" Exament "33" .If "y=1+(c_(1))/(x-c_(1))+(c_(2)x)/((x-c_(1))(x-c_(2)))+(1)/((x-c_(1)))

If quad 1+(c_(1))/(x-c_(1))+(c_(2)x)/((x-c_(1))(x-c_(2)))+(c_(3)x^(2))/((x-c_(1))(x-c_(2))(x-c_(3))) show that (dy)/(dx)=((y)/(x))[(c_(1))/(c_(1)-x)+(c_(2))/(c_(2)-x)+(c_(3))/(c_(3)-x)]

y=e^(x)(c_(1)+c_(2)x)

IIf y=1+c_1/(x-c_1)+(c_2x)/((x-c_1)(x-c_2))+(c_3 x^2)/((x-c_1)(x-c_2)(x-c_3)) show that dy/dx=(y/x)[c_1/(c_1-x)+c_2/(c_2-x)+c_3/(c_3-x)]

"Tan"^(-1)(c_(1)x-y)/(c_(1)y+x)+"Tan"^(-1)(c_(2)-c_(1))/(1+c_(2)c_(1))+"Tan"^(-1)(c_(3)-c_(2))/(1+c_(3)c_(2))+..."Tan"^(-1)1/c_(n)=

Tan^(-1)((c_(1)x-y)/(c_(1)y+x))+Tan^(-1)((c_(2)-c_(1))/(1+c_(2)c_(1)))+Tan^(-1)((c_(3)-c_(2))/(1+c_(3)c_(2)))+…+Tan^(-1)(1/c_(n))=

The solution of ((dy)/(dx))^(2)+(2x+y)(dy)/(dx)+2xy=0, is (i) (y+x^(2)-c_(1))(x+log y+y^(2)+c_(2))( (ii) (y+x^(2)-c_(1))(x-log y-c_(2))( iii) (y+x^(2)-c_(1))(x+log y-c_(2))( iv) (y+x^(2)-c_(1))(3x+log y-c_(2))

The area of the triangle formed by the lines y=m_(1x)+c_(1),y=m_(2),x+c_(2) and x=0 is (1)/(2)((c_(1)+c_(2))^(2))/(|m_(1)-m_(2)|) b.(1)/(2)((c_(1)-c_(2))^(2))/(|m_(1)+m_(2)|) c.(1)/(2)((c_(1)-c_(2))^(2))/(|m_(1)-m_(2)|) d.((c_(1)-c_(2))^(2))/(|m_(1)-m_(2)|)

The shortest distance between the lines (x-x_(1))/(a_(1)) =(y-y_(1))/(b_(1))=(z-z_(1))/(c_(1)) and (x-x_(2))/(a_(2))=(y-y_(2))/(b_(2))=(z-z_(2))/(c_(2)) is |{:(,x_(1)-x_(1),y_(2)-y_(1),z_(2)-z_(1)),(,a_(1),b_(1),c_(1)),(,a_(2),b_(2),c_(2)):}| /A Here, A refers to

Derivative of y=e^(x)(c_(1)+c_(2)x)