Question
Consider the following for the two (02) items that follow: The slope of the tangent to the curve y = f(x) at (x, f(x)) is 4 for every real number x and the curve passes through the origin. What is the nature of the curve?
NDA
Mathematics
Application of Derivatives
2025
NDA 2025 (Phase 1)
NDA Mathematics Question (2025) — Solution
Options
- A. A straight line passing through (1, 4)
- B. A straight line passing through (-1, 4)
- C. A parabola with vertex at origin and focus at (2, 0)
- D. A parabola with vertex at origin and focus at (1, 0)
Answer
A. A straight line passing through (1, 4)
Step-by-step solution
Given that the slope of the tangent to the curve y = f(x) is 4 for all x. dy dx = 4 Integrating both sides with respect to x: y = 4x + C Since the curve passes through the origin (0, 0), substituting x = 0 and y = 0 gives: 0 = 4(0) + C C = 0 The equation of the curve is y = 4x. This represents a straight line. Checking the given points, for x = 1, y = 4(1) = 4. Thus, the straight line passes through the point (1, 4).
Related: Mathematics — Application of Derivatives · All PYQ Banks