List of top Mathematics Questions on Calculus asked in GATE MA

Let \( \vec{F} = (y - z)\hat{i} + (z - x)\hat{j} + (x - y)\hat{k} \) be a vector field, and let \( S \) be the surface \( x^2 + y^2 + (z - 1)^2 = 9, 1 \leq z \leq 4 \).
If \( \hat{n} \) denotes the unit outward normal vector to \( S \), then the value of

\[ \frac{1}{\pi} \left| \iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS \right| \]

is equal to _________ (in integer).

Consider \[ I = \frac{1}{2\pi i} \int_C \frac{\sin z}{1 - \cos(z^3)} \, dz, \] where \( C = \{ z \in \mathbb{C} : z = x + iy, |x| + |y| = 1, x, y \in \mathbb{R} \} \) is oriented positively as a simple closed curve. Then, the value of \( 120I \) is equal to _________ (in integer).

Let \( \alpha, \beta, \gamma, \delta \in \mathbb{R} \) be such that the quadrature formula \[ \int_{-1}^{1} f(x) \, dx = \alpha f(-1) + \beta f(1) + \gamma f'(-1) + \delta f'(1) \] is exact for all polynomials of degree less than or equal to 3. Then, \( 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \) is equal to (in integer):
Let \( y(x) \) be the solution of the initial value problem \[ \frac{dy}{dx} = \sin(\pi(x + y)), \quad y(0) = 0. \] Using Euler's method, with the step-size \( h = 0.5 \), the approximate value of \( y(1.5) + 2y(1) \) is equal to (in integer):
Let \( \vec{F} = (y - z)\hat{i} + (z - x)\hat{j} + (x - y)\hat{k} \) be a vector field, and let \( S \) be the surface \( x^2 + y^2 + (z - 1)^2 = 9, 1 \leq z \leq 4 \). If \( \hat{n} \) denotes the unit outward normal vector to \( S \), then the value of \[ \frac{1}{\pi} \left| \iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS \right| \] is equal to _________ (in integer).
Consider \[ I = \frac{1}{2\pi i} \int_C \frac{\sin z}{1 - \cos(z^3)} \, dz, \] where \( C = \{ z \in \mathbb{C} : z = x + iy, |x| + |y| = 1, x, y \in \mathbb{R} \} \) is oriented positively as a simple closed curve. Then, the value of \( 120I \) is equal to _________ (in integer).
Let \( \alpha, \beta, \gamma, \delta \in \mathbb{R} \) be such that the quadrature formula \[ \int_{-1}^{1} f(x) \, dx = \alpha f(-1) + \beta f(1) + \gamma f'(-1) + \delta f'(1) \] is exact for all polynomials of degree less than or equal to 3. Then, \( 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \) is equal to (in integer):
Let \( y(x) \) be the solution of the initial value problem \[ \frac{dy}{dx} = \sin(\pi(x + y)), \quad y(0) = 0. \] Using Euler's method, with the step-size \( h = 0.5 \), the approximate value of \( y(1.5) + 2y(1) \) is equal to (in integer):
Consider the sequence of Lebesgue measurable functions \(f_n: \mathbb{R} \to \mathbb{R}\) given by
\[ f_n(x) = \begin{cases} n^2(x-n), & \text{if } x \in [n, n + \frac{1}{n^2}] \\ 0, & \text{otherwise} \end{cases} \]
For a measurable subset E of \(\mathbb{R}\), denote m(E) to be the Lebesgue measure of E.
Which of the following statements is/are correct?
Define the characteristic function \(\chi_E\) of a subset E in \(\mathbb{R}\) by
\[ \chi_E(x) = \begin{cases} 1, & \text{if } x \in E \\ 0, & \text{if } x \notin E \end{cases} \]
For \(1 \le p < 2\), let \(L^p[0,1] = \{f: [0,1] \to \mathbb{R} : f \text{ is Lebesgue measurable and } \int_0^1 |f(x)|^p dx < \infty\}\).
Let \(f: [0,1] \to \mathbb{R}\) be defined by
\[ f(x) = \sum_{n=1}^\infty \frac{2^n}{n^3} \chi_{[\frac{1}{2^{n+1}}, \frac{1}{2^n}]}(x). \]
Consider the following two statements:
P: \(f \in L^p[0,1]\) for every \(p \in (1, 2)\).
Q: \(f \in L^1[0,1]\).
Then
Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{2x^2y}{x^2 + y^2}, & (x, y) \neq (0,0) \\ 0, & (x, y) = (0,0) \end{cases} \] Then
Let \(L^2[-1, 1] = \{f: [-1,1] \to \mathbb{R} : f \text{ is Lebesgue measurable and } \int_{-1}^1 |f(x)|^2dx < \infty\}\) and the norm \(||f||_2 = \left(\int_{-1}^1 |f(x)|^2 dx\right)^{1/2}\) for \(f \in L^2[-1,1]\).
Let \(F: (L^2[-1,1], ||.||_2) \to \mathbb{R}\) be defined by \[ F(f) = \int_{-1}^1 f(x)x^2dx \quad \text{for all } f \in L^2[-1,1]. \] If \(||F||\) denotes the norm of the linear functional F, then \(5||F||^2\) is equal to ................
Let C[0,1] denote the set of all real valued continuous functions defined on [0,1] and \(||f||_\infty = \sup\{|f(x)| : x \in [0,1]\}\) for all \(f \in C[0,1]\). Let
\[ X = \{ f \in C[0,1] : f(0) = f(1) = 0 \}. \]
Define \(F : (C[0,1], ||.||_\infty) \to \mathbb{R}\) by \(F(f) = \int_0^1 f(t)dt\) for all \(f \in C[0,1]\).
Denote \(S_X = \{f \in X : ||f||_\infty = 1\}\).
Then the set \(\{f \in X : F(f) = ||F||\} \cap S_X\) has
Let \( f, g: \mathbb{R}^2 \to \mathbb{R} \) be defined by
\[ f(x, y) = x^2 - \frac{3}{2}xy^2 \quad \text{and} \quad g(x, y) = 4x^4 - 5x^2y + y^2 \] for all \( (x, y) \in \mathbb{R}^2 \).
Consider the following statements:
P: \( f \) has a saddle point at (0, 0).
Q: \( g \) has a saddle point at (0, 0).
Then
Let C be the curve of intersection of the cylinder \( x^2 + y^2 = 4 \) and the plane \( z - 2 = 0 \). Suppose C is oriented in the counterclockwise direction around the z-axis, when viewed from above. If \[ \int_C ( \sin x + e^x ) dx + 4x dy + e^z \cos^2 z dz = \alpha\pi, \] then the value of \( \alpha \) equals ..................
If for some \(a \in \mathbb{R}\), \[ \int_1^4 \int_{-x}^x \frac{1}{x^2 + y^2} dydx = \int_{-\pi/4}^{\pi/4} \int_{\sec\theta}^{a \sec\theta} \frac{1}{r} drd\theta, \] then the value of \(a\) equals ................
Let S be the portion of the plane \(z = 2x + 2y - 100\) which lies inside the cylinder \(x^2 + y^2 = 1\). If the surface area of S is \(a\pi\), then the value of a is equal to ................