Question

How many positive numbers x satisfy the equation cos (97 x) = x?
A) 1 B) 15 C) 31 D) 49​

Answer 1

Given:

• cos (97 x ) = x.

To find :

• How many positive numbers satisfy x

Solution:

• We know that period of cos (x) = 2π
• Hence, period of cos ( 97 x ) = 2π / 97.
• Range of cos function for positive values is [0,1].
• Now, we see that at 15 * 2 π / 97 is 0.97.
• So, there are definitely, 15 values of x for which the equation remains in cosine region.
• However, during each period, a positive value of cosine  repeats twice.
• Hence, total possible values of x = 15 * 2 = 30.
• Also, 15 * 2 π / 97 is 0.97. So there might be another value left that lies between 0.97 and 1.
• Hence our answer is number of possible values = 31.

Answer:

Number of positive values of x that satisfy cos ( 97 x ) = x are (c) 31

Answer 2

Answer: 31

Step-by-step explanation:

Given: Cos (97 x ) = x.

To find : How many positive numbers satisfy x

Solution:

• We know that Cos function repeats itself after $2\pi$ so, cos (x) = 2π

• Now, according to question Cos ( 97 x ) = 2π / 97.

• Cos function values lies between [0,1].

• Now, $15*\frac{2\pi }{97} =0.97$

• So, there are 15 values of x for which the equation shows the positive values of Cos function.

• It will show two sets one negative and one positive.

• ∴ total possible values of x = 15 * 2 = 30.

• So there might be another value left that lies between 0.97 and 1.

• Hence the required answer is 31.

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