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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

f(x)=4x^2 - x + 1

g_t(x)=\left(t - \frac{3}{2}\right)x + \left(\frac{39}{64}\right)

The parameter t ranges over the real numbers

Find the parameter values that make the line and parabola touch at one point

(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 graphs have no common point

Use Δ(t)>0; the graphs meet at two distinct points

Use Δ(t)=0; the line and parabola touch at one point

Use Δ(t)≥0; the graphs meet at least once

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 every real value of t

Take t between -2 and 3; decide endpoint inclusion separately

Take t below -2 or above 3; decide endpoint inclusion separately

Take t equal to one of the two critical values -2 or 3

(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