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GUESS PAPER – 2012
Subject – Mathematics
Class –xii
Time: 3 hours
M.M. 100
General Instructions:
(i) All questions are compulsory.
(ii) There are three sections in this paper.
SECTION-A: Question number 1 to 10 carry 1 mark each.
SECTION-B: Question number 11 to 22 carry 4 marks each.
SECTION-C: Question number 23 to 29 carry 6 marks each.
(iii) Use of calculators is not permitted.
SECTION – A






1.
If a  2i  2 j  3k , b  - i  2 j  k , find the cross product of the given vectors.
2.
Find values of x for which
3.
Find the value of tan-1 [ 2cos (2sin–1 ½)
4.
Write fog, if f : RR and g: RR are given by f(x) = 8x and g(x)=x
5.
3 x
=
x 1
3 2
0 1
]
3
Evaluate :

1/3
.
2
sec (log x)
dx .
x
1
7.
2x 

dx
2
1  x 
0
If y=log(sinx), find dy/dx.
8.
Find a - b if two vectors a and b
9.
Find the direction cosines of the vector a  2i  2 j  3k .
10.
Reduce the equation of the plane 3x+4y-z+7=0 in the normal form and hence find its
distance from origin.
6.
Evaluate
1 
 sin
are such that a  2 ; b  3 and a .b  2



SECTION-B
11.
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by
 x 1 
f(x)  
 . Show that f is bijective.
 x  2
12.
Using properties of determinants, prove that :
1 a2
ab
ac
ab
1 b
bc
ac
2

 1  a 2  b2  c 2
bc
1 c

2
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13.
 x  1
1  2x  1 
1  33 
Solve for x : tan1 
  tan 
  tan 
.
 x  1
 2x  1 
 36 
14.
For what value of k, is the following function continuous at x=2?
2 x  1; x  2

f ( x)   k ; x  2
3x  1; x  2

15.
If y 
log x
d 2 y 2 log x  3
, Show that

x
dx 2
x3
OR
If y = (sin-1x)2 , show that (1  x 2 )
16.
d2 y
dy
x
2  0
2
dx
dx
Find the intervals in which the function f(x) = sin 3x; 0  x 

2
(i) is increasing (ii) is decreasing
17.

1
 loglog x   log x 
Evaluate :
2

 dx

OR
3
Evaluate :
 x
2

 5 x  1 dx as a limit of a sum
1
18.
 dy 
Solve the following differential equation : sin1 
xy
 dx 
19.
Find the particular solution of the differential equation 1  x 2

y(0) = 2.
20.
dy
 dx
 xy  x
2
, given that
OR
Form the differential equation corresponding to y2 =m(a2-x2) y eliminating m and a.
If with reference to the right handed system of mutually perpendicular unit
 
 

 

vectors i , j and k ,   3 i  j,   2 i  j  3k then express  in the form of  =
1 +  2 , where 1 is parallel to  and  2 is perpendicular to  .
21. Find the equation of the plane that contains the point (1, – 1, 2) and is
perpendicular to each of the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8.
OR
Find the shortest distance between the lines:

  



 
  
r (2i  j  k )   (3i  5 j  2k ), r  (i  2 j  k )   (i  j  k ) .
22. A football match may be either won, drawn or lost by the host country’s team, So
there are three ways of forecasting the result of any one match, one correct and two
incorrect. Find the probability of forecasting at least three correct results for four
matches.
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SECTION-C
23.
1 3
2

Find A 1 , where A  4  1 0 . Hence, Solve the


 7 2 1
system of linear equations :
2x+y+3z=3, 4x-y= 3, and -7x+2y+z=2
1  1 2 
ORUsing elementary row transformations, find inverse of the matrix 0 2  3


3  2 4 
24.
The sum of the perimeter of a circle and a square is k, where k is a constant. Prove
that the sum of their areas is least when the side of a square is double the radius of
the circle.
OR
A helicopter is flying along a curve y=x2+2. A soldier is placed at the point (3,2). Find
the nearest distance between the soldier and the helicopter.
25.
Evaluate:
26.
Using integration, find the area of the region{ (x,y); x-1  y  5  x 2 }
OR
Sketch the region common to the circle x2 + y2 = 16 and the parabola x2 = 6y. Also find
the area of the region using integration.
27.
Find the equation of the plane through the point P(1, 2, 1) and perpendicular joining
the points(1,4,2) and (2,3,5). Also, find the perpendicular distance of the plane from
the origin.
28.
Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates. The
corresponding values for rice are 0.05 gm and 0.5 gm 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 50 gms and 200 gms 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 L.P.P. and
solve it graphically.

cot xdx
29. A card from a pack of 52 cards is lost. From the remaining cards of the pack, 2
cards are drawn at random and are found to both clubs. Find the probability of the
lost card being of clubs.
Paper Submitted By:
Name RAKESH KUMAR SINGH
Email [email protected]
Phone No.
8439312215
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