Gate2017 ss Q31

0. Consider a quadratic equation x² - 13x + 36 = 0 with coefficients in a base b. The solutions of this equation in the same base b are x = 5 and x = 6. Then b = _____.

Note – Numerical Type question

Answer
Comment
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Option : A
Explanation :
Clearly 13 = 1 x 10 + 3 and 36 = 3 x 10 + 6 ⇒ base b = 10
The quadratic equation with solutions x = 5 and x = 6 is x² - 11x + 30 = 0
According to the given condition, we have b + 3 = 11 and 3b + 6 = 30 ⇒ b = 8
Answer is 8
Alternative solution:
x² - 13x + 36 = 0 (given quadratic equation)
In bae b, 13 = 1xb¹ + 3xb⁰ = b+3 and
36 = 3x¹ + 6x⁰ = 3b + 6
So the equation becomes x² - (b + 3)x + (3b + 6) = 0
Since x = 5 is a solution
∴ 5² - (b + 3)5 + (3b + 6) = 0 ⇒ b = 8
Similarly, by putting x = 6, we get b = 8

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