Question

For 𝑥∈ℝ, the curve 𝑦=𝑥² intersects the curve 𝑦=𝑥 sin 𝑥+cos 𝑥 at exactly 𝑛 points. Then, 𝑛 equals

• A. 1
• B. 2
• C. 4
• D. 8

Answer 1

The Correct Option is B

Let's solve the problem of finding the number of intersection points between the two curves represented by the functions:

• \(y = x^2\)
• \(y = x \sin x + \cos x\) 

Finding the intersection points involves solving the equation:

\(x^2 = x \sin x + \cos x\)

This equation can be rearranged to:

\(x^2 - x \sin x - \cos x = 0\)

Step-by-Step Solution

Check for potential intersection points by substituting simple values of \(x\).

Substitute \(x=0\):

\(0^2 = 0 \cdot \sin 0 + \cos 0\)

\(0 = 1\)

This is false, so \(x=0\) is not an intersection point.

Try \(x=1\):

\(1^2 = 1 \cdot \sin 1 + \cos 1\)

\(1 \neq \sin 1 + \cos 1\)

This is not an intersection point either.

Identify intersection points graphically or numerically, as analytic solutions might not be easily derivable.

Graphical analysis or a numerical solver would show that there are exactly two points where these curves intersect for \(x \in \mathbb{R}\).

Thus, the number of intersection points between the given curves is \(n = 2\).

Conclusion

The correct answer is 2, as detailed above.