The set of equations : λx-y+(cosθ) z=0 ; 3 x+y+2 z =0 ;(cosθ) x+y+2 z=0,0 ≤θ<2 π, has non-trivia

Name: Problem 81, Chapter 6: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-09-25T11:07:42-08:00
Duration: 1 min 59 s
Channel: Gaurav Kalra
Description: The set of equations : λx-y+(cosθ) z=0 ; 3 x+y+2 z =0 ;(cosθ) x+y+2 z=0,0 ≤θ<2 π, has non-trivial solutions. (A) for no values of λand θ(B) for all values of λand θ(C) for all values of λand only two values of θ(D) for only one value of λand all values of θ

step 1 I have given here this homogene equations, theta line between 0 and 2 pi, and non -tribal soluti

Question

The set of equations : $\lambda x-y+(\cos \theta) z=0 ; 3 x+y+2 z$ $=0 ;(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \pi$, has non-trivial solutions. (A) for no values of $\lambda$ and $\theta$ (B) for all values of $\lambda$ and $\theta$ (C) for all values of $\lambda$ and only two values of $\theta$ (D) for only one value of $\lambda$ and all values of $\theta$

   The set of equations : $\lambda x-y+(\cos \theta) z=0 ; 3 x+y+2 z$
$=0 ;(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \pi$, has non-trivial
solutions.
(A) for no values of $\lambda$ and $\theta$
(B) for all values of $\lambda$ and $\theta$
(C) for all values of $\lambda$ and only two values of $\theta$
(D) for only one value of $\lambda$ and all values of $\theta$

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 6, Problem 81

Instant Answer

Solved by Expert Gaurav Kalra

09/25/2021

Step 1

Step 1: We are given the system of equations: \[\begin{cases} \lambda x-y+(\cos \theta) z=0 \\ 3 x+y+2 z=0 \\ (\cos \theta) x+y+2 z=0 \end{cases}\] We are asked to find the values of $\lambda$ and $\theta$ for which this system has non-trivial solutions. Show more…

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The set of equations : $\lambda x-y+(\cos \theta) z=0 ; 3 x+y+2 z$ $=0 ;(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \pi$, has non-trivial solutions. (A) for no values of $\lambda$ and $\theta$ (B) for all values of $\lambda$ and $\theta$ (C) for all values of $\lambda$ and only two values of $\theta$ (D) for only one value of $\lambda$ and all values of $\theta$

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Key Concepts

Homogeneous Systems of Equations

A homogeneous system of linear equations is one in which all of the constant terms are zero. Such systems always have at least the trivial solution (where all the variables are zero), and non?trivial solutions exist when the coefficient matrix is singular.

Matrix Determinant and Singularity

The determinant of the coefficient matrix indicates whether the matrix is singular or non-singular. If the determinant is zero, the matrix is singular, and for homogeneous systems this means there are infinitely many solutions, including non-trivial ones.

Parameter Dependence in Linear Systems

When the coefficients of the system depend on parameters, conditions for the existence of non-trivial solutions are found by determining the values of the parameters that make the determinant of the coefficient matrix equal to zero. This analysis is crucial in understanding how the parameters affect the solution set.

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