Graphical/algebraic methods, consistency, applications MCQs

Q1. For what value of k do x + 2y = 3 and 2x + ky = 6 have unique solution?

• A.k ≠ 4
• B.k = 4
• C.k ≠ 2
• D.k = 2

Answer: A. k ≠ 4

From the stem we have: 2; 3; 2; 6. Required: the unknown asked in the stem. Formula: recall the relevant relation from this chapter (math, chapter 'Pair of Linear Equations in Two Variables') that ties the given data to the unknown. Why this works - Unique solution: 1/2 ≠ 2/k, so k ≠ 4. So the answer is A) k ≠ 4.

Q2. For what value of k does the system 2x + 3y = 7, kx + 9y = 21 have infinitely many solutions?

• A.6
• B.3
• C.2
• D.9

Answer: A. 6

From the stem we have: 2; 3; 7; 9; 21. Required: the unknown asked in the stem. Working tool - Coincident lines: 2/k = 3/9 = 7/21. This is the equation that links the given quantities to the unknown (math, chapter 'Pair of Linear Equations in Two Variables'). Carrying out the arithmetic: Coincident lines: 2/k = 3/9 = 7/21 → From 3/9 = 1/3 = 7/21, the ratio is 1/3 → So 2/k = 1/3 ⇒ k = 6. Putting it together the answer is A) 6.

Q3. The pair x + y = 2 and 2x + 2y = 4 has how many solutions?

• A.Infinitely many
• B.Exactly one
• C.None
• D.Exactly two

Answer: A. Infinitely many

Given 2, 2, 2, 4, asked for how many solutions?. Apply the standard chapter relation to the data. Second equation is just 2× the first. Consequently option A) Infinitely many.