Relations and Functions - Class 11

Let $A$ be a $3 \times 3$ matrix with real entries such that $\det(A) > 0$. The matrix of minors of $A$ is given by $M = \begin{pmatrix} -24 & -20 & -5 \\ -18 & -15 & -4 \\ 5 & 4 & 1 \end{pmatrix}$ If the ordered triple $(x, y, z)$ satisfies the matrix equation $\begin{pmatrix} x & y & z \end{pmatrix} A = \begin{pmatrix} 1 & -1 & 2 \end{pmatrix}$ then the value of $x + y + z$ is:

Let $R$ be the region in the $xy$-plane defined by the set of points $(x, y)$ that satisfy the following conditions: $0 \le x \le 64$ $0 \le y \le 64$ $[x+y] - [x-y] = 2[y]$ where $[t]$ denotes the greatest integer less than or equal to $t$. Find the area of the region $R$.

Let $t_1, t_2, t_3$ be the three distinct real roots of the equation $t^3 - 10t^2 + 29t - 22 = 0$. For $i = 1, 2, 3$, let $P_i(x_i, y_i)$ be points in the Cartesian plane such that the ordered pairs $(x_i, y_i)$ satisfy the equality: $(x_i + 2y_i, 2x_i - y_i) = (7t_i^3 - 7t_i^2 + 7t_i - 4, -t_i^3 + t_i^2 - t_i + 2)$ Let $\Delta$ be the area of the triangle formed by the points $P_1, P_2,$ and $P_3$. The value of $\Delta$ is:

Let $f(x) = \frac{Px + Q}{2x^2 - 5x + 2}$ be expressed as a series expansion $f(x) = \sum_{n=0}^{\infty} a_n x^n + \sum_{n=1}^{\infty} b_n x^{-n}$ valid in the region $\frac{1}{2} < |x| < 2$. If $a_2 = -1$ and $b_2 = 1$, find the ordered pair $(P, Q)$.

Let $S = \{1, 2, 3, 4, 5, 6\}$ and $A = \{1, 2, 3\}$. A function $f: A \to S$ is chosen at random. Given that $f(1) \neq f(3)$ and there exists at least one $x \in A$ such that $f(x) \neq x$, what is the probability that $f(x) \neq x$ for all $x \in A$?

Let $A = \{ z \in \mathbb{Z}[i] \mid |z|^2 \le 5 \}$ and $B = \{ w \in \mathbb{Z}[i] \mid |w|^2 \le 5 \}$, where $\mathbb{Z}[i]$ denotes the set of Gaussian integers. Let $S$ be the subset of the Cartesian product $A \times B$ defined by: $S = \left\{ (z, w) \in A \times B \mid \text{Im}(z) = \text{Re}(w) \text{ and } d(P, \Pi) \le \frac{1}{\sqrt{3}} \right\}$ where $d(P, \Pi)$ is the perpendicular distance from the point $P(\text{Re}(z), \text{Im}(z), \text{Im}(w))$ to the plane $\Pi: x+y+z=1$. Find the cardinality of $S$.

Let $S$ be the set of all $4 \times 4$ matrices with real entries. For any matrix $M \in S$, let $M^*$ be the matrix obtained by shifting the rows of $M$ cyclically downwards by one position (i.e., the $i$-th row of $M$ becomes the $(i+1)$-th row of $M^*$ for $i=1,2,3$, and the $4$-th row of $M$ becomes the $1$-st row of $M^*$). A relation $R$ is defined on $S$ such that for $A, B \in S$, $A \ R \ B$ if and only if $\det(A + B^*) = 0$. Which of the following statements is true regarding the relation $R$?

Let $S = \{1, 2, 3, 4, 5, 6\}$. Let $N$ be the number of relations $R$ on $S$ satisfying the following conditions: (i) $R$ is a bijective function from $S$ to $S$. (ii) $R \cap R^{-1} = \emptyset$, where $R^{-1} = \{(y, x) \mid (x, y) \in R\}$. Find the value of $N$.

