This is a Pre Board Question paper of Kendriya Vidyalaya,Kolkata Region

1. BLUE PRINT -2nd pre board- 2013
XII mathematics
VSA
SL. NO
TOPIC
(1
Mark)
(a)
SA
LA
(4 Mark)
(6 Mark)
Relations & Functions
1(1)
4(1)
-
Inverse Trigonometric Functions
1(1)
4(1)
-
Matrices
2(2)
-
6(1)
Determinants
1(1)
4(1)
-
Continuity & differentiability
8(2)
-
Applications of Derivatives
4(1)
6(1)
4(1)
TOTAL
6(1)
10(4)
(b)
(a)
13(5)
(b)
(a)
(b)
Integrals
2(2)
Applications of Integrals
(c)
6(1)
Differential Equations
44(11)
4(2)
(d)
(a)
Vectors
2(2)
4(1)
-
Three Dimensional Geometry
1(1)
4(1)
6(1)
Linear Programming
-
-
6(1)
Probability
-
4(1)
6(1)
TOTAL
10(10)
48(12)
42(7)
17(6)
(b)
CBSE –Annexure –F 2013-14
Pratima Nayak,KV,Fort William
6(1)
10(2)
100(29)

2. MATHEMATICS (041)
CLASS XII
Time allowed : 3hours
Max Marks: 100
General Instructions
1. All questions are compulsory.
2. The question paper consist of 29 questions divided into three sections A, B and C.
Section A comprises of 10 questions of one mark each, section B comprises of 12
questions of four marks each and section C comprises of 07 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 and 02 questions of six marks each. You have to
attempt only one of the alternatives in all such questions.
5. Use of calculators in not permitted. You may ask for logarithmic tables, if required.
Section A
Q1.
Let A  N  N and let * be the binary operation on A is defined by
a*b=
Find 3 * 2 .
5 

Q2. Find the principal value of cos 1  cos
.
3 

Q3. For what value of k, the matrix
is skew symmetric?
Q4. A is a non- singular square matrix of order 3 and
Q5. Find x and y, if
Pratima Nayak,KV,Fort William
+
=
A  4 .Find adjA .

3. Q6. If
are two unit vectors inclined to x–axis at angles 300 and 1200
and
respectively,
write the value of
|
sec 2 (log x )
dx
x
Q7. Evaluate:
.

Q8. Find the projection of
-
+
on
-2 +
.
Q9. Evaluate
Q.10 Find the value of p for which the following two lines are perpendicular to each
other.
x3 y 5 z 7
x 1  y 1 z 1


and


1
2
1
7
p
1
Section B
given by f(x) = x2 + 4.Show that f is invertible with the
Q11. Consider f : R +
inverse f-1 of given by
, where R + is the set of all non negative real
numbers.
Q12. Prove that
 x 1 
1  x  1 
tan 1 
 + tan 
=
 x 2
 x  2
OR
+
=
Q13. Using the properties of determinants, prove that –
1 x
1 y
1 z
x3
y 3  ( x  y )( y  z )( z  x)( x  y  z )
z3
Q14. Show that the function f (x) = |x + 2| is continuous at every x
differentiable at x = –2.
Q15. If
, find
OR
Pratima Nayak,KV,Fort William
R but fails to be

4. If x sin ( a + y ) + sin a cos ( a + y) = 0 prove that
=
Q16. Using differentials, find the approximate value of
Q17. . Evaluate:
dx
n
 1)
 x( x
OR
x 1
Evaluate:
Q18.
 ( x  3)
3
e x dx
Using vectors, fine the area of a triangle ABC whose vertices are
A (1, 1, 2), B (2, 3, 5) and C(1, 5, 5)
OR
 
 


If a , b and c are unit vectors such that a is perpendicular to the plane of b , c and
 
  

the angle between b , c is
then find a  b  c
3
Q19.
Solve the differential equation
x
dy
 y
 y  x tan 
dx
 x
Q20. Solve the following differential equation
OR
Form the differential equation of the family of circles of radii 3.
Q21. Find the shortest distance between the lines
 ˆ
ˆ j ˆ
r  i  ˆ   (2i  ˆ  k ) and
j

