KENDRIYA VIDYALAYA SANGATHAN HYDERABAD REGION
PRE BOARD –I
2022-2023
SUB: MATHEMATICS
MAX.MARKS: 80
CLASS: XII
MAX.TIME: 3 HOURS
General Instructions:
1. This paper contains five sections A, B, C, D, and E. Each section is compulsory. However
there are internal choices in some questions.
2. Section A has 18MCQ’s questions and 2 Assertion and Reasoning questions of 1 mark each.
3. Section B has 5 VERY SHORT ANSWER (VSA)-type questions of 2 marks each.
4. Section C has 6 SHORT ANSWER (SA)-type questions of 3 marks each.
5. Section D has 4 LONG ANSWER (LA)-type questions of 5 marks each.
6. Section E has 3 Source Based/Case based/Passage based/Integrated Units of assessment
(4 marks each) with sub parts.
SECTION A
(Multiple Choice Questions) Each question carries 1 mark
1.
[
] is a skew symmetric matrix, then the value of 2p+q+r is
A)1
2.
B) 2
| |
If
A)100
3.
4.
C) -1
B) 16
D) 0
|
| is
C) 80
D) 320
The function f(x) = x – [x] , where [.] denotes the greatest integer function is
A)Continuous everywhere.
B)Continuous at integer points only.
C) Continuous at non-integer points only
D)Differentiable everywhere
In a linear programming problem, the constraints on the decision variables x and y are
− 3𝑦 ≥ 0, 𝑦 ≥ 0, 0 ≤
≤ 3. The feasible region
A) is not in the first quadrant
B) is bounded in the first quadrant
C) is unbounded in the first quadrant
D) does not exist
5.
*
A) *
6.
+
*
A)72
+
B) *
+
C) *
+
D) *
+
+ is a singular matrix, then x=
B) 36
C) 6
D)
6
7.
8.
If A and B are symmetric matrices of the same order, then (
A) Skew symmetric matrix
(B) Null matrix
C) Symmetric matrix
(D) None of these
If ( )
( )
A)
9.
If the projection of ⃗
If
and
̂ on ⃗ = 2 ̂
̂
The value of ∫
⁄
̂) (
B)
̂ is 0,then
is
D)
| |
̂)
C)
y = , then
(
D)
C)
B)
D) 4
=
C)
D)
C) 0
D)-2
)
(
The sum of the order and degree of the differential equation
A) 1
B) 2
C)3
B) ̂
C)
̂
̂
√
̂ and ̂
̂ is
D)
Integrating factor of the differential equation
A) s
If
B) s
) is
D) 6
The unit vector perpendicular to both the vectors ̂
A) ̂
17.
̂
√
C)
B) 16
A)2
16.
√
̂ (
A)
15.
D)
is
B) 1
A)0
14.
C)
B)
11.
13.
)
The angle between the lines
A)0
12.
(
B)
A)
10.
( ⁄ )
) is a
C)
and ⃗ are unit vectors ,then the angle between
̂
̂
√
is
D)
and ⃗ such that √
⃗ to be a
unit vector is
A)450
18.
B) 300
C) 600
D) 900
Based on the given shaded region
as the feasible region in the
graph, at which point(s) is the
objective function
Z = 3x + 9y maximum?
A) Point B
B) Point C
C) Point D
D) every point on the line segment CD
19.
Assertion (A) : A relation R ={ (1,1),(1,2),(2,2),(2,3)(3,3)}defined in the set
A={1,2,3} is symmetric
Reason(R) : A relation R on the set A is symmetric if (a,b)
(
)
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
20.
Assertion:
Reason:
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
SECTION B
(This section comprises of very short answer type questions (VSA) of 2 marks each)
21.
Find the value of
(
)
(
√
).
Or
Is the function ( )
22.
| |, a one-one function? Justify your answer.
A particle moves along the curve 𝑦
. Find the points on the curve at which
the 𝑦-coordinate is changing 8 times as the -coordinate.
23.
(
𝑦
If
|⃗ |
24.
If | |
25.
If the lines
(
𝑦)
|⃗⃗ ⃗⃗ |
, then find the value of
)
⃗ .
are perpendicular to each other
then find the value of
Or
Find a vector of magnitude 6 and opposite to the direction of the ̅
̂
̂
̂.
SECTION C
(This section comprises of short answer type questions (SA) of 3 marks each)
26.
27.
∫(
)
Evaluate:
(
∫
)
.
Or
∫|
Evaluate:
28.
|
Subject to the constraints
29.
30.
𝑦
Solve graphically: Maximise :
Evaluate : ∫
𝑦
𝑦
𝑦
√
Given three identical boxes I,II and III,each containing two coins. In box I both coins are
gold coins, in box II ,both are silver coins and in box III there is one gold and one silver
coin. A person chooses a box at random and takes out a coin. If the coin is of gold,
what is the probability that the other coin in the box is silver?
Or
Find the probability distribution of the number of heads in three tosses of a fair coin and
find the mean number of heads.
31.
Solve:
Or
Solve :
SECTION D
(This section comprises of long answer type questions (LA) of 5 marks each)
| |.
32.
Find the area bounded by the curves
33.
Find the intervals in which the function f given by
( )
34.
is strictly increasing or decreasing.
Let L be the set of all lines XY-plane and let R be a relation defined in L as R={(L1,L2):
L1 || L2 }.Show that R is an equivalence relation .Find the set of all lines related to the
line y=2x+4.
Or
Let R be the relation defined on the set A={1,2,3,4,5,6,7} by R={(a,b):both a and b are
either even or odd}.Show that R is an equivalence relation. Further show that all
elements of the subset {1,3,5,7} are related to each other and all elements of the subset
{2,4,6} are related to each other ,but no element of {1,3,5,7} is related to any element of
{2,4,6}.
35.
If
[
]
[
]
. Hence solve the system of equations
.
SECTION E
The first two have three sub parts of 1,1 and 2 marks each and the third has two parts
of 3 and 1 mark each.
36.
Two girls Gita and Rita are involved in solving a problem given by their teacher. The
probability of Gita solving the problem is
and
Rita solving the problem is .If both try to solve
the problem independently, find the probability
that
37.
i)
The problem is not solved
ii)
Exactly one of them solves the problem
iii)
The problem is solved
An open toy box with a square base is to be made out of a
given quantity of metal sheet of area
2
. Based on the above
information answer following.
i)If x represents the side of square base and y represents the
height of the toy box then write the relation between the variables x,y and c
ii) Express the volume of the toy box V as a function of x.
iii) Find the maximum volume of the box.
38.
The paths of two aeroplanes one taking off and
the other just landing on the runway are
respectively along the lines
(̂
(̂
̂)
̂
̂
̂)
(̂
( ̂
̂ ) and
̂
̂
̂)
i) Find the shortest distance between these lines.
ii) Are the two paths intersecting?
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