Class 11MathematicsTrigonometrical EquationsMix Examples-Trigonometrical Equations and Inequations, Properties of Triangles, Height and Distance
DifficultIn $\triangle ABC$, right-angled at $A$, the circumradius, inradius, and radius of the excircle opposite to $A$ are respectively in the ratio $2:5:\lambda$. Then the roots of the equation $x^2-(\lambda-5)x+(\lambda-6)=0$ are:
- A$3, 4$
- B$5, 13$
- C$1, 3$
- D$8, 13$
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Similar Questions
Match the items of List-$I$ with those of List-$II$ (Here $\Delta$ denotes the area of $\triangle ABC$.)
List-$I$ List-$II$ $(A)$ $\sum \cot A$ $(i)$ $\frac{(a+b+c)^2}{4\Delta}$ $(B)$ $\sum \cot \frac{A}{2}$ $(ii)$ $\frac{a^2+b^2+c^2}{4\Delta}$ $(C)$ If $\tan A : \tan B : \tan C = 1 : 2 : 3$,then $\sin A : \sin B : \sin C =$ $(iii)$ $8 : 6 : 5$ $(D)$ If $\cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = 3 : 7 : 9$,then $a : b : c =$ $(iv)$ $12 : 5 : 13$ $(v)$ $\sqrt{5} : 2\sqrt{2} : 3$ $(vi)$ $4\Delta$
Then the correct match is
| List-$I$ | List-$II$ |
| $(A)$ $\sum \cot A$ | $(i)$ $\frac{(a+b+c)^2}{4\Delta}$ |
| $(B)$ $\sum \cot \frac{A}{2}$ | $(ii)$ $\frac{a^2+b^2+c^2}{4\Delta}$ |
| $(C)$ If $\tan A : \tan B : \tan C = 1 : 2 : 3$,then $\sin A : \sin B : \sin C =$ | $(iii)$ $8 : 6 : 5$ |
| $(D)$ If $\cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = 3 : 7 : 9$,then $a : b : c =$ | $(iv)$ $12 : 5 : 13$ |
| $(v)$ $\sqrt{5} : 2\sqrt{2} : 3$ | |
| $(vi)$ $4\Delta$ |
Then the correct match is
DifficultAP EAMCET 2024
View SolutionIf angles $A$,$B$,and $C$ of a $\Delta ABC$ are $75^o$,$45^o$,and $60^o$ respectively,then the ratio of the areas of $\Delta OBC$,$\Delta COA$,and $\Delta AOB$ respectively is [where $O$ is the circumcentre of the triangle].
If in $\triangle ABC$,with usual notations,$a^2, b^2, c^2$ are in $A$.$P$.,then $\frac{\sin 3B}{\sin B} =$
In $\triangle ABC$,if $a, b$ and $c$ are in arithmetic progression,then $\cos A + 2 \cos B + \cos C =$
The number of solutions to $\sin(\pi \sin^2 \theta) + \sin(\pi \cos^2 \theta) = 2 \cos(\frac{\pi}{2} \cos \theta)$ satisfying $0 \leq \theta \leq 2\pi$ is
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