 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/6/a/solution

Complete the square in the quadratic form:

Q_{1} = (x_{1} + 2 x_{2})^{2} + x_{2}^{2} + 6 x_{3}^{2} .

x = L y, L = 100 - 2 10 00 1/ 6

x_{1} + 2 x_{2} = y_{1}

x_{2} = y_{2}

6 x_{3} = y_{3}

Q_{1} = y_{1}^{2} + y_{2}^{2} + y_{3}^{2}

. The same transformation gives

Q_{2} = y_{1}^{2} + y_{2}^{2} - y_{3}^{2} .

Its signature is

(2, 1)

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