Linear Inequalities - Class 11

Evaluate the following limit: $\lim_{x \to 0^-} \frac{e^{x|x|} - \cos x + x^3 \sin\left(\frac{1}{x}\right)}{x |\sin x|}$

Find the complete set of real values of $x$ that satisfy the inequality: $\frac{\cos^{-1}(|x - 1| + |x + 1| - 3)}{\sin^{-1}(|x - 1| + |x + 1| - 3)} < 1$

Let $f: (0, \infty) \to \mathbb{R}$ be a continuous function satisfying the differential equation $x f'(x) + f(x) = 15x^2 \frac{|x-2|}{x-2}$ for all $x \in (0, \infty) \setminus \{2\}$. If $f(1) = 27$, then the value of $f(8)$ is

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix where $a_{ij}$ is the number of integers $k$ satisfying the inequality $k^2 - (i+2j)k + 2ij \le 0$ for $1 \le i, j \le 3$. Let $B$ be the $3 \times 3$ matrix such that for any $3 \times 3$ matrix $Y$, the product $BY$ represents the matrix obtained by first swapping the first and second rows of $Y$, and then swapping the second and third rows of the resulting matrix. Consider the system of linear equations $M X = C$, where $M = (A+B)(A-B) - A^2 + B^2$ and $C = \begin{pmatrix} -4 \\ -2 \\ 0 \end{pmatrix}$. If $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is the unique solution to the system, find the value of $x + y + z$.

Find the complete solution set for $x$ satisfying the inequality: $\frac{|\log_2(x-2)| + |\log_2(x-2) - 2| - 2}{|x^2 - 6x + 8| - 3} \le 0$

Let $y = f(x)$ be a real-valued continuous function on its maximal domain $D$, satisfying the differential equation $y^3 \frac{dy}{dx} = \text{sgn}(x) (x - \alpha)$ where $\alpha > 0$ is a constant. If $D = [-\beta, 6] \cup [20, \infty)$ for some positive constant $\beta$, find the value of $f(0)^4 + \alpha + \beta$.

Let $y = f(x)$ be a real-valued continuous function defined on $\mathbb{R}$ satisfying the implicit equation $e^y + y^3 = \int_0^x \frac{t^2 - t - 2 + (t^2 - 4)|t - 1|}{t^2 + 1} dt$ If $S$ is the set of all $x \in \mathbb{R}$ for which $f(x)$ is strictly decreasing, then $S$ is equal to:

Find the complete solution set for $x$ satisfying the inequality: $\frac{\sin^{-1}(x) \cdot \left(3\cos^{-1}(x) - \pi\right)}{1 - 4x^2} \ge 0$

Let $f: (0, 5) \to \mathbb{R}$ be defined as $f(x) = (x-1)^2 (x-4)^2 \{x\}$, where $\{x\}$ denotes the fractional part of $x$. Let $S$ be the set of all points $x \in (0, 5)$ such that $f$ is differentiable at $x$ and $f'(x) = 0$. Find the sum of all rational elements in $S$.

Let $S$ be the set of all integers $x$ that satisfy the inequality: $\lim_{t \to x} \frac{(1 - \cos(2t - 6)) \cdot (|t^2 - 4t - 5| - 9)}{(t-3)^2 (t^2 + 1)} \le 0$ Find the number of elements in $S$.

Let $S$ be the set of all points $(x, y) \in \mathbb{R}^2$ satisfying the following constraints: $(y - x)^2 \le (x-1)^2 (x-6)^2 (2-x)$ $x \ge 1$ Find the maximum value of the objective function $Z = 3x + 4y$ for $(x, y) \in S$.

Let $I_n = \int_2^\infty \frac{dx}{(x-1)^2 (x+1)^n}$ for $n \in \mathbb{N}$. If $S_n = 4I_n - 4I_{n-1} + I_{n-2}$ for $n \ge 3$, evaluate the infinite series $\sum_{n=3}^\infty S_n$.

Let $f: [0, 3] \to [0, 4]$ be a function defined by $f(x) = x^4 - 6x^3 + 13x^2 - 12x + 4$. Let $g(x)$ denote the inverse function of $f(x)$. Determine the value of $g'(4)$, if it exists.

Let $f(x) = \lim_{y \to 0} \left( \frac{x^2 - 5x + 4 + y}{x^2 - 5x + 4} \right)^{\frac{[x]}{y}}$ for $x \in (1, 4)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Find the number of points of local extrema of $f(x)$ in the interval $(1, 4)$.

