# 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

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

**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).

**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

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

**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).

**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).

**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).