If `x_1 and x_2` are the real and distinct roots of `a x^2+b x+c=0,` then prove that `lim_(n->x_1){1+"sin"(a x^2+b x+c)}^(1/(x-x_1))=e^(a(x_1-x_2))`

If x_(1) and x_(2) are the real and distinct roots of ax^(2)+bx+c=0 then prove that lim_(xtox1) (1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))=e^(a(x_(1)-x_(2))).

Let f(x)=a x^3+b x^2+c x+d , a!=0 If x_1 and x_2 are the real and distinct roots of f prime(x)=0 then f(x)=0 will have three real and distinct roots if

If x_1 and x_2 are two distinct roots of the equation acosx+bsinx=c , then tan((x_1+x_2)/2) is equal to (a) a/b (b) b/a (c) c/a (d) a/c

lim_(x->1)(a x^2+b x+c)/(c x^2+b x+a),a+b+c!=0

If f(x)=sin^(-1)x then prove that lim_(x->1/2)f(3x-4x^3)=pi-3lim_(x->1/2)sin^(-1)x

If alpha,beta are the roots of the equation a x^2+b x+c=0, then find the roots of the equation a x^2-b x(x-1)+c(x-1)^2=0 in term of alphaa n dbetadot

If the image of the point (x_1, y_1) with respect to the mirror ax+by+c=0 be (x_2 , y_2) then. (a) (x_2-x_1)/a = (a x_1 + b y_1+ c)/(a^(2)+b^(2)) (b) (x_2-x_1)/a = (y_2 - y_1)/b (c) (x_2-x_1)/a = -2 ((a x_1 + b y_1+ c)/(a^(2)+b^(2))) (d) None of these

If alpha,beta are the roots of the equation a x^2+b x+c=0, then find the roots of the equation a x^2-b x(x-1)+c(x-1)^2=0 in term of alpha and betadot

The roots of a_1x^2+b_1x + c_1=0 are reciprocal of the roots of the equation a_2x^2 +b_2x+c_2=0 if

lim_(x rarr0)((1+x)^(1/x)-e(1-(x)/(2)))/((1-cos x))