Let $y = f(x)$ be a differentiable function on the interval $\left(-\frac{1}{2}, \infty\right)$ that satisfies the differential equation $(x^2 - 1) \frac{dy}{dx} = y^2 - 1$ and the initial condition $f(0) = 2$. The value of $f(2)$ is:

Let $f(x) = \frac{|x^2 - 34x + 271|}{x^2 - 34x + 245}$. Let $S_1$ be the sum of all $x$ where $f(x)$ attains a local maximum, and $S_2$ be the sum of all $x$ where $f(x)$ attains a local minimum. If $f(x)$ is non-differentiable at exactly $k$ points in its domain, find the value of $S_1 - S_2 + k$.

Let $S = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$. A subset $\{a, b, c\}$ of three distinct elements is chosen from $S$. Find the number of such subsets for which the lines $L_a, L_b,$ and $L_c$ are concurrent, where the family of lines is given by $L_m : x + \frac{2}{m}y + m^2 - m = 0$

Consider the linear programming problem: Minimize $Z = 353x + 57y$ subject to the constraints: $\frac{3x + y - 8}{y - x} \ge 2$ $\frac{x + 3y - 8}{x - y} \ge 2$ $x \ge 0, y \ge 0$ The minimum value of $Z$ is:

Let $f(x) = \int_0^x \left( [t] - \frac{3}{2} \right) \left( [t] - \frac{7}{2} \right) (t-2)^2 (t-4)^4 \, dt$, where $[ \cdot ]$ denotes the greatest integer function. Let $S$ be the sum of the squares of the $x$-coordinates of all points of local extremum of $f(x)$ in the interval $(0, 6)$. Then the value of $S$ is

Evaluate the following limit: $L = \lim_{x \to 0} \frac{x^2 \sin x - x \ln(1+x^2)}{(\tan x - \sin x)^2}$

Let $A$ be the total area of the regions bounded by the curve $y = 60 \left( x^4 - 2x^3 + x^2 - |x^5 - 4x^4 + 5x^3 - 2x^2| \right)$ and the $x$-axis. Find the value of $A$.

Let $f(x) = \frac{1}{x-k} \int_{0}^{x} (t^2 - a[t]) dt$, where $k \in \mathbb{Z}^+$, $a \in \mathbb{R}$, and $[\cdot]$ denotes the greatest integer function. If the right-hand limit $\lim_{x \to k^+} f(x) = 0$ and the left-hand limit $\lim_{x \to k^-} f(x) = L$ (where $L$ is a finite real number), find the value of $a + k + L$.

Find the set of all real values of $x$ satisfying the inequality: $\frac{x^2 - x - 2}{[x]} \ge x + 1$ where $[ \cdot ]$ denotes the greatest integer function.

Let $S$ be the set of all real values of $x$ satisfying the equation $\tan^{-1}\left(\frac{x+1}{x-1}\right) + \tan^{-1}\left(\frac{x-1}{x}\right) = \tan^{-1}(7) - \pi$ Find the sum of the squares of all elements in $S$.

Let $f: \mathbb{R} \setminus \{-2, 2\} \to \mathbb{R}$ be defined as $f(x) = \frac{|x^2 - 1| - 2}{x^2 - 4}$ If $S$ is the set of all points of local extrema of $f$, then the value of $\sum_{x \in S} (x^2 + 2)$ is:

Let $S$ be the set of all $x \in [0, 3]$ satisfying the inequality $\int_0^x ||t-1| - 1| \, dt \ge \frac{x}{2}$ Find the set $S$.

Let $M$ be a $2 \times 2$ matrix given by $M = \begin{pmatrix} |x^2 - 4x - 2| - 2 & -1 \\ 1 & |x-2| - 4 \end{pmatrix}$. If there exists a $2 \times 2$ matrix $N$ with real entries such that $M N - N M = M$, find the sum of all possible real values of $x$.

