If A, B, C are the angles of a triangle and l | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the determinant: $\left|\begin{array}{ccc}1 & 1 & 1 \ 1+\sin A & 1+\sin B & 1+\sin C \ \sin A+\sin ^{2} A & \sin B+\sin ^{2} B & \sin C+\s

Vedclass · Accepted Answer

an isosceles triangle

Vedclass · Answer

(B) Given the determinant: $\left|\begin{array}{ccc}1 & 1 & 1 \ 1+\sin A & 1+\sin B & 1+\sin C \ \sin A+\sin ^{2} A & \sin B+\sin ^{2} B & \sin C+\sin ^{2} C\end{array}ight|=0$Applying column operations $C_2 ightarrow C_2 - C_1$ and $C_3 ightarrow C_3 - C_1$:$\left|\begin{array}{ccc}1 & 0 & 0 \ 1+\sin A & \sin B-\sin A & \sin C-\sin A \ \sin A+\sin^2 A & (\sin B-\sin A)(1+\sin B+\sin A) & (\sin C-\sin A)(1+\sin C+\sin A)\end{array}ight|=0$Taking $(\sin B - \sin A)$ common from $C_2$ and $(\sin C - \sin A)$ common from $C_3$:$(\sin B - \sin A)(\sin C - \sin A) \left|\begin{array}{ccc}1 & 0 & 0 \ 1+\sin A & 1 & 1 \ \sin A+\sin^2 A & 1+\sin B+\sin A & 1+\sin C+\sin A\end{array}ight|=0$Expanding along the first row:$(\sin B - \sin A)(\sin C - \sin A) [ (1+\sin C+\sin A) - (1+\sin B+\sin A) ] = 0$$(\sin B - \sin A)(\sin C - \sin A)(\sin C - \sin B) = 0$This implies $\sin B = \sin A$ or $\sin C = \sin A$ or $\sin C = \sin B$.Since $A, B, C$ are angles of a triangle,this implies $A=B$ or $C=A$ or $C=B$.Therefore,the triangle is an isosceles triangle.

If $A, B, C$ are the angles of a triangle and $\left| {\begin{array}{*{20}{c}}1&1&1\\{1 + \sin A}&{1 + \sin B}&{1 + \sin C}\\{\sin A + {{\sin }^2}A}&{\sin B + {{\sin }^2}B}&{\sin C + {{\sin }^2}C} \end{array}} \right| = 0$,then the triangle is

Similar Questions

Let $f(\theta) = \left| \begin{array}{ccc} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{array} \right|$. Suppose $A$ and $B$ are respectively the maximum and minimum values of $f(\theta)$. Then $(A, B)$ is equal to

If $\left|\begin{array}{lll}a & a^3 & a^4 \\ b & b^3 & b^4 \\ c & c^3 & c^4\end{array}\right|=k(a-b)(b-c)(c-a)$ then $k=$

If one of the roots of $\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0$ is $-10$, then the other roots are

Find the area of the triangle with vertices at the points $(2,7), (1,1), (10,8)$.

If $\left| \begin{array}{ccc} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{array} \right| = x + iy$,then $(x, y)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module