Let and be respectively the sets of all for which the system of linear equations: has unique solution and infinitely many solutions. Then find the nature of sets and .

is an infinite set and Explanation: Solution: System of equations ke unique solution ke liye coefficient matrix ka determinant ( ) zero nahi hona chahiye. Step 1: Determinant ( ) calculate karna Pehle row ( ) se common lene par: Operations apply karne par: aur Isse expand karne par: Step 2: (Unique Solution) ke liye condition Unique solution ke liye : Quadratic equation ka discriminant ( ) check karte hain: Kyunki , iska matlab yeh quadratic expression hamesha positive rahega aur kabhi zero nahi hoga. Isliye, sirf tabhi hoga jab . Lekin question mein diya hai ki , yani zero nahi ho sakta. Athe: Sabhi ke liye hoga. Step 3: (Infinitely many solutions) ke liye condition Infinitely many solutions ke liye hona anivarya hai. Lekin upar humne dekha ki ke liye kabhi zero nahi hota. Athe: Aisi koi ki value nahi hai jiske liye infinit

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $S_1$ and $S_2$ be respectively the sets of all $a \in \mathbb{R} - \{0\}$ for which the system of linear equations:

$ax + 2ay - 3az = 1$

$(2a + 1)x + (2a + 3)y + (a + 1)z = 2$

$(3a + 5)x + (a + 5)y + (a + 2)z = 3$

has unique solution and infinitely many solutions. Then find the nature of sets $S_1$ and $S_2$ .

JEE 2023 — Mathematics PYQ

Choose the correct answer:

Explanation

Solution:

Correct Option: (4) $S_1 = \mathbb{R} - \{0\}$ and $S_2 = \Phi$

Explanation

Solution:

Correct Option: (4) $S_1 = \mathbb{R} - \{0\}$ and $S_2 = \Phi$