Let $C_1$ be the circle $x^2 + y^2 = 25$ and $C_2$ be the circle $x^2 + y^2 = 16$. Let $S$ be the set of all points $P$ on $C_1$ such that the chord of contact of $P$ with respect to $C_2$ intersects $C_2$ at points $A$ and $B$. If the coordinates of both $A$ and $B$ must satisfy the strict inequality $|x| + |y| < 4\sqrt{2}$, then the number of points on $C_1$ that are NOT in $S$ is:

Let $y(x)$ be a real-valued differentiable function defined on the interval $\left[ \frac{1}{2}, \frac{\sqrt{3}}{2} \right]$ satisfying the differential equation $\frac{dy}{dx} = \frac{4x^3 - 4x}{x^4 - 2x^2} e^{-y}$ If $y\left(\frac{1}{\sqrt{2}}\right) = \ln \alpha$, then the set of all possible values of $\alpha$ is:

Let $S = \{1, 2, \dots, 10\}$. An integer $x$ is chosen uniformly at random from $S$. Given $x$, an integer $y$ is chosen uniformly at random from the set $S_x = \{z \in S : |x - z| \le 2\}$. The probability that $y \ge x$ is

Let $g(x) = |x^2 - 4x + 3|$. Find the solution set $S$ of the inequality $\frac{g'(x)}{[x] - 2} \le 0$ in the domain $x \in [-2, 4]$, where $[x]$ denotes the greatest integer less than or equal to $x$, and $g'(x)$ is the derivative of $g(x)$.

Let $R$ be the set of points $(x, y)$ in the Cartesian plane satisfying the following conditions: $\frac{x - y}{x + y} \ge 1, \quad \frac{x + y}{x - y} \le -1, \quad \text{and} \quad x^2 + y^2 \le 16$ What is the area of the region $R$?

Let $\alpha$ be the greatest real number satisfying the inequality $\frac{\lambda^2 - 7\lambda + 10}{\lambda - 5} \le 0$ Consider the following system of linear equations in variables $x, y, z$: \begin{align*} x + y + z &= 1 \ x + 2y + 2z &= 2 \ x + 3y + (\alpha + 1)z &= 4 \end{align*} Which of the following statements is true regarding the system?

Find the complete set of values of $x$ for which the function $f(x) = \frac{x^2 + 7x - 9}{x^2 + 6x - 7}$ is monotonically increasing.

Let $f(x) = \lim_{n \to \infty} e^{-n x^2}$. Let $g: \mathbb{R} \to \mathbb{R}$ be a function satisfying the inequality $\cos x - x^2 - 3 f(x) \le g(x) \le \cos x + x^2 - 3 f(x)$ for all $x \in \mathbb{R}$. If $L = \lim_{x \to 0} g(x)$ and $V = g(0)$, then the ordered pair $(L, V)$ is:

Let $y = y(x)$ be the solution of the differential equation $x \ln x \frac{dy}{dx} + (1 - \ln x) y = x^2 (\ln x)^2$ on its maximal interval of existence containing $x = e^{-1}$, satisfying $y(e^{-1}) = 5e^{-2}$. Find the number of roots of the equation $y(x) = 0$.

A variable plane $\Pi$ has $x$, $y$, and $z$ intercepts all equal to $a$, where $a = t^2 - 4t + 5$ and $t \in (0, 3]$. Let $S$ be the locus of the image of the point $M(1, 2, 3)$ reflected across the plane $\Pi$. If $\bar{S}$ denotes the closure of the set $S$, which of the following points belongs to the set $\bar{S} \setminus S$?

A point $P$ is generated in $\mathbb{R}^3$ by the following random process: With probability $\frac{1}{3}$, $P$ is chosen uniformly from the surface of the unit sphere $S = \{(x,y,z) : x^2+y^2+z^2 = 1\}$. With probability $\frac{2}{3}$, $P$ is chosen uniformly from the interior of the unit sphere $V = \{(x,y,z) : x^2+y^2+z^2 < 1\}$. Let $O$ be the origin. Let $A$ be the event that the volume of the sphere centered at $O$ with radius $OP$ is at least $\frac{\pi}{6}$. Let $B$ be the event that $P$ lies strictly inside $S$. Find the probability $P(\overline{A} \cup \overline{B})$.

Consider the feasible region $R$ in the $xy$-plane determined by the following system of inequalities: $|x - 2y| \le 4$ $x + y \le 6$ $x \ge 0, y \ge 0$ Find the maximum value of the objective function $Z = 4x + 7y$ over the region $R$.

Let $S$ be the feasible region in the $xy$-plane defined by the inequalities: $\frac{x + 2y - 8}{y - 3} \le 2, \quad 0 \le x \le 6, \quad 0 \le y \le 6$ If $[ \cdot ]$ denotes the greatest integer function, find the maximum value of the objective function $Z = 7x + 3[y]$ on the region $S$.

