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Determine parameter values from a discriminant condition

QUESTION TYPEMix of allChange

Question 1 of 8

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Let t be a real parameter

q_t(x)=-4x^2 + \left(t - 2\right)x + \left(-\frac{25}{16}\right)=0

For every real value of t, the displayed equation is quadratic in x

For which real values of \(t\) does \(q_t(x)=0\) have exactly one real solution?

(a)3 marks

Calculate \(\Delta(t)\) and enter the coefficients A, B and C in \(At^2+Bt+C\) form

\(A\) \(B\) \(C\)

(b)1 mark

Select the correct condition on \(\Delta(t)\) for the requested result

Use Δ(t)=0; the equation has one repeated real solution

Use Δ(t)<0; the equation has no real solutions

Use Δ(t)≥0; the equation has at least one real solution

Use Δ(t)>0; the equation has two distinct real solutions

Enter the exact critical parameter values from \(\Delta(t)=0\) in ascending order

\(t_1\) \(t_2\)

(d)1 mark

Select the topology of the parameter solution set

Take no real value of t

Take t equal to one of the two critical values -3 or 7

Take every real value of t

Take t below -3 or above 7; decide endpoint inclusion separately

Take t between -3 and 7; decide endpoint inclusion separately

(e)1 mark

Decide whether both critical parameter values are included in the final solution set

No. At least one critical value is excluded by the strict relation or the degree restriction

Yes. Both critical parameter values are included in the final solution set