Engineering Mathematics GATE-2000

Q 1: A pair of fair dice is rolled simultaneously. The probability that the sum of the numbers from the dice equals six is

A) 1/6
B) 7/36
C) 5/36
D) 1/12

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Q 2: For an even function f(x),

A) \int_{-a}^af(x)dx=0
B) \int_{-a}^af(-x)dx=0
C) f(x)=-f(-x)
D) f(x)=f(-x)

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Q 3: The integrating factor for the differential equation, \left(\cos^3 x\right)\frac{dy}{dx}+y=\tan x , is

A) e^{\tan x}
B) \cos2x
C) e^{-\tan x}
D) \sin2x

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Q 4: The line integral of \int_{c}\left[\frac y{x^2+y^2}dx-\frac x{x^2+y^2}dy\right] , where C is the unit circle around the origin traversed once in the counter-clockwise direction, is

A) -2π
B) 0
C) 2π
D) π

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Q 5: The inverse of the matrix \begin{bmatrix}1&-1\\-1&-1\end{bmatrix}

A) does not exist
B) is \begin{bmatrix}1&1\\1&-1\end{bmatrix}
C) is \begin{bmatrix}0.5&-0.5\\-0.5&-0.5\end{bmatrix}
D) is \begin{bmatrix}-0.5&0.5\\0.5&0.5\end{bmatrix}

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Q 6: The complex conjugate of \frac1{1+i}  is

A) \frac1{1-i}
B) 1-i
C) 0.5\left(1-i\right)
D) in the 1st quadrant of the complex plane

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Q 7: The general solution of \frac{d^4y}{dx^4}+2\frac{d^2y}{dx^2}+y=0  is

A) (C_1x+C_2)e^x+(C_3+C_4x)e^{-x}
B) C_1\cos x+C_2\sin x+C_3e^x+C_4e^{-x}
C) C_1e^{ix}+C_2e^{-ix}
D) (C_1+C_2x)\cos x+(C_3+C_4x)\sin x

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