A factory produces three chemicals $A, B,$ and $C$ in quantities $x, y,$ and $z$ (in kg) respectively, where $x, y, z \ge 0$. The total production must be exactly $12$ kg. Due to safety regulations, the production state $(x,y,z)$ must satisfy a stability constraint: its perpendicular distance from the plane $x - 2y + 2z = 0$ cannot exceed $2$ units. The profit per kg for chemicals $B$ and $C$ are $4$ and $3$ respectively. The profit per kg for chemical $A$ is $\lambda$, where $\lambda$ is the maximum possible value of $|z_1 - z_2|$ for complex numbers $z_1, z_2$ satisfying $|z_1| = 2$ and $|z_2 - 3 - 4i| = 1$. Find the maximum total profit the factory can achieve.

Let $f, g : [0, 2] \to \mathbb{R}$ be continuous functions such that both satisfy the differential equation $(x^3 - 1)\frac{dy}{dx} = 3x^2 y$ for all $x \in (0, 2) \setminus \{1\}$. Given that $f(0) = 2$, $f(2) = 14$, $g(0) = -1$, and $g(2) = 7$. Let $R$ be the region in the $xy$-plane defined by $R = \{(x, y) : 0 \le x \le 2, g(x) \le y \le f(x)\}$ The area of the region $R$ is:

Let $L: y = f(x)$ be the line tangent to the parabola $y = \frac{1}{4}x^2 - 1$ that is parallel to the line $y = x$. Let $R$ be the feasible region in the $xy$-plane defined by the system of inequalities: $y \le \sqrt{x}, \quad y \ge f(x), \quad \text{and} \quad \frac{x - 1}{x - 4} \ge 0$ What is the area of the region $R$?

Let $\{x\}$ denote the fractional part of $x$. Find the number of real solutions to the equation $\int_0^x \left( \left| 2\{t\} - 1 \right| - \frac{1}{2} \right) dt = \frac{x}{100}$

Let $\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^3$ be three vectors such that $|\vec{u}| \le 2$, $|\vec{v}| \le 3$, and $|\vec{w}| \le 4$. It is given that $\vec{u} \cdot \vec{v} \ge 3\sqrt{3}$ and $\vec{v} \cdot \vec{w} \ge 6\sqrt{2}$. If the matrix $M = [\vec{u}\ \vec{v}\ \vec{w}]$ is invertible, let $S$ be the set of all possible values of the dot product $\vec{u} \cdot \vec{w}$. Which of the following represents the exact set $S$?

Let $S$ be the region in the $xy$-plane defined by the system of inequalities $y \ge 0$ and $y \le -x^2 + 4x - 3$. Let $x_1$ and $x_2$ ($x_1 < x_2$) be the abscissae of the intersection points of the curves $y = 0$ and $y = -x^2 + 4x - 3$.

A function $f:\mathbb{R} \to \mathbb{R}$ is defined as $f(x) = \begin{vmatrix} |x^2 - 4x + 3| & x^2 - 4x + 3 & 1 \\ x^2 & 2x & 1 \\ 1 & 0 & x \end{vmatrix}$

Find the value of $f'(x_1^+) + f'(x_2^-)$.

Let $A$ be the set of all integer solutions to the system of inequalities: $\log_2(x^2 - 4x + 4) \le 4$ $2^{x+2} + 2^{4-x} \le 65$ Let $N$ be the number of elements in set $A$. If $S_n$ denotes the sum of the first $n$ terms of a sequence $\{a_n\}$ and is given by $S_n = 4n^2 - 3n + 7$, then the value of $\sum_{k=1}^{N} a_{2k-1}$ is:

Let $S$ be the solution set of the system of inequalities $|2x - 3| \le 1$ and $x^2 - 4 \le 0$. Let $f: S \to \mathbb{R}$ be defined by $f(x) = \begin{cases} (x-1)^2 \sin\left(\frac{1}{x-1}\right) + x^2, & x \in (1, 2] \\ 1, & x = 1 \end{cases}$ Let $A = f'(1^+)$ be the right-hand derivative of $f$ at $x=1$. Let $B$ be the value of $f'(c)$ for the point $c \in (1, 2)$ guaranteed by Lagrange's Mean Value Theorem for $f$ on $S$. If $A$ exists, find the value of $A + B$. (If $A$ does not exist, select the corresponding option).

Evaluate the integral: $I = \int_0^{90} (2x - 1) \ln | 2[x] - 2x + 1 | \, dx$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $f(x) = \int_{0}^{3x} |t^2 - 3xt + 2x^2| \,dt$ for $x > 0$. Find the value of $f'(2)$.