JEE Main 2022 · 25 July · evening · Single correct · +4 / −1
The number of real values of λ, such that the system of linear equations
2x - 3y + 5z = 9
x + 3y - z = -18
3x - y + (λ² - | λ |)z = 16
has no solutions, is
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Answer and solution
Answer (c) 2
Solution
Maths sir · 4 steps
Set D = 0, check that the constants break the row relation, then count the roots in |λ|.
- Condition for none. Write k = λ² - |λ|. A 3 × 3 system has no solution only when D = 0 and the equations contradict each other.
D = |2, -3, 5; 1, 3, -1; 3, -1, k| = 2(3k-1) + 3(k+3) + 5(-10) = 9k - 43
- Check inconsistency. D = 0 gives k = 43/9. Then the third left side equals 10/9 times the first plus 7/9 times the second. Those multiples need 10/9(9) + 7/9(-18) = -4 on the right, but the right side is 16, so the system has no solution.
- Solve for lambda. Put t = |λ| ≥ 0: 9t² - 9t - 43 = 0, so t = (9 ± √1629)/18. Only t = (9 + √1629)/18 ≈ 2.74 is positive.
- So. λ = ±(9 + √1629)/18, which gives 2 real values of λ. The answer is (c).
The idea to keep. For no solution, make D = 0 and then confirm the right-hand sides do not fit. With |λ|, solve for t = |λ|, keep only t ≥ 0, and each positive t gives two values of λ.
Answer from the NTA final answer key. Solution by PrepWiser.