UPSC NDA 2 2017 Mathematics Solved Paper

Concept:Kinematics: derivative gives slope, integration of acceleration yields velocity, and derivative of displacement gives velocity not acceleration.Explanation:Statement 1: The derivative $\frac{dy}{dx}$ at a point on a curve gives the slope of the tangent at that point.This is a fundamental definition from differential calculus.Thus statement 1 is correct.Statement 2: Acceleration $a(t)$ is the rate of change of velocity with respect to time: $a(t) = \frac{dv}{dt}$.Integrating both sides with respect to $t$ gives $\int a(t)\,dt + c = v(t)$, where $c$ is the constant of integration (initial velocity).Thus the integral of acceleration plus a constant yields velocity.Hence statement 2 is correct.Statement 3: If $s(t)$ is displacement, then $\frac{ds}{dt}$ gives instantaneous velocity, not acceleration.Acceleration is given by $\frac{d^2 s}{dt^2}$ or $\frac{dv}{dt}$.Therefore statement 3 is incorrect.Answer:Only statements 1 and 2 are correct.Thus option A (1 and 2 only) is the correct answer.