IPMAT Indore 2025 · Quantitative Ability · Question 4

Step 1 — recall the Remainder Theorem. When a polynomial P\left( x \right) is divided by \left( x-k \right), the remainder is simply P\left( k \right).

Step 2 — apply it twice. Let P\left( x \right)=a{x}^{2}+bx+5.

Divided by x-1, remainder =P\left( 1 \right)=a+b+5=3 \Rightarrow a+b=-2

Divided by x+1 (that is, x-\left( -1 \right)), remainder =P\left( -1 \right)=a-b+5=2 \Rightarrow a-b=-3

Step 3 — solve the pair.

Adding: 2a=-5 \Rightarrow a=-5/2

Subtracting: 2b=1 \Rightarrow b=1/2

Step 4 — evaluate what was asked.

2b-4a=2\left( 1/2 \right)-4\left( -5/2 \right)=1+10=11

Answer: 11