## GATE 1993

Question 1 |

The eigen vector(s) of the matrix

is (are)

(0,0,α ) | |

(α,0,0) | |

(0,0,1) | |

(0,α,0) | |

Both B and D |

So the question as has

(A - λI)X = 0

AX = 0

What x

_{1}, x

_{2}, x

_{3}are suitable?

Which means:

x

_{1}times column 1 + x

_{2}times column 2 + x

_{3}times column 3 = zero vector

Since α is not equal to zero, so x

_{3}must be necessarily zero to get zero vector.

Hence, only (B) and (D) satisfies.

Question 2 |

The differential equation

d^{2}y/dx^{2} + dy/dx + siny = 0 is:

linear | |

non-linear | |

homogeneous | |

of degree two |

__Note: Out of syllabus.__

d

^{2}y/dx

^{2}+ dy/dx + siny = 0

In this DE, degree is 1 then this represent linear equation.

Question 3 |

Simpson’s rule for integration gives exact result when f(x) is a polynomial of degree

1 | |

2 | |

3 | |

4 |

Question 4 |

Which of the following is (are) valid FORTRAN 77 statement(s)?

DO 13 I = 1 | |

A = DIM ***7 | |

READ = 15.0 | |

GO TO 3 = 10 |

Question 5 |

Fourier series of the periodic function (period 2π) defined by

But putting x = π, we get the sum of the series.

π ^{2}/4 | |

π ^{2}/6 | |

π ^{2}/8 | |

π ^{2}/12 |

Question 6 |

Which of the following improper integrals is (are) convergent?

Question 7 |

The function f(x,y) = x^{2}y - 3xy + 2y + x has

no local extremum | |

one local minimum but no local maximum | |

one local maximum but no local minimum | |

one local minimum and one local maximum |

Question 8 |

1 |

Question 9 |

The radius of convergence of the power series

Out of syllabus. |

Question 10 |

If the linear velocity is given by

The angular velocity at the point (1, 1, -1) is ________

Out of syllabus. |

Question 11 |

Given the differential equation, y′ = x − y with the initial condition y(0) = 0. The value of y(0.1) calculated numerically upto the third place of decimal by the second order Runga-Kutta method with step size h = 0.1 is ________

Out of syllabus. |

Question 12 |

For X = 4.0, the value of I in the FORTRAN 77 statement

1 = -2**2 + 5.0*X/X*3 + 3/4 is _______

Out of syllabus. |

Question 13 |

The value of the double integral is

1/3 |

Question 14 |

If the matrix A^{4}, calculated by the use of Cayley-Hamilton theorem or otherwise, is _________

A ^{4} = I |

(1-λ) (-1-λ) (i-λ) (-i-λ)

= (λ

^{2}-1) (λ

^{2}+1)

= λ

^{4}-1

Characteristic equation is λ

^{4}-1 = 0.

According to Cayley-Hamilton theorem, every matrix satisfies its own characteristic equation, so

A

^{4}= I

Question 15 |

Given and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of integral

Out of syllabus. |