# Engineering Mathematics GATE-2013

**Q 1: **The number of emails received on six consecutive days is 11, 9, 18, 18, 4 and 15, respectively. What are the median and the mode for these data?

**Q 2: **For two rolls of a fair die, the probability of getting a 4 in the first roll and a number less than 4 in the second roll, up to 3 digits after the decimal point, is _________.

**Q 3: **Which of the following statements are TRUE?

P. The eigenvalues of a symmetric matrix are real

Q. The value of the determinant of an orthogonal matrix can only be +1

R. The transpose of a square matrix A has the same eigenvalues as those of A

S. The inverse of an ‘n × n’ matrix exists if and only if the rank is less than ‘n’

**Q 4:** Evaluate \int\frac{dx}{e^x-1} (Note C is a constant of integration)

**Q 5:** For the function f(z)=\frac1{(2-z)(z+2)} , the residue at z = 2 is ________.

**Q 6:** The solution of the differential equation \frac{dy}{dx}-y^2=0 , given y = 1 at x = 0 is

**Q 7:** The solution of the differential equation \frac{d^2y}{dx^2}-\frac{dy}{dx}+0.25y=0 , given y = 0 at x = 0 and \frac{dy}{dx}=1 at x = 0 is

**Q 8: **The value of the integral \int_{0.1}^{0.5}e^{-x^3}dx evaluated by Simpson’s rule using 4 subintervals (up to 3 digits after the decimal point) is ____________.