Properties of Triangle

$\begin{aligned} & \text { Equation of lines } Q R=5 x+2 y+2=0 \\ & \text { Equation of lines } P R=10 x-3 y-38=0 \\ & \therefore \text { Point } R(2,-6) \\ & \text { Centroid }=\left(\frac{5-2+2}{3}, \frac{4+4-6}{3}\right) \\ & =\left(\frac{5}{3}, \frac{2}{3}\right) \\ & c+2 d=\frac{5}{3}+\frac{4}{3}=3 \end{aligned}$

$\begin{aligned} & m_{P A}=\frac{2}{-2}=-1 \\ & \therefore \quad m_{B C}=1 \end{aligned}$

$\begin{aligned} & B C: y=x+5 \\ & m_{B P}=\frac{2}{-3}=\frac{-2}{3} \\ & \therefore m_{A C}=\frac{3}{2} \\ & A C: y=\frac{3}{2} x-\frac{11}{2} \quad \Rightarrow 2 y=3 x-11 \\ & \therefore \quad C:(21,26) \end{aligned}$

Let the circumcentre be $(h, k)$

$\begin{aligned} & (h-1)^2+(k-1)^2=(h+2)^2+(k-3)^2 \quad \text{... (i)}\\ & (h-1)^2+(k-1)^2=(h-3)^2+(k+1)^2 \quad \text{... (ii)} \end{aligned}$

Solving (i) and (ii)

$\begin{aligned} & h=\frac{-19}{2}, k=\frac{-23}{2} \\ & \alpha+\beta+2(h+k) \\ & =21+26-19-23 \\ & =2+3=5 \end{aligned}$

$\begin{aligned} & A(a,-2), B(a, 6), C\left(\frac{a}{4},-2\right), O\left(5, \frac{a}{4}\right) \\ & A O=B O \\ & (a-5)^2+\left(\frac{a}{4}+2\right)^2=(a-5)^2+\left(\frac{a}{4}-6\right)^2 \\ & a=8 \\ & A B=8, A C=6, B C=10 \\ & \alpha=5, \beta=24, \gamma=24 \end{aligned}$

$ \begin{aligned} & \cos A+2 \cos B+\cos C=2 \\\\ & \cos A+\cos C=2(1-\cos B) \\\\ & 2 \cos \frac{A+C}{2} \cos \left(\frac{A-C}{2}\right)=2 \times 2 \sin ^2 \frac{B}{2} \\\\ & \cos \frac{A-C}{2}=2 \sin \frac{B}{2} \\\\ & 2 \cos \frac{B}{2} \cos \frac{A-C}{2}=4 \sin \frac{B}{2} \cos \frac{B}{2} \\\\ & 2 \sin \left(\frac{A+C}{2}\right) \cos \left(\frac{A-C}{2}\right)=2 \sin B \\\\ & \sin A+\sin C=2 \sin B \\\\ & a+c=2 b \quad(\because a=3, c=7) \\\\ & \Rightarrow b=5 \quad \end{aligned} $

$ \begin{aligned} \cos A & -\cos C=\frac{b^2+c^2-a^2}{2 b c}-\frac{a^2+b^2-c^2}{2 \mathrm{ab}} \\\\ & =\frac{25+49-9}{70}-\frac{9+25-49}{30} \\\\ & =\frac{65}{70}+\frac{1}{2}=\frac{20}{14}=\frac{10}{7} \end{aligned} $

If $\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}$ is minimum then $\mathrm{A}=$

$\mathrm{B}=\mathrm{C}=60^{\circ}$

So $\triangle \mathrm{ABC}$ is equilateral

Now in-radias $r=3$

So in $\triangle \mathrm{MBD}$ we have

$ \begin{aligned} & \operatorname{tan} 30^{\circ}=\frac{M D}{B D}=\frac{r}{a / 2}=\frac{6}{a} \\\\ & 1 / \sqrt{3}=\frac{1}{a}=a=6 \sqrt{3} \end{aligned} $

Perimeter of $\triangle \mathrm{ABC}=18 \sqrt{3}$

Area of $\triangle \mathrm{ABC}=\frac{\sqrt{3}}{4} a^2=27 \sqrt{3}$

${1 \over 2}ab = {{98\sqrt 3 } \over 3}$

$ \Rightarrow \sqrt 3 ab = 196$ ..... (i)

