If three lines whose equations are `y = m_(1) x + c_(1) , y = m_(2) x + c_(2) " and " 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 `.

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)|)

Choose the odd one out (1) y^(2) = 4ax (2) c= a/m (3) c^(2) = a^(2(1+m^(2))) (4) (a/m^(2) , (2a)/m)

If a_(1) x^(3) + b_(1)x^(2) + c_(1)x + d_(1) = 0 and a_(2)x^(3) + b_(2)x^(2) + c_(2)x + d_(2) = 0 a pair of repeated roots common, then prove that |{:(3a_(1)", "2b_(1) ", "c_(1)),(3a_(2)", " 2b_(2)", "c_(1)),(a_(2)","b_(1)- a_(1)b_(2)", "c_(2)a_(1)-c_(2)a_(1)", "d_(1)a_(2)-d_(2)a_(1)):}|=0

The line a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 are perpendicular if:

If (1 - x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) + ......... + C_(n)x^(n) , then the value of 1.C_(1) + 2.C_(3) + 3.C_(3) + ......... + n.C_(n) =

if (1 + C_(1)/C_(0))(1 + C_(2)/C_(1)) (1 + C_(3)/C_(2)) ...... (1 + C_(n)/C_(n - 1))is

Let each of the circles S_(1)-=x^(2)+y^(2)+4y-1=0 S_(1)-= x^(2)+y^(2)+6x+y+8=0 S_(3)-=x^(2)+y^(2)-4x-4y-37=0 touch the other two. Also, let P_(1),P_(2) and P_(3) be the points of contact of S_(1) and S_(2) , S_(2) and S_(3) , and S_(3) , respectively, C_(1),C_(2) and C_(3) are the centres of S_(1),S_(2) and S_(3) respectively. P_(2) and P_(3) are images of each other with respect to the line

A company has four categories of employees given by Assistants (A), Clerks(C), Mangagers (M) and an Excutive Officer(E). The company provide ₹10,000, ₹25,000, ₹50,000 and ₹1,00,000 as salaries to the people who work in the categories A, C, M and E respectively. If A_(1), A_(2), A_(3), A_(4) and A_(5) were Assistants, C_(1), C_(2), C_(3), C_(4) were Clerks, M_(1), M_(2) and M_(3) were managers and E_(1), E_(2) were Executive officers and if the relation R is defined by xRy, where x is the salary given to person y, express the relation R through an ordered pair and an arrow diagram/

Find the differential equation whose general solution is given by y=(c_(1)+c_(2))cos(x+c_(3))-c_(4)e^(x+c) , where c_(1),c_(2), c_(3), c_(4), c_(5) are arbitary constants.

Let each of the circles S_(1)-=x^(2)+y^(2)+4y-1=0 S_(1)-= x^(2)+y^(2)+6x+y+8=0 S_(3)-=x^(2)+y^(2)-4x-4y-37=0 touch the other two. Also, let P_(1),P_(2) and P_(3) be the points of contact of S_(1) and S_(2) , S_(2) and S_(3) , and S_(3) , respectively, C_(1),C_(2) and C_(3) are the centres of S_(1),S_(2) and S_(3) respectively. The ratio ("area"(DeltaP_(1)P_(2)P_(3)))/("area"(DeltaC_(1)C_(2)C_(3))) is equal to