it
correctly?
a) Abhay b) Bina c)
Chhayn d) Devesh
Solution : Matrix multiplication
is not commutative
4AB + 3(AB + BA) - 4BA=4AB+3AB+3BA-4BA=7AB-BA
11. A cylindrical tank of radius 10 cm is being filled with
sugar at the rate of 100Ï€ cm³/s. The rate, at which the height
of the sugar
inside the tank increasing, is:
a) 0.1 cm/s b) 0,5 cm/s c) 1 cm/s d)
1.1 cm/s
Solution :Volume of the sphere is
12. Let 
and be two-unit vectors and a be the angle between them. Then 
will be a unit vector
for what
value α?
a)
b)
c) d) 
Sol: - 
13. The line 
passes through which
of the following point?
a)
(1.-3,6) b) (1,
5, 6) c) (1, -5, -6) d) (-1, -5, 6)
sol: - 
passing through 
14. If A denotes the set of continuous functions and B
denotes set of differentiable functions, then which of the
following
depicts the correct relation between set A and B?
a)
b)
c)
d)
sol:-
set of all differentiable functions, are continuouse but all continuouse
functions are not differentiable
there for option (d) is suitable.
15. The area of the shaded region figure) represented by the
curves 
and y-axis is giver by
a) 
b) 
c)
d) 
sol:- from the given diagram area is along y-axis so option
(b) is satisfies.
16. A factory
produces two products X and Y. The profit earned by line X and Y is represented
by the
objective function 
, where are the
number of units of X and Y respectively and. Which of the
following statement is correct?
a) The objective function maximizes the difference of
the profit earned from product X and Y
b) The objective function measures the total
production of products X and Y.
c) The
objective function maximizes the combined profit earned from selling X and Y
d) The objective function ensures the company produces
more of product X than product Y.
sol: - Z is increases as x and y increases so by the
definition of objective function option (c) is
correct.
17. If A and B are
square matrices of order m such that 
=(A-B) (A+B). then
which of the following
is always correct?
a) A=B b) AB=BA c) A=0 or B=0 d) A=1or B=1
sol: - =(A-B) (A+B)
18. If p and q are
respectively the order and degree of the differential equation
then (p – q) is
a) 0 b) 1 c)2 d) 3
sol: - 
Questions number 19 and 20 are Assertion and
Reason based questions. Two statements are given, one labeled Assertion (A) and
the other labelled Reason (R). Select the correct answer from the codes (A)
(B), (C) and (D) as given below.
a) Both Assertion (A) and Reason (R) are true and
Reason (R) is the correct explanation of the
Assertion
(A)
b)
Both
Assertion (A) and Reason (B) are true, but Reason (R) is not the correct
explanation of the
Assertion (A)
c) Assertion (A) is true, but Reason (R) is false.
d) Assertion (A) is false, but Reason (R) is true.
19. Assertion (A):
A= diag [3 5 2] is a scalar matrix of order 3 x 3
Reason (R) If a diagonal matrix has a
non-zero elements equal, it is known as a scalar matrix
(D) Assertion (A) is false and Reason (R) is true,
Sol: - for scalar
matrix all the diagonals elements must be same so A is false
20. Assertion (A):
Every point of the feasible region of a Linear Programming Problem is an
optimal
solution.
Reason (R) The optimal solution for Linear
Programming Problem exists only at one or more corner
point(s) of the feasible region.
(D) Assertion (A) is false and Reason (R) is true,
Sol: - optimal
solution in the feasible region gives ether maximum or minimum for the
objective
Function so assertion A
is false
SECTION-B
2 MARKS QUESTIONS
21. 
f(x) to be increasing 
i.e, 
-----------------------------------------------1/2M
Since, 
-----------------------------------1/2M
Hence, 
(Also, accept 
----------------------------1M
22. Evaluate 
23.(a) 
--------------------------------------------------------1/2 M
Differentiating both sides
----------------------------------------------------1M
----------------------------------------------------------1/2M
23.(b) 
if 
= 
if 
At x=-2 
is not continuous so
it is not differentiable. -------------------------------2M
24.
25(a). Let α be
the angle which the vector 
makes with all three
axes.
------------------------------------------------------------------------1/2M
The unite vector along the vector 
-----------------------------1/2M
---------------------------------------------------------1M
25(b).
