
The number of solutions of the equation $\cos \left( {\pi \sqrt {x - 4} } \right) \cdot \cos \left( {\pi \sqrt x } \right) = 1$ is
$
{\text{A}}{\text{. 1}} \\
{\text{B}}{\text{. 2}} \\
{\text{C}}{\text{. More than two}} \\
{\text{D}}{\text{. None of these}} \\
$
Answer
511.8k+ views
Hint: In this question we have to find the number of solutions of the given equation. To solve this question the main point is that the value of $\cos x$ is less than or equal to 1. At $x=0$ the value of cos is always to be 1.
Complete step-by-step answer:
In this question we have been given the equation $\cos \left( {\pi \sqrt {x - 4} } \right) \cdot \cos \left( {\pi \sqrt x } \right) = 1$
The RHS is 1 and both terms in multiplication in LHS are in cosine.
Now both terms can only be less than or equal to 1. But if anything less than 1 is multiplied with something less than 1, then we can never get 1 in RHS.
So, $\cos \left( {\pi \sqrt {x - 4} } \right) = 1$ and $\cos \left( {\pi \sqrt x } \right) = 1$
$ \Rightarrow \sqrt {x - 4} = 0$ and $\sqrt x = 0$
$ \Rightarrow x = 4{\text{ and }}x = 0$
As x=4 is only given in the options
And hence, option A is correct.
Note: Whenever we face such types of problems the key point to remember is that we need to have a good grasp over trigonometric properties, some of which have been used above. We must remember that the maximum value of sine and cosine is 1. This helps in getting us the required condition and gets us on the right track to reach the answer.
Complete step-by-step answer:
In this question we have been given the equation $\cos \left( {\pi \sqrt {x - 4} } \right) \cdot \cos \left( {\pi \sqrt x } \right) = 1$
The RHS is 1 and both terms in multiplication in LHS are in cosine.
Now both terms can only be less than or equal to 1. But if anything less than 1 is multiplied with something less than 1, then we can never get 1 in RHS.
So, $\cos \left( {\pi \sqrt {x - 4} } \right) = 1$ and $\cos \left( {\pi \sqrt x } \right) = 1$
$ \Rightarrow \sqrt {x - 4} = 0$ and $\sqrt x = 0$
$ \Rightarrow x = 4{\text{ and }}x = 0$
As x=4 is only given in the options
And hence, option A is correct.
Note: Whenever we face such types of problems the key point to remember is that we need to have a good grasp over trigonometric properties, some of which have been used above. We must remember that the maximum value of sine and cosine is 1. This helps in getting us the required condition and gets us on the right track to reach the answer.
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE

Write an application to the principal requesting five class 10 english CBSE
