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CAT 2017 Slot 1 — QA Question 13

IndicesEasy

If a and b are integers of opposite signs such that (a + 3)2 : b2 = 9 : 1 and (a - 1)2: (b - 1)2 = 4 : 1, then the ratio a2 : b2 is:

Answer & solution

Correct answer: 25 : 4

  • A

    9 : 4

  • B

    81 : 4

  • C

    1 : 4

  • 25 : 4

Solution

Easy

Take square roots of both ratios, remembering each gives a ±\pm sign. That produces four sign-combinations; impose "a,ba,b integers of opposite signs" to keep only the valid one, then read off a2:b2a^2:b^2.

1

Square-root each given ratio. From the two proportions:

(a+3)2b2=9(given)⇒ a+3b=±3(a−1)2(b−1)2=4(given)⇒ a−1b−1=±2\begin{aligned} &\frac{(a+3)^2}{b^2}=9 \quad\text{(given)}\\ &\Rightarrow\ \frac{a+3}{b}=\pm 3\\ &\frac{(a-1)^2}{(b-1)^2}=4 \quad\text{(given)}\\ &\Rightarrow\ \frac{a-1}{b-1}=\pm 2 \end{aligned}
2

List the four cases. Combining each sign choice gives a linear system. Only one yields integers of opposite signs.

a+3=3b,  a−1=2b−2⇒ a=3, b=2(same sign — reject)a+3=3b,  a−1=−2b+2⇒ b∉Z(reject)a+3=−3b, a−1=2b−2⇒ b∉Z(reject)a+3=−3b, a−1=−2b+2⇒ a=15, b=−6(opposite signs — keep)\begin{aligned} &a+3=3b,\ \ a-1=2b-2 &&\Rightarrow\ a=3,\ b=2 \quad\text{(same sign — reject)}\\ &a+3=3b,\ \ a-1=-2b+2 &&\Rightarrow\ b\notin\mathbb{Z} \quad\text{(reject)}\\ &a+3=-3b,\ a-1=2b-2 &&\Rightarrow\ b\notin\mathbb{Z} \quad\text{(reject)}\\ &a+3=-3b,\ a-1=-2b+2 &&\Rightarrow\ a=15,\ b=-6 \quad\text{(opposite signs — keep)} \end{aligned}
3

Form the required ratio. Using a=15, b=−6a=15,\ b=-6 from step 2:

a2b2=152(−6)2=22536=254(divide by 9)\begin{aligned} &\frac{a^2}{b^2}=\frac{15^2}{(-6)^2}=\frac{225}{36}=\frac{25}{4} \quad\text{(divide by }9\text{)} \end{aligned}
a2:b2=25:4a^2:b^2 = 25:4

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CAT 2017 Slot 1 QA Q13: If a and b are integers of opposite signs such that (a + 3) 2 : b 2 = 9 : 1 and (a - 1) 2 : (b - 1) 2 = 4 : 1, — Solution | TheCATExam