1. Xtra Edge February 2012 - Career Point

= 2π ∫ π
2 / 3
2 t
sec dt
2
t
1+
3tan
2
π/
3
2
3 2du
= 2π∫ =
1/
3
2
1+
3u
3
4π . { } 3
−1
3
3 tan u
4π –1 –1 4π –1
= (tan 3 – tan 1) = tan
3
3
3
⎛ 1 ⎞
⎜ ⎟
⎝ 2 ⎠
π/
3 π + 4x
4π –1 ⎛ 1 ⎞
∴ ∫ dx = tan ⎜ ⎟ .
−π
/ 3 ⎛ π ⎞
2 − cos⎜|
x | +
3 ⎝ 2 ⎠
⎟
⎝ 3 ⎠
14. An unbiased die, with faces numbered 1, 2, 3, 4, 5,
6, is thrown n times and the list on n numbers
showing up is noted. What is the probability that
among the numbers 1, 2, 3, 4, 5, 6 only three
numbers appear in this list ? [IIT-2001]
Sol. Let us define at onto function F from A : [r1, r2 ...
rn] to B : [1, 2, 3] where r1r2 .... rn are the readings
of n throws and 1, 2, 3 are the numbers that appear
in the n throws.
Number of such functions,
M = N – [n(1) – n(2) + n(3)]
where N = total number of functions and
n(t) = number of function having exactly t elements
in the range.
Now, N = 3 n , n(1) = 3.2 n , n(2) = 3, n(3) = 0
⇒ M = 3 n – 3.2 n + 3
Hence the total number of favourable cases
= (3 n – 3.2 n + 3). 6 C3
⇒ Required probability =
Nuclear Physics
1/
n n 6
( 3 − 3.
2 + 3)
× C3
n
6
3
Physics Facts
15. Show that the value of tanx/tan3x, wherever defined
never lies between 1/3 and 3. [IIT-1992]
tan x
Sol. y = =
tan 3x
x – tan
XtraEdge for IIT-JEE 13 FEBRUARY 2012
=
=
3tan
tan x(
1–
3tan
3tan
1–
3
x – tan
tan
2
2
x
3 – tan x
⇒ x ≠ 0
⇒ tan x ≠ 0
0 ∞
tan x
1–
3tan
2
3
x)
+ – +
1/3 3
Let tan x = t
⇒ y =
3 – t
⇒ 3y – t 2 y = 1 – 3t 2
⇒ 3y – 1 = t 2 y – 3t 2
2
1–
3t
⇒ 3y – 1 = t 2 (y – 3)
⇒
2
3y
– 1
= t
y – 3
2
x
2
3
x
x
[Q tan 3x ≠ 0 ⇒ 3x ≠ 0]
3y
– 1
⇒ ≥ 0, t
y – 3
2 ≥ 0 ∀ t ∈ R
⇒ y ∈ (– ∞, 1/3) ∪ (3, ∞)
Therefore, y is not defined in between (1/3, 3).
1. Alpha particles are the same as helium nuclei and have the symbol .
1. The atomic number is equal to the number of protons (2 for alpha)
2. Deuterium ( ) is an isotope of hydrogen ( )
3. The number of nucleons is equal to protons + neutrons (4 for alpha)
4. Only charged particles can be accelerated in a particle accelerator such as a cyclotron or Van Der Graaf
generator.
5. Natural radiation is alpha ( ), beta ( ) and gamma (high energy x-rays)
6. A loss of a beta particle results in an increase in atomic number.
7. All nuclei weigh less than their parts. This mass defect is converted into binding energy. (E=mc 2 )
8. Isotopes have different neutron numbers and atomic masses but the same number of protons (atomic
numbers).
9. Geiger counters, photographic plates, cloud and bubble chambers are all used to detect or observe radiation.