$OP = OB\cos 30^\circ = OA\cos 60^\circ $

$ \Rightarrow {{b\sqrt 3 } \over 2} = {a \over 2}$

$ \Rightarrow \sqrt 3 b = a$ ..... (ii)

By (i) and (ii)

${a^2} = 196$

$a = 14$

${b^2} = {{{a^2}} \over 3}$

${a^2} - {b^2} = {{2{a^2}} \over 3} = {{392} \over 3}$

$CD = \sqrt {{{(10 + {x^2})}^2} - {{(10 - {x^2})}^2}} = 2\sqrt {10} |x|$

Area $ = {1 \over 2} \times CD \times AB = {1 \over 2} \times 2\sqrt {10} |x|(20 - 2{x^2})$

$A = \sqrt {10} |x|(10 - {x^2})$

${{dA} \over {dx}} = \sqrt {10} {{|x|} \over x}(10 - {x^2}) + \sqrt {10} |x|( - 2x) = 0$

$ \Rightarrow 10 - {x^2} = 2{x^2}$

$3{x^2} = 10$

$x = k$

$3{k^2} = 10$

${{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9} = \lambda $

$a + b = 7\lambda $

$b + c = 8\lambda $

$c + a = 9\lambda $

$a + b + c = 12\lambda $

$\therefore$ $a = 4\lambda ,\,b = 3\lambda ,\,c = 5\lambda $

$S = {{4\lambda + 3\lambda + 5\lambda } \over 2} = 6\lambda $

$\Delta = \sqrt {S(s - a)(s - b)(s - c)} = \sqrt {(6\lambda )(2\lambda )(3\lambda )(\lambda )} = 6{\lambda ^2}$

$R = {{abc} \over {4\Delta }} = {{(4\lambda )(3\lambda )(5\lambda )} \over {4(6{\lambda ^2})}} = {5 \over 2}\lambda $

$r = {\Delta \over s} = {{6{\lambda ^2}} \over {6\lambda }} = \lambda $

${R \over r} = {{{5 \over 2}\lambda } \over \lambda } = {5 \over 2}$

${{\sin A} \over {\sin B}} = {{\sin (A - C)} \over {\sin (C - B)}}$

As A, B, C are angles of triangle.

A + B + C = $\pi$

A = $\pi$ $-$ (B + C) ...... (1)

Similarly sinB = sin(A + C) ..... (2)

From (1) and (2)

${{\sin (B + C)} \over {\sin (A + C)}} = {{\sin (A - C)} \over {\sin (C - B)}}$

$\sin (C + B).\sin (C - B) = \sin (A - C)\sin (A + C)$

${\sin ^2}C - {\sin ^2}B = {\sin ^2}A - {\sin ^2}C$

$\because$ $\{ \sin (x + y)\sin (x - y) = {\sin ^2}x - {\sin ^2}y\} $

$2{\sin ^2}C = {\sin ^2}A + {\sin ^2}B$

By sine rule

$2{c^2} = {a^2} + {b^2}$

$\Rightarrow$ b², c² and a² are in A.P.

As, $\cos B = {3 \over 5} \Rightarrow B = 53^\circ $

As, $R = 5 \Rightarrow {c \over {\sin c}} = 2R$

$ \Rightarrow {5 \over {10}} = \sin c \Rightarrow C = 30^\circ $

Now, ${b \over {\sin B}} = 2R \Rightarrow b = 2(5)\left( {{4 \over 5}} \right) = 8$

Now, by cosine formula

$\cos B = {{{a^2} + {c^2} - {b^2}} \over {2ac}}$

$ \Rightarrow {3 \over 5} = {{{a^2} + 25 - 64} \over {2(5)a}}$

$ \Rightarrow {a^2} - 6a - 3g = 0$

$\therefore$ $a = {{6 \pm \sqrt {192} } \over 2} = {{6 \pm 8\sqrt 3 } \over 2}$

$ \Rightarrow 3 + 4\sqrt 3 $ (Reject $a = 3 - 4\sqrt 3 $)

Now, $\Delta = {{abc} \over {4R}} = {{(3 + 4\sqrt 3 )(8)(5)} \over {4(5)}} = 2(3 + 4\sqrt 3 )$

$ \Rightarrow \Delta = (6 + 8\sqrt 3 )$

$\Rightarrow$ Option (3) is correct.