Let $S$ be the set of all differentiable functions $f: (1, \infty) \to (1, \infty)$. A relation $R$ on $S$ is defined as $(f, g) \in R$ if for every homogeneous function $H(x, y)$ of degree $k > 0$ (where $H(x, y) > 1$), the composite function $U(x, y) = f(g(H(x, y)))$ satisfies the partial differential equation: $x \frac{\partial U}{\partial x} + y \frac{\partial U}{\partial y} = 2k U \ln U$ It is given that $(f, g) \in R$ and $(g, h) \in R$. If $g(x) = e^{x^2}$, $h(x) = 2x$, and $f(e) = e^3$, what is the value of $f(h(2))$?

Let $f(x) = \frac{1}{\sqrt{[x]^2 - 3[x] + 2}}$ and $g(x) = \frac{12}{x^2 + 2x + 5}$, where $[ \cdot ]$ denotes the greatest integer function. Let $D$ be the domain of the composite function $h(x) = f(g(x))$. Find the sum of all integers that do NOT belong to $D$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = (|x-1| - 3) e^{-|x|}$. The range of $f(x)$ is:

Let $S$ be the set of all integers in the domain of the real-valued function $f(x) = \sin^{-1}\left(\frac{|x^2 - 9x + 20| + |x^2 - 9x + 18|}{2}\right) + \ln(x-3) + \ln(6-x)$ The total number of relations that can be defined on the set $S$ is:

Let $S = \{-3, -2, -1, 0, 1, 2, 3\}$. A relation $R$ on $S$ is defined such that it represents a bijective function $f: S \to S$ (i.e., $R = \{(x, f(x)) \mid x \in S\}$). Find the total number of such relations $R$ where no element $(x, y) \in R$ satisfies $y = x$ or $y = -x$.

Let $A$ be a set containing $n$ elements, where $n$ is the number of distinct terms in the expansion of $(x + y + z)^4 (x - y + z)^4$. If the total number of reflexive relations on set $A$ is $2^k$, then the value of $k$ is:

Let $A$ be the set of all points $(x, y)$ in the $xy$-plane satisfying the equations $x^2 + y^2 = 25$ and $xy = 12$. Let $B$ be the range of the function $f(t) = \left[ \frac{12}{t^2 - 2t + 4} \right]$ for $t \in \mathbb{R}$, where $[ \cdot ]$ denotes the greatest integer function. Find the number of injective mappings from $A$ to $B$.

Let $f: [-1, 1] \to \mathbb{R}$ be defined by $f(x) = \int_0^x \ln\left(t + \sqrt{t^2+1}\right) dt$ Which of the following represents the value of $f^{-1}\left( 1 - \sqrt{2} + \ln(1 + \sqrt{2}) \right)$?

Let $y = f(x)$ be a real-valued function defined on its maximal domain $D \subset \mathbb{R}$ such that $f$ is continuous on $D$, differentiable on the interior of $D$, and satisfies the equation $y^3 - x^3 = 3(y - x)$ for all $x \in D$. If $f(0) = \sqrt{3}$, let $I \subseteq D$ be the largest interval containing $x = 0$ such that the restriction of $f$ to $I$ is a one-to-one function. What is the length of the interval $I$?

Let $f: [k, 5] \to \left[-5, \frac{5}{4}\right]$ be a bijective function defined by $f(x) = |x^2 - 6x + 5| - x$. If $g(x)$ is the inverse function of $f(x)$, find the value of $g(-1) + g'(-1)$.

Let $S = \{ (x,y) \in \mathbb{R}^2 \mid y - x \le 1, \; x + y < 5, \; x > 0, \; y \ge 0 \}$. Let $f: S \to \mathbb{R}$ be a function defined by $f(x,y) = x - y$. If $E$ represents the range of $f$ on the domain $S$, then $E$ is equal to:

Let $F(x) = \int_{x}^{x^2} \frac{dt}{\sqrt{3 - |[t]|}}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Let $D$ be the maximal domain of $F(x)$ in $\mathbb{R}$, and $S$ be the set of points in $D$ where $F(x)$ is not differentiable. Find the number of elements in $S$.

