1. A. Nagaraj’S II P.U.C.MATHEMATICS- ANNUAL EXAM-2014
Trin : 9845222682
Max. Marks:100
PART-A
ANSWER ALL THE QUESTIONS
10X1=10
1. Give example of relation which is symmetric but neither reflexive nor transitive.

 1 
 
 1


2. Evaluate sin cos −1  −  − sin −1   
2
2

cos α
 sin α
3. If A = 
− sin α 
, find the value of α , if A is Identity matrix .
cos α 

101 102 103
4. Evaluate 104 105 106
107 108 109
without using direct expansion .
5. Differentiate sin [ cos(tan x) ] w.r.t.x
6. Evaluate
x2
∫ 1 + x 2 dx
(


) (


7. Evaluate 3a − 5b • 2a + 7b
)
8. If a line has direction ratios 2 , -1 , -2 , determine its direction cosines .
9. Define “optimal solution “to a linear programming problem.
10. If P ( A) =
1
2
and P ( B ) = 0
then find P ( A | B ) .
PART-B
Answer any Ten question
10x2=20
ab
then find the identity element .
4
11. If ∀a, b ∈ Q and * is defined by a ∗ b =


12. Find the value of sin −1  sin
3π 

5 
π
13. Solve ta n −1 2 x + ta n −1 3 x =
4
14. If each element of a row ( or a column ) of a determinant is multiplied by a constant K , then show that
Its value gets multiplied by K.

2. A. Nagaraj’S Trin : 9845222682
 x + 3, if x ≤ 2
2 x − 3, if x > 2
15. Find all points of discontinuity of f ,where f is defined by f ( x ) = 
3
16. Verify Mean Value Theorem for the function f ( x) = x 2 in the interval [ 2, 4]
17. Find the equation of the tangent to the curve y
=
18. Evaluate
3 x − 2 which is  to line 4 x − 2 y + 5 =
0
∫ xSinxCosx dx
e
log x 2
19. Evaluate ∫
dx
x
1
2
20. Find the order and degree of D . E
d2 y
dy
 dy 
xy 2 + x   − y
=
0
dx
dx
 dx 
(
)
ˆ
ˆ
21. If the position vectors of the points A and B respectively are i + 2 ˆ − 3k and
j
( ˆj − kˆ )
find the direction
cosines of AB.
ˆ
ˆ j
22. Find a vector of magnitude 8 units in the direction of the vector, 5i − ˆ + 2k

ˆ 9
ˆ
23. Find the distance of the point ( 2, 3, -5 ) from the plane r.(i + 2 ˆ − 2k ) =
j
24. A Die marked 1, 2, 3 in Red and 4,5,6 in Green is tossed. Let A be the event “ the number is even “
and B be the event “ the number is Red “. Are “ A and B “ independent .
Answer any Ten questions.
10x3=30
25. On Z * is detined by a* b = a-b , determine whether * is Commutative or Associative
26. Show that ta n −1
1
1
1
1 π
+ ta n −1 + ta n −1 + ta n −1 =
5
7
3
8 4
1 3 

2 7
27. Using elementary transformation, find inverse of 
28. If e x + e y = y ,then show that
e x+
dy
= −e y − x .
dx
dy
 1 
29. If y
= s ec −1  2
 , 0 < x < 1/ 2 then find
dx
 2x − 1 
30. Use differential to find approximate value of
31. Evaluate
∫ tan
4
x dx
36.6

3. A. Nagaraj’S Trin : 9845222682
3
32. Evaluate
x
∫ (1 + x )dx
2
2
33. Find the Area of the region bounded by the curve y 2 = x and the lines= 1, x 4 and the x-axis.
x =
34. From the Differential equation of the family of circles having centre on Y- axis and radius 3 units.
(
) (
)
(
)
ˆ
ˆ
ˆ
ˆ
ˆ ˆ
35. Show that the points A −2i + 3 ˆ + 5k , B i + 2 ˆ + 3k , &C 7i − k are collinear.
j
j






36. If= 3, b 4, c 5 and each one of them being ⊥ to the other two , find a + b + c
a
= =
37. Find the vector and the Cartesian equations of the line through the point (5,2,-4) and which is parallel to
ˆ
ˆ
the vector 3i + 2 ˆ − 8k
j
38. From a lot of 30 bulbs which includes 6 defectives, a sample of 4 bulbs is drawn at random with
Replacement. Find the probability distribution of the number of defective bulbs.
6×5 =
30
PART - D
Answer any SIX questions
39. Let f : R → R be defined by f ( x) 3 x − 7 . show that f is invertible. Find f −1 : R → R
=
40. If A
1 2 −3
=

5 0 2  , B
1 −1 1 


 3 −1 2 
4 2 5 & C
=

 2 −0 3 


4 1 2
 0 3 2  verify that A + ( B − C ) = ( A + B) − C .


1 −2 3 


41.Solve by using Matrix method: 3 x − 2 y + 3= 8, 2 x + y − = 1& 4 x − 3 y + 2= 4
z
z
z
42.If y ae mx + be nx ,then Show that y2 − (m + n) y1 + mny =
0
=
43. A Man of height 2mt walks at a uniform speed 5km/h away from a Lamp post which is 6mt high.Find the
rate at which the length of his shadow increases.
44. Prove that
∫
1
x
dx
= sin −1   + C and hence evaluate
a
a2 − x2
∫
sec 2 x
4 − tan 2 x
dx .
45. Find the Area of the circle x 2 + y 2 =by integration method.
a2
46. Derive the equation of a plane in Normal form (both in vector and Cartesian form).
47. Solve the differential equation, x
dy
+ 2y = e .
x 2 log x
dx
48. If a Fair coin is tossed 10 times .Find the Probability of a) exactly 6 heads
most 6 heads.
b) At least 6 heads c) At

4. A. Nagaraj’S Trin : 9845222682
Part- E
Answer any One question
1x10=10
49. (a) An Aero plane can carry a maximum of 200 passengers, A Profit of Rs.1000 is made on each executive class
ticket and a profit of Rs.600 is made on each economy class ticket.The Airline reserves at least 20 seats for executive
class. However at least 4 times as many passengers prefer to travel by economy class than by the executive class.
Determine how many tickets of each type must be sold in order to maximize the Profit for the airline .What is the
Maximum Profit?
 x + 2, if x < 1

(b) Find all the points of discontinuity of the function f defined by f ( x ) = 0,
if x = 1
 x − 2, if x > 1

a
50.
(a) Prove that
f ( x)
∫=
0
π
a
∫
f (a − x) dx ,
xSinx
∫ (1 + Cos x )dx .
hence evaluate
2
0
0
a 2 + 1 ab
ac
2
(b) Prove that ab
b + 1 bc = + a 2 + b 2 + c 2 `
1
2
ca
bc
1+ c
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10-March-2014
&
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Shree susheela tutorials,
Trin:9845222682
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