or

# For which value (s) of lambda, do the pair of linear equations lambdax + y = lambda^(2) and x + lambda y = 1 have <br> (i) no solution ? (ii) infinitely many solutions ? <br> (iii) a unique solution ?

Question from  Class 10  Chapter Pair Of Linear Equations In Two Variables

Apne doubts clear karein ab Whatsapp par bhi. Try it now.
CLICK HERE

Loading DoubtNut Solution for you

Watch 1000+ concepts & tricky questions explained!

6.6 K+ views | 6.6 K+ people like this

Share
Share

Answer Text

Solution :
The given pair of linear equations is <br> `lambdax + y = lambda^(2)` and `x + lambda y = 1` <br> Here, `" " a_(1) = lambda, b_(1) = 1, c_(1) = - lambda^(2)` <br> `a_(2) = 1, b_(2) = lambda, c_(2) = -1` <br> (i) For no solution, <br> `(a_(1))/(a_(2)) = (b_(1))/(b_(2)) != (c_(1))/(c_(2))` <br> `rArr " " (lambda)/(1) = (1)/(lambda) != (-lambda^(2))/(-1)` <br> `rArr " " lambda^(2) - 1 = 0` <br> `rArr " " (lambda - 1) (lambda + 1) = 0` <br> `rArr " " lambda = 1, -1` <br> Here, we take only `lambda = -1` because at `lambda = 1` the system of linear equations has infinitely many solutions. <br> (ii) For infinitely many solutions, <br> `(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))` <br> `rArr " " (lambda)/(1) = (1)/(lambda) = (lambda^(2))/(1)` <br> `rArr " " (lambda)/(1) = (lambda^(2))/(1)` <br> `rArr " " lambda (lambda - 1) = 0` <br> when `lambda != 0, ` then ` lambda = 1` <br> For a unique solution, <br> ` (a_(1))/(a_(2))!=(b_(1))/(b_(2)) rArr (lambda)/(1) != (1)/(lambda)` <br> `rArr " " lambda^(2) != 1 rArr lambda != +- 1` <br> So, all real values of `lambda ` except `+-1`.

Related Video

2:05

60.2 K+ Views | 231.6 K+ Likes

2:11

500+ Views | 58.1 K+ Likes

9:10

400+ Views | 300+ Likes

1:00

30.5 K+ Views | 92.1 K+ Likes

4:05

67.1 K+ Views | 201.1 K+ Likes

2:50

41.7 K+ Views | 13.1 K+ Likes

1:26

134.0 K+ Views | 480.3 K+ Likes
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}