Sets - Class 11

Let $A = \begin{pmatrix} |x| & 2 \\ 2 & |x|-3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$. Let $S$ be the set of all real values of $x$ such that the system of linear equations $A \mathbf{y} = \mathbf{c}$ has a unique solution for every $2 \times 1$ column matrix $\mathbf{c}$, and the matrices satisfy the equation: $(A+B)^2 = A^2 + B^2 + \begin{pmatrix} x^2-4|x|+8 & x^2-4|x|+9 \\ -x^2+4|x|-1 & x^2-4|x|+2 \end{pmatrix}$ Find the number of elements in $S$.

Evaluate the definite integral: $I = \int_{0}^{3/2} \left( \frac{1}{2x^2 - 3x + 1} - \frac{4x^2 - 3x}{(2x^2 - 3x + 1)^2} \right) dx$

Find the complete solution set for the inequality $\log_{0.5} \left( \frac{(x-4)^2}{x^2 - 9} \right) \ge 0$

Let $\vec{u} = (k^3 + k^2 - 4k - 4)\hat{i} + (k^3 - 2k^2 - 4k + 8)\hat{j} + (k^3 - 3k^2 - 4k + 12)\hat{k}$ and $\vec{v} = 2\hat{i} - \hat{j} - 2\hat{k}$. Let $S$ be the set of all real values of $k$ for which the vector $\vec{w} = \frac{\vec{u}}{|\vec{u}|} + \frac{\vec{v}}{|\vec{v}|}$ does NOT represent a valid direction for the internal angle bisector of $\vec{u}$ and $\vec{v}$. Find the sum of the squares of the elements of $S$.

Consider the following system of equations: $x + \sqrt{y} - 2^z = -1$ $2x + 3\sqrt{y} - 4 \cdot 2^z = 1$ $3x + 4\sqrt{y} - (a^2 - 3a + 7) 2^z = a^2 + a - 2$ Find the number of integers $a$ for which the system has exactly one real solution $(x, y, z)$.

A subset of 6 distinct integers is chosen uniformly at random from the set $S = \{1, 2, 3, \dots, 20\}$. Let $A$ be the event that the chosen subset contains at least one pair of consecutive integers. Let $B$ be the event that the difference between the largest and the smallest integers in the chosen subset is strictly greater than 9. What is the probability that neither event $A$ nor event $B$ occurs?

Let $f: D \to \mathbb{R}$ be a function defined by $f(x) = \frac{\ln(1 + x^2)}{\text{sgn}(x) + [x]}$ where $D \subset (-1, 1)$ is the maximal domain of $f$ in the interval $(-1, 1)$. Here, $[x]$ denotes the greatest integer less than or equal to $x$, and $\text{sgn}(x)$ denotes the signum function. Let $S$ be the set of all critical points of $f(x)$ in $D$. Which of the following represents the set $S$?

Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually orthogonal unit vectors in $\mathbb{R}^3$. Let $S$ be the set of all vectors $\vec{r}$ satisfying the following conditions: (i) $[\vec{r}-\vec{a} \quad \vec{a}-\vec{b} \quad \vec{b}-\vec{c}] = 0$ (ii) $|\vec{r} - \vec{a}|^2 + |\vec{r} - \vec{b}|^2 + |\vec{r} - \vec{c}|^2 = \frac{9}{5}$ Find the number of elements in the set $S$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = |x^2 - 4x| - 2x$. Let $S$ be the set of all closed intervals $A \subset \mathbb{R}$ such that the restricted function $f: A \to [-5, 0]$ is bijective. Which of the following intervals is an element of $S$?

