Application of Derivatives — NDA Mathematics PYQs

Consider the following statements in respect of the function $f(x)=x$ in the interval $(-1,1)$ :
I. The function attains maximum value.
II. The function attains minimum value.
Which of the statements given above is/are correct ?

What is the length of the longest interval in which the function $f(x)=2{\cos }^{2}x-1$ is decreasing ?

Directions : For the next two (02) items that follow :
Let S and T be the sets where $f(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+7$ decreases and increases respectively.

What is T equal to ?

Directions : For the next two (02) items that follow :
Let S and T be the sets where $f(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+7$ decreases and increases respectively.

What is S equal to ?

Consider the following for the two (02) items that follow :
Let $ABC$ be a triangle right-angled at $B$ and $AB+AC=3$ units.

What is $\angle A$ equal to if the area of the triangle is maximum?

Consider the following for the two (02) items that follow :
Let $ABC$ be a triangle right-angled at $B$ and $AB+AC=3$ units.

What is the maximum area of the triangle?

Consider the following for the two (02) items that follow :
Let the function $f(x)=x^2+9$.

Consider the following statements :
I. $f(x)$ is an increasing function.
II. $f(x)$ has local maximum at $x=0$.
Which of the statements given above is/are correct?

Directions : For the following three (03) items :
Let $y=f(x)=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+\ln \sqrt{1-x^2}$.

What is the slope of the tangent to the curve $y=f(x)$ at $x=0.5$?

Directions : For the following two (02) items :
Consider the function $f(x)=1-\sqrt[3]{(x-1{)}^2}$.

The function has

A wire of length $20\text{ cm}$ is to be bent into a rectangle. Which of the following statements is/are correct?
I. The rectangle of the largest area is the square.
II. It is possible to form a rectangle of an area of $27{\text{ cm}}^2$.
Select the answer using the code given below.