ˆ
r  2ˆ  ˆ - k   (3ˆ - 5ˆ  2k)
i j ˆ
i j
Pratima Nayak,KV,Fort William

5. Q22. A target is displayed as ‘’ Be truthful ‘’ the probability of A’s hitting a target is
4/5 and that of B’s hitting is 2/3.They both fire the target .Find the probability
at least one of them will hit the target
Only one of them will hit the target .
Which value is emphasized in the question?
OR
Assume that each child born is equally likely to be a boy or a girl. If a family has
two children, what is the conditional probability that both are girls given that
the youngest is a girl ?
At least one is a girl ?
Pre-natal sex determination is a crime. What will you do if you come to know that
some of our known is indulging in pre-natal sex determination?
SECTION -C
Q23.
Two schools A and B want to award prizes their students for the values of honesty
(X) , punctuality ( Y ) and obedience( Z ) .The sum of all the awardees is 12.Three
times of the sum of awardees for obedience and punctuality added to two times of
the number of awardees for honesty is 33.The sum of the number of awardees for
honesty and obedience is twice the number of awardees for punctuality, using
matrix method, find the number of awardees for each category. Apart from these
values suggest one more other value which could be considered for award?
Q24
A window in the form of a rectangle is surmounted by a semi circular
opening. The total perimeter of the window is 30 m. find the dimensions of the
rectangle part of the window to admit maximum light through the whole opening.
OR
Show that the volume of greatest cylinder that can be inscribed in a cone of
height h and semi vertical angle α is,
Pratima Nayak,KV,Fort William
4 3
h tan 2  .
27

6. Q25. Using properties of definite integrals, evaluate:
Q26. Find the equation of plane passing through the line of intersection of the
planes
x + 2y + 3z = 4 and 2x + y – z + 5 =0 and perpendicular to the plane
5x + 3y - 6z + 8 = 0.
Q27.
Find the area bounded by the curves
OR
Find the area of the region bounded by the two parabolas y = x2 and y2 = x.
Q28.
In a group of 400 people,160 are smokers and non vegetarians,100 are
smokers and vegetarians and remaining are non smokers and vegetarian. The
probability of getting a special chest pain disease are 35% , 20% and 10%
respectively. A person is chosen from the group at random and found to be suffering
from the disease. What is the probability is that the selected person is smoker and
no vegetarian? What value is reflected in the question?
Q29.
A dietician wishes to mix two types of foods in such a way that vitamin
contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C.
Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘II’ contains
1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase
Food ‘I’ and Rs 70 per kg to purchase Food ‘II’.
Formulate this problem as a linear programming problem to minimize the cost of
such a mixture. In what way a balanced and healthy diet is helpful in performing your
day-to-day activities?
Pratima Nayak,KV,Fort William

7. Answer ans Marking Scheme
Q1. 4 / 3
Q2. .
Q3. K =
.Q4 16
(sin 1 x) 2
C
Q6. 2 Q7. Tan ( log x) Q8. 7 Q9.
2
Q5. . x = 3, y = 3
Q10. P = - 4
Q11. For one- one
For onto
1
Use the formula for tan-1x + tan-1y
1
Correct solution
12
3
OR
1
1
tan-1x + tan-1y
13. Applying R1
R1 – R2 , R2
and result
R2 – R3
2
1
For taking common
1
For expansion along C1
1
For getting required result
1
Q14. L.H.L , R.H.L ,
L.H.L =, R.H.L ,
Continuous
L.H.D, RHD
½ + ½ +½
½
LHD ≠ RHD
Not differentiable
Q15 u = xy v = yx
½ + ½ +½
½
=0
Putting the values and simplifying
1
1+1
Answer
1
OR
x = -[sin a cos ( a + y) ]/ sin ( a + y )
1
dx/dy = sin a/ sin2(a+y)
2
=
Q16. Given
Pratima Nayak,KV,Fort William
1

8. 1
Ans.
1
=54+68(0.01)=54.68.
2
Ans.17
I=
dx
n
 1)
 x( x
Substitute xn = t  dx =
 I=
1
dt
n x n 1
1
1
dt
 t (t  1)
n
1
1
xn
c
= log n
n
x 1
2
OR
I=
x 1
 ( x  3)
3
e x dx = 
x 32 x
e dx
( x  3) 3
1