P divides RQ in the ratio 1:2 internally 
------------------1M
----------------------------------------------------------1M
SECTION-C
3 MARKS QUESTIONS
26.(a) The given function
can be written as
----------------------------------------1 M
26. (b) 
------------------------------------------1/2M
-------------------------------------1M
---------------------------------------------------1/2M
Getting 
-------------------------------------------1M
27. Let 
(Domain) such that 
-------------------------------------------------------------------1
½ M
Therefore, f is one-one
Let 
(codomain). Then f(x) = y
If, ax + b
=y
i.e., if, 
which may not belong
to N (domain)
Therefore, f
is not onto. ----------------------------------------------------1 ½ M
28.
----------------------------------------------------2 1/2M
-------------------------------------------------------1/2M
29.(a) Given differential equation can be written as
(3 marks questions)
-------------------------------------------------1M
--------------------------------------------1 1/2M
when 
Hence,
the required particular solution is
----------------------------------------------1/2M
29.(b) Given differential equation can be written as
which is linear in y
I.F 
---------------------------------------------1M
The
solution is given by
----------------------------------------------------------1M
or 
which is the required
general solution ------------------------1M
30. 
(any other number) 
---------------------------------1/2 M
Let X
represent the Random Variable “the number of 2’ s”.
The X = 0,
1, 2-----------------------------------------------------------------1/2 M
The
probability distribution is
|
X |
P(X) |
XP(X) |
|
0 |
|
0 |
|
1 |
|
|
|
2 |
|
|
--------------------------------1 ½ M
Mean = 
---------------------------------------------------------1/2
M
OR
--------------------------------------------------1M
----------------------------------------------------1/2 M
-------------------------------------------------------------1M
Therefore, A and B are not independent.
A and B
are not mutually exclusive as 
---------------------------------------1/2M
31.
……………………………………………..1M
……………………………………………1M
……………………………………………………………..1M
SECTION-D
5 MARKS QUESTIONS
32. (a)
Let the
image of A in the line be 
The
point P, which is the point of intersection of lines 
and 
, will have coordinates ----------1/2M
for some 
Drs of
AP are 
--------------------------------------------1/2 M
------------------------------------------------------------------------1M
Therefore, the coordinates of P are 
-----------------------------------1/2 M
P is
the mid-point of 
-------------------------------------------------------------1
½ M
The
coordinates of the image are 
The
equation of is
32. (b) The vector equations of the lines are
----------------------------------------------1/2 M
--------------------------------------1/2 M
---------------------------------------------------------1M
--------------------------------------2M
S. D =
-------------------------------------------1M
33.
------------------------------------------2M
---------------------------------------------1 ½ M
Given
integral
---------------------------------------------1 ½ M
34. (a) 
---------------------------------------------------------------2M
The system
of equations is equivalent to the matrix equation:
where 
-----------------------------------------------------1/2M
----------------------------------------------------------------------1M
------------------------------------------------------1 ½ M
34. (b) 
exists---------------------------------------------1M
----------------------------------------------------------1 ½
M
The given
system of equations is equivalent to the matrix equation
where 
--------------------------------------------------------1/2 M
------------------------------------------------------------------1/2
M
------------------------------------------1 ½
M
35.
The
required area
----------------------------------1M
---------------------------------------1M
-------------------------------------------------------1M
SECTION-E
3
CASE STUDY BASED QUESTIONS
36.
(i) The number of relations 
(ii) Since,
and 
have been
assigned the same judge 
the function is
not one-one. Hence, it is
not
bijective.
(iii) There
cannot exist any one-one function from S to J as n(S) > n(J) Hence, the
number of one-one
functions
from S to J is 0.
OR
(iii) To make 
reflexive and
not symmetric we need to add the following ordered pairs:
37. (i) (a) Let 
Amber manufactures the care
B = Bonzi manufactures the car
C = Coment manufactures the car
E = The selected car is electric
----------------------------------1/2 M
--------------------------------------------1M
-----------------------------------1/2 M
OR
(b) Let Amber manufactures the care
B = Bonzi manufactures the car
C = Coment manufactures the car
E = The selected car is electric
-------------------------1/2M
------------------------1M
=
----------------------------------1/2M
(ii)
---------------------------------------------------------1M
(iii) 
--------------------------------------1M
38.
(i) 
=>
For
critical points, 
-----------------------------------------------------------1/2M
For x to
be a critical point 
hence, 
-----------------------------1/2M
f is increasing 
---------------------------------------1/2M
f is decreasing 
------------------------1/2M
Note: If a
student concludes the answer in any of the following intervals, full marks may
be awarded:
(ii) is critical point
-------------------------------1M
--------------------------------------------------1/2M
Hence, 
is a point of local
maximum. -------------------------1/2M