CAT 2017 Question Paper | Quants Slot 1

The best questions to practice for CAT Exam are the actual CAT Question Papers. 2IIM offers you exactly that, in a student friendly format to take value from this. If you would like to take the same inside a testing engine (for Free) head out here: CAT Official Question Mocks. In CAT 2017 we saw some beautiful questions that laid emphasis on Learning ideas from basics and being able to comprehend more than remembering gazillion formulae and shortcuts. Original CAT Question paper is the best place to start off your CAT prep practice. This page provides exactly that. To check out about 1000 CAT Level questions with detailed video solutions for free, go here: CAT Question Bank

Question 13 : 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 :

1. 9 : 4
2. 81 : 4
3. 1 : 4
4. 25 : 4

Video Explanation

Explanatory Answer

Method of solving this CAT Question from Algebra

Let us consider the two ratios given to us.
For (a + 3)² : b² = 9 : 1
⟹ \\frac{(a + 3)^2}{b^2}) = 9
⟹ \\frac{(a + 3)}{b}) = ± 3 ------- (1)
Similarly, we can say that, (a - 1)² : (b - 1)² = 4 : 1,
⟹ \\frac{(a - 1)^2}{(b - 1)^2}) = 4
⟹ \\frac{(a - 1)}{(b - 1)}) = ± 2 ------(2)
From (1), we can say that, a + 3 = ± 3b, So
a = 3b – 3 ------- (3) Or,
a = -3b – 3 ------- (4)
Sub (3) in (2)
⟹ (a – 1) = 2(b – 1) or (a – 1) = -2(b – 1)
⟹ (3b – 4) = 2b – 2 or (3b – 4) = -2b + 2
⟹ b = 2 or b = \\frac{6}{5})
⟹ a = 3 or a = \\frac{3}{5})
Both these cases are not possible since a and b are said to be of opposite signs.
Let’s try condition (4).
Sub (4) in (2), we get
⟹ (a – 1) = 2(b – 1) or (a – 1) = -2(b – 1)
⟹ (-3b – 4) = 2b – 2 or (-3b – 4) = -2b + 2
⟹ b = -\\frac{2}{5}) or b = – 6
⟹ a = -\\frac{9}{5}) or a = 15
Here a = 15 and b = – 6 are possible.
Let’s find a² : b²
⟹ \\frac{a^2}{b^2}) = \\frac{15^2}{-6^2})
⟹ \\frac{a^2}{b^2}) = \\frac{225}{36})
⟹ \\frac{a^2}{b^2}) = \\frac{25}{4})
Hence a² : b² is equal to 25 : 4

The question is "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 :"

Hence, the answer is 25 : 4

Choice D is the correct answer.