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GUESS PAPER-2013
Class-XII
Subject : - MATHEMATICS
Section – A (10 x 1 = 10 marks)
 1
1   1 
  sin  
 2 
 2 
1.
Find the value of tan 1 1  cos 1 
2.
On the set Q+ of all positive rational numbers a binary operation * is defined by a  b 
ab
2
for all a,b Q+, Find the identity element.
1 3 2
3.
Write the cofactors of the elements of the third row of the determinant   4
3
4.
5.
6.
1 2
5
2
3 2 
 1 0
and 2X+Y = 


 3 2
1 4
2
 1 10 
If x    y      , find the value of x and y.
3 
1 5 
Find X, if Y = 
Evaluate  ex (tan x  sec 2 x)dx
 tan1 x  dx
3

7.
Evaluate
8.
In a triangle ABC, the sides AB and BC are represented by vectors

 



3i  2 j  k ; and 3i  4 j  3k then find the vector represented by AC.
9.
Find the intercepts made by the plane r  (iˆ  3 jˆ  kˆ )  5 on the co-ordinate axes.
(1  x2 )
10. Find the angle between the lines
x 1 y  2 z  3
x  2 y 1 z  2


=
=
and
.
2
1
1
2
2
3
Section – B (12 x 4 = 48 marks)
11. (a)Let * be a binary operation on the set Q of rational numbers as follows:
(i)a * b = a2 + b2, (ii) a * b = a+ab,. Which of these binary operation are commutative and which are
associative.
(OR)
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(b) Let * be a binary operation defined on NXN, by (a,b)*(c,d) = (ac, bd). Show that * is commutative
and associative. Also find the identity element for * on NxN.
 a cos x  b sin x 
 (OR)
 b cos x  a sin x 
12. Simplify tan1 
Solve the following equations simultaneously: sin1 x  sin1 y 
abc
c
b
c
bc a
a
b
a
c ab
13. Show that
2

and cos 1 x  cos 1 y 
3
3
 2(a  b)(b  c )(c  a)
1
14.
(a)Differentiate y  (xcos x)x  (x sin x) x w.r.t.x
(b) If y x2  1  log
15.
(OR)
 x2  1  x  , Show that (x2  1) dy  xy  1  0


dx
 1  sin3 x

;x

2
2
 3cos x

Find the values of a and b if
 is continuous at x 

f(x)   a
;x
4
2


 b(1  sin x)
 (  2x)2 ; x  2

16.
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line
x + 14y + 4 = 0
17.
Evaluate 
18.
Solve cosx(1+cosy)dxsiny(1+sinx)dy=0.
19.
x
20.
Prove that [ a  b b  c c  a ]  2[a b c ] (OR)
x2  x  1
dx
(x  2)(x 2  1)
dy
 y  x  xy cot x  0; x  0
dx
Let a  ˆi  ˆj, b  3ˆj  kˆ and c  7iˆ  kˆ find a vector d which perpendicular to a and b and also
given c d  1.
21.
Find the equation of the plane passing through the intersection of the planes
2x8y+4z = 3 and 3x5y+4z+10 = 0 and perpendicular to the plane 3xy2z4=0
22.
If a fair coin is tossed 10 times, find the probability of (i) exactly six heads (ii) at least six heads
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Section – C (7 x 6 = 42 marks)
23.
 3 4 
 then,
 1 1 
Prove the following by principle of mathematical induction, If A  
 1  2n 4n 
An  
 for every positive integer n
1  2n 
 n
24.
(a)An open tank with square base and vertical sides is to be constructed from metal sheet so as to
hold a given quantity of water. Show that the cost of material will be least when the depth of the tank
is half the width
(OR)
(b) Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cms
25.
Evaluate   cot x  tan x  dx
26.
Find the area lying above the x-axis and included between the circle x2+y2=16 and the parabola y2=6x
27.
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is
transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be
red in colour. Find the probability that the transferred ball is black
28.
There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and
F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds
that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs 6/kg
and F2 costs Rs 5/kg, determine how much of each type of fertiliser should be used so that nutrient
requirements are met at a minimum cost. What is the minimum cost?
29.
Find the distance of the point (2,3,4) from the line


x  2 2y  3 3z  4


measured parallel to
3
4
5
the plane 4x+12y3z+1 = 0.
WISH YOU ALL THE BEST
D. VASU RAJ, CHENNAI.
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