Practice GATE Aerospace Engineering Engineering Mathematics previous year questions from 8 papers, topic by topic, with the official answer key where available.
Engineering Mathematics - Calculus
Consider the contour $C$ shown in the figure below. For the vector $\vec{F} = (x+2y)\hat{e}_x + (2x+4y)\hat{e}_y$, the integral $\oint_C \vec{F}\cdot d\vec{l} =$ ________. Here $d\vec{l}$ represents an infinitesimal length along the contour $C$.
Official answer: A
Engineering Mathematics - Complex Analysis & Fourier Series
The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near $x = \pm 1$ are named after which one of the following scientists?
Official answer: C
Engineering Mathematics - Differential Equations
The following equation with respect to $\varphi(x,t)$, where $a$ is a non-zero constant, represents ________. $$\frac{\partial \varphi}{\partial t} + a\frac{\partial \varphi}{\partial x} = 0$$
Official answer: A
Engineering Mathematics - Linear Algebra
An $n \times n$ square matrix $A$ satisfies $A^T = A^{-1}$. The determinant of this matrix may take which of the following value(s)?
Official answer: A, B
Engineering Mathematics - Linear Algebra
If a matrix can be written as $A = uv^T$, where both $u$ and $v$ are $n$-dimensional real-valued non-zero column vectors, then the rank of the matrix $A$ is ________ (answer in integer).
Official answer: 1
Engineering Mathematics - Linear Algebra
For the matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if the relation $a + b = c + d$ holds and $a, b, c, d \neq 0$, then which one of the following statements about $A$ is FALSE?
Official answer: D
Engineering Mathematics - Differential Equations
Consider the differential equation with the initial conditions given below. If $y(x)$ is the solution of the equation, the value of the slope, $\dfrac{dy}{dx}$, at $x = \ln(2)$ is ________ (rounded off to three decimal places). $$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0 \text{ with } y|_{x=0} = 0 \text{ and } \frac{dy}{dx}\Big|_{x=0} = 1$$
Official answer: 0.151 to 0.155
Engineering Mathematics - Differential Equations
An object of mass 1 kg is launched with an initial speed of $v_o$ into a large tank filled with a viscous liquid. The liquid exerts a resistive force (drag) of the form $D = \alpha v$ on any object that is moving inside it, where $v$ is the instantaneous speed of the object and $\alpha = 1$ kg/s. If the effect of gravity is ignored, the time taken by the object to slow down to the speed $v_o/2$ is ________ s (rounded off to 2 decimal places). Assume that the tank is sufficiently large for the above deceleration to happen inside the tank.
Official answer: 0.67 to 0.71
Engineering Mathematics - Calculus
The minimum value of the function $f(x) = |x| + |2x + 3|$ for real $x$ is ________ (rounded off to 1 decimal place).
Official answer: 1.5
Engineering Mathematics - Linear Algebra
For any real symmetric matrix $A$, the transpose of $A$ is _____ .
Official answer: D
Engineering Mathematics - Linear Algebra
The system of equations $$\begin{aligned} 2x + 3y + z &= 0 \\ x + y &= 0 \\ y + z &= 0 \end{aligned}$$ The given system of equations has
Official answer: B
Engineering Mathematics - Linear Algebra
The eigenvalues of the matrix $\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ are _____.
Official answer: C
Engineering Mathematics - Differential Equations
The partial differential equation $\dfrac{\partial^2 u}{\partial x^2} + 4\dfrac{\partial^2 u}{\partial x\,\partial y} + 2\dfrac{\partial^2 u}{\partial y^2} = 0$ is _____.
Official answer: B
Engineering Mathematics - Calculus
$f(x) = \begin{cases} A + x, & \text{if } x < 2 \\ 1 + x^2, & \text{if } x \geq 2 \end{cases}$ If the function $f(x)$ is continuous at $x = 2$, the value of $A$ is _____.
Official answer: C
Engineering Mathematics - Complex Analysis & Fourier Series
If $i = \sqrt{-1}$, $\dfrac{(i+1)^3}{i-1} =$ _______.
