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51.

If the vectors a, b and c are coplanar, then `|{:(a, b, c),(a*a, a*b,a*c),(b*a,b*b,b*c):}|` is equal toA. 1B. 0C. `-1`D. None of these 3

Answer» Correct Answer - B
Since, a, b and c are coplanar, there must exists three scalars c, y and z are not all zero such that
`xa+yb+zc=0" "…(i)`
Multiplaying both sides of Eq. (i) by a and b respectivelt, we get
`xa*aya*b+za*x=0" "...(ii)`
`ab(a+yb*b+zb*x=0" "...(ii)`
Eliminating x, y and z from Eqs. (i) ,(ii) and (iii), we get
`|{:(a,b,c),(a*a,a*b,a*c),(b*a,b*b,b*c):}|=0`
Discussion
52.

A vector v is equally inclined to the X-axis, Y-axis and Z-axis respectively the dirction cosines areA. `lt (1)/(sqrt3),(1)/(sqrt3),(1)/(sqrt3)gt`B. `lt -(1)/(sqrt3), -(1)/(sqrt3), -(1)/(sqrt3)gt`C. `lt (1)/(sqrt3), (1)/(sqrt3), (1)/(sqrt3)gt or lt -(1)/(sqrt3), -(1)/(sqrt3), -(1)/(sqrt3)gt`D. None of the above

Answer» Correct Answer - C
Let the vecto v make an angle `alpha` with each of the three axes, then diractin consine of v are
`lt cos alpha, cos alpha, cos alphagt`
Also, `cos ^(2)alpha +cos ^(2)alpha+cos ^(2)alpha=1`
`impliescos ^(2)alpha =1//3`
`impliescos alpha =pm(1)/(sqrt3)`
Hence, direction cosine of v are
`lt (1)/(sqrt3),(1)/(sqrt3), (1)/(sqrt3)gt`
or `lt-(1)/(sqrt3), -(1)/(sqrt3),-(1)/(sqrt3)gt`
Discussion
53.

The ratio of moment of an electron and an `alpha`-particle which are accelerated from rest by a potential difference of `100V` isA. 1B. `sqrt((2m_(2))/(malpha))`C. `sqrt((m_(e))/(malpha))`D. `sqrt((m_(e))/(2m_(alpha))`

Answer» Correct Answer - D
Momentum `p=mvand v=sqrt((2qV)/(m))`
`p=sqrt(2qmV)`
`p=sqrt(qm)`
`(p_(e))/(p_(alpha))=sqrt((em_(e))/(2em_(alpha)))`
`therefore (p_(e))/(p_(alpha))=sqrt((m_(e))/(2malpha))`
Discussion
54.

Energy per unit volume for a capacitor having area A and separation d kept at potential diffeence V is given by : -A. `1/2 epsi_(0)(V^(2))/(d^(2))`B. `(1)/(2epsi_(0)) (V^(2))/(d^(2))`C. `1/2CV^(2)`D. `(Q^(2))/(2C)`

Answer» Correct Answer - A
Energy density `=1/2 epsi_(0)E^(2)`
Since `E=V/d`
Therefore energy density `=1/2epsi_(0)((V_(2))/(d_(2)))`
Discussion