# Engineering Mathematics GATE-2006

**Q 1:** The ordinary differential equation \frac{dY}{dt}=f(Y) is solved using the approximation Y(t+\triangle t)=Y(t)+f\lbrack Y(t)\rbrack\triangle t . The numerical error introduced by the approximation at each step is

**Q 2:** The trapezoidal rule of integration, when applied to \int_a^bf(x)\operatorname dx will give the exact value of the integral

**Q 3: **The value of α for which the following three vectors are coplanar is

a=i+2j+k

b=3j+k

c=2i+\alpha j

**Q 4:** The derivative of \vert x\vert with respect to x when x\neq0 is

**Q 5:** If the following represents the equation of a line \begin{vmatrix}x&2&4\\y&8&0\\1&1&1\end{vmatrix}=0 , then the line passes through the point

**Q 6:** If A=\begin{bmatrix}2&1\\2&3\end{bmatrix} , then the eigen values of A^{3} are

**Q 7: ** With y=e^ax , if the sum S=\frac{dy}{dx}+\frac{d^2y}{dx^2}+\cdots+\frac{d^ny}{dx^n} approaches 2y as n\rightarrow\infty , then the value of a is

**Q 8:** Determine the following integral I=\int_s\overrightarrow r.d\overrightarrow s where, \overrightarrow r is the position vector field \overrightarrow r=\widehat ix+\widehat jy+\widehat kz and S is the surface of a sphere of radius R.

**Q 9:** The solution to the following equation x^2\frac{d^3y}{dx^3}+2x\frac{d^2y}{dx^2}-2\frac{dy}{dx}=0 is given by

**Q 10:** The value of the contour integral I=\frac1{2\mathrm{πi}}\oint_C\frac{e^z}{\left(z+1\right)\left(z+3\right)}dz where C is the circle \vert z\vert=2 is

**Q 11:** The Newton Raphson method is used to solve the equation, (x-1)^2+x-3=0 . The method will fail in the very first iteration, if the initial guess is

**Q 12:** A pair of fair dice is rolled three times. What is the probability that 10 (the sum of the numbers on the two faces) will show up exactly once?

**Q 13:** A company purchased components from three firms P, Q, and R as shown in the table below.

Firm | Total number of components purchased | Number of components likely to be defective |
---|---|---|

P | 1000 | 5 |

Q | 2500 | 5 |

R | 500 | 2 |

The components are stored together. One of the components is selected at random and found to be defective. What is the probability that it was supplied by firm R?