Determinants - Class 12

Let $A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$. Let $C_{ij}$ and $M_{ij}$ denote the cofactor and minor of the element $a_{ij}$ of matrix $A$, respectively. Define vectors $\vec{u}_i = \sum_{j=1}^3 \left( C_{ij} + \text{sgn}(x^2 - 4x + 3) M_{ij} \right) \hat{e}_j$ for $i = 1, 2, 3$, where $\hat{e}_1, \hat{e}_2, \hat{e}_3$ are the standard orthogonal unit vectors. If $V(x)$ is the volume of the parallelepiped formed by the vectors $\vec{u}_1, \vec{u}_2, \vec{u}_3$, find the value of $\int_{0}^{3} V(x) dx$.

Let $A(t) = \begin{bmatrix} 1 & 1 & 1 \\ [t-1] & [t] & [t+1] \\ 2 & 3 & -1 \end{bmatrix}$, where $[x]$ denotes the greatest integer less than or equal to $x$. Let $C_{11}(t), C_{12}(t), C_{13}(t)$ be the cofactors of the elements of the first row of $A(t)$. A point $P(t)$ in 3D space is defined by the coordinates $(C_{11}(t), C_{12}(t), C_{13}(t))$. Consider another matrix $B(t) = \begin{bmatrix} C_{11}(t) & C_{12}(t) & C_{13}(t) \\ 2 & -1 & 3 \\ 1 & 4 & -2 \end{bmatrix}$. Let $M_{11}, M_{12}, M_{13}$ be the minors corresponding to the elements $C_{11}(t), C_{12}(t), C_{13}(t)$ respectively in $B(t)$. Define a modified determinant function $\Delta^*(t)$ based on the octant location of $P(t)$ as: $\Delta^*(t) = \operatorname{sgn}(C_{11}(t)) M_{11} + \operatorname{sgn}(C_{12}(t)) M_{12} + \operatorname{sgn}(C_{13}(t)) M_{13}$ where $\operatorname{sgn}(\cdot)$ is the signum function. Find the value of $\lim_{t \to 0^+} \Delta^*(t) - \lim_{t \to 0^-} \Delta^*(t)$.

Let $A(\theta) = \begin{pmatrix} 1+\sin^2\theta & \cos^2\theta & \sin 2\theta \\ \sin^2\theta & 1+\cos^2\theta & \sin 2\theta \\ \sin^2\theta & \cos^2\theta & 1+\sin 2\theta \end{pmatrix}$ and $f(\theta) = |\text{adj}(2 A(\theta))|$. The maximum value of $f(\theta)$ for $\theta \in \mathbb{R}$ is

Let $f(n)$ and $g(n)$ denote the exponent of the highest power of $2$ and $3$ dividing $n!$, respectively. Let $S$ be the set of all integers $n \in [3, 15]$ for which the matrix $M_n$ is singular, where $M_n = \begin{pmatrix} \frac{1}{n-5} & \frac{1}{n-5} & \frac{1}{n-5} \\ \frac{f(n-2)}{n-7} & \frac{f(n-1)}{n-7} & \frac{f(n)}{n-7} \\ g(n-2) & g(n-1) & g(n) \end{pmatrix}$ Find the sum of all elements in $S$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$. Three new vectors are defined as: $\vec{u} = \vec{a} + 2\vec{b} - \vec{c}$ $\vec{v} = 3\vec{a} + \lambda\vec{b} + \vec{c}$ $\vec{w} = \vec{a} + 4\vec{b} + \lambda\vec{c}$ where $\lambda \in \mathbb{R}$. A $3 \times 3$ matrix $M$ is constructed such that its elements are given by: $M = \begin{pmatrix} \vec{u}\cdot\vec{a} & \vec{u}\cdot\vec{b} & \vec{u}\cdot\vec{c} \\ \vec{v}\cdot\vec{a} & \vec{v}\cdot\vec{b} & \vec{v}\cdot\vec{c} \\ \vec{w}\cdot\vec{a} & \vec{w}\cdot\vec{b} & \vec{w}\cdot\vec{c} \end{pmatrix}$ If $M$ is a singular matrix, what is the absolute difference between the possible values of $\lambda$?

