Engineering Mathematics GATE-2018

Q 1: Consider the following two equations:

$\frac{dy}{dx}+x+y=0$ $\frac{dy}{dt}-x=0$

The above set of equations is represented by

A) $\frac{d^2y}{dt^2}-\frac{dy}{dt}-y=0$
B) $\frac{d^2x}{dt^2}-\frac{dx}{dt}-y=0$
C) $\frac{d^2y}{dt^2}+\frac{dy}{dt}+y=0$
D) $\frac{d^2x}{dt^2}+\frac{dx}{dt}+y=0$

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Q 2: The fourth-order Runge-Kutta (RK4) method to solve an ordinary differential equation $\frac{dy}{dx}=f(x,y)$ is given as

$y(x+h)=y(x)+\frac16(k_1+2k_2+2k_3+k_4)$

$k_1=hf(x,y)$ $k_2=hf\left(x+\frac h2,\;y+\frac{k_1}2\right)$ $k_3=hf\left(x+\frac h2,\;y+\frac{k_2}2\right)$ $k_4=hf(x+h,\;y+k_3)$

For a special case when the function f depends solely on x, the above RK4 method reduces to

A) Euler’s explicit method
B) Trapezoidal rule
C) Euler’s implicit method
D) Simpson’s 1/3 rule

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Q 3: A watch uses two electronic circuits (ECs). Each has a failure probability of 0.1 in one year of operation. Both ECs are required for the functioning of the watch. The probability of the watch functioning for one year without failure is 

A) 0.99
B) 0.90
C) 0.81
D) 0.80

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Q 4: The figure which represents $y=\frac{\sin\left(x\right)}x$ for x > 0 (x in radians) is

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Q 5: A person is drowning in the sea at location R and the lifeguard is standing at location P. The beach boundary is straight and horizontal, as shown in the figure.

The lifeguard runs at a speed of V_L and swims at a speed of V_S. In order to reach to the drowning person in optimum time, the lifeguard should choose point Q such that

A) $\frac{\sin^2\left(\theta_L\right)}{\sin^2\left(\theta_S\right)}=\frac{V_S}{V_L}$
B) $\frac{\sin\left(\theta_L\right)}{\sin\left(\theta_S\right)}=\frac{V_S}{V_L}$
C) $\frac{\sin^2\left(\theta_L\right)}{\sin^2\left(\theta_S\right)}=\frac{V_L}{V_S}$
D) $\frac{\sin\left(\theta_L\right)}{\sin\left(\theta_S\right)}=\frac{V_L}{V_S}$

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Q 6: The decay ratio for a system having complex conjugate poles as $\left(-\frac1{10}+j\frac2{15}\right)$ and $\left(-\frac1{10}-j\frac2{15}\right)$ is

A) 7×10^-1
B) 8×10^-2
C) 9×10^-3
D) 10×10^-4

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Q 7: If $y=e^{-x^2}$ then the value of $\lim\limits_{x \to \infty}\frac1x\frac{dy}{dx}$ is __________

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Q 8: For the matrix $A=\begin{pmatrix}\cos\left(\theta\right)&-\sin\left(\theta\right)\\sin\left(\theta\right)&\cos\left(\theta\right)\end{pmatrix}$ if det stands for the determinant and A^T is the transpose of A then the value of det(A^T A) is _______

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