IIT JEE 2009 Mathematics Paper2 Code 1 Solutions

18.    Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 lim_x->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

        lim_x->0 (x²+p(x)/x²) = 2

        => P(0) = 0 Þ e = 0

        lim_x->0 (2x+p'(x)/2x) = 2

        => P'(0) = 0 Þ d = 0

        lim_x->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.

<< Back || Next >>