Let $S$ be the set of all real values of $x$ for which the points $A(1, |x|)$, $B(2, x)$, and $C(3, 1)$ form a non-degenerate triangle whose area is at most $1$. Then $S$ is equal to:

Let $S$ be the set of all pairs of $3 \times 3$ real matrices $(A, B)$ such that $A$ is an orthogonal matrix, $B$ is a nilpotent matrix of index $3$ (i.e., $B^3 = O$ and $B^2 \neq O$), and $A + B = I$, where $I$ is the $3 \times 3$ identity matrix and $O$ is the null matrix. The number of elements in the set $S$ is

Let $A$ be a $3 \times 3$ matrix with complex entries such that $A A^\dagger = I$ and $\det(A) = i$, where $A^\dagger$ is the conjugate transpose of $A$. Let $B = \begin{pmatrix} 1 & 1+i & x \\ 1-i & -1 & i \\ x & -i & 1 \end{pmatrix}$ where $x$ is a real number. If $\det(A B A^T) = -3$, then the product of all possible real values of $x$ is:

Let $A = \begin{pmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & 1 \end{pmatrix}$ and $B$ be a $3 \times 3$ real matrix such that $\det(B) = 2$. If $x \in \mathbb{R}$ satisfies the equation $\det(A^T B A) = \det(\det(A) B^{-1})$, then the sum of all distinct real values of $x$ is:

Let $\Delta(x) = \begin{vmatrix} \cos x & x^2 & 1 \\ x^3 & x & x \\ [x] & [-x] & [x]+[-x] \end{vmatrix}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Evaluate the integral $I = \int_{-\pi/2}^{\pi/2} \Delta(x) \, dx$

For any integer $m \ge 1$, let $A_{m,1}, A_{m,2}, \dots, A_{m,m}$ be $m$ arithmetic means inserted between $-1$ and $2m+1$. Let $S_m = \sum_{k=1}^m A_{m,k}$ and $f_m(r) = \sum_{k=1}^m A_{m,k} r^{k-1}$. Let $d_m$ be the common difference of the arithmetic progression formed by inserting $m$ arithmetic means between $0$ and $m+1$. Define the matrix $M_n(r) = \begin{pmatrix} f_n(r) & f_{n+1}(r) & f_{n+2}(r) \\ S_n & S_{n+1} & S_{n+2} \\ d_n & d_{n+1} & d_{n+2} \end{pmatrix}$ If $\lim_{r \to 1} \frac{\det(M_n(r))}{1-r} = 399$, then the value of $n$ is:

Let $A(x) = \begin{pmatrix} 1 & \sin x & x + \cos x \\ -\sin x & 1 & x^2 + e^x \\ x - \cos x & x^2 - e^x & 4x^4 - 8x^3 + 7x^2 + 5 \end{pmatrix}$.

Define $f(x) = \det\left(\frac{A(x) + A^T(x)}{2}\right) + \det\left(\frac{A(x) - A^T(x)}{2}\right)$.

If $S$ is the sum of all local extreme values of the function $f(x)$, then the value of $S$ is:

Let $\vec{a} = \left(\sqrt{x^2} - 5\right)\hat{i} + x\hat{j} + \hat{k}$, $\vec{b} = x\hat{i} + \left(\sqrt{x^2} - 5\right)\hat{j} + \hat{k}$, and $\vec{c} = 2\hat{i} + \hat{j} + \hat{k}$. Let $S$ be the set of all real values of $x$ for which the vectors $\vec{a}, \vec{b}$, and $\vec{c}$ are coplanar. Find the value of $4 \sum_{x \in S} x^2$.

Consider the system of linear equations: $x + y + z = \lambda$ $x + 2y + 3z = \mu$ $14x + 22y + 30z = \lambda^2 + \mu^2 - 11$ If the system has infinitely many solutions, then the maximum value of $3\lambda + 4\mu$ is:

Let $\omega = e^{i \frac{2\pi}{5}}$. Define a $4 \times 4$ matrix $A$ with entries $a_{jk} = (\omega^j + \omega^{-k})^3$ for $1 \le j, k \le 4$, and a $5 \times 5$ matrix $B$ with entries $b_{jk} = (\omega^j + \omega^{-k})^3$ for $1 \le j, k \le 5$. Find the value of $\frac{\det(A) + \det(B)}{125}$.

