Show that is an isosceles triangle, if the determinant = 1&1&1 1+… - Vidyadip

Question

Show that $\triangle\text{ABC}$ is an isosceles triangle, if the determinant $\triangle=\begin{vmatrix}1&1&1\\1+\cos\text{A}&1+\cos\text{B}&1+\cos\text{C}\\\cos^2\text{A}+\cos\text{A}&\cos^2\text{B}+\cos\text{B}&\cos^2\text{C}+\cos\text{C}\end{vmatrix}=0.$

✓

Answer

We have, $\triangle=\begin{vmatrix}1&1&1\\1+\cos\text{A}&1+\cos\text{B}&1+\cos\text{C}\\\cos^2\text{A}+\cos\text{A}&\cos^2\text{B}+\cos\text{B}&\cos^2\text{C}+\cos\text{C}\end{vmatrix}=0$
$ \triangle=\begin{vmatrix}0&0&0\\\cos\text{A}-\cos\text{C}&\cos\text{B}-\cos\text{C}&1+\cos\text{C}\\\cos^2\text{A}+\cos\text{A}-\cos^2\text{C}-\cos\text{C}&\cos^2\text{B}+\cos\text{B}-\cos^2\text{C}-\cos\text{C}&\cos^2\text{C}+\cos\text{C}\end{vmatrix}=0.$
$[\text{C}_1\rightarrow\text{C}_1-\text{C}_3\text{ and C}_2\rightarrow\text{C}_2+\text{C}_3]$
$=\begin{vmatrix}0&0&1\\1 & 1&1+\cos\text{C}\\\cos\text{A}+\cos\text{C}+1&\cos\text{B}+\cos\text{C}+1&\cos^2\text{C}+\cos\text{C}\end{vmatrix}=0$
$\big[\text{Taking }(\cos\text{A}\cos\text{C})\text{ common from C}_1\text{ and }(\cos\text{B}-\cos\text{C})\text{ common from C}_2\big]$
$\Rightarrow\ (\cos\text{A}-\cos\text{C}).(\cos\text{B}-\cos\text{C})\big[(\cos\text{B}+\cos\text{C}+1)-(\cos\text{A}+\cos\text{C}+1)\big]=0$
$\Rightarrow\ (\cos\text{A}-\cos\text{C}).(\cos\text{B}-\cos\text{C})(\cos\text{B}+\cos\text{C}+1-\cos\text{A}-\cos\text{C}-1)=0$
$\Rightarrow\ (\cos\text{A}-\cos\text{C}).(\cos\text{B}-\cos\text{C})(\cos\text{B}-\cos\text{A})=0$
$\text{i.e., }\cos\text{A}=\cos\text{C or }\cos\text{B}=\cos\text{C or}\cos\text{B}=\cos\text{A}$
$\Rightarrow\ \text{A = C or B = C or B = A}$
Hence, ABC is an isosceles triangle.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from DETERMINANTS→DETERMINANTS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\text{y}=\sin^{-1}\Bigg(\frac{{1}-\text{x}^{2}}{1+\text{x}^{2}}\Bigg), 0 <\text{x}<1$

Show that the function $x^2 - x + 1$ is neither increasing nor decreasing on $(0, 1).$

A straight line is drawn through a given point $P(1, 4).$ Determine the least value of the sum of the intercepts on the coordinate axes.

If $\text{f(x)}=\begin{cases}\text{ax}^2-\text{b}, & \text{if |x|}<1\\\frac{1}{|\text{x}|}, & \text{if |x|}\geq1\end{cases}$ is differentiable at x = 1, find a, b.

If $\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}$, find $A^{-1}.$ Using $A^{-1}$, solve the system of linear equations: $x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7$

Evaluvate the following intregals:
$\int\frac{8\cot\text{x}+1}{3\cot\text{x}+2}\ \text{dx}$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\frac{2}{\text{x}}-\frac{2}{\text{x}^{2}}, \text{x}>0$

Differentiate the following functions with respect to x:
$\text{x}\sin2\text{x}+5^{\text{x}}+\text{k}^\text{k}+(\tan^2\text{x})^3$

A manufacturer has three machine I, II, III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.
The number of hours required for producing 1 unit each of M and N on the three machines are given in the following table:

Item	Number of hours required on machines
	I	II	III
M	1	2	1
N	2	1	1.25

She makes a profit of Rs. 600 and Rs. 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10,000. If the profit is Rs. 60 per unit for the product A and Rs. 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.