Vector Calculus Integration Quiz

The lengths of portions of tangents inside the triangle ABC, parallel to the sides BC, CA, AB are denoted by λ1, λ2, and λ3. If AF = 3, BD = 5, CE = 4, then what is the value of [λ1 + λ2 + λ3]?

If g(x + y) = g(x) + g(y) + 1 and g(3f(x)) = 0 for all x, y in real numbers, what is the value g(f^2(x) + 2) + g(x^2f^2(x)) - g(x^2f(x)) - x?

If A, B are matrices of size 2x2 and A + B = 2x[−x f(x + f(1))] / 1− x2, then what is the value of f(1) - f(1-1) when A = B^T?

What is the scalar triple product of the vectors $ f{a} = egin{pmatrix} 1 \ 2 \ -1 \ f{b} = egin{pmatrix} -3 \ 0 \ 4 \ f{c} = egin{pmatrix} 2 \ -2 \ 5$?

If a function f is continuous on R and f(2023) = f(2024) ≠ f(0), what is the possible value of f(0)?

If g'(0) ≠ 0, then for function g: R R and f: R R, where g(f(x)) = 0, what can be inferred about the function f?

What is the value of the scalar triple product $\int_0^x f(t) dt$, where $f(x) = \cos x \hat{i} + \sin x \hat{j} + \cos 2x \hat{k}$?

Let $a_1 = a\hat{i} + b\hat{j}$, $a_2 = a_1 + 2\hat{i} - 5\hat{j}$, and $a_3 = a_1 + 2\hat{i} - 5\hat{j}$. What is the value of the matrix $\begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{bmatrix}$?

If $f(x) = \cos x \hat{i} + \sin x \hat{j} + \cos 2x \hat{k}$, $g(x) = \tan x \hat{i} + \sin 3x \hat{j} + \cos 4x \hat{k}$, and $h(x) = \cos 3x \hat{i} + \sin 5x \hat{j} + \cos 6x \hat{k}$, then the scalar triple product $\int_0^{\pi/2} [f(x) \times g(x)] \cdot h(x) dx$ belongs to which of the following intervals?

If $f(x) = \cos x \hat{i} + \sin x \hat{j} + \cos 2x \hat{k}$, $g(x) = \tan x \hat{i} + \sin 3x \hat{j} + \cos 4x \hat{k}$, and $h(x) = \cos 3x \hat{i} + \sin 5x \hat{j} + \cos 6x \hat{k}$, which of the following statements about the vector $f(x) + g(x) + h(x)$ is true?

If $f(x) = \cos x \hat{i} + \sin x \hat{j} + \cos 2x \hat{k}$, $g(x) = \tan x \hat{i} + \sin 3x \hat{j} + \cos 4x \hat{k}$, and $h(x) = \cos 3x \hat{i} + \sin 5x \hat{j} + \cos 6x \hat{k}$, which of the following statements about the vector field $\nabla \cdot (f(x) + g(x) + h(x))$ is true?

If $f(x) = \cos x \hat{i} + \sin x \hat{j} + \cos 2x \hat{k}$, $g(x) = \tan x \hat{i} + \sin 3x \hat{j} + \cos 4x \hat{k}$, and $h(x) = \cos 3x \hat{i} + \sin 5x \hat{j} + \cos 6x \hat{k}$, which of the following statements about the vector field $\nabla \times (f(x) + g(x) + h(x))$ is true?

Given the function $f(x) = x^3 - x - 1$, suppose $g(x)$ is a cubic polynomial such that $g(0) = -1$ and the roots of $g(x)$ are the square of the roots of $f(x)$. What is the relationship between $g(x^2)$ and $f(x)?

In a $ riangle ABC$, if $ ext{sec} A = anrac{ ext{pi}}{4}$, $ ext{sec} B = anrac{ ext{pi}}{3}$, and $ ext{sec} C = anrac{ ext{pi}}{6}$, what is the value of $\frac{1}{2} \rac{ ext{sin}^2 A + ext{sin}^2 B + ext{sin}^2 C}{2}?$

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors. Which of the following expressions represents the scalar triple product of these vectors?

If $f(x) = x^3 - 2x + 1$ and $g(x) = x^2 - 3x + 2$, which of the following expressions represents the composition of $f$ and $g, i.e., $f(g(x))?