NDA 2026 — Mathematics PYQ

1. Isolate the derivative:

Given the differential equation:

$$\cos \left(\frac{dy}{dx}\right)=p$$

Taking the inverse cosine on both sides, we get:

$$\frac{dy}{dx}={\cos }^{-1}(p)$$

2. Recognize the form:

Note that ${\cos }^{-1}(p)$ is simply a constant value. Let $C={\cos }^{-1}(p)$. The equation simplifies to:

$$\frac{dy}{dx}=C$$

3. Integrate:

Integrating both sides with respect to $x$:

$$\int dy=\int Cdx$$

$$y=Cx+K$$

where $K$ is the constant of integration. Substituting $C={\cos }^{-1}(p)$ back:

$$y=x\cdot {\cos }^{-1}(p)+K$$

4. Apply the initial condition:

We are given $y(0)=q$, which means when $x=0$, $y=q$:

$$q=(0)\cdot {\cos }^{-1}(p)+K$$

$$K=q$$

5. Form the final equation:

Substituting $K=q$ into our integrated equation:

$$y=x\cdot {\cos }^{-1}(p)+q$$

Rearranging to isolate the terms:

$$y-q=x\cdot {\cos }^{-1}(p)$$

$$\frac{y-q}{x}={\cos }^{-1}(p)$$

Taking the cosine of both sides:

$$\cos \left(\frac{y-q}{x}\right)=p$$

Final Answer:

The correct option is (a) $\cos \left(\frac{y-q}{x}\right)=p$.