Official answer: A
Engineering Mathematics - Calculus
The maximum value of the function $f(x) = (x-1)(x-2)(x-3)$ in the domain $[0, 3]$ occurs at $x =$ ______ (rounded off to two decimal places).
Official answer: 1.41 to 1.43
Engineering Mathematics - Calculus
$\lim_{x \to 0} \dfrac{1 - \cos(2x)}{x^2} =$ _____ (answer in integer).
Official answer: 2
Engineering Mathematics - Calculus
The equation of a closed curve in two-dimensional polar coordinates is given by $r = \dfrac{2}{\sqrt{\pi}}(1 - \sin\theta)$. The area enclosed by the curve is ________ (answer in integer).
Official answer: 6
Engineering Mathematics - Differential Equations
Consider the ordinary differential equation: $\dfrac{1}{2}\dfrac{dy}{dx} + \dfrac{y}{x} = 1$. If $y = 2/3$ at $x = 1$, then the value of $y$ at $x = 3$ is ______ (rounded off to the nearest integer).
Official answer: 2
Engineering Mathematics - Numerical Methods
An approximate solution of the equation $x^3 - 17 = 0$ is to be obtained using the Newton-Raphson method. If the initial guess is $x_o = 2$, the value at the end of the first iteration is $x_1 =$ _____ (rounded off to two decimal places).
Official answer: 2.75
Engineering Mathematics - Calculus
$\hat{i}$ and $\hat{j}$ denote unit vectors in the $x$ and $y$ directions, respectively. The outward flux of the two-dimensional vector field $\vec{v} = x\hat{i} + y\hat{j}$ over the unit circle centered at the origin is ______ (rounded off to two decimal places).
Official answer: 6.20 to 6.30
Engineering Mathematics - Linear Algebra
The following system of linear equations $$\begin{aligned} 7x - 3y + z &= 0 \\ 3x - y + z &= 0 \\ x - y - z &= 0 \end{aligned}$$ has:
Official answer: A
Engineering Mathematics - Differential Equations
The acceleration of a body travelling in a straight line is given by $a = -C_1 - C_2 v^2$ where $v$ is the velocity, and $C_1$, $C_2$ are positive constants. Starting with an initial positive velocity $v_o$, the distance travelled by the body before coming to rest for the first time is:
Official answer: A
Engineering Mathematics - Probability & Statistics
Two fair dice with numbered faces are rolled together. The faces are numbered from 1 to 6. The probability of getting odd numbers on both the dice is ________________ (rounded off to 2 decimal places).
Official answer: 0.24 to 0.26
Engineering Mathematics - Calculus
A particle acted upon by a constant force $4\hat{i} + \hat{j} - 3\hat{k}$ N is displaced from point $\mathbf{A}$ with position vector $\hat{i} + 2\hat{j} + 3\hat{k}$ m to point $\mathbf{B}$ with position vector $5\hat{i} + 4\hat{j} + \hat{k}$ m. The work done by this force is _______ J (answer in integer).
Official answer: 24
Engineering Mathematics - Numerical Methods
Using Trapezoidal rule with one interval, the approximate value of the definite integral: $$\int_1^2 \dfrac{dx}{1+x^2}$$ is _____ (rounded off to 2 decimal places).
Official answer: 0.35
Engineering Mathematics - Differential Equations
Given $y = e^{px}\sin qx$, where $p$ and $q$ are non-zero real numbers, the value of the differential expression $$\frac{d^2y}{dx^2} - 2p\frac{dy}{dx} + (p^2 + q^2)y$$ is
Official answer: A
Engineering Mathematics - Calculus
The volume of the solid formed by a complete rotation of the shaded portion of the circle of radius $R$ about the $y$-axis is $k\pi R^3$. The value of $k$ is:
Official answer: B
Engineering Mathematics - Calculus
Consider the function $$f(x) = \begin{cases} x^2 & x < 0 \\ x & x \geq 0 \end{cases}$$ where $x$ is real. Which of the following statements is/are correct?
