[latex]
\[
\textbf{Difficult MCQs on Engineering Mathematics with Answers}
\]
\[
\textbf{Q1: The solution to the differential equation } y” + 4y’ + 4y = 0 \text{ is:}
\]
\[
\text{(A) } y = (C_1 + C_2 e^{-4x})
\]
\[
\text{(B) } y = C_1 e^{-2x} + C_2 e^{-4x}
\]
\[
\text{(C) } y = C_1 e^{-2x} + C_2 e^{2x}
\]
\[
\text{(D) } y = (C_1 + C_2 e^{2x})
\]
\[
\textbf{Answer: (B) } y = C_1 e^{-2x} + C_2 e^{-4x}
\]
\[
\textbf{Q2: Which of the following is a condition for a function to be differentiable at a point?}
\]
\[
\text{(A) } \text{The function is continuous at that point.}
\]
\[
\text{(B) } \text{The function has a finite slope at that point.}
\]
\[
\text{(C) } \text{The function has no jump discontinuities.}
\]
\[
\text{(D) } \text{The function is smooth and continuous.}
\]
\[
\textbf{Answer: (A) The function is continuous at that point.}
\]
\[
\textbf{Q3: The inverse Laplace transform of } \frac{1}{s^2 + 1} \text{ is:}
\]
\[
\text{(A) } \sin t
\]
\[
\text{(B) } \cos t
\]
\[
\text{(C) } e^{-t}
\]
\[
\text{(D) } \frac{1}{s}
\]
\[
\textbf{Answer: (A) } \sin t
\]
\[
\textbf{Q4: The Fourier series expansion of } f(x) \text{ in the interval } [0, 2\pi] \text{ is given by:}
\]
\[
\text{(A) } a_0 + \sum_{n=1}^{\infty} a_n \cos(nx)
\]
\[
\text{(B) } \sum_{n=1}^{\infty} b_n \sin(nx)
\]
\[
\text{(C) } a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))
\]
\[
\text{(D) } \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx)
\]
\[
\textbf{Answer: (C) } a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))
\]
\[
\textbf{Q5: The eigenvalues of a symmetric matrix are:}
\]
\[
\text{(A) } \text{Always real}
\]
\[
\text{(B) } \text{Always imaginary}
\]
\[
\text{(C) } \text{Complex}
\]
\[
\text{(D) } \text{Real or Complex, depending on the matrix}
\]
\[
\textbf{Answer: (A) Always real}
\]
\[
\textbf{Q6: The Laplace transform of } \frac{1}{(s^2 + 1)^2} \text{ is:}
\]
\[
\text{(A) } \frac{t \sin t}{2}
\]
\[
\text{(B) } \frac{\sin t}{s^2 + 1}
\]
\[
\text{(C) } \frac{1}{2} \left( \frac{1}{s} – \frac{1}{s^2 + 1} \right)
\]
\[
\text{(D) } \frac{s}{(s^2 + 1)^2}
\]
\[
\textbf{Answer: (A) } \frac{t \sin t}{2}
\]
\[
\textbf{Q7: In a second-order linear differential equation, the discriminant } \Delta = b^2 – 4ac \text{ is greater than 0. What is the nature of the roots of the characteristic equation?}
\]
\[
\text{(A) } \text{Complex conjugate roots}
\]
\[
\text{(B) } \text{Two real and distinct roots}
\]
\[
\text{(C) } \text{Equal real roots}
\]
\[
\text{(D) } \text{Imaginary roots}
\]
\[
\textbf{Answer: (B) Two real and distinct roots}
\]
\[
\textbf{Q8: If } f(x) = x^2 + 4x + 3 \text{, what is the value of } f'(x) \text{ at } x = -2 \text{?}
\]
\[
\text{(A) } 0
\]
\[
\text{(B) } -2
\]
\[
\text{(C) } 4
\]
\[
\text{(D) } -4
\]
\[
\textbf{Answer: (A) } 0
\]
\[
\textbf{Q9: The method of characteristics is used to solve which type of partial differential equation?}
\]
\[
\text{(A) } \text{Elliptic PDEs}
\]
\[
\text{(B) } \text{Parabolic PDEs}
\]
\[
\text{(C) } \text{Hyperbolic PDEs}
\]
\[
\text{(D) } \text{Non-linear PDEs}
\]
\[
\textbf{Answer: (C) Hyperbolic PDEs}
\]
\[
\textbf{Q10: The solution to the differential equation } y” + 2y’ + y = 0 \text{ is:}
\]
\[
\text{(A) } y = C_1 e^{-x} + C_2 e^{-2x}
\]
\[
\text{(B) } y = C_1 e^{-x} + C_2 e^{x}
\]
\[
\text{(C) } y = C_1 e^{x} + C_2 e^{2x}
\]
\[
\text{(D) } y = C_1 e^{2x} + C_2 e^{-x}
\]
\[
\textbf{Answer: (A) } y = C_1 e^{-x} + C_2 e^{-2x}
\]
\[
\textbf{Q11: Which of the following is the general form of the Fourier transform?}
\]
\[
\text{(A) } F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx
\]
\[
\text{(B) } F(k) = \int_{0}^{\infty} f(x) e^{ikx} \, dx
\]
\[
\text{(C) } F(k) = \int_{0}^{\infty} f(x) \, dx
\]
\[
\text{(D) } F(k) = \int_{-\infty}^{\infty} f(x) \, dx
\]
\[
\textbf{Answer: (A) } F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx
\]
\[
\textbf{Q12: The solution to the partial differential equation } \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \text{ is:}
\]
\[
\text{(A) } u(x, y) = e^{x} \sin(y)
\]
\[
\text{(B) } u(x, y) = x^2 + y^2
\]
\[
\text{(C) } u(x, y) = x \cdot y
\]
\[
\text{(D) } u(x, y) = A e^{x} + B e^{-x}
\]
\[
\textbf{Answer: (A) } u(x, y) = e^{x} \sin(y)
\]