Let $S$ be the set of all distinct coefficients in the expansion of $(x_1 + x_2 + x_3 + x_4)^4$. Let $A$ be the set of all polynomials of the form $P(x) = x^4 - ax^3 + bx^2 - cx + d$ such that $a, b, c, d \in S$ and the equation $P(x) = 0$ has roots of even multiplicity. If $K$ is the set of all distinct coefficients $(a, b, c, d)$ that appear in the polynomials of $A$, find the number of proper subsets of $K$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$ such that the dot product of any two of them is non-zero. A set $S$ is defined as $S = \{\vec{a}, \vec{b}, \vec{c}, \vec{p}, \vec{q}, \vec{r}\}$, where $\vec{p} = \vec{a} \times (\vec{b} \times \vec{c})$, $\vec{q} = \vec{b} \times (\vec{c} \times \vec{a})$, and $\vec{r} = \vec{c} \times (\vec{a} \times \vec{b})$. What is the number of proper subsets of $S$ that span $\mathbb{R}^3$?

Let $S = \{1, 2, 3, \dots, 10\}$. A subset $A$ of $S$ is chosen uniformly at random from the set of all subsets of $S$ that do not contain any two consecutive integers. Let $E$ be the event that $A$ is a proper subset of $\{1, 3, 5, 7, 9\}$, and $F$ be the event that $5 \in A$. The conditional probability $P(E \mid F)$ is:

Let the universal set be $U = \left\{ \vec{r} \in \mathbb{R}^3 : |\vec{r}|^2 \le \frac{11}{4} \right\}$. Two sets $S_1$ and $S_2$ are defined as: $S_1 = \left\{ \vec{r} \in U : \vec{r} \times (\hat{i} + \hat{j} + \hat{k}) = (\hat{i} - \hat{j}) \times (\hat{i} + \hat{j} + \hat{k}) \right\}$ $S_2 = \left\{ \vec{r} \in U : \vec{r} \times (\hat{i} - \hat{j} + \hat{k}) = (2\hat{i} + \hat{k}) \times (\hat{i} - \hat{j} + \hat{k}) \right\}$ If $d$ is the shortest distance between any point in $S_1$ and any point in $S_2$, then the value of $d$ is:

Let $S = \{1, 2, 3, 4, 5, 6\}$. A permutation $f: S \to S$ is called \textit{acceptable} if it satisfies at least one of the following two conditions:

(I) $f(i) \neq i$ for all $i \in \{1, 2, 3\}$ (II) $f(j) \neq j$ for all $j \in \{4, 5\}$

Find the number of acceptable permutations of $S$.

Let $D = [a, b)$ be the domain of the function $f(x) = \sqrt{ \log_{1/2} \left( \frac{ [\sin^{-1} x] + 2 }{ [\cos^{-1} x] } \right) }$ where $[ \cdot ]$ denotes the greatest integer function. Then the value of the integral $\int_{a}^{b} \frac{x}{\sqrt{1 - x^2}} \, dx$ is:

Let $S$ be the set of all real values of $\alpha$ for which the improper integral $\int_{-1}^{1} \frac{x \cos x \cdot |x|^{\alpha-2}}{1 + e^x} dx$ converges. Which of the following represents the set $S$?

Let $S$ be the set of all odd polynomial functions $f(x)$ of degree 5 such that $f(x)$ is strictly increasing on $\mathbb{R}$. If $f(1) = 20$ and $f'(1) = 262$, find the number of elements in the set $S$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$. Let $S = \{\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4, \vec{v}_5\}$ be a set of 5 distinct vectors in $\mathbb{R}^3$ such that $|\vec{v}_i| = r > 0$ for all $i=1, \dots, 5$. Define the sets: $A = \{\vec{v} \in S \mid [\vec{v}, \vec{a}, \vec{b}] = 0\}$ $B = \{\vec{v} \in S \mid [\vec{v}, \vec{b}, \vec{c}] = 0\}$ $C = \{\vec{v} \in S \mid [\vec{v}, \vec{c}, \vec{a}] = 0\}$ Suppose $|A| = 4$, $|B| = 3$, and $|C| = 3$. If the maximum absolute value of the scalar triple product of any three vectors in $S$ is 9, find the value of $\sum_{1 \le i < j < k \le 5} [\vec{v}_i, \vec{v}_j, \vec{v}_k]^2$