Let $A = \begin{pmatrix} x & y & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} x & y & 1 \\ -1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$. Let $M = A B^T$. A point $P(x,y)$ moves in the $xy$-plane such that it satisfies the following conditions:

(i) $\text{sgn}(\det(M)) \cdot |\det(M) - 8| \ge 1$ (ii) $|x + y - 1| + |x - y + 1| \le 4$

Find the area of the region consisting of all such points $P(x,y)$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that their scalar triple product $[\vec{a} \ \vec{b} \ \vec{c}] = V$. Consider the determinant $\Delta = \begin{vmatrix} (\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{a}) & (\vec{a}\times\vec{b})\cdot(\vec{b}\times\vec{c}) & |\vec{a}\times\vec{b}|^2 \\ (\vec{b}\times\vec{c})\cdot(\vec{c}\times\vec{a}) & |\vec{b}\times\vec{c}|^2 & (\vec{b}\times\vec{c})\cdot(\vec{a}\times\vec{b}) \\ |\vec{c}\times\vec{a}|^2 & (\vec{c}\times\vec{a})\cdot(\vec{b}\times\vec{c}) & (\vec{c}\times\vec{a})\cdot(\vec{a}\times\vec{b}) \end{vmatrix}$ Which of the following represents the value of $\Delta$?

Let $A(x)$ be a $3 \times 3$ matrix defined as: $A(x) = \begin{pmatrix} \frac{x}{x-5} & \frac{x^2}{x-5} & \frac{x^3}{x-5} \\ \frac{4}{x-4} & \frac{16}{x-4} & \frac{64}{x-4} \\ \frac{6}{x-8} & \frac{36}{x-8} & \frac{216}{x-8} \end{pmatrix}$ Find the sum of all integers $x$ that satisfy the inequality $\det(A(x)) \le 0$.

Let $\vec{u}(t) = \hat{i} + t\hat{j} + \frac{t^2}{2}\hat{k}$, $\vec{v}(t) = \hat{i} + \sin(t)\hat{j} + (1-\cos t)\hat{k}$, and $\vec{w}(t) = \hat{i} + \tan(t)\hat{j} + \ln\left(1+\frac{t^2}{2}\right)\hat{k}$ be three vectors in $\mathbb{R}^3$ for $t \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Let $V(t)$ denote the volume of the parallelepiped determined by $\vec{u}(t)$, $\vec{v}(t)$, and $\vec{w}(t)$. Let $\vec{n}(t) = \vec{u}(t) \times \vec{v}(t) = n_x(t)\hat{i} + n_y(t)\hat{j} + n_z(t)\hat{k}$ be the normal vector to the plane containing $\vec{u}(t)$ and $\vec{v}(t)$. If $m(t) = \frac{n_y(t)}{n_x(t)}$ represents the slope of the projection of $\vec{n}(t)$ onto the $xy$-plane, evaluate the limit: $\lim_{t \to 0} \left( m(t) \cdot \frac{V(t)}{t^6} \right)$

Let $f(x) = \begin{vmatrix} x+1 & x^2 & x \\ x+1+\cos\left(\frac{1}{x}\right) & x^2 + \sin\left(\frac{1}{x}\right) & x + \cos\left(\frac{1}{x}\right) \\ x+2 & x^2 - 2x & x + 1 \end{vmatrix}$ for $x \neq 0$. Evaluate the limit: $L = \lim_{x \to 0} \frac{1}{x} \int_0^x f(t) dt$

Let $n \ge 2$ be an integer. Consider the determinant $\Delta(y) = \begin{vmatrix} \binom{n}{0}y^2 & \binom{n}{1} & \binom{n}{2} \\ \binom{n+1}{0}y^2 & \binom{n+1}{1} & \binom{n+1}{2} \\ \binom{n+2}{0}y^2 & \binom{n+2}{1} & \binom{n+2}{2} + \frac{1}{y^3} \end{vmatrix}$ Let $T$ be the term independent of $y$ in the binomial expansion of $(\Delta(y))^9$. For an integer $x \in [1, T]$, a matrix $M(x)$ is defined as: $M(x) = \begin{pmatrix} \sin\left(\frac{\pi x}{2}\right) & 1 - \cos(\pi x) & x \sin\left(\frac{\pi x}{2}\right) \\ \sin\left(\frac{\pi x}{3}\right) & \cos\left(\frac{2\pi x}{3}\right) - 1 & \sin\left(\frac{2\pi x}{3}\right) \\ \sin\left(\frac{\pi x}{7}\right) & \sin\left(\frac{2\pi x}{7}\right) & 1 - \cos\left(\frac{2\pi x}{7}\right) \end{pmatrix}$ Find the number of valid values of $x$ such that no row of $M(x)$ is a zero vector.

