KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION MODEL EXAMINATION 2012-13 MATHEMATICS CLASS:XII

KENDRIYA VIDYALAYA SANGATHAN, ERNAKULAM REGION
MODEL EXAMINATION 2012-13
MATHEMATICS
CLASS:XII
Time: 3Hours
Max.Marks:100
General Instructions
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A,
B, C. Section A comprises of 10 questions of one mark each, section B
comprises of 12 questions of 4 marks each and section C comprises of o7
questions of six marks each.
3. All questions in Section A are to be answered in one word, one sentence or
as per the exact requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
04 questions of four marks each and02 questions of six marks each. You
have to attempt only one of the alternatives in all such questions.
SECTION A
1. Let E={1,2,3,4} and F={1,2}. Then find the number of onto functions from
E to F.
2. Find the numerical value of tan[2tan-1(1/5)-π/4]
3. For what value of k, the matrix
4. If A=
has no inverse?
then A+AT =I, Find the value of
.
5. Give an example of two nonzero 2x2 matrices A,B such that AB=O
6. Evaluate:
7. Evaluate:
where f(x) = x-[x]; [x] is the integral part of x.
8. Let
be vectors of magnitudes 3,4,5 respectively . Let
Then, find
9. The points with position vectors
are
collinear. Find the value of a
10.Find the equation of the line parallel to x axis and passing through the
origin
SECTION B
11.Let a relation R on the Set N of natural numbers be defined as (x,y)εR if
and only if x2-4xy+3y2=0 for all x,yεN. Verify that R is reflexive but not
symmetric and transitive.
12.
OR
Find x if
13.Show that
14.Find the values of a and b so that the function
f(x)=
is continuous for 0≤x≤π
15.If cos y =x cos(a+y), with cos a ≠±1, prove that
OR
If x=a(t+sint) and y=a(1+cost) . Find
16.Find the intervals in which xex(1-x) is strictly increasing or decreasing .
17.Evaluate
OR
Evaluate
18.Form the differential equation of the function (a+bx)
OR
Form the differential equation of the family of circles in the second
quadrant and touching the co-ordinate axes.
19.Solve the differential equation [x sin2(y/x)-y] dx+xdy = 0
20.For any two vectors
21.State whether the lines
intersect
or not. If intersecting find the point of intersection.
22.There are 5 cards numbered 1 to 5. One number on one card. Two cards
are drawn at random without replacement. Find the probability
distribution of the sum of the numbers on the two cards.
SECTION C
23.If A=
Find A-1 and hence solve the system of linear
equations
X+2y+z=4 ,-x+y+z=0, x-3y+z=2
OR
Using elementary transformations, find the inverse of the
matrix
24.A window of perimeter(including the base of the arc ) is in the form of a
rectangle surrounded by a semi circle. The semi-circular portion is fitted
with coloured glass while the rectangular part is fitted with clear glass.
The clear glass transmits three times as much lighrt per square metre as
the coloured glass does. Show that the ratio of the length and breadth
of the rectangle is 6:6+π, sothat the window transmits maximum light.
25.Sketch the region bounded by the curves y=
and y=
and
find its area
26.Evaluate
OR
Evaluate
27.Find the equation of the plane passing through the point(1,1,1) and
containing the line
)
Also, show that the plane contains
the line
28.Every gram of wheat provides 0.1 gm of proteins and 0.25gm of
carbohydrates. The corresponding values for rice are 0.05gm and 0.5gm
respectively. Wheat costs Rs.4 per kg and rice Rs.6 per kg. The minimum
daily requirements of proteins and carbohydrates for an average child
are 50gms and 200gms respectively. In what quantities should wheat
and rice be mixed in the daily diet to provide minimum daily
requirements of proteins and carbohydrates at minimum cost. Frame an
LPP and solve it graphically.
29.Bag A contains 3 red and 4 black balls and bag B contains 4 red and 5
black balls. One ball is transferred from bag A to bag B and then a ball is
drawn from bag B. The ball so drawn is found to be red in colour. Find
the probability that the transferred ball is black.
MODEL EXAMINATION 2012-13
MATHEMATICS
CLASS:XII
BLUE PRINT
S.No
1.(a)
(b)
2.(a)
(b)
3.(a)
(b)
(c)
(d)
(e)
4.(a)
(b)
5.
6.
Topics
VSA
(1 mark each)
Relations and Functions
1(1)
Inverse Trigonometric Functions 1(1)
Matrices
2(2)
Determinants
1(1)
Continuity and differentiability
Applications of derivatives
Integration
2(2)
Applications of integrals
Differential equations
Vectors
2(2)
3-Dimensional geometry
1(1)
Linear programming
Probability
Total
10(10)
SA
LA
(4 marks each) (6 marks each)
4(1)
4(1)
6(1)
4(1)
8(2)
4(1)
6(1)
4(1)
6(1)
6(1)
8(2)
4(1)
4(1)
6(1)
6(1)
4(1)
6(1)
48(12)
42(7)
Total
10(4)
13(5)
44(11)
17(6)
6(1)
10(2)
100(29)
Marking Scheme
Q.NO
1
2
3
4
5
Value points/Ans
14
-7/17
K=3/2
α=π/3
A=
6.
