Continuity and Differentiability - Class 12

Evaluate the following limit: $\lim_{x \to 0} \frac{\cos^{-1}\left( e^{x^2} + [x^2] - x^2 \right)}{|x|}$ where $[ \cdot ]$ denotes the greatest integer function.

Let $f(x) = \frac{|(x-2)(x-4)(x-a)|}{(x-2)(x-4)(x-a)} \sum_{k=1}^5 \frac{k^2 - a^2}{x-k}$. Find the sum of all real values of $a$ for which $\lim_{x \to a} f(x)$ exists.

Let $y = y(x)$ be the solution curve of the differential equation $x \frac{dy}{dx} = y + |x| \sqrt{1 - \left(\frac{y}{x}\right)^2}$ satisfying the initial condition $y(-1) = 0$. Then the value of $y(-e^{\pi/4})$ is:

Let $A$ be a $3 \times 3$ non-zero matrix with real entries such that $A^2 = \mathbf{0}$. Let $M(\lambda) = (\lambda - 2)A + (\lambda^2 - 4)A^2$. A new matrix $B(\lambda)$ is formed by first exchanging the first and third rows of $M(\lambda)$, and then exchanging the first and second columns of the resulting matrix. If $\lambda \in \mathbb{R}$ satisfies the equation $\det(B(\lambda)) + (\lambda - 2)^2 (\lambda - 1) = (\lambda - 2)^2 (\lambda^2 - 9\lambda + 20)$ then the sum of all possible values of $\lambda$ is:

Let $S$ be the set of all distinct complex numbers $z$ satisfying the equation $(z^5 - 1)^2 = z^4 (z - 1)^2$ Find the value of $\sum_{z \in S} |z - 3|^2$.

Let $f(x) = (x^2 - 6x + 8)^{4/5}$ be a function defined on the interval $[0, 6]$. Let $m$ be the number of points in the open interval $(0, 6)$ where $f(x)$ is not differentiable. Let $n$ be the number of points $c \in (0, 6)$ such that $f'(c) = 0$. Let $k = 1$ if Rolle's Theorem is applicable to $f(x)$ on $[0, 6]$, and $k = 0$ otherwise. Find the value of $2^m + 3^n + 5^k$.

Evaluate the limit: $\lim_{x \to \frac{\pi}{4}^+} \frac{\sqrt{2 - 2\sin 2x} + \sqrt{2}(\cos x - \sin x) + 2\ln\left(1 + \tan\left(x - \frac{\pi}{4}\right)\right)}{\cos 2x}$

Consider the function $f(x) = \\left[ \\frac{6}{\\sqrt{x^2 - 2x} + 1} \\right]$, where $[ \\cdot ]$ denotes the greatest integer function. Let $D$ be the domain of $f(x)$. Find the number of points of discontinuity of $f(x)$ in the set $D \\cap [-3, 5]$.

Let $f(x) = \ln\left(\frac{2x}{1-x^2}\right)$ and $g(x) = \tan x$. Consider the composite function $h(x) = f(g(x))$. If $S$ is the largest subset of $(0, \pi)$ on which the derivative $h'(x)$ exists and is continuous, then $S$ is equal to:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{cases} \lim_{n \to \infty} \frac{x^2(3+\sin(1/x))}{1 + x^{2n}}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ Let $g(x) = \text{sgn}(f'(x))$ for $x \in (-1, 1) \setminus \{0\}$. Evaluate the pair $(L_1, L_2)$, where $L_1 = \limsup_{x \to 0^-} g(x) \quad \text{and} \quad L_2 = \liminf_{x \to 0^-} g(x)$

Let $M(x) = \begin{pmatrix} \sin x & \cos x & \cos(x-\frac{\pi}{6}) \\ \sin 2x & \cos 2x & \cos(2x-\frac{\pi}{6}) \\ \sin 3x & \cos 3x & \cos(3x-\frac{\pi}{6}) + \sqrt{3}\sin x \end{pmatrix}$ and $N(x) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \sin x \end{pmatrix}$.