Let $A = \{x \in \mathbb{R} : x^2 - 4x + 3 \le 0\}$ and $B = \{x \in \mathbb{R} : x^2 - 6x + 8 \le 0\}$. A function $g: A \cup B \to \mathbb{R}$ is defined as: $g(x) = \begin{cases} x^2 - 5 & \text{if } x \in A \setminus B \\ 2x - 5 & \text{if } x \in A \cap B \\ 2x - 7 & \text{if } x \in B \setminus A \end{cases}$ Let $f(x) = \int_{2}^{x} g(t) \, dt$. If $S$ is the set of all interior points in the domain of $f$ where $f(x)$ attains a local extremum, then the sum of all elements in $S$ is:

Let $A, B,$ and $C$ be three subsets of a universal set $U$ where $|U| = 100$. It is given that $|A| + |B| + |C| = 120$, $|A \cap B| + |B \cap C| + |C \cap A| = 37$, and $|A \cap B \cap C| = 5$. Let $n_k$ (for $k=1,2,3$) denote the number of elements of $U$ belonging to exactly $k$ of these sets, and $n_0$ denote the number of elements belonging to none of the sets. If $K$ is the coefficient of $x_0^{n_0} x_1^{n_1} x_2^{n_2} x_3^{n_3}$ in the expansion of $(x_0 + x_1 + x_2 + x_3)^{100}$, find the exponent of $5$ in the prime factorization of $K$.

Let $y = f(x)$ be a continuously differentiable function defined on $(0, \infty)$ which satisfies the differential equation $\frac{dy}{dx} = \frac{y}{x} - \sqrt{\left(\frac{y}{x}\right)^2 - 1}$ with the initial condition $f\left(\frac{1}{2}\right) = \frac{5}{8}$. Find the value of $f(2)$.

Let $S$ be the set of all real numbers $x$ satisfying the inequality $\sqrt{x^2 - 4x + 3} \le 2 - |x - 2|$ If $S$ is expressed as the union of disjoint intervals, let $L$ be the sum of the lengths of these intervals. Find the value of $L$.

Let $A$ be a $7 \times 7$ invertible real matrix with $\det(A) = 2$, and let $B = \text{adj}(A)$ be its adjugate matrix. For any index sets $X, Y \subset \{1, 2, \dots, 7\}$, let $M_{X, Y}$ denote the submatrix of a matrix $M$ formed by taking the rows indexed by $X$ and columns indexed by $Y$, both in increasing order. Let $S = \{1, 2, 3, 4\}$ and $T = \{1, 2, 4, 6\}$. If $\det(A_{T^c, S^c}) = 5$ and $\det(A_{S^c, T^c}) = 7$, where $X^c$ denotes the complement of $X$ in $\{1, 2, \dots, 7\}$, what is the value of $\det(B_{S, T})$?

Let $A = \{x \in \mathbb{R} : ||x-2| - 1| \ge 3 \}$ and $B = \left\{x \in \mathbb{R} : \frac{|x+1| - 2}{|x-3| - 1} \le 0 \right\}$. If $S = (A \Delta B)^c$, where $A \Delta B$ denotes the symmetric difference of sets $A$ and $B$, find the sum of all integers in $S$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as: $f(x) = \begin{cases} \frac{\sin(\pi x)}{x-8} + \frac{\ln(1 + (x-10)^2)}{(x-10)^2} + \frac{|x-13|}{x-13}, & x \notin \{8, 10, 13\} \\ 2024, & x \in \{8, 10, 13\} \end{cases}$ Let $S$ be the set of all points $c \in \mathbb{R}$ where $f(x)$ is discontinuous at $x=c$ but $\lim_{x \to c} f(x)$ exists. Let $T$ be the set of all integers $k$ such that the equation $3x^4 - 8x^3 - 6x^2 + 24x = k$ has exactly three distinct real roots. Find the sum of the elements in the set $S \Delta T$, where $\Delta$ denotes the symmetric difference of two sets.

