Question

The equation of the normal to the curve y = (1 + x)^y + sin^-1(sin²x) at x = 0 is:

• A. \( x + y = 1 \)
• B. \( x - y = 1 \)
• C. \( x + y = -1 \)
• D. \( x - y = -1 \)

Hint:

The equation of normal can be derived using the derivative of the curve at the given point.

Answer 1

The Correct Option is A

The equation of normal to a curve at a point is given by the formula: \[ y - y_0 = -\frac{1}{f'(x_0)} (x - x_0) \] where \( f'(x_0) \) is the derivative of the curve at the point \( (x_0, y_0) \). 1. First, differentiate the equation of the curve implicitly to find the slope of the tangent at \( x = 0 \). For the given equation, \( y = (1 + x)^y + \sin^{-1}(\sin^2 x) \), we need to differentiate both sides with respect to \( x \), considering implicit differentiation: \[ \frac{d}{dx} \left( (1 + x)^y \right) + \frac{d}{dx} \left( \sin^{-1}(\sin^2 x) \right) \] 2. After differentiating and solving, substitute \( x = 0 \) to get the slope of the normal. Finally, we can determine the equation of the normal, which is \( x + y = 1 \).