Let $S = \{ x \in [0, 2\pi] : \det(M(x)) \neq 0 \}$. For $x \in S$, define $g(x) = \det(M(x)^{-1} N(x))$.

A function $h: [0, 2\pi] \to \mathbb{R}$ is defined as: $h(x) = \begin{cases} \lim_{n \to \infty} \frac{1}{1 + |g(x)|^n} & \text{if } x \in S \\ 0 & \text{if } x \notin S \end{cases}$

Find the number of points in $[0, 2\pi]$ where $h(x)$ is discontinuous.

Let $y = f(x)$ be the unique real-valued continuous function satisfying the implicit equation $y^3 + y^2 \sin(|x|) + y = x^2 + 2$ for all $x \in \mathbb{R}$. If $L = \lim_{x \to 0^+} f'(x)$ and $R = \lim_{x \to 0^-} f'(x)$, then the value of $16|L \cdot R| + 8|L - R|$ is

Let $f(x) = \begin{vmatrix} \sin(\pi x) \cos(\pi \{x\}) & (x-2)^2 \\ \sin(\pi x) & x-2 \end{vmatrix}$, where $\{x\}$ denotes the fractional part of $x$. Let $S$ be the set of all points in the interval $(0, 5)$ where $f(x)$ is not differentiable. Find the sum of all elements in $S$.

Let $y = y(x)$ be a real-valued continuous function defined on $\mathbb{R}$ satisfying the implicit equation: $y^3 + 2y + \cos(xy) = \int_0^x \max(2, 2e^{2t}) \, dt + 1$ Find the value of $y''(0)$.

Consider the differential equation $\frac{dy}{dx} = \frac{2x - x^3 - x y^3}{3y^2}$ Let $y(x)$ be the real-valued solution satisfying $y(0) = \sqrt[3]{4}$. If $(a, b)$ is the maximal open interval on which $y(x)$ is a valid differentiable solution, find the value of $b - a$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = |x^2 - 4|^{|x-2|} + |x^2 - 2x + 2|^{|x^2 - 1|} + |x^2 - 9|^{|x+3|}$ Let $S$ be the set of all points $x \in \mathbb{R}$ where $f(x)$ is not differentiable. Find the sum of all elements in $S$.

Let $y = f(x) > 0$ be a real-valued function defined on its maximal domain $D \subset \mathbb{R}$ by the implicit relation $y^{x \ln x} = e^{x^2 - x}$. Let $L = \lim_{x \to 1} f'(x)$. Which of the following represents the ordered pair $(L, f'(1))$?

Let $A(x) = \begin{pmatrix} x & 0 & 2 \\ 0 & x & 0 \\ 2 & 0 & x \end{pmatrix}$. Let $f(x) = (\det(A(x)))^{\det(A(x))}$ be a real-valued function defined on its maximal domain $D \subset \mathbb{R}$. Find the number of distinct real solutions to the equation $f'(x) = 0$.

Let $y = f(x)$ be a continuously differentiable function defined on the entire interval $(0, \infty)$ that satisfies the differential equation $x^2 \frac{dy}{dx} = y^2 - 5xy + 9x^2$ Then the value of $f(e^2)$ is:

Let $f: (1, \infty) \to \mathbb{R}$ be a function defined as $f(x) = \lim_{n \to \infty} \prod_{k=1}^{n} \left( 1 + \frac{k^2}{n^2} \ln x \right)^{\frac{k}{n^2 \ln x}}$ Find the value of $e \sqrt{e} f'(e)$.

Let $S$ be the set of all $x \in (0, 4\pi)$ for which there exists at least one real number $y$ such that the point $(x, y)$ lies on the curve $2\cos(y) + \cos(2x) = \sqrt{2}$ and the tangent to the curve at $(x, y)$ is parallel to the $x$-axis. The sum of all elements in $S$ is

Let $y = f(x)$ be a differentiable function defined for $x > 0$ that satisfies the equation $\begin{vmatrix} x^2 & y & 1 \\ 2x & y' & \frac{1}{x} \\ a & b & 0 \end{vmatrix} + \begin{vmatrix} y & x^2 & 1 \\ y' & 2x & \frac{1}{x} \\ -a & -b & 0 \end{vmatrix} = 0$ where $a$ and $b$ are distinct positive integers. If the curve passes through the points $(1, a+b-2)$ and $(2, ab)$, find the value of $f(3)$.