Let $S = \{1, 2, 3, \dots, 20\}$. Consider the vectors $\vec{a} = 2\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b}_n = n\hat{i} + n^2\hat{j} + n^3\hat{k}$ for $n \in S$. Two sets $A$ and $B$ are defined as follows: $A = \{ n \in S \mid (\vec{a} \times \vec{b}_n) \cdot \hat{k} \le 80 \}$ $B = \{ n \in S \mid (\vec{b}_n \times \hat{a}) \cdot \hat{k} \ge -20 \}$ where $\hat{a}$ is the unit vector in the direction of $\vec{a}$. If $A \Delta B$ denotes the symmetric difference of sets $A$ and $B$, find the sum of all elements in $A \Delta B$.

Let $A$ be the set of all possible values of $k$ such that a selection of exactly $k$ items can be made from a pool of $38$ identical objects. Let $B$ be the set of all possible values for the number of linear arrangements of $k$ distinct objects, where $k \in A$. Find the cardinality of the symmetric difference, $|A \Delta B|$.

Let $M$ and $N$ be two matrices of order $3 \times 3$ with real entries such that $\det(M) = 4$, and $N^2 = N$ where $N$ is not the identity matrix. Define the following sets: $A = \{ k \in \mathbb{R} \mid \det(kM) = k \det(M) \}$ $B = \{ k \in \mathbb{R} \mid kM \text{ is a singular matrix} \}$ $C = \{ k \in \mathbb{R} \mid \det(kN) = k \det(N) \}$ $D = \{ k \in \mathbb{R} \mid kN \text{ is a singular matrix} \}$ Which of the following represents the set $(A \Delta B) \cap (C \Delta D)$? (where $\Delta$ denotes the symmetric difference of two sets)

Let $A, B,$ and $C$ be three exhaustive events in a sample space such that $P(A) = \frac{1}{2}$, $P(B) = \frac{2}{3}$, $P(C) = \frac{3}{4}$, and $P(A \cap B \cap C) = \frac{1}{6}$. Find the probability of the event $E = (A^c \cup B^c) \cap (B^c \cup C^c) \cap (C^c \cup A^c)$.

Let $U = \{x \in \mathbb{Z} : -5 \leq x \leq 5\}$ be the universal set. Let $f: U \to U$ be a function defined by $f(x) = -|x|$, and let $R_f$ denote the range of $f$. Let $X = \{x \in U : x \text{ is an even integer}\}$. Define set $B = (X \cup R_f) \cap (X \cup R_f^c) \cap R_f$, where $R_f^c$ is the complement of $R_f$ in $U$. Let $A = \{ x \in \mathbb{Z} : (x^2 - 9)(x^2 - 4x + 3) = (x^2 - 9)(x - 1) \}$. Find the cardinality of the Cartesian product $A \times B$.

Let $A = \{ z \in \mathbb{C} : \text{Re}(z) \ge 1 \}$, $B = \left\{ z \in \mathbb{C} : \log_{|z|/2} (|z - 3|) \le 1 \right\}$, and $C = \{ z \in \mathbb{C} : |z - 4| \le 2 \}$. Let $\Omega = (B \cup C) \cap (A \cup C) \cap (B \cup A^c)$. If the area of the region $\Omega$ in the complex plane is $\Delta$, then the value of $\Delta$ is:

Let $S$ be the set of all points $(x,y) \in \mathbb{Z}^2$ that simultaneously satisfy the following four inequalities: I. $\max(y - x, 4 - y) \ge 0$ II. $\max(y + x, 4 - y) \ge 0$ III. $\max(y - x, y - 5) \ge 0$ IV. $\max(y + x, y - 5) \ge 0$

