Determinants Questions - JEE

Which of the following is an orthogonal matrix?

If the following system of equations is consistent,$(a+1)^3x + (a+2)^3y = (a+3)^3$$(a+1)x + (a+2)y = a+3$$x + y = 1$,then find the value of $a$.

If det(A)=3, then det(2$A$^{-1}) for 3 \times 3 matrix equals:

If $\alpha, \beta, \gamma$ are the roots of $ax^3 + bx^2 + cx + d = 0$ and \[\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix} = 0,\] $\alpha \neq \beta \neq \gamma$, then find the equation whose roots are $\alpha+\beta-\gamma$, $\beta+\gamma-\alpha$, and $\gamma+\alpha-\beta$.

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 0 & 5 \\ 0 & 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 \\ -3 \\ 1 \end{bmatrix}$. Which of the following is true?

If $ P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix} $, $ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ and $ Q = PAP^T $ and $ X = P^T Q^{2005} P $, then $ X $ is equal to

There are two numbers x making the \[ \begin{vmatrix} 1 & -2 & 5 \\ & 2 & x & -1 \\ & 0 & 4 & 2x \end{vmatrix} \] equal to 86. The sum of these numbers is:

If determinant has one column entirely zero, then its value is:

If the system of linear equations,$x + ky + 3z = 0$$3x + ky - 2z = 0$$2x + 4y - 3z = 0$has a non-zero solution $(x, y, z)$, then $\dfrac{xz}{y^2}$ is equal to

If A = $$\begin{bmatrix} 3 & 4 \\ 5 & 7 \end{bmatrix}$$, then A × (adj A) is equal to

The value of the determinant $\begin{vmatrix} ^nC_{r-1} & ^nC_r & (r+1)^{n+2}C_{r+1} \\ ^nC_r & ^nC_{r+1} & (r+2)^{n+2}C_{r+2} \\ ^nC_{r+1} & ^nC_{r+2} & (r+3)^{n+2}C_{r+3} \end{vmatrix}$ is

If $\det(A)$=4, then det(A\)^6A⁻^4A^{-1}) equals:

Let $A = \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix}$ and $(A+I)^{50} - 50A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then the value of $a + b + c + d$ is

If $\det(A)$=2, then det(A\)^4A⁻^2A^{-1}) equals:

If A = $\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where I is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

Consider a matrix $A = [a_{ij}]$ of order $3 \times 3$ such that $a_{ij} = (k)^{i+j}$ where $k \in I$.Match List I with List II and select the correct answer using the codes given below the lists.List Ia. $A$ is singular ifb. $A$ is null matrix ifc. $A$ is skew-symmetric which is not null matrix ifd. $A^2 = 3A$ ifList IIp. $k \in \{0\}$q. $k \in \phi$r. $k \in I$s. $k \in \{-1, 0, 1\}$Codes:(1) a-r, b-p, c-s, d-q(2) a-s, b-p, c-q, d-r(3) a-r, b-p, c-q, d-s(4) a-q, b-p, c-r, d-s

If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, then the value of the determinant\[\begin{vmatrix} y+1 & \beta & \alpha \\ \beta & y+\alpha & 1 \\ \alpha & 1 & y+\beta \end{vmatrix}\]is equal to

Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1/√3 [[1, 1, 1], [1, α, α2], [1, α2, α4]], then the matrix A31 is equal to:

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

We have $p = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$. If $|\text{adj}\, A| = |A|^2 \Rightarrow |\text{adj}\, A| = 16$, find $\alpha$.

If $ P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix} $, $ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ and $ Q = PAP^T $, then $ P^T Q^{2015} P $ is

We have \[BC = \begin{bmatrix}3 & 4\\2 & 3\end{bmatrix}\begin{bmatrix}3 & -4\\-2 & 3\end{bmatrix} = I.\] Find \[\text{tr}(A) + \text{tr}\!\left(\frac{A(BC)}{2}\right) + \text{tr}\!\left(\frac{A(BC)^2}{4}\right) + \text{tr}\!\left(\frac{A(BC)^3}{8}\right) + \cdots\infty,\] given that $A = \begin{bmatrix}2 & 1\\1 & 1\end{bmatrix}$ (so that $\text{tr}(A)=3$ and the series converges).

