1. Xtra Edge February 2012 - Career Point

9. Point P (x, 1/2) under the given condition are length
PB = OB
O
(t – 1)
P
A
θ
C
B (t, 1)
rθ = t ; so θ = t
PB θ
from ∆PAB : = PA sin
2 2
t
⇒ PB = 2 sin ........(1)
2
θ t
Now ∠ PBC = = ;
2 2
θ t
so from ∠ PCB ; =
2 2
1/
2 t
so from ∆ PCB ; = sin
PB 2
........(2)
from (1) & (2) PB = 1 ; so θ = t = π/3
thus | PB | 2 = (t − x) 2 1
+ = 1.
4
3 3
| t − x | = ; t − x = ; as t > x
2
2
π 3
so x = −
3 2
10. Let xn = n −1
+ n + 1 be rational, then
1
=
xn
1
n −1 +
is also rational
n + 1
1
=
xn
n + 1 −
2
n −1
is also rational
n + 1 − n −1
is also rational
as n + 1 + n −1
& n + 1 − n −1
are rational
so n + 1 + n −1
must be rational
i.e. (n + 1) & (n – 1) are perfect squares.
This is not possible as any two perfect squares differe
at least by 3. Hence there is not positive integer n for
which n −1
+ n + 1 is a rational.
Regents Physics
Modern Physics :
You Should Know
• The particle behavior of light is proven by the
photoelectric effect.
• A photon is a particle of light {wave packet}.
• Large objects have very short wavelengths when
moving and thus can not be observed behaving
as a wave. (DeBroglie Waves)
• All electromagnetic waves originate from
accelerating charged particles.
• The frequency of a light wave determines its
energy (E = hf).
• The lowest energy state of a atom is called the
ground state.
• Increasing light frequency increases the kinetic
energy of the emitted photo-electrons.
• As the threshold frequency increase for a photocell
(photo emissive material) the work function
also increases.
• Increasing light intensity increases the number of
emitted photo-electrons but not their KE.
Mechanics :
• Centripetal force and centripetal acceleration
vectors are toward the center of the circle- while
the velocity vector is tangent to the circle.
• An unbalanced force (object not in equilibrium)
must produce acceleration.
• The slope of the distance-tine graph is velocity.
• The equilibrant force is equal in magnitude but
opposite in direction to the resultant vector.
• Momentum is conserved in all collision systems.
• Magnitude is a term use to state how large a
vector quantity is.
XtraEdge for IIT-JEE 39 FEBRUARY 2012