For `x >0,l e th(x)={1/q ,ifx=p/q0,ifxi si r r a t ion a l` where `p ,q >0` are relatively prime integers. Then prove that `f(x)` is continuous for all irrational values of `xdot`

Express 1.bar(62) in (p)/(q) from where qne0,p, q are integers.

Find the value of x where f(x)={x ,ifxi sr a t ion a l1-x ,ifxi si r r a t ion a li scon t inuou sdot

If lines p x+q y+r=0,q x+r y+p=0a n dr x+p y+q=0 are concurrent, then prove that p+q+r=0(w h e r ep ,q ,r are distinct )dot

Find the value of a for which f(x)={x^2,x in Q x+a ,x !in Q is not continuous at any xdot

Consider three lines as follows. L_1:5x-y+4=0 L_2:3x-y+5=0 L_3: x+y+8=0 If these lines enclose a triangle A B C and the sum of the squares of the tangent to the interior angles can be expressed in the form p/q , where pa n dq are relatively prime numbers, then the value of p+q is 500 (b) 450 (c) 230 (d) 565

Consider three lines as follows. L_1:5x-y+4=0 L_2:3x-y+5=0 L_3: x+y+8=0 If these lines enclose a triangle A B C and the sum of the squares of the tangent to the interior angles can be expressed in the form p/q , where pa n dq are relatively prime numbers, then the value of p+q is (a) 500 (b) 450 (c) 230 (d) 465

If 2 + isqrt3 is a root of x^(3) - 6x^(2) + px + q = 0 (where p and q are real) then p + q is

Column I: Function, Column II: Type of function f(x)="{"sgnx")"^(sgn)}; x!=0,n is an odd integer, p. odd function f(x)=x/(e^x-1)+x/2+1 , q. even function f(x)={0,ifxi sr a t ion a l1,ifxi si r r a t ion a l , r. neither odd nor even function. f(x)=max{tanx ,cotx} , s. periodic

Express the following decimal expansion is the form p/q , where p and q are integers and q ne 0 . 0.75

Express the following decimal expansion is the form p/q , where p and q are integers and q ne 0 . 0.28