Properties of Triangle — NDA Mathematics PYQs

In a triangle $ABC$, $\sin A=\cos B+\cos C$ then what is $\tan \left(\frac{B}{2}\right)+\cot \left(\frac{B}{2}\right)$ equal to ?

The angles A, B and C of a triangle are in the ratio $1:1:4$. If the longest side of the triangle is $3$ units, then what is the perimeter of the triangle ?

In a triangle $ABC$, if $a$, $b$ and $c$ are the lengths of the sides opposite to the angles $A$, $B$ and $C$ respectively, then what is $\frac{\sin (A-B)}{\sin (A+B)}$ equal to ?

What is the length of the chord of a unit circle which subtends at the centre of the circle an angle of $45^{\circ }$ ?

Consider the following for the three (03) items that follow :
The sides of a triangle $ABC$ are $AB=3$ cm, $BC=5$ cm and $CA=7$ cm.

Consider the following statements :
I. The triangle is obtuse-angled triangle.
II. The sum of acute angles of the triangle is also acute.
Which of the statements given above is/are correct?

Consider the following for the three (03) items that follow :
The sides of a triangle $ABC$ are $AB=3$ cm, $BC=5$ cm and $CA=7$ cm.

What is $\angle B$ equal to?

Consider the following for the three (03) items that follow :
The sides of a triangle $ABC$ are $AB=3$ cm, $BC=5$ cm and $CA=7$ cm.

What is the area of the triangle?

Consider the following for the two (02) items that follow :
In a triangle $ABC$, two sides $BC$ and $CA$ are in the ratio $2:1$ and their opposite corresponding angles are in the ratio $3:1$.

One of the angles of the triangle is

Consider the following for the two (02) items that follow :
In a triangle $ABC$, two sides $BC$ and $CA$ are in the ratio $2:1$ and their opposite corresponding angles are in the ratio $3:1$.

Consider the following statements :
I. The triangle is right-angled.
II. One of the sides of the triangle is $\sqrt{3}$ times the other.
III. The angles $A$, $C$ and $B$ of the triangle are in AP.
Which of the statements given above is/are correct?

If in a triangle $ABC$, ${\sin }^{3}A+{\sin }^{3}B+{\sin }^{3}C=3\sin A\sin B\sin C$, then what is the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ where $a$, $b$, $c$ are sides of the triangle?