Properties of Triangle — Class 12 Mathematics PYQs

In a triangle $ABC$, $\angle A=75^{\circ }$ and $\angle B=45^{\circ }$. What is $2a-b$ equal to?

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?

$ABC$ is a triangle such that angle
$C=60^{\circ }$, then what is $\frac{\cos A+\cos B}{\cos \left(\frac{A-B}{2}\right)}$
equal to?

In a triangle $ABC$, $AB=16$ cm, $BC=63$ cm and $AC=65$ cm. What is the value of $\cos 2A+\cos 2B+\cos 2C$?

Consider the following statements:

1. In a triangle $ABC$, if
   $\cot A\cdot \cot B\cdot \cot C>0$, then the triangle is an acute angled triangle.
2. In a triangle $ABC$, if
   $\tan A\cdot \tan B\cdot \tan C>0$, then the triangle is an obtuse angled triangle.

Which of the statements given above is/are correct?

Consider the following for the next two (02) items that follow :

The perimeter of a triangle $ABC$ is $6$ times the AM of sine of angles of the triangle. Further $BC=\sqrt{3}$ and $CA=1$.

What is the perimeter of the triangle?

Consider the following for the next two (02) items that follow :

The perimeter of a triangle $ABC$ is $6$ times the AM of sine of angles of the triangle. Further $BC=\sqrt{3}$ and $CA=1$.

Consider the following statements :

1. $ABC$ is right angled triangle
2. The angles of the triangle are in AP

Which of the statements given above is/are correct?

Consider the following for the next two (02) items that follow :

In the triangle $ABC$,
$a^2+b^2+c^2=ac+\sqrt{3}bc$

What is the nature of the triangle?

Consider the following for the next two (02) items that follow :

In the triangle $ABC$,
$a^2+b^2+c^2=ac+\sqrt{3}bc$

If $c=8$, what is the area of the triangle?

Consider the following for the next two (02) items that follow :

In a triangle $PQR$, $P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the in-circle of the triangle touches the sides $PQ,QR$ and $RP$ at $N,L$ and $M$ respectively such that the lengths $PN,QL$ and $RM$ are $n,n+2,n+4$ respectively where $n$ is an integer.

What is the value of $n$?