GATE Engineering Mathematics & Analytical Skills 1 — Questions and Answers

Question 1: What is the derivative of sin(x)?

cos(x) (Correct answer)
-sin(x)
-cos(x)
sin(x)

Correct answer: cos(x)

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?

Scaling a row
Swapping two rows (Correct answer)
Adding a row to another
Multiplying a row by 0

Correct answer: Swapping two rows

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?

s
1/s (Correct answer)
s^2
s+1

Correct answer: 1/s

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?

det(A)
1/det(A) (Correct answer)
det(A)^2
-det(A)

Correct answer: 1/det(A)

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?

dy/dx = x^2
d²y/dx² = 3x (Correct answer)
y = mx + c
∫x dx

Correct answer: d²y/dx² = 3x

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.

Question 6: What is the probability of getting a sum of 7 when two dice are thrown?

1/12
1/6 (Correct answer)
1/8
1/10

Correct answer: 1/6

When two fair dice are thrown, there are 36 possible outcomes (6 sides on each die, so 6x6). The combinations that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), which totals 6 favorable outcomes. Therefore, the probability of getting a sum of 7 is 6/36, which simplifies to 1/6.

Question 7: Which numerical method is used for solving ordinary differential equations?

Gaussian Elimination
Runge-Kutta method (Correct answer)
Newton-Raphson method
Simpson’s Rule

Correct answer: Runge-Kutta method

The Runge-Kutta method is a family of iterative numerical methods widely used to approximate solutions of ordinary differential equations (ODEs) with a given initial value. It is particularly valued for its accuracy and stability, making it a cornerstone in computational mathematics and engineering for modeling dynamic systems.

Question 8: What is the general solution of a homogeneous linear differential equation?

Only particular solution
Only complementary function
Sum of complementary function and particular solution (Correct answer)
Zero always

Correct answer: Sum of complementary function and particular solution

The general solution of a linear differential equation, whether homogeneous or non-homogeneous, is composed of two parts: the complementary function and a particular solution. The complementary function solves the associated homogeneous equation, while the particular solution addresses the non-homogeneous part. For a strictly homogeneous equation, the particular solution is typically zero, making the complementary function the complete solution.

Question 9: In probability, what does a cumulative distribution function (CDF) represent?

Probability mass
Probability density
Cumulative probability up to a value (Correct answer)
Mean of a distribution

Correct answer: Cumulative probability up to a value

In probability theory, a cumulative distribution function (CDF) represents the probability that a random variable X will take a value less than or equal to a specific value x. It essentially accumulates the probabilities from the lowest possible value up to x. The CDF provides a comprehensive view of the probability distribution, showing the likelihood of observing values within a certain range.

GATE Engineering Mathematics & Analytical Skills 1

What is the derivative of sin(x)?