InterviewSolution
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InterviewSolution
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If `a` and `b` are randomly chosen from the set `{1,2,3,4,5,6,7,8,9}`, then the probability that the expression `ax^(4)+bx^(3)+(a+1)x^(2)+bx+1` has positive values for all real values of `x` isA. `(34)/(81)`B. `(31)/(81)`C. `(32)/(81)`D. `(10)/(27)` |
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Answer» Correct Answer - C `(c )` The expression `ax^(4)+ax^(2)+bx^(3)+bx+x^(2)+1` `=(x^(2)+1)(ax^(2)+bx+1)` `:.x^(2)+1` is positive for all real `x` For `ax^(2)+bx=1` to be positive for all real `x` `a gt 0`, `b^(2)-4a lt 0` If `b=1`, a can take `9` value from `1` to `9` `b=2`, a can take `8` value from `2` to `9` `b=3`, a can take `7` value from `3` to `9` `b=4`, a can take `5` value from `5` to `9` `b=5`, a can take `3` value from `7` to `9` `b` cannot take the values `6,7,8,9`. `:.` Number of exhaustive cases `=9+8+7+5+3=32` For each of the `9` values of `a`, there are `9` corresponding values for `h`. `:.` Number of exhaustive cases `=9xx9=81` `:.` The required probability `=(32)/(81)` |
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