Application of Derivatives — NDA Mathematics PYQs

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?

Consider the following for the three (03) items that follow :
Consider the function $f(x)=x|x|$.

Consider the following statements :
I. The function is increasing in the interval $(-\infty ,\infty )$.
II. The function is differentiable at $x=0$.
Which of the statements given above is/are correct?

Consider the following for the three (03) items that follow :
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 $x=0.5$?

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

The function has

A wire of length $20$ 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.

Consider the following statements regarding the function $f(x)=\frac{1}{x-5}$ :
Statement-I :
$f(x)$ is decreasing on the intervals $x<5$ and $x>5$.
Statement-II :
$f^{\prime }(x)>0$ for all $x\ne 5$.
Which one of the following is correct in respect of the above statements?

Consider the following statements :
Statement-I :
The function $f(x)=\frac{x^3+128}{x}$ has a minimum value $48$ at $x=4$.
Statement-II :
As $x$ increases through $4$, $f^{\prime }(x)$ changes sign from positive to negative.
Which one of the following is correct in respect of the above statements?