UPSC NDA I 2025 Mathematics Paper

Given,
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.
The slope of the tangent is the derivative of the function, so we have:
f′(x)=4
Integrating f′(x)=4 with respect to x :
f(x)=4x+C
The curve passes through the origin, so when x=0,y=0. Substituting these values into the equation f(x)=4x+C :
0=4(0)+C‌‌⇒‌‌C=0
Therefore, the equation of the curve is:
f(x)=4x
This is the equation of a straight line with a slope of 4 , passing through the origin.
∴ The curve is a straight line with a slope of 4 , passing through the origin.