# Engineering Mathematics GATE-1999

**Q 1:** The system of equations,

2x+4y=10

5x+10y=25

**Q 2:** Four fair coins are tossed simultaneously. The probability that at least one head turns up is

**Q 3:** The rank of the matrix \begin{pmatrix}3&0&1&2\\4&7&3&3\\1&7&2&1\end{pmatrix} is

**Q 4:** The harmonic series {\textstyle\sum_{n=1}^\infty}\frac1{n^P}

**Q 5:** A box contains 8 balls, 2 of which are defective. The probability that none of the balls drawn are defective when two are drawn at random without replacement is

**Q 6:** The gradient of xy^2+yz^3 at the point (-1, 2, 1) is

**Q 7:** Evaluate \int_0^1\left(4x^3+x\right)dx by the trapezoidal rule. Use a step size of 0.2. Obtain the error bounds for this solution. Compute the absolute error of the numerical solution by evaluating the integral analytically.

**Q 8:** Solve \frac{dy}{dx}-6xy=-6x by the following methods: a. variation of parameters b. separation of variables.

**Q 9:** (a) At what points are the Cauchy-Reiman equations satisfied for the function, F(z)=xy^2+ix^2y ? Where is F(z) analytic? (b) Compute the distinct cube roots of (1+i).