GATE AE (Aerospace Engineering) Calculus previous year questions from the Engineering Mathematics section, with figures and 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Calculus
The value of definite integral $\int_0^{\pi} (x\sin x)\,dx$ is _______.
Official answer: 3.13 to 3.15
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 - 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 - 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