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    Taylor Series, Matrix Algebra and Complex Numbers

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    A. Basic Mathematics - Please view the attachment for proper formatting of this question.

    1. Given that the Taylor series for the function

    1/(1-x) = 1 + x + x^2 + x^3 + ...,

    use this to show the following

    x/(1+x^2)^2 = x - 2x^3 + 3x^5 - ...

    log(1-x) = -x - x^2/2 - x^3/3 - x^4/4 - ...

    2. Consider complex number division (a + ib)/(c + id) which we express as re^ia. Using Euler's identity only, work out the precise form for the quotient's modulus r and argument alpha. Your calculations should not involve division of complex numbers.

    3. Using row operations evaluate the following determinant

    y-z z-x x-y
    z-x x-y y-z
    x-y y-z z-x

    4. Find the eigenvalues and eigenvectors of the following matrix

    3 3 3
    3 -1 1
    3 1 -1

    Verify that the eigenvectors are mutually orthogonal and hence diagonalise the matrix. Show all working.

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