Graphical method / consistency MCQs

Q1. For what value of k does the pair 1x + 2y = 3 and 2x + ky = 7 have NO solution?

• A.-4
• B.4
• C.8
• D.5

Answer: B. 4

Given 1, 2, 3, 2, 7, asked for the unknown. By for a1x+b1y+c1=0, a2x+b2y+c2=0 compare a1/a2, b1/b2, c1/c2. for a1x+b1y+c1=0, a2x+b2y+c2=0 compare a1/a2, b1/b2, c1/c2 → unique solution iff a1/a2 != b1/b2, infinite iff all three ratios are equal, and no solution iff a1/a2 = b1/b2 != c1/c2 → hence the answer = 4. match the ratios with the given coefficients. Putting it together option B) 4. Others fail: option A) '-4' fails since Wrong: uses an incorrect intermediate value; option C) '8' misses the point - Wrong: a common multi-step slip gives this.

Q2. For what value of k does the pair 1x + 4y = 2 and 3x + ky = 7 have NO solution?

• A.12
• B.-12
• C.13
• D.24

Answer: A. 12

Given 1, 4, 2, 3, 7, asked for the unknown. By for a1x+b1y+c1=0, a2x+b2y+c2=0 compare a1/a2, b1/b2, c1/c2. for a1x+b1y+c1=0, a2x+b2y+c2=0 compare a1/a2, b1/b2, c1/c2 → unique solution iff a1/a2 != b1/b2, infinite iff all three ratios are equal, and no solution iff a1/a2 = b1/b2 != c1/c2 → hence the answer = 12. match the ratios with the given coefficients. Thus option A) 12. Others fail: option B) '-12' doesn't hold - Wrong: uses an incorrect intermediate value; option C) '13' misses the point - Wrong: a step was skipped or HCF/LCM swapped.

Q3. For what value of k does the pair 1x + 3y = 5 and 5x + ky = 26 have NO solution?

• A.30
• B.-15
• C.15
• D.16

Answer: C. 15