Official answer: A, B, C
Engineering Mathematics - Linear Algebra
Consider the matrix $A = \begin{bmatrix} 5 & -4 \\ k & -1 \end{bmatrix}$, where $k$ is a constant. If the determinant of $A$ is 3, then the ratio of the largest eigenvalue of $A$ to the constant $k$ is ___________ (rounded off to 1 decimal place).
Official answer: 1.4 to 1.6
Engineering Mathematics - Calculus
The direction in which a scalar field $\phi(x,y,z)$ has the largest rate of change at any point with position vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ is the same as that of the vector
Official answer: A
Engineering Mathematics - Numerical Methods
If a monotonic and continuous function $y = f(x)$ has only one root in the interval $x_1 < x < x_2$, then
Official answer: C
Engineering Mathematics - Differential Equations
Consider the one-dimensional wave equation $\dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x} = 0$ for $-\infty < x < \infty$, $t \geq 0$. For an initial condition $u(x,0) = e^{-x^2}$, the solution at $t = 1$ is
Official answer: A
Engineering Mathematics - Linear Algebra
Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?
Official answer: A, C, D
Engineering Mathematics - Linear Algebra
The system of equations $$\begin{aligned} x - 2y + \alpha z &= 0 \\ 2x + y - 4z &= 0 \\ x - y + z &= 0 \end{aligned}$$ has a non-trivial solution for $\alpha =$ ______. (Answer in integer)
Official answer: 3
Engineering Mathematics - Calculus
Given the function $y(x) = (x+3)(x-2)$, for $-4 < x < 4$. What is the value of $x$ at which the function has a minimum?
Official answer: B
Engineering Mathematics - Differential Equations
Consider the equation $\dfrac{dy}{dx} + ay = \sin\omega x$, where $a$ and $\omega$ are constants. Given $y = 1$ at $x = 0$, select all correct statement(s) from the following as $x \to \infty$.
Official answer: C, D
Engineering Mathematics - Linear Algebra
Given the vectors $\vec{A} = 9\hat{i} - 5\hat{j} + 2\hat{k}$, $\vec{B} = 11\hat{i} + 4\hat{j} + \hat{k}$, $\vec{C} = -7\hat{i} + 14\hat{j} - 3\hat{k}$, which of the following statement(s) is/are TRUE?
Official answer: A, B
Engineering Mathematics - Differential Equations
Consider the differential equation $$x^2\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} + 2y = 0 \quad \text{for } x \geq 1$$ with initial conditions $y = 0$, $\dfrac{dy}{dx} = 1$ at $x = 1$. The value of $y$ at $x = 2$ is ____. (round off to two decimal places)
Official answer: 0.24 to 0.26
Engineering Mathematics - Calculus
The equation of the straight line representing the tangent to the curve $y = x^2$ at the point $(1,1)$ is
Official answer: C
Engineering Mathematics - Linear Algebra
Let $\hat{i}, \hat{j}$, and $\hat{k}$ be the unit vectors in the x, y and z directions, respectively. If the vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^\circ$, one gets
Official answer: D
Engineering Mathematics - Complex Analysis & Fourier Series
Let $x$ be a real number and $i = \sqrt{-1}$. Then the real part of $\cos(ix)$ is
Official answer: B
Engineering Mathematics - Linear Algebra
If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s)
Official answer: B, C
Engineering Mathematics - Calculus
The arc length of the parametric curve: $x = \cos\theta,\ y = \sin\theta,\ z = \theta$ from $\theta = 0$ to $\theta = 2\pi$ is equal to ________ (round off to one decimal place).
Official answer: 8.6 to 9.1
Engineering Mathematics - Calculus
The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius $R$ is
Official answer: C
Engineering Mathematics - Differential Equations
Consider the differential equation $\dfrac{d^2y}{dx^2} - 2\dfrac{dy}{dx} + y = 0$. The boundary conditions are $y = 0$ and $\dfrac{dy}{dx} = 1$ at $x = 0$. Then the value of $y$ at $x = 1/2$ is
Official answer: C
Engineering Mathematics - Differential Equations
Consider the partial differential equation $\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} = 0$ where $x, y$ are real. If $f(x,y) = a(x)b(y)$, where $a(x)$ and $b(y)$ are real functions, which one of the following statements can be true?
