IIT JEE Maths Practice Paper Part 13 Calculus (Previous Years’ Questions)

1. The derivative of \( e^{\tan x} \) is:

\( \sec^2 x \cdot e^{\tan x} \)
\( \tan x \cdot e^{\tan x} \)
\( \sec x \cdot e^x \)
\( \sec x \cdot \tan x \)

2. If \( f(x) = \ln(\sin x) \), then \( f'(x) \) equals:

\( \cot x \)
\( \frac{1}{\sin x} \)
\( \frac{\cos x}{\sin x} \)
\( \cos x \cdot \ln x \)

3. \( \int_0^1 x e^x \, dx \) is:

\( e – 2 \)
\( 1 \)
\( 2e \)
\( e – 1 \)

4. Limit \( \lim_{x \to 0} \frac{\sin x}{x} \) equals:

1
0
∞
Does not exist

5. If \( f(x) = x^2 \), then \( \int f'(x) dx \) is:

\( x^2 + C \)
\( 2x + C \)
\( 2x^2 + C \)
\( x^3 + C \)

6. If \( \int_1^2 f(x)dx = 3 \), what is \( \int_1^2 5f(x)dx \)?

15
5
8
1.5

7. If \( y = x^x \), then \( \frac{dy}{dx} \) equals:

\( x^x(1 + \ln x) \)
\( x^x \ln x \)
\( x \ln x \)
\( \ln x + 1 \)

8. \( \int \frac{1}{x^2 + 1} dx \) is:

\( \tan^{-1}(x) + C \)
\( \ln(x^2 + 1) + C \)
\( \frac{1}{x^2 + 1} + C \)
\( \tan(x) + C \)

9. If a function is differentiable, it is always:

Continuous
Discontinuous
Constant
None

10. The area under the curve \( y = x^2 \) from 0 to 2 is:

\( \frac{8}{3} \)
\( 4 \)
\( 2 \)
\( \frac{4}{3} \)