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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 + 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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IIT JEE 2009 Mathematics Paper2 Code 1 Solutions...