Engineering Mathematics GATE-2019

Q 1: A system of n homogeneous linear equations containing n unknowns will have non-trivial solutions if and only if the determinant of the coefficient matrix is

A) 1
B) -1
C) 0
D) ∞

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Q 2: The value of the expression $\lim\limits_{x\rightarrow\frac{\mathrm\pi}2}\;\left|\frac{\tan\left(x\right)}x\right|$

A) ∞
B) 0
C) 1
D) -1

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Q 3: The product of the eigenvalues of the matrix $\begin{pmatrix}2&3\\0&7\end{pmatrix}$ is _______ (rounded off to one decimal place).

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Q 4: The solution of the ordinary differential equation $\frac{dy}{dx}+3y=1$, subject to the initial condition y = 1 at x = 0, is

A) $\frac13\left(1+2e^{-x/3}\right)$
B) $\frac13\left(5-2e^{-x/3}\right)$
C) $\frac13\left(5-2e^{-3x}\right)$
D) $\frac13\left(1+2e^{-3x}\right)$

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Q 5: The value of the complex number $i^{-1/2}$ (where $i=\sqrt{-1}$ ) is

A) $\frac1{\sqrt2}\left(1-i\right)$
B) $-\frac1{\sqrt2}i$
C) $\frac1{\sqrt2}i$
D) $\frac1{\sqrt2}\left(1+i\right)$

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Q 6: If x, y and z are directions in a Cartesian coordinates system and I, j and k are the respective unit vectors, the directional derivative of the function $u(x,y,z)=x^2-3yz$ at the point (2, 0, -4) in the direction $(i+j-2k)/\sqrt6$ is ______ (rounded off to two decimal places).

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Q 7: Two unbiased dice are thrown. Each dice can show any number between 1 and 6. The probability that the sum of the outcomes of the two dice is divisible by 4 is ______ (rounded off to two decimal places).

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Q 8: The Newton-Raphson method is used to determine the root of the equation $f(x)=e^{-x}-x$. If the initial guess for the root is 0, the estimate of the root after two iterations is ______ (rounded off to three decimal places).

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