March 2011 - Career Point

MATHEMATICS
11. Show that the value of tanx/tan3x, wherever defined
never lies between 1/3 and 3.
[IIT-1992]
tan x tan x
Sol. y = =
3
tan 3x
3tan x – tan x
=
tan x(1– 3tan
3tan x – tan
1– 3tan x
=
2
3 – tan x
⇒ x ≠ 0
⇒ tan x ≠ 0
0 ∞
2
1– 3tan
2
3
x)
+ – +
1/3 3
Let tan x = t
2
1– 3t
⇒ y =
2
3 – t
⇒ 3y – t 2 y = 1 – 3t 2
⇒ 3y – 1 = t 2 y – 3t 2
⇒ 3y – 1 = t 2 (y – 3)
3y
–1
⇒ = t 2
y – 3
x
2
x
[Q tan 3x ≠ 0 ⇒ 3x ≠ 0]
3y
–1
⇒ ≥ 0, t 2 ≥ 0 ∀ t ∈ R
y – 3
⇒ y ∈ (– ∞, 1/3) ∪ (3, ∞)
Therefore, y is not defined in between (1/3, 3).
12. Find all maxima and minima of the function
y = x (x – 1) 2 , 0 ≤ x ≤ 2.
Also determine the area bounded by the curve
y = x(x – 1) 2 , the y-axis and the line y = 2 [IIT-1989]
y
Sol.
y = x(x–1) 2
4
27
max
O 1/2
1 min
x
+ – +
1/ 3 1
1 ⎛ 2 ⎞ 4
∴ maximum at x = 1/3 ⇒ y max = ⎜ – ⎟ =
3 ⎝ 3 ⎠ 27
minimum at x = 1 ⇒ y min. = 0
Now, to find the area bounded by the curve
y = x(x – 1) 2 , The y-axis and line y = 2
y =2 C
4
27
O
1
B
A
x=2
2
⇒ Req. Area = Area of square OABC –
∫
0
2
= 2 × 2 –
∫
( x – 2x
+ x)
dx
0
⎛
4 3 2
2
= 4 – –
4 3 2 ⎟ ⎞
⎜
x x
+ x
⎝
⎠
3
2
2
0
2
x ( x –1) 2 . dx
⎛ 16 ⎞ 16 10
= 4 – ⎜4 – + 2⎟ = –2 = sq. units
⎝ 3 ⎠ 3 3
13. Find the intervals of values of a for which the line
y + x = 0 bisects two chords drawn from a point
⎛ ⎞
⎜
1+
2a
1– 2a
, ⎟ to the circle
⎝ 2 2 ⎠
2x 2 + 2y 2 – (1 + 2 a)x – (1 – 2 a) y = 0
[IIT-1996]
Sol. 2x 2 + 2y 2 – (1 + 2 a)x – (1 – 2 a) y = 0
⇒
⎛ ⎞
x 2 + y 2 – ⎜
1+
2a ⎛
⎟ x – ⎝ 2
⎟ ⎞
⎜
1–
2a
y = 0
⎠ ⎝ 2 ⎠
Since, y + x = 0 bisects two chords of this circle,
mid-points of the chords must be of the form
(α, – α).
y
⇒
y = x(x – 1) 2
dy = x.2(x – 1) + (x –1)
2
dx
= (x – 1) . {2x + x –1}
= (x – 1) (3x – 1)
y + x = 0
O
(α,–α)
(α,–α)
x
⎛ ⎞
⎜
1+
2a
1 – 2a
, ⎟
⎝ 2 2 ⎠
XtraEdge for IIT-JEE 15 MARCH 2011