Find the value(s) of k for which the pair of linear equations k x + y = k 2 and x + k y = 1 have infinitely many solutions.

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Question 9 Find the value(s) of k for which the pair of linear equations kx + y = k2 and x + ky = 1 have infinitely many solutions. kx + y = k2 i.e. kx + y – k2 = 0 x + ky = 1 i.e. x + ky – 1 = 0 kx + y – k2 = 0 Comparing with a1x + b1y + c1 = 0 ∴ a1 = k , b1 = 1 , c1 = –k2 x + ky – 1 = 0 Comparing with a2x + b2y + c2 = 0 ∴ a2 = 1 , b2 = k , c2 = –1 Since equation has infinite number of solutions So, 𝑎1/𝑎2 = 𝑏1/𝑏2 = 𝑐1/𝑐2 Putting in values 𝑘/1 = 1/𝑘 = (−𝑘^2)/(−1) 𝑘/1 = 1/𝑘 = 𝑘^2/1 Solving 𝒌/𝟏 = 𝟏/𝒌 k2 =1 k = ± √1 k = ± 1 So, k = 1, –1 Solving 𝟏/𝒌 = 𝒌^𝟐/𝟏 1 = k3 k3 = 1 k3 = 13 k = 1 Therefore, k = 1 satisfies both equations. Hence k = 1 is the answer

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