The following system of linear equations
2x + 3y + 2z = 9
3x + 2y + 2z = 9
x - y + 4z = 8
2x + 3y + 2z = 9
3x + 2y + 2z = 9
x - y + 4z = 8
Show Hint
- does not have any solution
- has a unique solution
- has infinitely many solutions
- has a solution ($\alpha, \beta, \gamma$) satisfying $\alpha + \beta^2 + \gamma^2 = 12$
The Correct Option is B
Solution and Explanation
To determine the nature of the solution for a system of linear equations, we first calculate the determinant of the coefficient matrix, denoted by $\Delta$. 
$\Delta = \det(A) = 2(2 \cdot 4 - 2 \cdot (-1)) - 3(3 \cdot 4 - 2 \cdot 1) + 2(3 \cdot (-1) - 2 \cdot 1)$.
$\Delta = 2(8 + 2) - 3(12 - 2) + 2(-3 - 2)$.
$\Delta = 2(10) - 3(10) + 2(-5)$.
$\Delta = 20 - 30 - 10 = -20$.
Since the determinant $\Delta$ is non-zero ($\Delta = -20 \neq 0$), the system of linear equations is consistent and has a unique solution.
Therefore, option (B) is the correct statement.
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Matrices and Determinants Questions
Let $$ B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix} $$ and $A$ be a $2 \times 2$ matrix such that $$ AB^{-1} = A^{-1}. $$ If $BCB^{-1} = A$ and $$ C^4 + \alpha C^2 + \beta I = O, $$ then $2\beta - \alpha$ is equal to:
- Consider the matrices: \[ A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, \quad B = \begin{bmatrix} 20 \\ m \end{bmatrix}, \quad \text{and} \quad X = \begin{bmatrix} x \\ y \end{bmatrix}. \] Let the set of all \( m \), for which the system of equations \( AX = B \) has a negative solution (i.e., \( x < 0 \) and \( y <0 \)), be the interval \( (a, b) \). Then \[ 8 \int_a^b |\det(A)| \, dm \] is equal to _____ .
- If the system of equations
\( x + \left( \sqrt{2} \sin \alpha \right) y + \left( \sqrt{2} \cos \alpha \right) z = 0 \)
\( x + \left( \cos \alpha \right) y + \left( \sin \alpha \right) z = 0 \)
\( x + \left( \sin \alpha \right) y - \left( \cos \alpha \right) z = 0 \)
has a non-trivial solution, then \( \alpha \in \left( 0, \frac{\pi}{2} \right) \) is equal to: - Let \( \alpha \in (0, \infty) \) and \( A = \begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix} \). If \( \det(\text{adj}(2A - A^\top) \cdot \text{adj}(A - 2A^\top)) = 2^8 \), then \( (\det(A))^2 \) is equal to:
- Let \( A \) be a square matrix of order 2 such that \( |A| = 2 \) and the sum of its diagonal elements is \(-3\). If the points \((x, y)\) satisfying \( A^2 + xA + yI = 0 \) lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is \( e \) and the length of the latus rectum is \( \ell \), then \( e^4 + \ell^4 \) is equal to _____
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.