Maths Q4 – JEE-MAIN 2025

If the system of equations

2 x + λ y + 3 z = 5

3 x + 2 y - z = 7

4 x + 5 y + μ z = 9

has infinitely many solutions, then

(λ^{2} + μ^{2})

is equal to :

Clue 1 of 3: Condition for Infinitely Many Solutions

🧠 Full Solution Path

Step 1: Apply the condition for infinitely many solutions✦ Active

For a system of linear equations to have infinitely many solutions, the rank of the coefficient matrix must be equal to the rank of the augmented matrix, and this rank must be less than the number of variables. This implies that the rows of the augmented matrix are linearly dependent. Let the rows of the augmented matrix be $R_{1} = (2, λ, 3, 5)$ , $R_{2} = (3, 2, - 1, 7)$ , and $R_{3} = (4, 5, μ, 9)$ . We can assume $R_{3} = a R_{1} + b R_{2}$ for some scalars $a$ and $b$ .

Step 2: Solve for the scalar coefficients

a

and

b

○ Expand

Equating the corresponding components of the rows, we get a system of equations:

4 = 2 a + 3 b (1)

5 = a λ + 2 b (2)

μ = 3 a - b (3)

9 = 5 a + 7 b (4)

From equations (1) and (4), we can solve for $a$ and $b$ . Multiply (1) by 7 and (4) by 3:

28 = 14 a + 21 b

27 = 15 a + 21 b

Subtracting the second modified equation from the first gives $1 = - a$ , so $a = - 1$ . Substitute $a = - 1$ into (1): $4 = 2 (- 1) + 3 b ⟹ 4 = - 2 + 3 b ⟹ 6 = 3 b ⟹ b = 2$ .

Step 3: Determine

λ

and

μ

and calculate the final expression○ Expand

Substitute $a = - 1$ and $b = 2$ into equations (2) and (3):

5 = (- 1) λ + 2 (2) ⟹ 5 = - λ + 4 ⟹ λ = - 1

μ = 3 (- 1) - 2 ⟹ μ = - 3 - 2 ⟹ μ = - 5

Finally, calculate $(λ^{2} + μ^{2})$ :

λ^{2} + μ^{2} = (- 1)^{2} + (- 5)^{2} = 1 + 25 = 26

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Maths Question 4 – JEE-MAIN 2025

Clue 1 of 3: Condition for Infinitely Many Solutions

🧠 Full Solution Path