Engineering Mathematics GATE-2003

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

A) 1/15
B) 2/25
C) 2/15
D) 4/25

Show Answer

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

A) 0
B) 1
C) \sqrt2
D) 2\sqrt2

Show Answer

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

A) x+x^2+x^3+x^4\dots\dots\dots
B) 1+x+x^2+x^3+x^4\dots\dots\dots
C) 2x+4x^2+8x^3+16x^4\dots\dots\dots
D) x-x^2+x^3-x^4\dots\dots\dots

Show Answer

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

A) 0 < K < 7
B) 6.25 < K < 10
C) -6 < K ≤ 6.25
D) 0 ≤ K ≤ 7

Show Answer

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

A) 1
B) 1/2
C) 1/4
D) 1/8

Show Answer

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

A) z
B) z²
C) z³
D) 1/z²

Show Answer

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

A) (C_1+C_2t)e^{-5t}
B) C_1e^{-2t}
C) C_1e^{-5t}+C_2e^{5t}
D) C_1e^{-5t}+C_2e^{2t}

Show Answer