# Engineering Mathematics GATE-1994

**Q 1:** The inverse of a matrix \begin{bmatrix}a&0\\0&b\end{bmatrix}

**Q 2:** The limit of f(x)=\frac x{\sin x} as x→0 is

**Q 3:** Integrating factor for the differential equation \frac{dy}{dx}+P(x)y=Q(x) is

**Q 4:** If \underline i,\underline j,\underline k are the unit vectors in rectangular coordinates, then the curl of the vector i\underline y+jy+\underline kz

**Q 5:** The solution for the differential equation \frac{d^2y}{dx^2}+5\frac{dy}{dx}+6y=0 is

**Q 6:** The Taylor’s series expansion of f(x) around x = a is ______________.

**Q 7:** For a differential function f(x) to have a maximum, \frac{df}{dx} should be ________ and \frac{d^2f}{dx^2} should be ____________.

**Q 8:** Mdx+Ndy is an exact differential when __________.

**Q 9:** The integral of x\sin x is ____________.

**Q 10:** The Green’s theorem relates _________ integrals to surface integrals.

**Q 11:** If ‘a’ is a scalar and \underline b is a vector, then \nabla\times a\underline b= _________.

**Q 12:** The differential equation \frac{d^2y}{dx^2}+y=0 , with the conditions y(0) = 0 and y(1) = 1 is called a _______ value problem.

**Q 13:** State with reasons whether the statement is true or false

The series 1+x+x^2+x^3+ for x < 1 is divergent.

**Q 14:** Match the items in the left column with the appropriate items in the right column.

(I) \cosh at | (A) a/\left(s^2+a^2\right) |

(II) \sinh at | (B) a/\left(s^2-a^2\right) |

(C) s/\left(s^2-a^2\right) | |

(D) s/\left(s^2+a^2\right) |

**Q 15:** Match the items in the left column with the appropriate items in the right column.

(I) \frac{dy}{dx}=x^2+y^2 | (A) linear 1^{st} order ODE with constant coefficient |

(II) \frac{dy}{dx}=x^2+y | (B) linear ODE with variable coefficient |

(C) 1^{st} order nonlinear ODE | |

(D) linear 2^{nd} order ODE |

**Q 16:** Find the eigenvalues of the matrix \begin{bmatrix}0&2\\-1&-1\end{bmatrix}