Let [ ] denote the greatest integer function. Let g(x) = (x^2 - a)[x^2] be defined on the interval (1, 3) , where a is a constant. Let S be the set of points in (1, 3) where g(x) is discontinuous. Let f(x) = x^2 + [x] . If _ x S f(x) = 39 ,

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Let [ ] denote the greatest integer function. Let g(x) = (x^2 - a)[x^2] be defined on the interval (1, 3) , where a is a constant. Let S be the set of points in (1, 3) where g(x) is discontinuous. Let f(x) = x^2 + [x] . If _ x S f(x) = 39 , then the value of a is

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Correct answer: C. 6

Points of discontinuity of [x^2] in (1, 3) are where x^2 is an integer. Since 1 The integer values of x^2 are 2, 3, 4, 5, 6, 7, 8 . So, x 2 , 3 , 2, 5 , 6 , 7 , 8 . At these points, the values of f(x) = x^2 + [x] are: f( 2 ) = 2 + 1 = 3 f( 3 ) = 3 + 1 = 4 f(2) = 4 + 2 = 6 f( 5 ) = 5 + 2 = 7 f( 6 ) = 6 + 2 = 8 f( 7 ) = 7 + 2 = 9 f( 8 ) = 8 + 2 = 10 Sum of f(x) for all these points is 3 + 4 + 6 + 7 + 8 + 9 + 10 = 47 . The function g(x) = (x^2 - a)[x^2] will be continuous at x = a because the factor (x^2 - a) becomes

Let [ ] denote the greatest integer function. Let g(x) = (x^2 - a)[x^2] be defined on the interval (1, 3) , where a is a constant. Let S be the set of points in (1, 3) where g(x) is discontinuous. Let f(x) = x^2 + [x] . If _ x S f(x) = 39 , then the value of a is

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