Number of solutions of a modulus equation with a parameter
Number of solutions of a modulus equation with a parameter
IP 3
8 marks
- Sketch the graph of \(y=\left|x^2-4x\right|\), showing the coordinates of the points where it meets the axes and of its maximum point between them.[3]
- Hence state the number of real solutions of \(\left|x^2-4x\right|=k\) for every real value of the constant \(k\).[3]
- Find, in exact form, all the solutions when the equation has exactly three solutions.[2]
Solution:
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Answer:(b) none if \(k<0\); 2 if \(k=0\); 4 if \(0<k<4\); 3 if \(k=4\); 2 if \(k>4\) (c) \(x=2-2\sqrt{2},\ 2,\ 2+2\sqrt{2}\)
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