Evaluate the sum: $\sum_{(x,y) \in S} 2^{-y}$

Let $N$ be the number of ways to distribute 6 distinct objects into 3 identical boxes such that no box remains empty. Let $M$ be the number of ways to distribute the same 6 distinct objects into 3 distinct boxes such that no box remains empty. Evaluate the sum $\sum_{k=1}^{N} \left[ \frac{k \cdot M}{N^2} + \frac{1}{2} \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $P, Q,$ and $R$ be three finite sets such that $|P \cup Q \cup R| = 40$, $|P| = 20$, $|Q| = 22$, $|R| = 25$, $|P \cap Q| = 8$, $|Q \cap R| = 7$, and $|P \cap R| = 15$. Let $B = \begin{bmatrix} |P \cap Q| \\ |Q \cap R| \\ |P \cap Q \cap R| \end{bmatrix}$ and $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 3 & 1 \end{bmatrix}$. If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is the unique solution to the matrix equation $AX = B$, then the value of $\det(A^3) + x + y + z$ is:

For each $x \in \mathbb{R}$, define the following sets: $A_x = \{ y \in \mathbb{R} : 0 \le y \le x^2 - 10x + 29 \}$ $B_x = \{ y \in \mathbb{R} : 0 \le y \le 29 - 4x \}$ $C_x = \{ y \in \mathbb{R} : 0 \le y \le 2x - 1 \}$ Let $f(x) = \mu(A_x \cup B_x \cup C_x)$, where $\mu(S)$ denotes the Lebesgue measure (length) of the set $S$. Find the global minimum value of $f(x)$ for $x \in \mathbb{R}$.

Let $C$ be the curve $y = x^3 - 9x^2 + 24x - 18$ and $L$ be the tangent to $C$ at its point of inflection. Let $R$ be the closed region bounded by $C$, $L$, and the vertical lines $x = 0$ and $x = 7$. If $S$ is the set of all points $(x, y)$ with integer coordinates lying strictly inside the region $R$ (i.e., not on any of its boundaries), find the number of elements in $S$.

Let $S$ be the domain of the function $f(x) = \ln \left( \frac{3\{x\} - 1}{[x]^2 - 7[x] + 10} \right) + \arccos\left(\frac{x-4}{4}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}$ denotes the fractional part of $x$. If $S$ is expressed as the union of disjoint intervals, then the sum of the lengths of these intervals is:

Two distinct numbers are chosen at random from the set $S = \{1, 2, 3, \dots, 21\}$. Let $A$ be the event that the sum of the two chosen numbers is even, and $B$ be the event that the product of the two chosen numbers is divisible by $3$. What is the probability of the event $A \cup B$?

For any two $3 \times 3$ matrices $X$ and $Y$, define the operation $X \oplus Y = X + Y - XY$. Let $A$ be a symmetric $3 \times 3$ real matrix such that $A \oplus A = A$ and $\text{Tr}(A) = 2$. If $B = I - 2A$, then the value of $\det(\text{adj}(B^T B + 2B + 3I))$ is

A number $x$ is chosen at random from the set $S = \{1, 2, 3, \dots, 361\}$. Let $E_1$ be the event that $x$ is a perfect square, and $E_2$ be the event that $x$ is a multiple of $19$. If $E_1 \cup E_2$ occurs, a biased coin with probability of heads $\frac{20}{37}$ is tossed. If $E_1^c \cap E_2^c$ occurs, a different biased coin with probability of heads $\frac{5}{54}$ is tossed. Given that the coin toss results in heads, what is the probability that the chosen number $x$ was neither a perfect square nor a multiple of $19$?

Let $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Let $S$ be the set of all real numbers $x$ for which there exists an invertible matrix $A$ of order 2 with real entries such that $A B - B^2 A = ([x]^2 - 5[x] + 6)I$, where $I$ is the identity matrix of order 2 and $[ \cdot ]$ denotes the greatest integer function. Find the set $S$.

Let $U$ be a universal set of $60$ students. Let $A$ and $B$ be the sets of students who like Mathematics and Physics, respectively. It is given that $n(A \cap B) = 10$. Let $x = n(A \cap B')$ and $y = n(A' \cap B)$. The school administration imposes the constraints $2x + 3y \le 60$ and $x + y \ge 15$. To optimize a resource metric, the function $Z = 4x + 6y$ is to be maximized. If $M$ is the maximum possible value of $Z$, and $N$ is the number of distinct valid pairs $(x,y)$ that achieve this maximum, find the value of $M + N$.