Official answer: C
Engineering Mathematics - Calculus
The real function $y = \sin^2(|x|)$ is
Official answer: A, B
Engineering Mathematics - Calculus
$\vec{v} = x^3\hat{i} + y^3\hat{j} + z^3\hat{k}$ is a vector field where $\hat{i}, \hat{j}, \hat{k}$ are the base vectors of a cartesian coordinate system. Using the Gauss divergence theorem, the value of the outward flux of the vector field over the surface of a sphere of unit radius centered at the origin is ________ (rounded off to one decimal place).
Official answer: 7.4 to 7.7
Engineering Mathematics - Linear Algebra
The largest eigenvalue of the given matrix is _____. $$\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$
Official answer: 2
Engineering Mathematics - Linear Algebra
Let $\vec{a}, \vec{b}$ be two distinct vectors that are not parallel. The vector $\vec{c} = \vec{a} \times \vec{b}$ is
Official answer: C
Engineering Mathematics - Calculus
Consider the function $f(x,y) = \dfrac{x^2}{2} + \dfrac{y^2}{3} - 5$. All the roots of this function
Official answer: B
Engineering Mathematics - Calculus
Consider a vector field given by $x\hat{i} + y\hat{j} + z\hat{k}$. This vector field is
Official answer: B
Engineering Mathematics - Linear Algebra
The determinant of the matrix $\begin{bmatrix} 1 & 1 & -1 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{bmatrix}$ is ______ (accurate to one decimal place).
Official answer: 0
Engineering Mathematics - Differential Equations
The solution of the differential equation $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} = 0$, given that $y = 0$ and $\dfrac{dy}{dx} = 1$ at $x = 0$ is
Official answer: B
Engineering Mathematics - Calculus
Consider the vector field $\vec{v} = -\dfrac{y}{r^2}\hat{i} + \dfrac{x}{r^2}\hat{j}$; where $r = \sqrt{x^2 + y^2}$. The contour integral $\oint \vec{v}\cdot\vec{ds}$, where $\vec{ds}$ is tangent to the contour that encloses the origin, is __________ (accurate to two decimal places).
Official answer: 6.25 to 6.35
Engineering Mathematics - Calculus
The magnitude of the $x$-component of a unit vector at the point (1, 1) that is normal to equi-potential lines of the potential function $\phi(r) = \dfrac{1}{r^2+4}$, where $r = \sqrt{x^2+y^2}$, is _______ (accurate to two decimal places).
Official answer: 0.68 to 0.73
Engineering Mathematics - Linear Algebra
Consider an eigenvalue problem given by $\mathbf{A}x = \lambda_i x$. If $\lambda_i$ represent the eigenvalues of the non-singular square matrix $\mathbf{A}$, then what will be the eigenvalues of matrix $\mathbf{A}^2$?
Official answer: B
Engineering Mathematics - Linear Algebra
If $\mathbf{A}$ and $\mathbf{B}$ are both non-singular $n \times n$ matrices, then which of the following statement is NOT TRUE. Note: det represents the determinant of a matrix.
Official answer: B
Engineering Mathematics - Calculus
Let $x$ be a positive real number. The function $f(x) = x^2 + \dfrac{1}{x^2}$ has its minima at $x =$ ______.
Official answer: 1
Engineering Mathematics - Calculus
The vector $\vec{u}$ is defined as $\vec{u} = y\hat{e}_x - x\hat{e}_y$, where $\hat{e}_x$ and $\hat{e}_y$ are the unit vectors along x and y directions, respectively. If the vector $\vec{\omega}$ is defined as $\vec{\omega} = \vec{\nabla} \times \vec{u}$, then $\left|(\vec{\omega}\cdot\vec{\nabla})\vec{u}\right| =$ ______.
