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Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Roots of Equation.

## Roots of Equations (B.Stat Objective Question )

The number of roots of the equation \(x^2+sin^2{x}-1\) in the closed interval \([0,\frac{\pi}{2}]\) is

- 0
- 2
- 53361
- 5082

**Key Concepts**

Equation

Roots

Algebra

## Check the Answer

But try the problem first…

Answer:2

Source

Suggested Reading

B.Stat Objective Problem 711

Challenges and Thrills of Pre-College Mathematics by University Press

## Try with Hints

First hint

\(x^2+sin^2{x}-1=0\)

\(\Rightarrow x^{2}=cos^{2}x\)

we draw two graphs \(y=x^{2} and y=cos^{2}x\)

where intersecting point gives solution now we look for intersecting points

Second Hint

we get two intersecting points

Final Step

so number of roots is 2.

Another wrong solution! The proof is wrong as it only shows that there is at least one solution. It does not show that there is exactly one solution. To complete the proof, show that the derivative of \(f(x)\) is non-negative in the interval \([0, \pi/2]\).