7
8
9
10
Marks
1
1
1
1
1
B=
log | x + log sin x | + c
1
5√2
a= -40
1
1
1
1
1
For xεN, x2-4xx+3x2=0
(x,x)εR
R is reflexive
(3,1)εR as 32+4.3.1+3.12=0
But 12-4.1.3+3.32≠0
So(1,3)ε R
R is not symmetric
(9,3)εR as 92-4.9.3+3.32 =0
(3,1)εR as 32+4.3.1+3.12=0
11
1
1
½
1
½
But (9,1) R as 92-4.9.1+3.12≠0
R is not transitive
12
1
1
1
1
(OR)
Sin-1(5/x)+sin-1(12/x)=π/2
12/x=sin(π/2-sin-15/x)=cos(sin-15/x)
Let sin-1x=t
5/x=sint
Cost=
1
1/2
1/2
t=cos-1
1/2
12/x=
1/2
144/x2=(x2-25)/x2
X=±13
But x=-13 does not satisfy the equation
X=13
1/2
1/2
13
14
15
Applying R1→aR1,R2→bR2,R3→cR3 and multiply whole determinant by 1/abc
Taking out common factors a,b,c from C1,C2,C3 respectively
Applying R1→R1+R2+R3
Taking out 2 from R1
Applying R2→R2-R1,R3→R3-R1
Applying R1→R1+R2+R3
On expanding along Ist row 2{-c2(-b2a2)+b2(c2a2)}=4a2b2c2
f(x) is cont. at x=π/4
LHL=RHL
π/4+a√2sin π/4=2x π/4cot π/4+b
a-b= π/4------------------------(1)
f(x) is cont.at x= π/2
2x π/2cot π/2+b=a(-1)b
a+2b=0-------------------------(2)
soving (1)&(2) getting a=3 π/2&b=-3 π/4
Getting
1/2
½
1
1
½
½
½
1
½
1
1
1
1
1
1
OR
Getting dx/dt=a(1+cost) &dy/dt=-asint
dy/dx+-asint/a(1+cost)=-tan(t/2)
1
1
1
1
16
f’(x)=ex(1-x)+x ex(1-x)(1-2x)
=-ex(1-x)(x-1)(2x+1)
F’(x)=0 gives x=1 or -1/2
In (-∞,-1/2) ,function is decreasing
(-1/2,1),function is increasing
(1, ∞),function is decreasing
1
1/2
1
½
½
½
17
Put 1+xex=t
I=
implies ex(x+1)dx=dt
Let
Getting A=1,B=-1,C=-1
I=
1/2
1/2
1
1
1
OR
2
I=
1
1
18
Correct differentiation w.r.t. x
1
1
Obtaining
Again differentiating w.r.t x
1
1
OR
Obtaining eqn. of circle as (x+a)2+(y-a)2=a2
Diiferentiating w.r.t x
1
1
½
½
Obtaining(x+y)2(y’2+1)=(x+yy’)2
1
19
Getting dy/dx=y/x - sin2(y/x)
½
Put y=vx then dy/dx=v+x(dv/dx)
Getting
1
1
½
½
-cosec2vdv=dx/x
Integrating cotv=logx+c
Cot(y/x)-logx=c
½
1
20
R.H.S=
2
Simplifying and put
1
21
½
½
½
½
1
½
Let
If they intersect, then for some and
3 -1= +2,---------(1)
5 -3=3 +4,-----------(2)
7 -5=5 +6-------------(3)
Solving (1)&(2) and getting =1/2, =-3/2
Which satisfies (3)
Hence lines intersect
Point of contact(1/2,-1/2,-3/2)
22
½
Getting random variable as 3,4,5,6,7,8,9
X
P(X)
23
3
1/10
4
1/10
A is invertible
Finding A-1=1/10
Writing
ATX=B WhereX=
5
1/5
½
6
1/5
7
1/5
8
1/10
9
1/10
For each
Correct
Probability
½
½
2
½
½
½
½
½
X=(AT)-1B
X=
1
OR
A=IA
1
1
½
½
½
½
1
1
24
1
a
a
2b
Perimeter P=2a+4b+πb--------------------(1)
Let the transmission rate of coloured glass be L and Q be total transmitted light
Q=2ab(3L)+1/2πb2(L)
1/2
1
Q=L/2(6Pb-24b2-5πb2) using (1)
1/2
dQ/db=L/2(6P-48b-10πb)
applying dQ/db=0 implies
1/2
1
1/2
so Q is maximum
now (48+10π)b=6P=6(2a+4b+πb)
2b:a=6:(6+π)
1
25
2
(-√5,0)
-1
2 (√5,0)
Solving the equations and obtaining x=2,-1
1
Required area=
1
Evaluation and simplification
Obtain area as 5π/4-1/2
26
1
I=
1
1
1
½
1
(1)+(2)
1
2I=
1
Put π/2cosx = t
so
sinx dx = -2/π dx
½
1
I=
Integrating and simplifying
I=
OR
2
1
2
=
27
1
=
Let A(1,1,1)
P(-3,1,5) pt on the line
1
11/2
11/2
1
1
28
Suppose xgms ofwheat and y gms of rice are mixed in the daily diet
Constraints 0.1x+0.05y≥50
0.25x+0.5y≥200
x/4+y/2≥200, x≥0,y≥0
Objective function
Minimize Z=
2
1/2
2
Feasible region is unbdd and has vertices A(0,1000) B(800,0) P(400,200)
Point
Z
A
(0,1000)
6
B
(800,0)
3.2
P
(400,200)
2.8
1
Min. Z=2.8
Wheat 400gms rice 200gms
½
29
E1:Red ball is transferred from A to B
E2:Black ball is tranferred from A to B
E:Red ball is drawn from B
P(E1)=3/7 P(E2)=4/7
P(E/E1)=5/10=1/2
P(E/E2)=4/10=2/5
P(E2/E)=
=16/31
1
3
1
1