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CAT 2021 Slot 2 — QA Question 7

Number TheoryEasy

Consider the pair of equations: x2 – xy – x = 22 and y2 – xy + y = 34. If x > y, then x – y equal.

Answer & solution

  • 8

  • B

    7

  • C

    6

  • D

    4

Solution

Easy

Adding the two equations makes the messy xyxy terms combine into a perfect square in (x−y)(x-y). Set a=x−ya=x-y, solve the resulting quadratic, and keep the positive root since x>yx>y.

1

Add the equations.

x2−xy−x=22y2−xy+y=34⇒ x2+y2−2xy−(x−y)=56(add; −x+y=−(x−y))\begin{aligned} &x^2-xy-x=22\\ &y^2-xy+y=34\\ &\Rightarrow\ x^2+y^2-2xy-(x-y)=56 \quad\text{(add; }{-x}+y=-(x-y)\text{)} \end{aligned}
2

Recognise the perfect square. Since x2−2xy+y2=(x−y)2x^2-2xy+y^2=(x-y)^2.

(x−y)2−(x−y)=56\begin{aligned} &(x-y)^2-(x-y)=56 \end{aligned}
3

Solve the quadratic. Let a=x−ya=x-y, with a>0a>0 because x>yx>y.

a2−a−56=0⇒ (a−8)(a+7)=0⇒ a=8(a=−7 rejected, a>0)\begin{aligned} &a^2-a-56=0\\ &\Rightarrow\ (a-8)(a+7)=0\\ &\Rightarrow\ a=8 \quad\text{(}a=-7\text{ rejected, }a>0\text{)} \end{aligned}
x−y=8x-y=8
CAT 2021 Slot 2 QA Q7: Consider the pair of equations: x 2 – xy – x = 22 and y 2 – xy + y = 34. If x > y, then x &n — Solution | TheCATExam