NDA 1 2025 Maths, question 77

Topic: Differential equations

Passage · Q77 to Q78

The slope of the tangent to the curve $y = f(x)$ at $\left( x,f(x) \right)$ is 4 for every real number $x$ and the curve passes through the origin.

Q77

What is the nature of the curve?

(a) A straight line passing through $(1,4)$ Correct answer
(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 $(1,0)$

Solution.

Answer: (a) A straight line passing through $(1,4)$

The constant slope means $f\prime(x) = 4$ for every $x$ .
Integrating gives $f(x) = 4x + C$ .
The curve passes through the origin, so $C = 0$ .
Therefore it is the straight line $y = 4x$ , which passes through $(1,4)$ .