Let `f(x)={((alpha cotx)/x+beta/x^2 ,, 0<|x|lt=1),(1/3 ,, x=0):}`. If `f(x)` is continuous at `x=0` then the value of `alpha^2+beta^2` is

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

Let f(x)={{:((1-cos2ax)/(xsin2x),"when " xne0),(" " (1)/(2),"when " x =0):} If f (x) is continuous at x = 0 , find the value of alpha .

Let f(x)={(absx, "for",0ltabsxlt=2),(1, "for",x=0):} Then ,at x = 0, f(x) has

Let the roots of f(x)=xbe alpha and beta where f(x) is quadratic polynomial ax^2+bx+c.alpha and beta are also the roots of f(f(x))=x. Let the other two roots of f(f(x))=x be gamma and lamda Statement I : if alpha and beta are real unequal then gamma and lamda are also real. Statement II : if alpha and beta are imaginary then gamma and lamda are also imaginary. The correct statement are

Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots alpha,alpha,beta,… or alpha root is repreated root, then f(x) = 0 is equivalent to (x-alpha)^(2)(x-beta)…=0, from which we can conclude that f(x)=0 or 2(x-alpha)[(x-beta)...]+(x-alpha)^(2)[(x-beta)...]'=0 or (x-alpha) [2 {(x-beta)...}+(x-alpha){(x-beta)...}']=0 has root alpha . Thus, if alpha root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if alpha root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x)=0, and f''(x)=0. If x-c is a factor of order m of the polynomial f(x) of degree n (1ltmltn) , then x=c is a root of the polynomial [where f^(r)(x) represent rth derivative of f(x) w.r.t. x]

Let alpha,beta in Rdot If alpha,beta^2 are the roots of quadratic equation x^2-p x+1=0. a n dalpha^2,beta are the roots of quadratic equation x^2-q x+8=0, then find p ,q,alpha,betadot

Let f(x) :R->R,f(x)={(2x+alpha^2,xge2),((alpha x)/2+10,x lt 2)) If f(x) is onto function then alpha belongs to (A) [1,4] (B) [-2,3] (C) [0,3] (D) [2,5]

underset(x rarr 0)lim (sin alpha x)/(e^(beta x) - 1) (alpha, beta != 0) equal to -

Find cos(alpha+beta) if tan(alpha/2) and tan (beta/2) are roots of the equiation 8x^2-26x+15=0

Let f (x+(1)/(x))= x^(2) +(1)/(x^(2)) , x ne 0, then the value of f (x) is-