Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \leq 0$. Which of the following is true? (A) $f$ is not differentiable at $x = 0$ (B) $f$ is differentiable at $x = 0$ but $f'$ is not differentiable at $x = 0$ (C) $f$ is differentiable at $x = 0$ and $f'$ is differentiable at $x = 0$ (D) $f$ is differentiable everywhere and $f'$ is also differentiable everywhere
Which of the following is true about $\tan(\sin x)$? (A) $\tan(\sin x) = 1$ has solutions (B) $\tan(\sin x) \geq 1$ for some $x$ (C) $\tan(\sin x) < 1$ for all $x$ (D) $\tan(\sin x)$ never attains the value $1$
Let $f(x) = \dfrac{x^2}{x-1}$. Which of the following is true? (A) $f$ is neither one-one nor onto (B) $f$ is one-one and onto (C) $f$ is one-one but not onto (D) $f$ is onto but not one-one
Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$? (A) $h(t)$ depends on $t$ (B) $h(t)$ is not defined for all $t$ (C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$ (D) None of the above
Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true? (A) $f$ has exactly two real roots (B) $f$ has no real roots (C) $f$ has four real roots (D) $f$ has exactly two real roots, one of which is $-1$
In how many ways can $10$ A's and $6$ B's be arranged in a row such that no two B's are adjacent and no two A's are adjacent? (More precisely: find the number of arrangements of $10$ A's and $6$ B's such that between any two consecutive B's there is at least one A, and between any two consecutive A's there is at least one B.)
Let $f(x) = x|x|^n$ for $n \geq 1$ a positive integer. Which of the following is true? (A) $f$ is differentiable everywhere except at $x = 0$ (B) $f$ is continuous but not differentiable at $x = 0$ (C) $f$ is differentiable everywhere (D) None of the above
Let $x_n = \dfrac{1}{(2a)^n}$ for $a > 0$. Which of the following is true? (A) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$ (B) $x_n \to 0$ if $0 < a < 1/2$ and $x_n \to \infty$ if $a > 1/2$ (C) $x_n \to 1$ for all $a > 0$ (D) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$
For $\theta \in (0, \pi/2)$, which of the following is true? (A) $\cos(\sin\theta) < \cos\theta$ (B) $\cos(\sin\theta) < \sin(\cos\theta)$ (C) $\cos(\sin\theta) > \cos\theta$ (D) $\cos(\sin\theta) > \sin(\cos\theta)$
Let $A = \{(x,y) : x^4 + y^2 \leq 1\}$ and $B = \{(x,y) : x^6 + y^4 \leq 1\}$. Which of the following is true? (A) $A = B$ (B) $A \subset B$ (A is a proper subset of B) (C) $B \subset A$ (B is a proper subset of A) (D) Neither $A \subset B$ nor $B \subset A$
Using AM-GM inequality, find a lower bound for $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right)$, and determine which of the following holds: (A) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) < \left(\dfrac{2^{10}}{11}\right)^{11}$ (B) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) = \left(\dfrac{2^{10}}{11}\right)^{11}$ (C) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) > \left(\dfrac{2^{10}}{11}\right)^{11}$ (D) None of the above