KENDRIYA VIDYALAYA SANGATHAN : CHENNAI REGION
CLASS X I I COMMON PRE-BOARD EXAMINATION 2015 – 16
Subject : Mathematics
Time Allotted : 3 hours
Max. Marks : 100
General Instructions:
( i ) All questions are compulsory.
(ii) Please check that this question paper contains 26 questions.
(iii) Questions 1 – 6 in Section A are very short answer type questions carrying 1 mark each.
(iv) Questions 7 – 19 in Section B are long-answer I type questions carrying 4 marks each.
(v) Questions 20 – 26 in Section C are long-answer I I type questions carrying 6 marks each.
(vi) Please write down the serial number of the question before attempting it.
SECTION A
Question numbers 1 to 6 carry 1 mark each.
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1. Find | x | if for a unit vector a , ( x – a ) . ( x + a ) = 15.
1
2. Evaluate : 
0
dx
2x  3
.
3. Evaluate :  e x (tan x  1) sec x dx.
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4. Write the value of p for which a  3 i  2 j  9 k and b  i  p j  3 k are parallel vectors.
5. Write the vector equation of the line
x5 y 4 6 z


.
3
7
2
6. If A is a square matrix of order 3 such that adjA = 625 , find ǀ A ǀ .
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SECTION B
Question numbers 7 to 19 carry 4 marks each.
7. Using matrix method, solve the following equations:
2 x + 3 y + z = 11
–3x+2y +z =4
5x–4y–2z = -9
a
8. Using properties of determinants, prove that
ab
2a 3a  2b
abc
4a  3b  2c = a3
3a 6a  3b 10a  6b  3c
 1 3 5


9. Express the matrix A =   6 8 3  as sum of a symmetric and a skew symmetric matrix.
  4 6 5


10. Prove that
2 2
9
9 9 1  1 
.
sin 1 
 sin   =

8 4
4
3
 3


OR
Solve for x : tan-13x + tan-12x =

.
4
11. Solve the differential equation : x dy – y dx =
x 2  y 2 dx.
OR
Solve the differential equation : 1  x 2  y 2  x 2 y 2  xy
dy
= 0.
dx
 e 2 x
y  dx

12. Solve the differential equation : 
= 1. ( x ≠ 0 ).

x  dy
 x
13. If y =
sin 1 x
1 x2
, show that ( 1 – x2 )
d2y
dy
–3x
– y = 0.
2
dx
dx
tan x
14. Evaluate :
 sin x cos x dx
OR
Evaluate : 
5
dx
( x  2)( x 2  9)
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15. Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the
ground in such a way that the height of the cone is always one-sixth of the radius of the base.
How fast is the height of the sand cone increasing when the height is 4 cm ?
OR
Find the intervals in which the function f given by f ( x ) = x3 +
1
, x ≠ 0 is (i) increasing
x3
(ii) decreasing
 5 x  12 1  x 2
If y = sin 1 

13

16.

 , find dy .

dx

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17.
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The volume of the parallelopiped whose edges are  12 i   k , 3 j  k and 2 i  j  15 k
is 546 cubic units. Find the value of  .
18.
There are 40 hard working scholars in a class. Out of which 10 are sports-persons. Three
scholars are selected at random out of them. Write the probability distribution for selected
persons who are sports person. Explain the importance of sports in education.
19. Find the distance of the point ( – 1 , – 5 , – 10 ) from the point of intersection of the line
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  
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r   2 i  j  2 k     3 i  4 j  2 k  and the plane r . i  j  k  = 5.
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SECTION C
Question numbers 20 to 26 carry 6 marks each.
20.
Show that the lines
x  3 y 1 z  5
x 1 y  2 z  5




and
are co-planar. Also, find the
3
1
5
1
2
5
equation of the plane containing the lines.
OR
Find the image of the point ( 1 , 2, 3 ) in the plane x + 2 y + 4 z = 38.
21. An open box with a square base is to be made out of a given quantity of card board of area
C2
sq. units. Show that the maximum volume of the box is
C3
6 3
cubic units.
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22.
In a shop P, 30 tin pure ghee and 40 tin adulterated ghee are kept for sale while in shop Q,
50 tin pure ghee and 60 tin adulterated ghee are there. One tin of ghee is purchased from one
of the shop randomly and it is found to be adulterated. Find the probability that it is purchased
from shop Q.
23. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs.5760 to invest
and he has space for at most 20 items. A fan cost him Rs.360 and a sewing machine cost him
Rs.240. He expects to sell fan at a profit of Rs.22 and a sewing machine at a profit of Rs.18.
Assuming that he can sell all the items he buys, how much should he invest his money to maximize
his profit. Solve it graphically.
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24. Evaluate :
2
 2 log sin x  log sin 2 x dx
0
OR
3
Evaluate :  (3x 2  2 x) dx as limit of sums.
1
25. Using the method of integration, find the area of the triangle ABC, co-ordinates of whose
vertices are A ( 2 , 0 ) , B ( 4 , 5 ) and C ( 6 , 3 ).
26. Let A = N × N and * be the binary operation on A defined by ( a , b ) * ( c , d ) = ( a + c , b + d ).
Show that * is commutative and associative. Find the identity element for * on A if any.
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