Let $S$ be the set of all integers in the domain of the real-valued function $f(x) = \frac{\sqrt{25 - x^2}}{\ln(|x^3 - 3x^2 - 4x + 12|)} + \ln(|x^2 - x - 2|)$. Find the sum of all elements in $S$.

The coordinates $(x, y, z)$ of a point on a line $L$ are given by the relation: $\frac{3x - 6}{2} = \frac{2 - y}{3} = \frac{2z - 4}{4} = t$ where $t \in \mathbb{R}$ is a parameter. A real-valued function $f$ is defined as: $f(t) = \sin^{-1}\left( \frac{x + y + z - 6}{2} \right) + \sqrt{4 - t} + \sqrt{t + 10}$ If the domain of $f(t)$ is the interval $[a, b]$, find the value of $a^2 + b^2$.

Let $y: (0, \infty) \to \mathbb{R}$ be a differentiable function such that $y(x) \neq x$ for all $x \neq 1$, $y(1) = 1$, and $x \frac{dy}{dx} = y + 161 x \left| \frac{y}{x} - 1 \right|^{\frac{297}{298}}$ for all $x > 0$. If $y(e) > e$ and $y(1/e) < 1/e$, evaluate the limit: $\lim_{x \to 1} \frac{y(x) - x}{(x-1)^{298}}$

Let $A = \begin{pmatrix} 1 & -1 & 2 \\ 2 & 1 & -1 \\ 4 & -1 & 3 \end{pmatrix}$ and $M(x) = A + x I$ for $x \in \mathbb{R}$, where $I$ is the $3 \times 3$ identity matrix. Let $P(x)$ be a matrix such that $M(x) P(x) = I$ for all $x > 0$. Evaluate the limit: $\lim_{x \to 0^+} \frac{1}{x} \det(I - A P(x))$

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 6x - x^2$. A geometric constraint function is defined as $D(x) = |f(x) - x|$, representing the vertical distance to the identity line. Let $H(x) = D(D(x))$. Let $S$ be the set of all real numbers $x$ where $H(x)$ is not differentiable. Find the sum of the squares of the elements in $S$.

Let $f(x) = \sqrt{1 - \sin 2x} + \sqrt{1 + \sin 2x}$ for $x \in \left[0, \frac{\pi}{2}\right]$. Let $S$ be the set of all points $c \in \left(0, \frac{\pi}{2}\right)$ such that $f'(c) = \frac{f(\pi/2) - f(0)}{\pi/2 - 0}$. Find the number of elements in $S$.

Let $f(x) = \cos^{-1}\left( \frac{1-x^2}{1+x^2} \right) + \csc^{-1}\left( \frac{1+x^2}{2x} \right)$ for $x > 0$. Evaluate the integral: $I = \int_{0}^{\sqrt{3}} f(x) [x] \,dx$ where $[t]$ denotes the greatest integer less than or equal to $t$.

Let $f(x) = \begin{vmatrix} x^2 + \sqrt{x^2} - 2 & x^2 - \sqrt{x^2} & 1 \\ x^3 + x\sqrt{x^2} - 2x & x^3 - x\sqrt{x^2} & 2 \\ -3 & -4 & 1 \end{vmatrix}$.

If $S$ is the sum of the squares of all distinct real roots of the equation $f(x) = 0$, find the value of $S$.