Let $y = f(x)$ be a real-valued continuous function defined on its maximal domain $D \subset (0, 2\pi)$ satisfying the implicit equation $y^{\sin x} = (\sin x)^y$ with $y > 0$. Let $x_0 \in D$ be the point where $f(x)$ attains its global maximum. Find the value of $\lim_{x \to x_0} \frac{f(x) - 1}{(2x - \pi)^2}$

Let $C$ be a curve defined parametrically by $x = t + 1$ and $y = t^4 - 15t^2 - 12$ for $t \in \left[-\frac{3}{2}, \frac{5}{2}\right]$. The origin of the coordinate system is shifted to the point $(1, 0)$ without rotating the axes, establishing a new $(X, Y)$ coordinate system. Let $S$ be the sum of the slopes of all lines passing through the new origin $(0,0)$ that are tangent to $C$. Find the value of $S$.

A curve is defined parametrically by $x(t) = t - \sin t$ and $y(t) = \frac{1}{3}(1 - \cos t)^{3/2}$ for $t \in (-\pi, \pi)$. Let $y = f(x)$ be the continuous real-valued function representing this curve. If $L$ and $R$ denote the left-hand and right-hand derivatives of $f(x)$ at $x = 0$ respectively, find the value of $R - L$.

Let $z_1, z_2, z_3, z_4$ be the roots of the equation $z^4 - 2z^3 + 3z^2 - 4z + 5 = 0$. Let $A(x)$ be a $4 \times 4$ matrix whose entries are given by $A_{ij} = z_i z_j$ for $i \neq j$ and $A_{ii} = \sqrt{|x^3 - 3x + 2|} + z_i^2$ for $1 \le i, j \le 4$. If $f(x) = \det(A(x))$, find the number of points $x \in \mathbb{R}$ where $f(x)$ is NOT twice differentiable.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \int_0^x (t^2 - 2at + 3a + 4) e^{t^2} dt$ where $a$ is a real parameter. Let $S$ be the set of all integers $a$ such that $f(x)$ is invertible and its inverse function $g(x)$ has a continuous second derivative for all $x$ in the domain of $g$. Find the sum of all elements in $S$.

Let $y = f(x)$ be the unique continuous real-valued function defined implicitly by $y^3 - 3y - x = 0$ for $x \in [-2, 2]$ such that $f(0) = 0$. Let $g(x) = (x^2-4) f''(x)$. Consider the applicability of Rolle's Theorem to $g(x)$ on the interval $[-2, 2]$ and the value of the limit $L = \lim_{x \to 0} \frac{g(x)}{x}$. Which of the following statements is correct?

Let $\vec{a}(t) = (t^2 - 3)\hat{i} + k\hat{j}$ and $\vec{b}(t) = t\hat{i} + \hat{j}$, where $k$ is a real parameter. A scalar function $f(t)$ is defined as: $f(t) = (\vec{a}(t) \cdot \vec{b}(t)) \sqrt{|\vec{a}(t)|^2 |\vec{b}(t)|^2 - |\vec{a}(t) \times \vec{b}(t)|^2}$ Let $S$ be the set of all real values of $k$ for which $f(t)$ is twice differentiable for all $t \in \mathbb{R}$. The number of elements in $S$ is:

Let $k = 2\sqrt{2}$ and consider the function $F(x) = \int_{-k}^{x} \frac{4 - t^{4/3}}{2 t^{1/3}} dt$ defined for $x \in [-k, k]$. Let $m$ be the number of points $c \in (-k, k)$ such that $F'(c) = 0$, and let $n$ be the number of points in $(-k, k)$ where $F(x)$ is not differentiable. Find the value of $2m + 3n$.

