1. Xtra Edge February 2012 - Career Point

1. If
MATHS
4
sin α
8
a
+
4
cos α
8
b
=
1
a + b
sin α cos α 1
+ =
3
3
a b ( a + b)
3
, show that
a + b 4 a + b 4
Sol. Here sin α + cos α = 1
a
b
or sin 4 α + cos 4 b 4 a 4
α + sin α + cos α = 1
a b
or (sin 2 α + cos 2 α) 2 – 2 sin 2 α . cos 2 α
b 4 a 4
+ sin α + cos α = 1
a b
or
or
∴
⎛
⎜
⎝
⎛
⎜
⎝
b
a
b
a
2
sin
sin
2
⎞
α⎟
⎟
⎠
2
α –
.
b 2 a 2
– 2 . sin α . cos α
a b
a
b
cos
b 2 a 2
sin α = cos α
a b
2
⎞
α⎟
⎟
⎠
2
⎛
+ ⎜
⎝
= 0
or sin 2 a 2
α = cos α
b
2
2
2
sin α cos α sin α + cos
∴ = =
a b a + b
∴ sin 2 a
α =
a + b
, cos2 b
α =
a + b
8
sin α cos α
∴ + =
3
3
a b
=
a
( a + b
4
)
+
8
b
( a + b
1
3
a .
4
)
a
4
( a + b)
=
4
a + b
( a + b
2
4
)
+
a
b
α
2
cos
1
3
b .
=
2
⎞
α⎟
= 0
⎟
⎠
b
4
( a + b)
1
( a + b)
2. Let [x] stands for the greatest integer function find
2
3 x + sin x
the derivative of f(x) = ( x + [ x + 1])
, where it
exists in (1, 1.5). Indicate the point(s) where it does
not exist. Give reason(s) for your conclusion.
Students' Forum
Expert’s Solution for Question asked by IIT-JEE Aspirants
3
4
Sol. The greatest integer [x 3 + 1] takes jump from 2 to 3 at
3 3
2 and again from 3 to 4 at 3 in [1, 1.5] and
therefore it is discontinuous at these two points. As a
result the given function is discontinuous at 3 2 and
hence not differentiable.
To find the derivative at other points we write :
in (1, 3 2 ), f(x) = ( x + 2)
⇒ f ´(x) = ( x + 2)
2
x + sin x−1
2
x + sin x
{x 2 + sin x + (x + 2) (2x + cos x) log (x + 2)}
2
x + sin x
in ( 3 2, 3 3 ), f(x) = ( x + 3)
,
f ´(x) = (
2
x + sin x−1
x + 3)
{x 2 + sin x
+ (2x + cos x) (x + 3) × loge (x + 3)}
2
x + sin x
in ( 3 5 , 1.5), f(x) = ( x + 4)
,
2
x + sin x−1
x , {x 2 + sin x + (2x + cos x)
f ´(x) = ( + 4)
(x + 4) × loge(x + 4)}
3. The decimal parts of the logarithms of two numbers
taken at random are found to six places of decimal.
What is the chance that the second can be subtracted
from the first without "borrowing"?
Sol. For each column of the two numbers,
n(S) = number of ways to fill the two places by the
digits 0, 1, 2, ... , 9
= 10 × 10 = 100.
x
× × × × × ×
y
× × × × × ×
Let E be the event of subtracting in a column without
borrowing. If the pair of digits be (x, y) in the column
where x is in the first number and y is in the second
number then
E = {(0, 0), (1, 0), (2, 0), .. ,(9, 0),
(1, 1), (2, 1), ..., (9, 1),
(2, 2), (3, 2), ..., (9, 2),
(3, 3), (4, 3), ..., (9, 3),
......
(8, 8), (9, 8),
(9, 9)}
XtraEdge for IIT-JEE 40 FEBRUARY 2012