Let $A$ and $B$ be $3 \times 3$ real matrices satisfying the idempotent property $X^2 = X$ (analogous to the set identity $S \cup S = S$). Suppose $AB = BA = O$, $\det(I - 2A) = -1$, and $\det(I - 2B) = 1$. If $B$ is a non-null matrix, find the value of $\log_2(\det(A - B + 3I))$.

Let $A$ be a $3 \times 3$ non-scalar matrix with real entries such that $A^2 = A$ and $\text{trace}(A) = 2$. Let $B = I_3 - 2A$, where $I_3$ is the $3 \times 3$ identity matrix. Which of the following statements is \textbf{INCORRECT}?

Let $A$ be a $2n \times 2n$ skew-symmetric matrix with real entries. Assume that for any subset $I \subset \{1, 2, \dots, 2n\}$ with $|I| = n$, the $n \times n$ submatrix $A_{I, I^c}$ (formed by the rows indexed by $I$ and columns indexed by $I^c = \{1, 2, \dots, 2n\} \setminus I$) is non-singular. Define $f(n) = \prod_{|I|=n} \det(A_{I, I^c})$. Let $S_n \in \{+, -\}$ denote the sign of $f(n)$. Which of the following represents the sequence $(S_1, S_2, S_3, S_4)$?

Let $S$ be the set of all real numbers $a$ such that the function $f(x) = \frac{(x^2-1)(x-3)^2}{|x-1|(x+1)(x-3)(x-5)}$ is discontinuous at $x=a$, but $\lim_{x \to a} f(x)$ exists. Let $g: [0, 3] \to \mathbb{R}$ be defined by $g(x) = x^3 - 3x^2 + 2$. Let $T$ be the set of all integers $k$ such that the equation $g(x) = k$ has more than one distinct real solution. Find the sum of the elements in the set $S \Delta T$, where $S \Delta T$ denotes the symmetric difference of sets $S$ and $T$.

A number $x$ is chosen at random from the set $S = \{1, 2, \dots, 361\}$. Let $A$, $B$, and $C$ be the events that $x$ is divisible by 2, 3, and 19, respectively. A diagnostic test is performed on $x$ according to the following rules: \begin{itemize} \item If $x \in C$, the test yields a positive result with probability $\frac{10}{19}$. \item If $x \notin C$ and $x \in A^c \cap B^c$, the test yields a positive result with probability $\frac{5}{19}$. \item If $x \notin C$ and $x \notin A^c \cap B^c$, the test yields a positive result with probability $\frac{7}{19}$. \end{itemize} Given that the test yields a positive result, what is the probability that $x$ is NOT divisible by any of 2, 3, or 19?

Let $U$ be the set of all strictly increasing functions on $\mathbb{R}$. Consider the family of odd polynomials defined by $f(x) = x^{2n+1} - (2n+1) a x^{2n-1} + (2n+1) b x$ where $n \in \mathbb{N}, n \ge 2$, and $a, b \in \mathbb{R}$. For a fixed $a > 0$, which of the following is the necessary and sufficient condition on $b$ such that $f \in U$?

Let $A = \{x \in \mathbb{R} : |x - 2| \le 1\}$, $B = \{x \in \mathbb{R} : |x - 3| \le 1\}$, and $C = \{x \in \mathbb{R} : e^x > x^3\}$. Let $f: A \cup B \to \mathbb{R}$ be defined as: $f(x) = \begin{cases} x^2 - 4x + 5 & \text{if } x \in A \setminus B \\ -x^2 + 6x - 7 & \text{if } x \in A \cap B \\ x^2 - 8x + 17 & \text{if } x \in B \setminus A \end{cases}$ Let $S$ be the sum of the $x$-coordinates of all points of local extrema of $f(x)$ in the interior of its domain. Find the value of $S \int_1^3 f(x) dx$.