Engineering Mathematics GATE-2003

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Q 1: A box contains 6 red balls and 4 green balls; one ball is randomly picked and then a second ball is picked without replacement of the first ball. The probability that both are green is





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Q 2: The directional derivative of f(x,y,z)=x^2+y^2+z^2  at the point (1, 1, 1) in the direction \underline i-\underline k  is





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Q 3: The Taylor series expansion of the function: F(x)=x/(1+x)  around x = 0 is





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Q 4: The range of values for a constant ‘K’ to yield a stable system in the following set of time-dependent differential equations is

\frac{dy_1}{dt}=-5y_1+(4-K)y_2 \frac{dy_2}{dt}=y_1-2y_2




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Q 5: The value of y as t → ∞ for the following differential equation for an initial value of y(1) = 0 is

(4t^2+1)\frac{dy}{dt}+8yt-t=0




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Q 6: The most general complex analytical function f(z)=u(x,y)+i\;v(x,y)  for u=x^2-y^2  is





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Q 7: The differential equation \frac{d^2y}{dt^2}+10\frac{dy}{dt}+25x=0  will have a solution of the form NOTE: C1 and C2 are constants





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