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IIT JEE 2009 Mathematics Paper2 Code 1 Solutions

18.    Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 limx->0(1+p(x)/x2) = 2. Then the value of p(2) is

Sol.   0

        Let P(x) = ax4 + bx3 + cx2 + dx + e

        P'(1) = P'(2) = 0

        limx->0 (x2+p(x)/x2) = 2

        => P(0) = 0 Þ e = 0

        limx->0 (2x+p'(x)/2x) = 2

        => P'(0) = 0 Þ d = 0

        limx->0 (2+p''(x)/2) = 2

        => c = 1

        On solving, a = 1/4, b = -1

        So P(x) = x4/4 - x3 + x2

        => P(2) = 0.

19.    Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:

                     3x - y - z = 0

                       - 3x + z = 0

                -3x + 2y + z = 0

        Then the number of such points for which x2 + y2 + z2 < 100 is

Sol.   7

        3x - y - z = 0

        -3x + z = 0

    =>    y = 0

       and z = 3x

   => x2 + y2 + z2 = x2 + z2 = x2 + 9x2 = 10x2 < 100

   => x2 < 10

   => x = 10, +1, +2, +3

   Hence , there are such seven points.

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