1
2  x

  ( x  3) 2 ( x  3) 3  e dx


= 

1
=  e  f ( x)  f ( x)  dx
x
= ex f(x) + c = ex
Q18.
1
1
+c
( x  3) 2
1
=1 +2 +3
=0 +4 +3
Area of triangle=
1
=-6 -3 +4
ans.
1+1
OR
  
a  b  c 1
  


a.b  0 ; a.c  0 , b .c  b c cos
3
  2      
a  b  c  (a  b  c ).(a  b  c )
Pratima Nayak,KV,Fort William
Or
cos
3
=
1
2
.
½ + ½ +½ +1/2

9. =1+1+1+2
1
2
.
=4
1+1
Q19.
dy y
y
=  tan( )
x
dx x
dy
dv
Put y = vx 
=v + x
dx
dx
dv
v+x
= v+tan v
dx
dx
 cot vdv   x
&
1
=cx
1+1
Q20.
dy
1+1
Or,
1+1
OR
( x - a )2 + ( y - b )2 = 9
1
Formation of equation
3
Q21.
ˆ ˆ k
i j ˆ
ˆ ˆ
 a 2  a1  i  k , b1  b2  2 - 1 1
3
-5 2
ˆ
ˆ j
 3i  ˆ  7k

b1  b 2  59
shortest distance =
Ans 22. P(A) =4/5
3
(b1  b2 ).(a 2  a 2 )
b1  b2
=
10
59
P(B) = 2/3
1
1
(i)
P(at least one) =1 – P(0)= 1- P( . .)
1
(ii)
P (only one) = P(A + . B)
1
(iii)
Truthfulness
1
OR
Ans22.
(i). A: Both are girls ={GG},
Pratima Nayak,KV,Fort William
B: Youngest is the girl  BG, GG
1

10. 1
A
p( A  B)
1
P( ) 
 4
2
B
p( B)
2
4
(ii). A: Both are girls
B: At least one of them is girl  BG, GG, GB
1
A
p( A  B)
1
P( ) 
 4
3
B
p ( B)
3
4
1
(iii) Every valuable answer given by the student
1
Q23.
x + y + z = 12, 3(y + z ) + 2x = 33 , x + z -2y =0
Matrix multiplication form
1
1
|A| =3
Cofactors
2
x = 3,y = 4 z = 5
1½
One appropriate value
½
Q24.Figure
1
30= 2x+2y+2y+
1
A=2x
1
=
1
=-(
Length =
1
m ,breath =
1
OR
Can be marked in similar way.
Q25. Use of property
,I=
2I =
Use of property
Pratima Nayak,KV,Fort William
1
1½
1

11. 2I =
1½
tanx = t sec2xdx =dt & Correct result I =
2
Ans26
The required plane is (x+2y + 3z ) + k (2x + y – z +5 )= 0
1
Or (1+2k)x +(2+k)y +(3-k)z-4+5k=0
1
5(1+2k) +3 (2+k) -6 (3-k)=0, i.e k= 7/19,
3
The equation of the plane is : 33x+45y +50z = 41
Q27.
1
Correct figure
1½
Point of intersection
Required area = 2( Area of shaded portion)
1+2
Finding integral and getting answer
sq.unit
2½
OR
Figure
1
Intersection points ( 0,0) and ( 1,1).
1
Area of the shaded region
=
dx =
dx= [ 2/3 x3/2 – x3 /3 ]
3
= 1/3 sq. unit.
Ans28.
P(E1) = 2/5
P ( E2) = ¼
1
p(E3) = 7/20
P(A/E1) = 35% P(A/E2) = 20% P(A/E3) = 10%
Formula for P( E1/A) and expression
Correct answer
1
1
1+1
1
Ans 29
Let the mixture contain x kg of Food ‘I’ and ‘y’ kg of Food ‘II’
Min Z  50 x  70 y
2x  y  8
x  2 y  10
x0 y0
Drawing the graph feasible region has no point in common.
x=2 & y=4
Pratima Nayak,KV,Fort William
½ +1/2 +1/2+1/2
2

12. MinZ=380
Marking may be done for all alternative correct answer.
Pratima Nayak,KV,Fort William
2