Let $A(x)$ be a $3 \times 3$ matrix with differentiable real-valued functions as entries. For all $x \in \mathbb{R}$, the traces of the powers of $A(x)$ are given by: $\text{tr}(A(x)) = x$ $\text{tr}(A^2(x)) = 5x^2 - 22$ $\text{tr}(A^3(x)) = 10x^3 - 18x^2 - 18$ Let $f(x) = \text{tr}(\text{adj}(A(x)) \cdot A'(x))$. If $\alpha$ and $\beta$ are the roots of the equation $f(x) = 0$, find the value of $3|\alpha - \beta|^2$.

Let $S_n(x) = \sum_{k=1}^n \frac{x(1 - k(k-1)x^2)}{(1+(k-1)^2 x^2)(1+k^2 x^2)}$.

Since $S_n(0) = 0$ and $\lim_{x \to \infty} S_n(x) = 0$, by Rolle's Theorem, $S_n(x)$ has at least one critical point in $(0, \infty)$. Let $R_n$ be the unique positive root of $S_n'(x) = 0$.

Evaluate the limit: $\lim_{n \to \infty} n \int_0^{R_n} S_n(x) dx$

Let $f(x) = \lim_{t \to 0} \left( \cos t + \sin\left(t \sqrt{x^4 - 2x^2 + 1}\right) \right)^{\csc t}$ for $x \in [-2, 2]$. Let $N$ be the number of points in $(-2, 2)$ where $f(x)$ is non-differentiable, and $M$ be the number of points in $(-2, 2)$ where $f'(x) = 0$. Find the value of $N + M$.

Let $H$ be the right branch of the hyperbola $x^2 - y^2 = 1$. For a point $P(x,y)$ on $H$ with $y \ge 0$, let $d(x)$ be the perpendicular distance from $P$ to the asymptote $y=x$. Define the function $f(x) = d(x) \ln x$ for $x \ge 1$. Let $S$ be the set of points $x \in (1, \infty)$ where the tangent to the curve $y = f(x)$ is horizontal. Which of the following statements is TRUE?

Let $F(x) = \int_1^x \frac{[t]}{t^2} dt$, where $[ \cdot ]$ denotes the greatest integer function. Let $L_1 = \lim_{x \to 2^+} \frac{F(x) - F(2)}{x - 2}$ and $L_2 = \lim_{x \to 2^-} \frac{F(x) - F(2)}{x - 2}$. If $L = \frac{L_1}{L_2}$, consider the sequence $I_n = \int_1^n \frac{[t]}{t^L} dt$ for integers $n \ge 2$. Which of the following inequalities strictly holds true for all $n \ge 2$?

Let $\omega_k = \exp\left(\frac{i 2\pi k}{13}\right)$ for $k=0, 1, \dots, 12$. Define the function $g: \mathbb{R} \to \mathbb{R}$ as $g(x) = \frac{1}{13} \sum_{k=0}^{12} (x - \omega_k)^{13}$ Let $f(x) = [g(x)] \ln(1 + 3x) + \text{sgn}(g(x)) \sin(2\pi x)$, where $[ \cdot ]$ denotes the greatest integer function and $\text{sgn}(\cdot)$ denotes the signum function. If $L$ and $R$ are the left-hand and right-hand derivatives of $f(x)$ at $x=0$ respectively, find the value of $L + R$.

A curve $C$ is defined parametrically by $x(t) = t^3 + t$ $y(t) = -2t^3 + 9t^2 - 12t + 7[t]$ for $t \in [0, 4]$, where $[ \cdot ]$ denotes the greatest integer function. What is the absolute maximum value of $y$ on the curve $C$?

Let $A(x) = \begin{pmatrix} 2\{x\} & [x]^2 - 4[x] + 3 \\ [x]^2 - 8[x] + 15 & -2\{x\} \end{pmatrix}$, where $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}$ denotes the fractional part of $x$. Let $S$ be the set of all $x \in [0, 10]$ such that $A(x)$ is a nilpotent matrix. Find the sum of all elements in $S$.

Evaluate the sum $S = \sum_{n=1}^{120} \left( \left[ \sqrt{n} + \frac{1}{\sqrt{n}} \right] - \left[ \sqrt{n} \right] \right)$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $A(x)$ be a $3 \times 3$ matrix with differentiable entries such that $A(1) = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{pmatrix} \quad \text{and} \quad A'(1) = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & 3 \\ 2 & 0 & 1 \end{pmatrix}$ Let $f(x) = \det(A(x))$. If $g(x)$ is the local inverse function of $f(x)$ in a neighborhood of $x=1$, what is the value of $g'(-8)$?

