GATE Engineering Mathematics & Analytical Skills

Question 1

What is the derivative of sin(x)?

Accepted Answer

cos(x)

Answer

In calculus, the derivative of a function represents its instantaneous rate of change. For the fundamental trigonometric function sin(x), its derivative is cos(x). This is a standard and essential differentiation rule that forms the basis for solving many problems in physics, engineering, and mathematics.

Question 2

Which matrix operation results in a change in the determinant's sign?

Accepted Answer

Swapping two rows

Answer

When two rows (or columns) of a matrix are swapped, the determinant of the resulting matrix changes its sign. For instance, if the original determinant was D, performing a single row swap will yield a new determinant of -D. This is a fundamental property of determinants in linear algebra, crucial for understanding matrix transformations.

Question 3

What is the Laplace transform of 1?

Accepted Answer

1/s

Answer

The Laplace transform is an integral transform that converts a function of a real variable (often time, t) into a function of a complex variable (frequency, s). For the constant function f(t) = 1, its Laplace transform is 1/s. This is a fundamental result used extensively in solving linear differential equations and analyzing systems in engineering.

Question 4

If A is an invertible matrix, what is the determinant of A inverse?

Accepted Answer

1/det(A)

Answer

For any invertible square matrix A, a fundamental property in linear algebra states that the determinant of its inverse, denoted as det(A⁻¹), is equal to the reciprocal of the determinant of the original matrix A. This relationship, det(A⁻¹) = 1/det(A), is crucial for calculations involving matrix inverses and understanding their properties.

Question 5

Which of the following is a second-order differential equation?

Accepted Answer

d²y/dx² = 3x

Answer

The order of a differential equation is determined by the highest derivative present in the equation. In the equation d²y/dx² = 3x, the highest derivative is the second derivative (d²y/dx²). Therefore, this equation is classified as a second-order differential equation.

(GATE) Graduate Aptitude Test in Engineering Practice Test

(GATE) Graduate Aptitude Test in Engineering Practice Test

GATE Engineering Mathematics & Analytical Skills