List of top Engineering Mathematics Questions

Solve the differential equation:
\[ \frac{d^2y}{dx^2} - 4y = 0 \]
Use Newton-Raphson method to find the root of $f(x) = x^3 - x - 2 = 0$ starting with $x_0 = 1$ after one iteration.
A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
Evaluate the integral:
\[ \int_0^1 x e^x \, dx \]
Determine the rank of the matrix:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \]
Find the inverse of the matrix:
\[ \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \]
Use the bisection method to find the root of $f(x) = x^2 - 2 = 0$ in $[1, 2]$ with an error less than 0.01.
If a fair die is rolled twice, what is the probability that the sum is at least 10?
Evaluate the limit:
\[ \lim_{x \to 0} \frac{\sin x - x}{x^3} \]
Solve the differential equation:
\[ \frac{dy}{dx} + y = e^x \]
Find the residue of the function \[ f(z) = \frac{1}{z^2 + 1} \] at the pole $z = i$.
If a die is rolled twice, what is the probability that the sum of the numbers is 7?
Find the eigenvalues of the matrix: \[ A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \]
Evaluate the integral: \[ \int_0^\pi \sin^2(x) \, dx \]
Solve the differential equation: \[ \frac{d^2y}{dx^2} + 4y = 0 \]
If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real and distinct, what is the condition on the discriminant?
Which of the following is a valid method for solving a system of linear equations?
Let \( A, B \) be two events of a sample space such that \( P(A) = \frac{1}{4} \), \( P(B|A) = \frac{1}{2} \), \( P(A|B) = \frac{1}{4} \). If \( \overline{B} \) is the complement of \( B \), then \( P(A | \overline{B}) \) is _____
In a town, the probability that a person attends a gym on weekdays is 0.7, the probability that a person attends the gym on weekends is 0.4, and the probability that a person attends the gym on weekdays or weekends, or both is 0.3. What is the probability that a person attends the gym on both weekdays and weekends?
In a workshop of 100 machines, 25 machines are defective. Assuming the Poisson law for the number of defective machines, the probability that a random sample of 5 machines will have no defective machine is
A complex function \( f(z) = u + iv \) with \( u = \operatorname{Re}(z) + \operatorname{Im}(z) \) is analytic if \( v = \) ________
Consider the system of equations: \[ \begin{aligned} x + y + z &= 6 \\ 2x + 2y + 3z &= 13 \\ 3x + 4y + 5z &= 20 \end{aligned} \] Which of the following statements is true about the solution to this system?
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function such that \( f(0) = 3 \) and \( |f'(x)| \leq 2 \), \( \forall x \in \mathbb{R} \). Then \( f(2) \) lies in which of the following intervals?
Let \( f(x) = x^3 - \dfrac{9}{2}x^2 + 6x - 2 \) be a function defined on the closed interval \( [0, 3] \). Then, the global maximum value of \( f(x) \) is
The directional derivative of the function \( f(x, y) = 2x^2 + y^2 \) along a line directed from \( (0, 0) \) to \( (1, 1) \), evaluated at the point \( x = 1, y = 1 \), is ___