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1. Marking Scheme
Class XII Mathematics
Section: A (Multiple Choice Questions- 1 Mark each)
Q. No: Answer Hints/Solution
1 (b) In a symmetric matrix, the (i, j)th
element is same as the (j, i)th
element.
Hence,
2 (c)
3 (c) Projection of on is zero
4 (a) .
5 (b) Given integral is of the form
6 (c)
7 (a)
8 (a)
sq. units
9 (d)
10 (c)
11 (d) .
12 (c)
13 (c) .
14 (b) Since and are independent
15 (d)
16 (b)
17 (d) ,
.
18 (b) .
19 (b) Both A and R are true but R is not the correct explanation of A.
Hence, Domain of is .
20 (c) A is true but R is false.

2. Hence A is true.
Skew lines are lines in space which are neither parallel nor
intersecting. Hence R is false.
SECTION B (VSA questions of 2 marks each)
Q. No: Value point Marks
21
.
OR
To prove injection
To prove surjection
1
1
1
22 At any time, t let x cm be the length of a side of the equilateral triangle,
and let A be its area.
1
1
23 Let ,
Diagonal =
Required unit vector
OR
Given line is
Direction Ratios
Direction cosines,
1
1
24 ,
1
25

3. SECTION C
(Short Answer Questions of 3 Marks each)
26
1
1
27 P(not obtaining an odd person in a single round) = P(All three of
them throw tails or All three of them throw heads)
P(obtaining an odd person in a single round)
The required probability
.
OR
Let X denote the number of selected scientists who never commit error
in the work and reporting. X takes values 0,1,2
X 0 1 2
P(X)
Mean=
1
2

4. 28
Using
Adding (1) and (2)
OR
1
1+1
29
Applying the condition and getting
Solution
OR

5. Substituting
30 We have subject to the constraints
, , ,
.
The corner points of the feasible region are
1

6. Corner Points
A(0,200) 400
B(50,100) 250
C(20,40) 100
D(0,50) 100
Max (= 400) at x = 0, y = 200
1
31
By Partial fractions,
SECTION D
(Long answer type questions (LA) of 5 marks each)
32 Point of intersection of circle and line.
Required Area
sq. units
(Correct
Fig: 1
Mark)
1

7. Redefining
Area
square units
Correct
Fig: 1
Mark)
1
33 Let ( , ) × . Then we have
a+b = b+a (by commutative property of addition of natural
numbers)
( , ) ( , )
Hence, R is reflexive.
Let ( , ), ( , ) × such that (a, b) R (c, d). Then
a+d = b+c
= (by commutative property of addition of
natural numbers
( , ) ( , )
Hence, R is symmetric.
Let ( , ), ( , ), ( , ) × such that
(a, b) R (c, d) and (c, d) R (e, f).
Then a+d = b+c, c+f = d+e
=
=
( , ) ( , )
Hence, R is transitive.
Since, R is reflexive, symmetric and transitive, R is an
equivalence relation on × .
34 Vector equations of the given lines,

8. and
,
,
,
units.
OR
Eliminating t between the equations, we obtain the equation of the
path , which are the equations of the line passing
through the origin having direction ratios .
This line is the path of the rocket.
When t = 10 seconds, the rocket will be at the point (20, -40, 40).
Hence, the required distance from the origin at 10 seconds
The distance of the point (20, -40, 40) from the given line
km
1
½
½
½
1
1+½
1
½
1
2
½
35
,
Given system can be written as
½
2
1
½
1
SECTION E
(Case Studies/Passage based questions of 4 Marks each)
36

9. (i) , being a polynomial
function, is differentiable everywhere, hence, differentiable
in (1,5)
(ii)
Since, 2 is the critical point,
(iii)
In the Interval Conclusion
+ve is increasing
-ve C(x) is decreasing
+ve is increasing
OR
(iii)
are the critical values
Hence, by second derivative test 2 is a point of local
maximum. The local maximum value is
.
Hence, by second derivative test 3 is a point of local
minimum. The local minimum value is
We have
Also
5 is the point of absolute maximum and the absolute
maximum value of the function=56
1 is the point of absolute maximum and the absolute
maximum value of the function=24
1
1
1+1
1/2
½
1/2
1/2
37
(i)
(ii)
1

10. (iii) For the values of x less than 5, and close to 5 and for
the values of x greater than 5 and close to 5 .
Since there is only one critical point is possible (x=18 is not
possible) the volume is maximum at x=5. The maximum
volume is 2450 cubic inches.
OR
(iii)
, hence volume is
maximum when x=5. The maximum volume is 2450 cubic
inches.
1
½
1 ½
1 ½
½
38 Let the events be defined as:
E1:the examinee guesses , E2: copies the answer, E3: knows
the answer
A: the examinee answers correctively.
.
.
(i)
(ii) Required probability
.
1
1
2