Let $y = f(x)$ be a differentiable function defined on $(-\infty, 0)$ satisfying the differential equation $x^2 \left(\frac{dy}{dx}\right)^2 - 2xy \frac{dy}{dx} + 2y^2 - x^2 = 0$ If $f(-1) = 0$ and $f'(-1) > 0$, then the value of $f(-e^{-\pi/6})$ is:

Let $a_r$ be the term independent of $t$ in the binomial expansion of $\left(t - \frac{1}{t}\right)^{84-r} \left(t + \frac{1}{t}\right)^r$ for $r = 0, 1, 2, \dots, 84$. A polynomial $f(x)$ is defined as $f(x) = \sum_{r=0}^{84} \binom{84}{r} a_r x^r$. Which of the following correctly describes the parity of $f(x)$ and the coefficient of $x^{42}$ in $f(x)$?

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying the inequality $2x^3 - 3x^4 \le f(x) \le 2x^3 + 3x^4$ for all $x \in (-1, 1)$. A parametric curve is defined by $x(t) = \tan t - t$ and $y(t) = f(t)$ for $t \in (-1, 1)$. Evaluate the limit: $\lim_{t \to 0} \frac{y(t)}{x(t)}$

Let $S$ be the set of all real solutions to the equation $\cos^{-1}(16x^5 - 20x^3 + 5x) = 2\cos^{-1}x$ Find the value of $\sum_{x \in S} x$.

Let $f(x) = \frac{x e^{x-1} - 1}{|x-1| + \sin(x-1)}$ for $x \neq 1$. If $L_1 = \lim_{x \to 1^+} f(x)$ and $L_2 = \lim_{x \to 1^-} f(x)$, then which of the following is correct?

Let $t \in \mathbb{R} \setminus \{0\}$ be a parameter. Define the determinants: $D_1(t) = \begin{vmatrix} t^2 & 3t & 5 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{vmatrix} \quad \text{and} \quad D_2(t) = \begin{vmatrix} t^2 & 3t & -3 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{vmatrix}$ Consider the rational functions $f_1(x) = \frac{t x^2 - D_1(t) x + 1}{x + 1}$ and $f_2(x) = \frac{x^2 + D_2(t) x + 1}{t x - t^2}$. Let $L_1$ and $L_2$ be the oblique asymptotes of $f_1(x)$ and $f_2(x)$ respectively. Let $S$ be the set of all integers $t$ such that $L_1$ and $L_2$ intersect at a unique point $P(x_t, y_t)$, and both $x_t$ and $y_t$ are integers. Find the sum of all elements in $S$.

Let $y(x)$ be a continuous function on $\mathbb{R}$ satisfying the differential equation $(x^2+1) \frac{dy}{dx} + 2x \text{sgn}(x) y = 2x$ for all $x \neq 0$, with the boundary condition $y(-1) = -1$. If $A = \lim_{x \to \infty} y(x)$, then the value of $\frac{1}{\pi} \int_0^\infty (A - y(x)) dx$ is equal to

Consider the function $f(x) = 2|x - 2| - x^2 + 4x - 3$ defined on the interval $[0, 4]$. Let $P$ be the number of points in $(0, 4)$ where $f(x)$ attains a local extremum. Let $Q$ be the number of points in $(0, 4)$ where $f'(x) = 0$. Let $R$ be the number of intervals from the set $S = \{[0, 2], [1, 3], [2, 4]\}$ on which Rolle's Theorem is applicable to $f(x)$. Find the value of $P^2 + Q^2 + R^2$.

Let $A = [a_{ij}]$ be a square matrix of order $235$, where $a_{ij} = |i-j|^{234} \cos(i-j) + |i-j|^{235} \sin(i-j)$ for all $1 \le i, j \le 235$. Let $A = P + Q$, where $P$ is a symmetric matrix and $Q$ is a skew-symmetric matrix. What is the value of $\det(Q^{2024} + Q^{2025}) + \text{Tr}(Q^{2025})$?

