SAMPLE PAPER – IX
General Instructions: As in Sample Paper – I
SECTION A
1. If iˆ, ˆj and kˆ are unit vectors along X,Y and Z axis respectively, find the value of
ˆ ˆ)  kˆ.(ixj
ˆ ˆ)
iˆ.( ˆjxkˆ)  ˆj.(ixk
2. If a  5, b  13, axb  25. Find a.b
3. Find the value of λ so that the two vectors 2iˆ  3 ˆj  kˆ & 4iˆ  6 ˆj   kˆ are (i) parallel (ii)
perpendicular to each other.
4. Evaluate
(tan 1 x) 2
 (1  x 2 ) dx

5. Evaluate
2
dx
0
1  cot 2 x

3
6. A matrix A, of order 3x3, has determinant value k. Find the value of kA .
 1 2

  5 3
7. Find the adjoint of a matrix 
8. Find the number of binary operations on the set A = {a, b}


9. Using principal value , evaluate Cos 1  Cos
 cos 
 sin 
10. If A  
3 
3 
1 
  Sin  Sin

5 
5 

sin  
, find A’.
cos  
SECTION: B
 12 
1  4 
1  63 
  Cos    Tan    
 13 
5
 16 
11. Prove that Sin 1 
(OR)
x 2 2 xy
y2
 x
 y
If Cos 1    Cos 1     , Pr ove that 2 
Cos  2  Sin 2
ab
a
b
a
b
12. using properties of determinant , prove that
a2
bc
c 2  ac
a 2  ab
b2
ca  4a 2 b 2 c 2
ab
b 2  bc
c2
13. If f(x) , defined by the following, is continuous at x=4, find the values of a and b.
 x4
 x  4 a

f ( x)   a  b
 x4

b
 x  4
if x  4
if x  4
if x  4
 x2  y2 
dy
y
  tan 1 a , Pr ove that
14. 14. If Cos 1  2

2 
dx x
x

y


(OR)
If
x  a sin
1
t
1
, y  a Cos t , a  0, Pr ove that
dy  y

dx
x
15. Verify Rolle’s theorem for f(x) = Sin4x + Cos4x on [0,∏/2]
(OR)
Show that the curves 4x=y2 and 4xy = k cut at right angles if k2= 512.
16. Show that the relation R defined by (a,b) R (c,d) → a+d=b+c on the set NXN is an equivalence
relation
/2
17. Evaluate
 2 log sin x  log sin 2 xdx
0
(OR)
log( 1  x)
dx
2
1

x
0
1
Evaluate

dy
 (Cosx) y  Cosx.Sin 2 x
dx
19. Find the vector projection of a vector a  2iˆ  3 ˆj  2kˆ on the vector b  iˆ  2 ˆj  kˆ
18. Solve the following differential equation Sinx
20. Solve: (x2+y2)dx =2xydy given that y(1)=0
21. 15 cards, numbered 1 to 15, are placed in a box mixed up thoroughly and then a card is drawn at
random from the box. If it is known that the number on the card drawn is more than 3, find the
probability that it is an odd number.
22. Find the shortest distance between the following skew lines
r  (  1)iˆ  (  1) ˆj  (1   )kˆ
r  (1   )iˆ  (2   1) ˆj  (   2) kˆ
SECTION: C
23. Sketch the rough graph of two parabolas 4y2 =9x and 3x2=16y and find the area bounded by the
two curves
2
3 
 1


5
7 
24. Using elementary operation, find the inverse of the matrix  2
  2  4  5


(OR)
 2 1 1 


For the matrix A =   1 2  1 Verify that A3-6A2+9A-4I = 0 and
 1 1 2 


Hence find A-1 also.

25. Evaluate
Cos 2 x
0 Cos 2 x  4Sin 2 x dx
26. An Open tank with a square base and vertical sides is to be made from a metal sheet so as to
hold a given quantity of water. Show that the cost of the material will be least when depth of
the tank is
half of its width.
(OR)
Find the dimensions of the rectangle of perimeter 36cm which will sweep out a volume as large
as possible when revolve about one of its sides. Also find the maximum volume.
27. Find the Position Vector of a point A if its image in the plane
r .(2iˆ  ˆj  kˆ)  4 is (iˆ  2 ˆj  kˆ)
28. A company produces 2 types of steel trunks. It has machines A and B. For completing I type of
the trunk, it requires 3 hours on machine A and 1 hrs on machine B, where as II type of the trunk
requires 3 hrs on machine A and 2 hours on machine B. Machine A can work 18 hrs and
machine B for 8hrs only per day. There is a profit of Rs.30/- on I type of the trunk and Rs.48/- on
the II type of the trunk. How many trunks of each type should be produced every day to earn
maximum profit? Solve the problem graphically.
Three bags contain balls as shown in the following table.
Bag
I
II
III
White
1
2
4
Number of
Black
2
1
3
Red
3
1
2
A bag is chosen at random and two balls are drawn. They happen to be white and red. What is
the probability that they come either from the bag I or the bag II.