Question

Let $p\left(x\right)=a^2+bx,q\left(x\right)=l{x}^2+mx+n$. If $p\left(1\right)-q\left(1\right)=0,p\left(2\right)-q\left(2\right)=1$ and $p\left(3\right)-q\left(3\right)=4$, then $p\left(4\right)-q\left(4\right)$ equals

• A. $0$
• B. $5$
• C. $6$
• D. $9$

Answer 1

Answer: (D)

$9$

We have 3 relations given, and have 5 unknown coefficients, a, b, l, m, n to be evaluated. This clearly does not have a unique solution. Thus, we need to modify the equation to be solved.

In this question x is only an input variable and is not to be evaluated.

We observe that the question only discusses p(x) - q(x), and not p(x) and q(x) individually. So it makes sense to solve for this quantity instead of the individual expressions.

Let, $h\left(x\right)=p\left(x\right)-q\left(x\right)$

$\Rightarrow h\left(x\right)=a^2+bx-l{x}^2-mx-n$

$\Rightarrow h\left(x\right)=A{x}^2+Bx+C$

Here, $A=-l,B=(b-m),C=(a^2-n)$

So we will solve for A, B , C thus reducing the number of variables.

The first equation says $h\left(1\right)=p\left(1\right)-q\left(1\right)=0$

$\Rightarrow A+B+C=0$ $.......\left(i\right)$

The second relation is $h\left(2\right)=p\left(2\right)-q\left(2\right)=1$

$\Rightarrow 4A+2B+C=1$ $........\left(ii\right)$

The third relation give in the problem is $h\left(3\right)=p\left(3\right)-q\left(3\right)=4$

$\Rightarrow 9A+3B+C=4$ $........\left(iii\right)$

Solving (ii), we get

$4A+2B+C=3A+B+(A+B+C)=3A+B=1$ $........\left(iv\right)$ $.....\left[From\right(i),(ii\left)\right]$

Solving (iii), we get

$9A+3B+C=3(3A+B)+C=3\left(1\right)+C=3+C=4$ $.....\left[From\right(iii),(iv\left)\right]$

$\Rightarrow C=1$

Substituting the value of C in (i), we get

$A+B=-1$

We also have, from (iv), the relation, $3A+B=1$

Subtracting the two, we get

$(A+B)-(3A+B)=-2A=-1-1=-2$

$\Rightarrow A=1$

Substituting the value of a in (iv), we get

$3A+B=3\left(1\right)+B=B+3=1$

$\Rightarrow B=-2$

Thus, $h\left(x\right)=p\left(x\right)-q\left(x\right)=x^2-2x+1=(x-1{)}^2$

$\Rightarrow p\left(4\right)-q\left(4\right)=h\left(4\right)=(4-1{)}^2=9$

$p\left(4\right)-q\left(4\right)=9$