Let $y = y(x)$ be a continuous solution to the differential equation $\frac{dy}{dx} + 3x|x| y = 3x^2$ for all $x \in \mathbb{R}$. If $y(1) = 1 + \frac{1}{e}$, then the value of $y(-1)$ is

Let $f(x) = \text{sgn}(x^2 - 5x + 6) \cdot |x^2 - ax + b|$, where $a, b \in \mathbb{R}$. If Rolle's Theorem is applicable to $f(x)$ on the interval $[1, 4]$, then the value of $a + b$ is:

Let $f(x) = \frac{e^x \cos x}{x}$ for $x \neq 0$. Evaluate the integral $I = \int_{-\pi/2}^{\pi/2} \left( \text{sgn}(x) f(x) + x \text{sgn}(x) f'(x) \right) \, dx$ where $\text{sgn}(x)$ denotes the signum function.

Let $y = f(x)$ be the real-valued differentiable solution to the differential equation $\text{sgn}(y) \frac{dy}{dx} = 3 - 2x$ satisfying the initial condition $f(1) = -2$. If $D = (a, b)$ is the maximum valid domain of $f(x)$ and $m$ is the local minimum value of $f(x)$ on $D$, find the value of $a + b - 4m$.

Let $f: [0, 4] \to \mathbb{R}$ be defined as $f(x) = \text{sgn}(x) \cdot \left( |x^2 - 4x + 3| + 2 \right)$ where $\text{sgn}(x)$ denotes the signum function. If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(x)$ on the interval $[0, 4]$ respectively, then the value of $M + m$ is:

Evaluate the following limit: $\lim_{x \to 0} \frac{\left[ 256 - x^2 \sin^2\left(\frac{1}{x}\right) \right]}{x^2 + 1}$ where $[ \cdot ]$ denotes the greatest integer function.

Let $S$ be the set of all ordered pairs $(x, y) \in \mathbb{R}^2$ such that the complex number $z = x + iy$ satisfies the following two equations: $z^2 (3 - 4i) + \bar{z}^2 (3 + 4i) = 0$ $z \bar{z} - (3-i)z - (3+i)\bar{z} = 0$ If the points in $S$ form the vertices of a triangle in the $xy$-plane, find the area of this triangle.

Let $f(x) = \frac{x^2 - 6x + 10}{x^2 - 6x + 13} + \sqrt{2 - |x-2| - |x-4|}$. Find the range of $f(x)$.

Let $f: D \to \mathbb{R}$ be a function defined by $f(x) = 2\text{sgn}\left( \frac{x^4 - 6x^3 + 13x^2 - 12x + 4}{x^3 - 4x^2 + 5x - 2} \right) + \text{sgn}(x^2 - 3x + 2)$ where $D$ is the maximal domain of definition of $f$ in $\mathbb{R}$. Which of the following sets represents the range of $f(x)$?

Let $B = \begin{bmatrix} x & 1 & 2 \\ 1 & x & -1 \\ 2 & -1 & x \end{bmatrix}$ and $A = [a_{ij}]$ be a $3 \times 3$ matrix where $a_{ij}$ is the minor of the element $b_{ij}$ in $B$. A mapping $f: \mathbb{R}^3 \to \mathbb{R}^3$ is defined by $f(V) = AV$, where $V$ is a $3 \times 1$ column vector. If $f$ is not a one-to-one mapping, then the sum of the squares of all possible real values of $x$ is

Let $f: \mathbb{R} \setminus \{2\} \to \mathbb{R}$ be defined as $f(x) = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} (2t + t^2 \sin^2(t-x)) \, dt + \frac{|x-2|}{x-2} (x^2 - 4x + c)$ Find the minimum integral value of $c$ for which $f(x)$ is an injective (one-to-one) function.

Let $f(x) = \sqrt{ \sin^{-1}(2x\sqrt{1-x^2}) - 2\sin^{-1}x } + \sec^{-1}(2x)$. Find the domain of $f(x)$.

Let $S$ be the domain of the real-valued function $f(x) = \sin^{-1}\left( \frac{x^3 - 1}{2(x - 1)} - \frac{x^2}{2} \right) + \cos^{-1}\left( \frac{x^2 - x}{x} \right)$ Find the number of integers in the set $S$.