Let $f: \left(-\frac{\pi}{2}, 0\right) \to \mathbb{R}$ be defined as $f(x) = 2x^2 + 6x + \cos^{-1}\left(\sin\left(|x| - \frac{\pi}{6}\right)\right)$. If $g(x)$ is the inverse function of $f(x)$, then the value of $g'\left(\frac{\pi^2}{18} - \frac{\pi}{2}\right)$ is equal to:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 15x^7 - 140x^6 + 504x^5 - 840x^4 + 560x^3 + 59$. Let $g(x)$ be the inverse function of $f(x)$. If $S$ is the set of all real numbers $y$ such that $g$ is not differentiable at $y$, then the sum of all elements in $S$ is

Let $f: [1, e] \to \mathbb{R}$ be defined by $f(x) = \int_1^x t \ln t \, dt + \int_1^x [t] \, dt$, where $[t]$ denotes the greatest integer less than or equal to $t$. If $g(x)$ is the inverse function of $f(x)$, then the value of $g'\left(2 \ln 2 + \frac{1}{4}\right)$ is:

Let $H$ be the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$. A point $P(x, y)$ moves on the right branch of $H$ such that $y > 0$. Let $L$ be the polar of $P$ with respect to the conjugate hyperbola of $H$. If the slope of $L$ is denoted by $m$, then $y$ can be expressed as a function of $m$, say $y = g(m)$. Find the value of $-64 \cdot g'\left(\frac{20}{9}\right)$.

Let $F(x) = \int_{x}^{x^2} [|273 - 90t|] \, dt$, where $[ \cdot ]$ denotes the greatest integer function. Find the value of $F'(2^+) - F'(2^-)$.

Let $A(x) = \begin{pmatrix} |x-1| & x^2 \\ 2 & |x| \end{pmatrix}$ and $B(x) = \begin{pmatrix} x & -|x-1| \\ |x| & 2x \end{pmatrix}$. Define the matrix $M(x) = A(x)B(x)$. Let $D_1 = \det(M'(1^+))$ and $D_2 = \det(M'(1^-))$, where $M'(1^+)$ and $M'(1^-)$ denote the right-hand and left-hand derivatives of the matrix $M(x)$ with respect to $x$ at $x=1$, respectively. Find the value of $D_1 + D_2$.

Let $f: [246, 248] \rightarrow \mathbb{R}$ be defined by $f(x) = |x^2 - 496x + 61503|$. Let $S$ be the set of all $c \in (246, 248)$ such that $f$ is differentiable at $c$ and $f'(c) = \frac{f(248) - f(246)}{2}$. The number of elements in $S$ is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f'(x) = \frac{|x^2 - 5x + 6|}{x^2 - 5x + 6} \cdot \frac{2x - 5}{x^2 - 5x + 8}$ for all $x \in \mathbb{R} \setminus \{2, 3\}$. If $f\left(\frac{5}{2}\right) = 0$, then the value of $e^{f(1)} + e^{f(4)}$ is

Let $S_n = \sum_{k=1}^n (-1)^{k-1} \left[ k^2 + (k+1)^2 - k^3 \sin\left(\frac{1}{k}\right) - (k+1)^3 \sin\left(\frac{1}{k+1}\right) \right]$. Evaluate the limit: $\lim_{n \to \infty} n^2 \left| S_n - 1 + \sin(1) + \frac{(-1)^n}{6} \right|$

Let $\{ \cdot \}$ denote the fractional part function. Evaluate the limit: $\lim_{x \to \pi} \frac{\int_{\pi}^{x} \left( \left( \left\{ \frac{t}{\pi} \right\} + \left\{ -\frac{t}{\pi} \right\} \right) \sec^2\left(\frac{t}{4}\right) + \tan\left(\frac{t}{4}\right) \right) dt}{\sin(x - \pi)}$