Practice Problems in Mathematics

Let \(\vec{u}, \vec{v}\) and \(\vec{w}\) be vectors in three-dimensional space, where \(\vec{u}\) and \(\vec{v}\) are unit vectors which are not perpendicular to each other and \[\vec{u}\cdot \vec{w} = 1,\quad \vec{v}\cdot \vec{w} = 1\quad \vec{w}\cdot \vec{w} = 4\] If the volume of the parallelopiped, whose adjacent sides are represented by the vectors \(\vec{u}, \vec{v}\) and \(\vec{w}\), is \(\sqrt{2}\), then the value of \(|3\vec{u} + 5\vec{v}|\) is ______.

In a triangle ABC, let \(AB = \sqrt{23}\), and \(BC = 3\) and \(CA = 4\). Then the value of \(\dfrac{\cot A + \cot C}{\cot B}\) is ______.

For \(x\in\mathbb{R}\), thi number of real roots of the equation \(3x^2-4|x^2 - 1| + x - 1 = 0\) is ______.

For any complex number \(w = c + id\), let \(arg(w)\in (-\pi,\pi]\), where \(i=\sqrt{-1}\). Let \(\alpha\) and \(\beta\) be real numbers such that for all complex numbers \(z = x + iy\) satisfying \(arg\Big[\dfrac{z + \alpha}{z + \beta}\Big]=\dfrac{\pi}{4}\), the ordered pair \((x, y)\) lies on the circle \(x^2 + y^2 + 5x - 3y + 4 = 0\). Then, which of the following statements is (are) True?

1. \(\alpha = -1\)
2. \(\alpha\beta = 4\)
3. \(\alpha\beta = -4\)
4. \(\beta = 4\)

For any positive integer \(n\), let \(S_n : (0,\infty)\to \mathbb{R}\) be defined by \[S_n(x) = \sum_{k=1}^{n}\text{cot}^{-1}\Big(\frac{1 + k(k + 1)x^2}{x}\Big)\] where for any \(x\in\mathbb{R}, \text{cot}^{-1}(x)\in (0, \pi)\) and \(\text{tan}^{-1}(x)\in \Big(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\Big)\). Then which of the following statements is (are) True?

1. \(S_{10}(x) = \dfrac{\pi}{2}-\tan^{-1}\Big(\dfrac{1 + 11x^2}{10x}\Big)\), for all \(x>0\)
2. \(\lim_{n\to \infty}\cot(S_n(x)) = x\), for all \(x>0\)
3. The equation \(S_3(x)=\dfrac{\pi}{4}\) has a root in \((0,\infty)\)
4. \(\tan(S_n(x))\leq \dfrac{1}{2}\), for all \(n\geq 1\) and \(x > 0\)

For any \(3\times 3\) matrix \(M\), let \(|M|\) denote the determinant of \(M\). Let \(I\) be the \(3\times 3\) identity matrix. Let \(E\) and \(F\) be two \(3\times 3\) matrices such that \(I-EF\) is invertible. If \(G = (I - EF)^{-1}\), then which of the following statements is (are) True?

1. \(|FE| = |I-FE||FGE|\)
2. \((I-FE)(I + FGE) = I\)
3. \(EFG = GEF\)
4. \((I - FE)(I-FGE) = I\)