Official answer: 0
Engineering Mathematics - Differential Equations
The partial differential equation $\dfrac{\partial u}{\partial t} = \alpha\dfrac{\partial^2 u}{\partial x^2}$, where $\alpha$ is a positive constant, is
Official answer: D
Engineering Mathematics - Differential Equations
Consider a second order linear ordinary differential equation $\dfrac{d^2y}{dx^2} - 4\dfrac{dy}{dx} + 4y = 0$, with the boundary conditions $y(0) = 1$; $\dfrac{dy}{dx}\Big|_{x=0} = 1$. The value of $y$ at $x = 1$ is
Official answer: A
Engineering Mathematics - Linear Algebra
Consider the following system of linear equations: $$\begin{aligned} 2x - y + z &= 1 \\ 3x - 3y + 4z &= 6 \\ x - 2y + 3z &= 4 \end{aligned}$$ This system of linear equations has
Official answer: A
Engineering Mathematics - Calculus
The value of definite integral $\int_0^{\pi} (x\sin x)\,dx$ is _______.
Official answer: 3.13 to 3.15
Engineering Mathematics - Numerical Methods
Use Newton-Raphson method to solve the equation: $xe^x = 1$. Begin with the initial guess $x_0 = 0.5$. The solution after one step is $x =$ _________.
Official answer: 0.56 to 0.58
Engineering Mathematics - Linear Algebra
For a real symmetric matrix $[A]$, which of the following statements is true:
Official answer: C
Engineering Mathematics - Complex Analysis & Fourier Series
The series $s = \sum_{m=1}^{\infty} \dfrac{m^2}{3^m}(x-2)^m$ converges for all $x$ with $|x-2| \le R$ given by
Official answer: B
Engineering Mathematics - Calculus
The function given by $f(x) = \begin{cases} \sin(1/x), & x \neq 0 \\ 0, & x = 0 \end{cases}$ is
Official answer: C
Engineering Mathematics - Differential Equations
Given the boundary-value problem $\dfrac{d}{dx}\left(x\dfrac{dy}{dx}\right) + ky = 0,\ 0 < x < 1$, with $y(0) = y(1) = 0$. Then the solutions of the boundary-value problem for $k = 1$ (given by $y_1$) and $k = 5$ (given by $y_5$) satisfy:
Official answer: A
Engineering Mathematics - Calculus
The value of $I = \int_0^1 1000x^4\,dx$, obtained by using Simpson's rule with 2 equally spaced intervals, is,
Official answer: D
Engineering Mathematics - Linear Algebra
If $[A] = \begin{bmatrix} 3 & -3 \\ -3 & 4 \end{bmatrix}$. Then $\det\left(-[A]^2 + 7[A] - 3[I]\right)$ is
Official answer: A
Engineering Mathematics - Complex Analysis & Fourier Series
For the periodic function given by $f(x) = \begin{cases} -2, & -\pi < x < 0 \\ 2, & 0 < x < \pi \end{cases}$ with $f(x + 2\pi) = f(x)$, using Fourier series, the sum $s = 1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \cdots$ converges to
Official answer: C
Engineering Mathematics - Calculus
Let $\Gamma$ be the boundary of the closed circular region $A$ given by $x^2 + y^2 \le 1$. Then $I = \oint_\Gamma (3x^3 - 9xy^2)\,ds$ (where $ds$ means integration along the bounding curve) is
Official answer: D
Engineering Mathematics - Differential Equations
Solution to the boundary-value problem $-9\dfrac{d^2u}{dx^2} + u = 5x,\ 0 < x < 3$ with $u(0) = 0,\ \dfrac{du}{dx}\Big|_{x=3} = 0$ is
Official answer: A
Engineering Mathematics - Differential Equations
The Laplace transform $\mathcal{L}(u(t)) = U(s)$, for the solution $u(t)$ of the problem $\dfrac{d^2u}{dt^2} + 2\dfrac{du}{dt} + u = 1,\ t > 0$ with initial conditions $u(0) = 0,\ \dfrac{du(0)}{dt} = 5$ is given by:
Official answer: B