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 1st order ODE with constant coefficient |
| (II) \frac{dy}{dx}=x^2+y | (B) linear ODE with variable coefficient |
| (C) 1st order nonlinear ODE | |
| (D) linear 2nd order ODE |
Q 16: Find the eigenvalues of the matrix \begin{bmatrix}0&2\\-1&-1\end{bmatrix}