
The number of real solutions of the equation, lying in the interval is,
Answer
507.9k+ views
Hint: To solve this question first we will reduce the given equation into simplest form by using inverse trigonometric identity , then we will expand the expansion and using sum of infinite G.P we will obtain the algebraic expression and then we will see how many root we obtain for that algebraic equation.
Complete step-by-step answer:
Now, we have
Let, and
So, we can write as,
We know that, ,
So, re – writing as,
Now, we know that is one - one function, so it is only true for ,
if , that is
Now, let us take
We can write f(x) as,
Or,
Now, is G.P whose first term is x and common ratio is x,
So, , as Sum of infinite G.P is , whose first term is a and common ratio is r.
Similarly, is G.P whose first term is and common ratio is ,
So, .
So,
Now, in and , there is only difference in sign,
So,
Or on simplifying, we get
As, we have
So,
Now, we have to solve this algebraic expression,
Taking all expression on left side,
Taking, x common, we get
So, x = 0 and
Now,
Taking L.C.M, we get
On solving we get
On cross – multiplying, we get
On opening brackets, we get
Let,
Now, at , we get
at , we get
Now, , which means there exists one root in interval and one is x = 0.
So, we have two solutions for equation
Note: To, solve such question one must know the inverse trigonometric function properties such as and also how to open summation and what is sum of infinite geometric expansion which is equals to , whose first term is a and common ratio is r.
Complete step-by-step answer:
Now, we have
Let,
So, we can write
We know that,
So, re – writing
Now, we know that
if
Now, let us take
We can write f(x) as,
Or,
Now,
So,
Similarly,
So,
So,
Now, in
So,
Or on simplifying, we get
As, we have
So,
Now, we have to solve this algebraic expression,
Taking all expression on left side,
Taking, x common, we get
So, x = 0 and
Now,
Taking L.C.M, we get
On solving we get
On cross – multiplying, we get
On opening brackets, we get
Let,
Now, at
at
Now,
So, we have two solutions for equation
Note: To, solve such question one must know the inverse trigonometric function properties such as
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