CAT 2020 Question Paper | Quants Slot 1

Solving questions from Linear and Quadratic equations is an integral part of your CAT preparation. In CAT Exam, one can generally expect to get 1~2 questions from Linear Equations and Quadratic Equations. In this question, we have to find the number of real-valued solutions. Would something conventional work or, do we have to take some non-formulaic approach? Find it out yourself. Time to have a go at this question, and then check out the video solution to find out how Rajesh solves this question!

Question 6 : The number of real-valued solutions of the equation 2^x + 2^-x = 2 - (x - 2)² is

Video Explanation

Explanatory Answer

2^x + 2^-x is of the form y + $\frac{1}{y}$
Minimum value the expression can take is ≥ 2
So 2^x + 2^-x ≥ 2
Now, 2 – (x – 2)² ----> (x – 2)² is ≥ 0
So 2 – (x – 2)2 ≤ 2 (From 2 we are subtracting a non-negative number)
Maximum value the expression can have is 2
The only possibility is both sides are = 2
If value = 2
Then x = 0
2⁰ + 2⁰ = 2 – (0 – 2)²
2 ≠ – 2
There is no case where LHS = RHS = 2
Hence 0 real-valued solutions

The question is "The number of real-valued solutions of the equation 2^x + 2^-x = 2 - (x - 2)² is"

Hence, the answer is, "0"

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