NDA 1 2025 Maths, question 78

Topic: Integral calculus

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.

Q78

What is the area bounded by the curve, the $x$ -axis and the line $x = 4$ ?

(a) 8 square units
(b) 16 square units
(c) 32 square unitsCorrect answer
(d) 64 square units

Solution.

Answer: (c) 32 square units

The curve is $y = 4x$ .
It meets the $x$ -axis at the origin, and at $x = 4$ its height is 16.
The enclosed region is a triangle with base 4 and height 16, so the area is $\frac{1}{2} \times 4 \times 16 = 32$ square units.