2023 AISS AM S3 EOY Q6
2023 AISS AM S3 EOY Q6
7 marks
Others>EOY
- The curve with equation \({y = ax^2 + bx + a}\), where \({a}\) and \({b}\) are constants, lies completely above the \({x}\)-axis.
- Write down the conditions which must apply to \({a}\) and \({b}\).[2]
- Give an example of possible values for \({a}\) and \({b}\) which satisfy the conditions in part (i).[2]
- Using a suitable substitution, explain why the equation \({3^{2x-1} = 3^{x+2} - k}\) has no solution if \({k > 60.75}\).[3]
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Answer:(a)(i) \(a>0\) \text{ and } \({{b}^{2}}-4{{a}^{2}} < 0\); \(-2a < b < 2a\)
(ii) Any positive value for a \text{ and } \(-2a < b < 2a\)
(b) Since discriminant < 0 when \(k>60.75\), the equation \({{u}^{2}}-27u+3k=0\) has no real solution. Thus \({{3}^{2x-1}}={{3}^{x+2}}-k\) has no solution.
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