JEE 2023 Mathematics PYQ — Let be a root of the equation where are distinct real numbers suc… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let α be a root of the equation (a−c)x2+(b−a)x+(c−b)=0 where a,b,c are distinct real numbers such that the matrix α21aamp;αamp;1amp;bamp;1amp;1amp;cis singular. Then, the value of (b−a)(c−b)(a−c)2+(a−c)(c−b)(b−a)2+(a−c)(b−a)(c−b)2is:
Choose the correct answer:
A.
12
B.
9
C.
3
(Correct Answer)
D.
6
Correct Answer:
3
Explanation
Step 1: Analyzing the equation
The given equation is (a−c)x2+(b−a)x+(c−b)=0.
Notice that the sum of the coefficients is:
(a−c)+(b−a)+(c−b)=0
This implies that x=1 is always a root of this equation. Thus, α=1.
Step 2: Using the Singularity Condition
Since the matrix is singular, its determinant must be zero:
α21aamp;αamp;1amp;bamp;1amp;1amp;c=0
If we put α=1, the first two rows become identical [1,1,1], which automatically makes the determinant 0. This confirms α=1 is consistent with the condition.
Step 3: Finding the value of the expression
Let:
x=a−c
y=b−a
z=c−b
We know from Step 1 that x+y+z=0.
The expression we need to evaluate is:
yzx2+xzy2+xyz2
Taking the LCM:
xyzx3+y3+z3
Using the algebraic identity: If x+y+z=0, then x3+y3+z3=3xyz.
Substituting this into our expression:
xyz3xyz=3
Correct Option: (C) 3
Explanation
Step 1: Analyzing the equation
The given equation is (a−c)x2+(b−a)x+(c−b)=0.
Notice that the sum of the coefficients is:
(a−c)+(b−a)+(c−b)=0
This implies that x=1 is always a root of this equation. Thus, α=1.
Step 2: Using the Singularity Condition
Since the matrix is singular, its determinant must be zero:
α21aamp;αamp;1amp;bamp;1amp;1amp;c=0
If we put α=1, the first two rows become identical [1,1,1], which automatically makes the determinant 0. This confirms α=1 is consistent with the condition.
Step 3: Finding the value of the expression
Let:
x=a−c
y=b−a
z=c−b
We know from Step 1 that x+y+z=0.
The expression we need to evaluate is:
yzx2+xzy2+xyz2
Taking the LCM:
xyzx3+y3+z3
Using the algebraic identity: If x+y+z=0, then x3+y3+z3=3xyz.
Substituting this into our expression:
xyz3xyz=3
Correct Option: (C) 3
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