Jinesh Jain posted an Question
February 08, 2022 • 00:25 am 30 points
  • IIT JAM
  • Mathematics (MA)

Let p be a polynomial of degree 2n+1 with real coefficients. we say that a real number a is a fixed point of p if p(a) =a. then, p 25. 32. consider the functíon

Let p be a polynomial of degree 2n+1 with real coefficients. We say that a real number a is a fixed point of p if p(a) =a. Then, p 25. 32. Consider the functíon f:R--R defined by 2x, ifx is rational Thenf is f)3-1 ifx is irational has (a.) exactly 2n+1 fixed points continuous (b.) at least one fixed point (a.) only at x = 1 (c.) at most one fixed point (b.) only at x= 2 (d.) n fixed points (c.) only at x = 0 29. Define the function f:R RR by (d.) at no point of R. ax+b. ifxsi cx+1. ifx>l We want to find appropriate values of a, b andc such that Let SRR be defined by 33. s)=x+1-. then the range of f is 1. fis increasing in the interval (0, ): (a.) The whole of R 2. f' is continuous on R b.) The closed interval[-1.) Which of the following is the corect statement about the values of (a, b, c) for (c.) The unbounded subset of R. (d.) None of the above. which both conditions (1) and (2) are satisfied? 34. How many zeroes are there for the function (a.) (3, 2, 6) is the only possible value -5l+6 ? (b.) There are finitely many values of (a,b, c) (a.) 3 (b.)4 (c.) (-2,-3,-4) is one of the values (c.) 6 (d.) There are infinitely many values of (a, b, c) (d.) None of these 35. Let f:RR be defined by f()=r and let 30. Let F:R >R be a monotone function. U be any non-empty open subset of R.Then Then (a.) f(U)is open (a.) F has no discontinuities. F has only finitely many discontinuities. (b.)f(U) is open (b.) (c.) s(U) is closed (c.) F can have at most countably many discontinuities. (d.)(U) is closed many (d.) F can have uncountably discontinuities follows as Define f:R-R 36. with continuous 1, if xis rational 31. Let f:RR be Then f0)= f(1) =0. Which of the following is not possible? f)=sin if xis irrational (a.)fis continuous everywhere (a.) f([o, 1) = to} (b.)fis continuous only at x= 0. (b.) s([0. 1]) =[0, 1) (c.)fis continuous all rational points (c.) f([0, 1) = [0, 1 (d.)fis continuous at all irrational points. (d.) s(0. 1)= rHT, New Delhi-110016, Ph.: (01)-26537527, Cell: 9999183434 &9899161734, 8588844789 W'abitr; www.dipsacadenmv.com 180 please provide us the solutions of all the sums, sir

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