The system of linear equations $x + y + z = 6$, $2x + 5y + az = 36$, $x + 2y + 3z = b$ has :
A) unique solution for $a = 8$ and $b = 16$
B) infinitely many solutions for $a = 8$ and $b = 14$
C) infinitely many solutions for $a = 8$ and $b = 16$
D) no solution for $a = 8$ and $b = 14$
A) unique solution for $a = 8$ and $b = 16$
B) infinitely many solutions for $a = 8$ and $b = 14$
C) infinitely many solutions for $a = 8$ and $b = 16$
D) no solution for $a = 8$ and $b = 14$
✓
Solution Steps
1
Write the Coefficient Matrix and find $\Delta$
The determinant of the coefficient matrix is:
$\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 5 & a \\ 1 & 2 & 3 \end{vmatrix}$
Expanding along Row 1: $1(15 – 2a) – 1(6 – a) + 1(4 – 5)$
$\Delta = 15 – 2a – 6 + a – 1 = 8 – a$
2
Condition for non-unique solutions
For the system to have either infinitely many solutions or no solution, $\Delta$ must be zero.
$8 – a = 0 \implies a = 8$.
3
Calculate $\Delta_z$ with the constant terms
Replace the third column with the constants $(6, 36, b)$:
$\Delta_z = \begin{vmatrix} 1 & 1 & 6 \\ 2 & 5 & 36 \\ 1 & 2 & b \end{vmatrix}$
Expanding: $1(5b – 72) – 1(2b – 36) + 6(4 – 5)$
$\Delta_z = 5b – 72 – 2b + 36 – 6 = 3b – 42$.
4
Condition for infinitely many solutions
For infinitely many solutions, all partial determinants must be zero. Setting $\Delta_z = 0$:
$3b – 42 = 0 \implies 3b = 42 \implies b = 14$.
5
Final Verification
When $a = 8$ and $b = 14$, checking $\Delta_x$ and $\Delta_y$ confirms they also vanish. The rank of both the coefficient and augmented matrices becomes 2, which is less than the number of variables (3).
Infinitely many solutions for $a = 8$ and $b = 14$ (Option B)
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Theory
1. Cramer’s Rule for 3×3 Systems
Cramer’s Rule is a fundamental method in linear algebra used to solve a system of $n$ linear equations with $n$ variables using determinants. For a 3-variable system, we define $\Delta$ as the determinant of the coefficient matrix. The variables $x, y, z$ are given by $x = \Delta_x/\Delta$, $y = \Delta_y/\Delta$, and $z = \Delta_z/\Delta$, where $\Delta_i$ is the determinant formed by replacing the $i$-th column with the constants.
2. Analysis of Consistency
Consistency refers to whether a system has a solution. If $\Delta \neq 0$, the system is consistent and has a unique solution. If $\Delta = 0$, the system is either inconsistent (no solution) or consistent with infinite solutions. To distinguish these, we check the partial determinants. If $\Delta=0$ and at least one $\Delta_i \neq 0$, there is no solution. If $\Delta=0$ and all $\Delta_i = 0$, there are usually infinitely many solutions.
3. Rank of a Matrix
The rank of a matrix is the number of linearly independent rows or columns. According to the Rouché–Capelli theorem, a system $AX = B$ has a solution if the rank of the coefficient matrix $A$ is equal to the rank of the augmented matrix $[A|B]$. If this rank is equal to the number of variables, the solution is unique. If the rank is less than the number of variables, there are infinitely many solutions.
4. Geometric Interpretation of Linear Systems
Each linear equation in three variables represents a plane in 3D space. A unique solution means the three planes intersect at a single point. No solution occurs when the planes are parallel or intersect in a way that there is no common point for all three. Infinitely many solutions occur when the three planes intersect along a common line or are coincident.
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FAQs
1
What if $a \neq 8$?
If $a \neq 8$, then $\Delta \neq 0$, and the system will always have a unique solution regardless of the value of $b$.
2
Does $\Delta=0$ and all $\Delta_i=0$ always mean infinite solutions?
In most JEE problems, yes. However, mathematically, there is a rare case where the planes are parallel and distinct but all $\Delta_i$ are zero. Rank analysis is more robust.
3
How to calculate the determinant quickly?
Using properties like $R_2 \to R_2 – 2R_1$ and $R_3 \to R_3 – R_1$ can simplify the matrix before expansion.
4
Is there a shortcut for $\Delta_z$?
One can substitute the condition $\Delta=0$ (i.e., $a=8$) directly into the row operations to see how the third row of the augmented matrix behaves.
5
What if $a=8$ and $b=16$?
Then $\Delta = 0$ but $\Delta_z = 3(16)-42 = 6 \neq 0$. This leads to no solution.
6
What is the significance of $a=8$?
It makes the coefficient of $z$ in the second equation such that the left-hand side of the equations becomes linearly dependent.
7
Can the system have exactly two solutions?
No, a system of linear equations can only have zero, one, or infinitely many solutions.
8
What are ‘trivial’ solutions?
A trivial solution is $x=y=z=0$, which only occurs in homogeneous systems (where the constants are all zero). This system is non-homogeneous.
9
Is this topic important for JEE?
Yes, Determinants and Systems of Equations is one of the highest-weightage topics in JEE Main Mathematics.
10
Can we solve this by eliminating variables?
Yes, you can eliminate $x$ from the 2nd and 3rd equations to get two equations in $y$ and $z$, then compare their ratios.
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