1. Xtra Edge February 2012 - Career Point

I1 I2
Since I1 is an odd function therefore I1 = 0
π/
4
I = π/
4 ∫
– π/
4
π/
4
= π/
4 ∫
2
1
– π/
4
π/
4
dx
⎛
⎜
1–
tan
2 –
⎜
⎝1
+ tan
+ (
sec
2
x
2
2
x ⎞
⎟
x ⎟
⎠
2
3 tan x)
sec x
= 2 π/
4 ∫
[Q f(x) is even function]
2
2
1 + ( 3 tan x)
0
2
Let 3 tan x = t ⇒ sec 2 dt
x dx =
3
dt / 3
= ∫ 2 2 1+
t
π 3
0
= ( ) 3 – 1
/ 2 3 tan t
π =
26. The given lines are
→
r = ( kˆ î – 2jˆ
+ 3 ) + t ( kˆ – î + jˆ – 2 )
and →
r = ( kˆ î – jˆ – ) + s ( kˆ î + 2jˆ
– 2 )
Here →
1
→
2 –
0
2
π
a = kˆ î – 2jˆ
+ 3 ; →
b 1 = kˆ – î + jˆ – 2
a = kˆ î jˆ – ;
→
b = kˆ î + 2 jˆ – 2
b × →
b =
∴ →
1
2
î
– 1
1
jˆ
1
2
k ˆ
– 2
– 2
The S.D. between the gives lines
=
=
→
→
( b1×
b2
).( a1
– a 2 )
| b × b |
2
2
1
4 + 12
+ 4
2
→
2
2
+ 3
→
=
8
=
29
2
6
3
= kˆ 2 î – 4 jˆ – 3
k | ˆ
k) | 2î
– 4jˆ
– 3
ˆ k).( jˆ – 4 ˆ ( 2î
– 4 jˆ – 3
27. Part I st
Let x kg and y kg. of fertilizer A and B be mixed by
the farmer. Then LPP is
Minimize : C = 5x + 8y
subject to the constraints
10 5
x + y ≥ 7 ⇔ 2x + y ≥ 140
100 100
6 10
x + y ≥ 7 ⇔ 3x + 5y ≥ 350
100 100
x ≥ 0, y ≥ 0
y
(1)
(0, 140)B
D
XtraEdge for IIT-JEE 100 FEBRUARY 2012
O
(50, 40)
P
A C
⎡350
⎤
⎢ , 0⎥
⎣ 3 ⎦
(2)
Draw the lines
2x + y = 140 ..... (1)
3x + 5y = 350 ..... (2)
The shaded unbounded region is the feasible region.
These line meet at P(50, 40)
Now value of c = 5x + 8y
⎡350
⎤ 1750 1
at C ⎢ , 0⎥
, is = 583
⎣ 3 ⎦ 3 3
at P(50, 40) is 570
at B(0, 140) is 1120
∴ The cost is minimum at P(50, 40)
This occurs when 50 kg of type & fertilizer and 40
kg of type B fertilizer are mixed to the meet the
requirement.
Part 2 nd
Let x and y represent the number of tables and chairs
respectively that the dealer sells and P be the profit
of the dealer. Then the required LPP is .
Maximize: P = 50x + 15y
Subject to the constraints
500x + 200 y ≤ 10,000 ⇔ 5x + 2y ≤ 100
x + y = 60 ....(1)
5x + 2y = 100 .... (2)
y
D
(0, 50)
B(0, 60)
A(60, 0)
x
O C (20, 0)
(1)
(2)
A shaded region OCD is the feasible region.
Now P = 50x + 15y
At O, value of P is O
At C, value of P is 50 × 20 = 1,000
At D, value of P is 15 × 50 = 750
∴ Profit is maximum at C (20, 0)
x