If three lines whose equations are `y=m_(1) x+c_(1), y=m_(2) x + c_(2) and y=m_(3) x + c_(3)` are concurrent, then show that `m_(1) (c_(2) - c_(3))+m_(2) (c_(3) - c_(1) ) + m_(3) ( c_(1) - c_(2) ) =0`.

The distance between the lines y=mx + c_(1) and y= mx+ c_(2) is

If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 are concurrent then prove that, (1)/( 1-a) + (1) /( 1-b) + (1) /( 1-c) = .

Show that the area of the triangle formed by the lines y= m_(1) x + c_(1) , y= m_(2) x + c_(2) and x = 0 is ((c_1 - c_2)^2)/( 2| m_1 - m_2 | ) .

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are Perpendicular if a_(1) a_(2) + b_(1) b_(2) =0 .

Solve the equation 3^(x+1)C_(2)+.^2P_(1)x=4^(x)P_(2),x in N .

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are (i) Parallel if (a_1)/( b_1) = (a_2)/( b_2) and

If |a_(1)|gt|a_(2)|+|a_(3)|,|b_(2)|gt|b_(1)|+|b_(3)| and |c_(2)|gt|c_(1)|+|c_(2)| then show that |{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|ne0.

Explain giving reasons which of the following sets of quantum number are not possible (a ) n=0, l =0 m_(l) = 0, m_(s ) =+ (1)/(2) ( b) n=1 , l = 0 m_(l) = 0, m_(s ) = - (1)/(2) ( c) n=1 , l = 1, m_(l ) = 0, m_(s ) = + (1)/(2) (d ) n= 2, l = 1, m_(l ) = 0, m_(s ) = (1) /(2) ( e) n=3, l = 3, m_(l) = 3, m_(s ) = + (1)/(2) (f ) n=3, l = 1, m_(l) = 0, m_(s) l = + (1)/(2)

If 2x= y^((1)/(m)) + y^(-(1)/(m)) (n ge 1) then prove that, (x^(2)-1) y_(2) + xy_(1) = m^(2)y

(a) If x + y + z=0, show that x ^(3) + y ^(3) + z ^(3)= 3 xyz. (b) Show that (a-b) ^(3) + (b-c) ^(3) + (c-a)^(3) =3 (a-b) (b-c) (c-a)