Consider the following system of linear equations for $\theta \in (0, \frac{\pi}{2}) \setminus \{\frac{\pi}{4}\}$: $x \cos \theta + y \sin \theta + z = \cos^2 \theta$ $x \sin \theta + y \cos \theta + z = \sin^2 \theta + \sqrt{1 - \sin 2\theta}$ $x + y + 2z = 1$ Let $(x(\theta), y(\theta), z(\theta))$ be the unique solution to this system. If $f(\theta) = x(\theta) - y(\theta)$, and $L_1 = \lim_{\theta \to \frac{\pi}{4}^-} f(\theta)$, $L_2 = \lim_{\theta \to \frac{\pi}{4}^+} f(\theta)$, then the value of $L_1^2 + L_2^2 + L_1 L_2$ is:

Let $\vec{a} = [x]\hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + [x]\hat{j} + \hat{k}$, and $\vec{c} = \hat{i} + 2\hat{j} + [x]\hat{k}$, where $[x]$ denotes the greatest integer less than or equal to $x$. Let $N$ be the $3 \times 3$ matrix whose rows are the vectors $\vec{a} \times \vec{b}$, $\vec{b} \times \vec{c}$, and $\vec{c} \times \vec{a}$. If $S$ is the set of all $x \in [-5, 5]$ such that $\det(\text{adj}(\text{adj}(N))) = 0$, then the value of $\int_S 2x \, dx$ is:

Let $M$ be a non-zero $3 \times 3$ real skew-symmetric matrix such that $M^3 + 9M = O$, where $O$ is the null matrix. For $x > 0$, let $A(x) = \frac{2}{x} I - M$, where $I$ is the $3 \times 3$ identity matrix. If $f(x) = \text{Tr}(\text{adj}(A(x)))$ and $g(x) = \det(A(x))$, then the value of $\lim_{x \to \infty} \frac{x(f(x) - 9)}{g(x)}$ is

Let $A$ be a $3 \times 3$ Skew-Hermitian matrix with complex entries such that $A^3 + 7A - 6iI = 0$, where $I$ is the $3 \times 3$ identity matrix and $i = \sqrt{-1}$. Which of the following is the value of $\det((adj(A))^* - A^2)$?

Consider the system of linear equations: $x + y + z = 1$ $2x + 3y + [\alpha]z = 2$ $8x + 6y + 3|\alpha|z = \beta$ where $\alpha, \beta \in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Let $A$ be the set of all values of $\alpha$ for which there exists a real number $\beta$ such that the system has infinitely many solutions. If $S = \sum_{\alpha \in A} |\alpha|$ and $\beta_0$ is the unique value of $\beta$ that makes the system consistent for these $\alpha$, then the value of $3S + \beta_0$ is:

Let $f(x) = \min(\sin 2x, x^2+x)$. Let $a = \lim_{x \to 0^+} \frac{f(x)}{x}$ and $b = \lim_{x \to 0^-} \frac{f(x)}{x}$. Consider the system of linear equations: $x + ay + z = 1$ $x + 2y + bz = 2$ $2x + 3y + 3z = \lambda$ Let $S$ be the set of all real values of $\lambda$ for which the system has infinitely many solutions. Find the sum of the elements of $S$.

Consider the linear programming problem: Maximize $Z = \begin{pmatrix} 4 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$ subject to the constraints: $\begin{pmatrix} 2 & 1 \\ 4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \le \begin{pmatrix} 5 \\ 13 \end{pmatrix}$ $x \ge 0, y \ge 0$ The maximum value of $Z$ is:

Let $z_r = \cos\left(\frac{2\pi r}{11}\right) + i\sin\left(\frac{2\pi r}{11}\right)$ for $r = 1, 2, \dots, 10$. Consider the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3 \in \mathbb{C}^3$ such that the $k$-th component of $\vec{v}_j$ is given by: $A_{jk} = \sum_{r=1}^{10} i^{r|j-k|} \left( z_r^{j-k} - z_r^{k-j} \right)$ where $i = \sqrt{-1}$. If $V$ represents the volume of the parallelepiped formed by the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$, which is given by the modulus of their scalar triple product, find the value of $V$.

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix where its elements are given by the integral: $a_{ij} = \int_{0}^{\pi} \frac{(i^2 - j^2) \sin x}{1 + e^{x - \pi/2}} dx$ Find the value of $\det(I - A^2)$, where $I$ is the $3 \times 3$ identity matrix.

Let $S_n = \sum_{r=1}^n r^{104}$. Consider the real number $L$ defined by $L = \lim_{n \to \infty} \left( \frac{105 S_n}{n^{105}} \right)^{\frac{n}{105}}$ Let $S$ be the set of all $4 \times 4$ real matrices $A$ satisfying $A^T = -A$ and $A A^T = \left( \ln L - \frac{1}{2} \right) I$, where $I$ is the $4 \times 4$ identity matrix. Find the number of elements in $S$.

