Engineering Mathematics GATE-2004

Q 1: The inverse Laplace transform of the function f(s)=\frac1{s\left(s+1\right)}  is

A) 1+e^t
B) 1-e^t
C) 1+e^{-t}
D) 1-e^{-t}

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Q 2: The function f(x)=3(x-2)  has a

A) Minimum at x = 1
B) Maximum at x = 1
C) Minimum at x = 2
D) Maximum at x = 2

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Q 3: The complex number 2(1+i)  can be represented in polar form is

A) 2\sqrt2e^{i\mathrm\pi/4}
B) \frac{\mathrm\pi}4e^{i2\sqrt2}
C) \sqrt2e^{i\mathrm\pi/4}
D) \frac{\mathrm\pi}4e^{i\sqrt2}

Q 4: The differential equation \frac{d^2y}{dx^2}+\sinh x\;\frac{dy}{dx}+ye^x=\sinh x  is

A) first order and linear
B) first order and non linear
C) second order and linear
D) second order and non linear

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Q 5: For the time domain function f (t) = t, the Laplace transform of \int_0^tf(t)dt  is given by

A) \frac1{2s^3}
B) \frac2{s^3}
C) \frac1{s^3}
D) \frac2{s^2}

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Q 6: The sum of the eigen values of the matrix \begin{pmatrix}3&4\\x&1\end{pmatrix}  for real and negative values of x is

A) Greater than zero
B) less than zero
C) zero
D) dependent on the value of x

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Q 7: The system of equations

4x+6y=8
7x+8y=9
3x+2y=1

has

A) No solution
B) Only one solution
C) Two solutions
D) Infinite number of solutions

Q 8: A box contains three blue balls and four red balls. Another identical box contains two blue balls & five red balls. One ball is picked at random from one of the two boxes and it is red. The probability that it comes from the first box is

A) 2/3
B) 4/9
C) 4/7
D) 2/7

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Q 9: The series \sum_{n=1}^\infty\frac{\left(z+2\right)^n}{n!}  Converges for

A) all z
B) z > 2
C) |z| > 2
D) |z| < 2

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Q 10: The differential equation for the variation of the amount of salt x in a tank with time t is given by \frac{dy}{dt}+\frac x{20}=10 . Where x is in kg and t is in minutes. Assuming that there is no salt in the tank initially, the time (in minutes) at which the amount of salt increases to 100 kg is

A) 10 ln 2
B) 20 ln 2
C) 50 ln 2
D) 100 ln 2

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Q 11: Value of the integral \int_{-2}^2\frac{dx}{x^2}  is

A) 0
B) 0.25
C) 1
D) ∞

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Q 12: The differential equation \left(\frac{dy}{dx}\right)^2+y\frac{d^2y}{dx^2}=0  can be reduced to (where α is a constant)

A) \left(\frac{dy}{dx}\right)^3=\alpha-\frac{3y^2}2
B) \left(\frac{dy}{dx}\right)^2=\alpha-2y
C) \frac{dy}{dx}=\frac\alpha{y^2}
D) \frac{dy}{dx}=\frac\alpha y

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Q 13: The value of \lim\limits_{x\rightarrow9}\frac{\sqrt x-3}{x^2-81}  is

A) 0
B) 1/27
C) 1/108
D) ∞

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