Engineering Mathematics GATE-2014

Q 1: The gradient of a scalar variable is always

A) a vector
B) a scalar
C) a dot product
D) zero

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Q 2: For the time domain function, f(t)=t^2 , which one of the following is the Laplace transform of \int_0^tf(t)\operatorname dt ?

A) \frac3{s^4}
B) \frac1{4s^2}
C) \frac2{s^3}
D) \frac2{s^4}

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Q 3: If f*(x) is the complex conjugate of f(x)=\cos\left(x\right)+i\sin\left(x\right) , then for real a and b, \int_a^bf^\ast(x)f(x)dx is ALWAYS?

A) positive
B) negative
C) real
D) imaginary

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Q 4: If f(x) is a real and continuous function of x, the Taylor series expansion of f(x) about its minima will NEVER have a term containing

A) first derivative
B) second derivative
C) third derivative
D) any higher derivative

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Q 5: The integral of the time-weighted absolute error (ITAE) is expressed as

A) \int_0^\infty\frac{\vert\varepsilon(t)\vert}{t^2}dt
B) \int_0^\infty\frac{\vert\varepsilon(t)\vert}tdt
C) \int_0^\infty t\vert\varepsilon(t)\vert dt
D) \int_0^\infty t^2\vert\varepsilon(t)\vert dt

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Q 6: Consider the following differential equation \frac{dy}{dx}=x+\ln\left(y\right)\;;\;\;\;y=2\;at\;x=0 . The solution of this equation at x = 0.4 using the Euler method with a step size of h = 0.2 is _________.

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Q 7: The integrating factor for the differential equation \frac{dy}{dx}-\frac y{1+x}=(1+x) is

A) \frac1{1+x}
B) 1+x
C) x(1+x)
D) \frac x{1+x}

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Q 8: The differential equation \frac{d^2y}{dx^2}+x^2\frac{dy}{dx}+x^3y=e^x is a

A) non-linear differential equation of first degree
B) linear differential equation of first degree
C) linear differential equation of second degree
D) non-linear differential equation of second degree

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Q 9: Consider the following two normal distributions

f_1(x)=exp(-\pi x^2)

f_2(x)=\frac1{2\pi}exp\left\{-\frac1{4\pi}(x^2+2x+1)\right\}

If μ and σ denote the mean and standard deviation, respectively, then

A) \mu_1<\mu_2\;and\;\sigma_1^2<\sigma_2^2
B) \mu_1<\mu_2\;and\;\sigma_1^2>\sigma_2^2
C) \mu_1>\mu_2\;and\;\sigma_1^2<\sigma_2^2
D) \mu_1>\mu_2\;and\;\sigma_1^2>\sigma_2^2

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Q 10: In rolling of two fair dice, the outcome of an experiment is considered to be the sum of the numbers appearing on the dice. The probability is highest for the outcome of ____________.

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