Let $A = [a_{ij}]$ be a $3 \times 3$ skew-symmetric matrix with real entries such that $A \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$. For $i=1,2,3$, consider the lines $L_i: a_{i1}x + a_{i2}y + a_{i3} = 0$. Assuming $a_{12} \neq 0$, the lines $L_1, L_2, L_3$ are concurrent at a point $P$. If the locus of $P$ as the entries of $A$ vary is a curve $S$, and $d$ is the perpendicular distance from the origin to $S$, find the value of $10d^2$.

Let $A = [a_{ij}]$ be a $4 \times 4$ matrix where its entries are given by the binomial sum: $a_{ij} = \frac{1}{10} \sum_{k=0}^2 (-1)^k \binom{2}{k} (i-j+1-k)^5$ for $1 \le i, j \le 4$. Let matrix $B$ be obtained from $A$ by applying the following sequence of elementary operations in order: $R_1 \leftrightarrow R_3$, $C_2 \leftrightarrow C_4$, and $R_2 \leftrightarrow R_4$. Find the value of $\left[ \frac{|B|}{1000} \right]$, where $[ \cdot ]$ denotes the greatest integer function.

Let $\vec{a} = (x^2-4)\hat{i} + x^2\hat{j} + e^x\hat{k}$, $\vec{b} = (x^2-4)[x]\hat{i} + (x^2[x]+[x])\hat{j} + (e^x[x]+[x])\hat{k}$, and $\vec{c} = (x^2-4)[x]^2\hat{i} + (x^2[x]^2+[x]^2)\hat{j} + (e^x[x]^2+[x]^2+[x])\hat{k}$, where $[x]$ denotes the greatest integer less than or equal to $x$. Let $V(x)$ be the volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b},$ and $\vec{c}$. Find the value of $\lim_{h \to 0^+} \frac{V(2+h) - V(2-h)}{h}$

Let $\omega = e^{i \frac{2\pi}{7}}$. Let $A$ be a $6 \times 6$ lower triangular matrix with diagonal entries $A_{kk} = 1 - \omega^k + \frac{1}{1 - \omega^k}$ for $k = 1, 2, \dots, 6$. Let $B$ be a $6 \times 6$ upper triangular matrix with diagonal entries $B_{kk} = 1 - \omega^{6k} + \frac{1}{1 - \omega^{6k}}$ for $k = 1, 2, \dots, 6$. Find the value of $49 \det(AB)$.

Let $P$ be the set of all polynomials $p(x) = ax^2 + bx + c$ of degree exactly 2 ($a \neq 0$) with integer coefficients. For each $p \in P$, define a $3 \times 3$ matrix $A_p$ derived from function composition as: $A_p = \begin{pmatrix} p(1) & p(p(1)) & p(p(p(1))) \\ 0 & p(2) & p(p(2)) \\ 0 & 0 & p(3) \end{pmatrix}$ Let $T_p : \mathbb{Z}^3 \to \mathbb{Z}^3$ be the linear transformation defined by $T_p(X) = A_p X$. If $S$ is the subset of $P$ containing all polynomials for which $T_p$ is a bijective mapping, find the value of $\sum_{p \in S} (a^2 + b^2 + c^2)$.

Let $A = [a_{i,j}]$ be a $15 \times 15$ matrix defined as: $a_{i,j} = \begin{cases} 0 & \text{if } i \neq j \\ 2^{\left| \frac{1}{30} - \frac{1}{i(i+1)} \right|} & \text{if } i = j \end{cases}$ The value of $48 \log_2 (\det A)$ is equal to:

Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually orthogonal vectors forming a left-handed system. Consider three lines $L_1, L_2, L_3$ defined as: $L_1: \vec{r} = \vec{a} \times \vec{b} + \lambda (\vec{b} \times \vec{c})$ $L_2: \vec{r} = \vec{b} \times \vec{c} + \mu (\vec{c} \times \vec{a})$ $L_3: \vec{r} = \vec{c} \times \vec{a} + \gamma (\vec{a} \times \vec{b})$ If the shortest distances between the pairs of lines $(L_1, L_2)$, $(L_2, L_3)$, and $(L_3, L_1)$ are $2$, $3$, and $6$ respectively, find the value of $[\vec{a}' + \vec{b}', \vec{b}' + \vec{c}', \vec{c}' + \vec{a}']$, where $\vec{a}', \vec{b}', \vec{c}'$ represent the reciprocal system of vectors for $\vec{a}, \vec{b}, \vec{c}$.