The number of diagonal matrices A of order $n$ for which $A^3 = A$ is

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and a, d, g are in A.P., then determinant is:

If $\begin{vmatrix} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{vmatrix} = 0$, then the line $ax + by + c = 0$ passes through the fixed point which is

Let $a, z, y, z$ be real numbers satisfying the equations $az + ay = 5$, $x - ay = z$, $x + ay = az$, where $x, y, z$ are not all zero, then the number of the possible values of $a$ is

If a3 + b3 + c3 − 3abc = −3 and A = bc − a2, B = ca − b2 and C = ab − c2, then the value of aA + bB + cC is

If $f(x) = \begin{vmatrix} \cos(x-\phi) & \cos(x-\psi) & \cos(x-\omega) \\ \sin(x-\phi) & \sin(x-\psi) & \sin(x-\omega) \\ \sin(\psi - \omega) & \sin(\omega - \phi) & \sin(\phi - \psi) \end{vmatrix}$, then $f(9) - 2f(6) + f(3)$ is equal to

Find the value of k such that the determinant \[\begin{vmatrix} 3^2 + k & 4^2 & 3^2 + 3 + k \\ 4^2 + k & 5^2 & 4^2 + 4 + k \\ 5^2 + k & 6^2 & 5^2 + 5 + k \end{vmatrix} = 0\]

Question 85: Statement-1: If $f(x) = \begin{vmatrix} (1-x)^{11} & (1-x)^{12} & (1-x)^{13} \\ (1-x)^{21} & (1-x)^{22} & (1-x)^{23} \\ (1-x)^{31} & (1-x)^{32} & (1-x)^{33} \end{vmatrix}$, then the coefficient of $x$ in $f(x) = 0$.Statement-2: If $P(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n$, then $a_1 = P'(0)$, where dash denotes the differential coefficient.

Given matrix $A = \begin{bmatrix} 1 & x & 1 \\ x & 2 & y \\ 1 & y & 3 \end{bmatrix}; B = \begin{bmatrix} 3 & -3 & z \\ -3 & 2 & -3 \\ z & -3 & 1 \end{bmatrix}$. Obtain $x, y, z$ if the matrix $AB$ is symmetric.

Let $f(x) = \begin{vmatrix} \sec x & x^2 & x \\ 2\sin x & x^3 & 4 \\ \tan 3x & x & x \end{vmatrix}$, then $\lim_{x \to 0} \frac{f(x)}{x^2}$ is equal to

Sum of real roots of the equation \[\begin{vmatrix} 1 & 4 & 20 \\ 1 & 2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} = 0\] is

If $f(x) = (x-1)^m x^n (x+1)^p$, where $m, n, p \in \mathbb{N}$, then the value of $m+n+p$ is:

If the system of equations ax + y = 1, x + 2y = 3, 2x + 3y = 5 are consistent, then a is given by

If A is a square matrix of order 2 such that A−1 = pmatrix1 & -1 \\ -1 & 2pmatrix and Apmatrix2 & 1 \\ 1 & -1pmatrix = pmatrix0pmatrix. The sum of elements and product of elements of A are S and P, then S + P is

If \det(A) = 3, then \det(k$A$^{-1}) for $3 \times 3$ matrix is:

Evaluate the determinant:$\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix}$

If I is a unit matrix of order 10, the determinant of I is equal to

If m = 2 and n = 5, then p equals to

Which of the following ordered triplet ( m, n, p ) is false?

The number of all possible values of θ, where 0 < θ < π, for which the system of equations(y + z)cosθ = (xyz)sinθxsinθ = 2cos3θ/y + 2sin3θ/z(xyz)sinθ = (y + 2z)cosθ + ysin3θhave a solution (x0, y0, z0) with y0z0 ≠ 0, is

Let $M = \begin{bmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{bmatrix}$ and $adjM = \begin{bmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?(A) $a + b = 3$(B) $\det(adjM^2) = 81$(C) $(adjM)^{-1} + adjM^{-1} = -M$(D) If $M \begin{bmatrix} \alpha \\ 1 \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $\alpha - \beta + \gamma = 3$

The determinant xxz+xzyzzxx+y equals -

The number of real values of x satisfying is -

The number of real values of x satisfying <mfenced open="|

Let $f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1+\cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1+\sin 2x \end{vmatrix}, x \in \left[ \frac{\pi}{6}, \frac{\pi}{3} \right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

Let a, λ, μ ∈ R. Consider the system of linear equationsax + 2y = λ3x - 2y = μWhich of the following statement(s) is(are) correct?(A) if a = -3, then the system has infinitely many solutions for all values of λ and μ(B) if a ≠ -3, then the system has a a unique solution for all values of λ and μ(C) if λ + μ = 0, then the system has infinitely many solutions for a = -3(D) if λ + μ ≠ 0, then the system has no solution for a = -3

For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?(A) NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric(B) MN − NM is skew symmetric for all symmetric matrices M and N(C) MN is symmetric for all symmetric matrices M and N(D) (adjM)(adjN) = adj(MN) for all invertible matrices M and N

Let A = 02y12xy-12x-y1, (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is :-

Determinants Questions (2072)