# CAT 2021 Set-2 | Quantitative Aptitude | Question: 1

1 vote
44 views

Consider the pair of equations: $x^{2} – xy – x = 22$ and $y^{2} – xy + y = 34.$ If $x>y,$ then $x – y$ equals

1. $7$
2. $8$
3. $6$
4. $4$

retagged

1 vote

Given that,

• $x^{2} – yx – x = 22\quad \longrightarrow(1)$
• $y^{2} – xy + y = 34\quad \longrightarrow(2)$
• $x>y$

From equation $(1),$

$x^{2} – yx – x = 22$

$\Rightarrow x(x-y-1) = 22\quad \longrightarrow(3)$

From equation $(2),$

$y^{2} – xy + y = 34$

$\Rightarrow y(y-x+1) = 34$

Multiply $‘-’$ on both the sides.

$-y(y-x+1) = -34$

$\Rightarrow y(-y+x-1) = -34$

$\Rightarrow y(x-y-1) = -34\quad \longrightarrow(4)$

Subtract equation $(4),$ from equation $(3).$

$x(x-y-1) - y(x-y-1) = 22-(-34)$

$\Rightarrow (x-y-1)(x-y) = 56 \quad \longrightarrow(5)$

Let, $x-y = m \quad {\color{Blue} {(x>y \Rightarrow x – y >0 \Rightarrow m>0)}}$

Now, $(m-1)m = 56$

$\Rightarrow m^{2}-m-56 =0$

$\Rightarrow m^{2}-8m+7m-56 =0$

$\Rightarrow m(m-8)+7(m-8) =0$

$\Rightarrow (m-8)(m+7) =0$

$\Rightarrow m=8\; \text{(or)}\; m=-7\;{\color{Red} {\text{(rejected)}}}$

$\Rightarrow \boxed{m=8}$

$\Rightarrow \boxed{x-y=8}$

Correct Answer $:\text{B}$

10.1k points 4 8 30
edited

## Related questions

1 vote
1
74 views
Anil, Bobby and Chintu jointly invest in a business and agree to share the overall profit in proportion to their investments. Anil's share of investment is $70 \%.$ His share of profit decreases by $₹ \; 420$ if the overall profit goes down from $18 \%$ to $15 \%.$ Chintu's share of ... goes up from $15 \%$ to $17 \%.$ The amount, $\text{in INR},$ invested by Bobby is $2400$ $2200$ $2000$ $1800$
1 vote
2
82 views
If a rhombus has area $12 \; \text{sq cm}$ and side length $5 \; \text{cm},$ then the length, $\text{in cm},$ of its longer diagonal is $\sqrt{13} + \sqrt{12}$ $\sqrt{37} + \sqrt{13}$ $\frac{\sqrt{37} + \sqrt{13}}{2}$ $\frac{\sqrt{13} + \sqrt{12}}{2}$
1 vote
3
73 views
The number of ways of distributing $15$ identical balloons, $6$ identical pencils and $3$ identical erasers among $3$ children, such that each child gets at least four balloons and one pencil, is
1 vote
4
68 views
For a sequence of real numbers $x_{1}, x_{2}, \dots , x_{n},$ if $x_{1} – x_{2} + x_{3} – \dots + (-1)^{n+1} x_{n} = n^{2} + 2n$ for all natural numbers $n,$ then the sum $x_{49} + x_{50}$ equals $2$ $-2$ $200$ $-200$
1 vote
For a real number $x$ the condition $|3x – 20| + |3x – 40| = 20$ necessarily holds if $9 < x < 14$ $6 < x < 11$ $7 < x < 12$ $10 < x < 15$