Area of triangle ABC



$A = {1 \over 2} \times BC \times AM$

$ = {1 \over 2} \times 2\sqrt {{r^2} - {x^2}} \times (r + x)$

$A = (r + x)\sqrt {{r^2} - {x^2}} $

${{dA} \over {dx}} = \sqrt {{r^2} - {x^2}} - {x \over {\sqrt {{r^2} - {x^2}} }} \times (r + x) $

$= {{{r^2} - {x^2} - rx - {x^2}} \over {\sqrt {{r^2} - {x^2}} }} = {{{r^2} - rx - 2{x^2}} \over {\sqrt {{r^2} - {x^2}} }} = {{ - (x + r)(2x - r)} \over {\sqrt {{r^2} - {x^2}} }}$

${{dA} \over {dx}} = 0 \Rightarrow x = {r \over 2}$

Sign change of ${{dA} \over {dx}}$ at $x = {r \over 2}$

$ \Rightarrow $ A has maximum at $x = {r \over 2}$

$BC = 2\sqrt {{r^2} - {x^2}} = \sqrt 3 r$,

$AM = r + {1 \over 2}r$ = ${3 \over 2}r$

$ \Rightarrow AB = AC = \sqrt 3 r$

2b = a + c

${{2a + 2} \over 3} = {{10} \over 3}$ and ${{2b + c} \over 3} = {7 \over 3}$

a = 4,

$\left\{ \matrix{ 2b + c = 7 \hfill \cr 2b - c = 4 \hfill \cr} \right\}$, solving

$b = {{11} \over 4}$

$c = {3 \over 2}$

$ \therefore $ Quadratic Equation is $4{x^2} + {{11} \over 4}x + 1 = 0$

$ \therefore $ The value of

${\alpha ^2} + {\beta ^2} - \alpha \beta $

= ${\alpha ^2} + {\beta ^2} + 2\alpha \beta - 3\alpha \beta $

= ${(\alpha + \beta )^2} - 3\alpha \beta $

$= {{121} \over {256}} - {3 \over 4} = - {{71} \over {256}}$

Distance between A and B

$ = \sqrt {{{(2)}^2} + {{( - 1)}^2}} = \sqrt 5 $

Area of triangle = $5\sqrt 5 $

$ \Rightarrow {1 \over 2} \times \sqrt 5 \times x = 5\sqrt 5 $

$ \Rightarrow x = 10$

Slope of AB, ${m_{AB}} = {{1 - 2} \over {3 - 1}} = - {1 \over 2}$

AC is perpendicular to AB.

$ \therefore $ ${m_{AB}}\,.\,{m_{AC}} = - 1$

$ \Rightarrow {m_{AC}} = 2 = \tan \theta $

So, $\sin \theta = {2 \over {\sqrt 5 }}$, $\cos \theta = {1 \over {\sqrt 5 }}$

$ \therefore $ $a = {x_A} + r\cos \theta $

$ = 1 + 10 \times {1 \over {\sqrt 5 }}$

$ = 1 + 2\sqrt 5 $

Centroid = $\left( {{{1 + 3 - 3} \over 3},{{2 + 0 + 4} \over 3}} \right) = \left( {{1 \over 3},2} \right)$

From the question 2B = A + C & A + B + C = $\pi $

$ \Rightarrow $ 3B = $\pi $

$ \Rightarrow $ B = ${\pi \over 3}$

$ \therefore A + C = {{2\pi } \over 3}\sigma $

${a \over b} = {1 \over {\sqrt 3 }}$

${{2R\sin A} \over {2R\sin B}} = {1 \over {\sqrt 3 }}$

sin A = ${1 \over 2}$

$ \therefore $ A = 30^o

$ \therefore $ a = 2, b = 2$\sqrt 3$, c = 4

$\Delta = {1 \over 2} \times 2\sqrt 3 \times 2 = 2\sqrt 3 $

Let smallest angle $\angle A$ = $\theta $

and largest angle $\angle C$ is double of $\angle A$.

$ \therefore $ $\angle C$ = 2$\theta $

$ \therefore $ $\angle B$ = $\pi $ - 3$\theta $

Let length of sides are

a, b, c

where a < b < c

As a, b, c are in A.P then

2b = a + c

From sin rule we can say,

2 sin B = sin A + sin C

$ \Rightarrow $ 2 sin ($\pi $ – 3$\theta $) = sin $\theta $ + sin 2$\theta $

$ \Rightarrow $ 2 sin (3$\theta $) = sin $\theta $ + sin 2$\theta $

$ \Rightarrow $ 2(3sin $\theta $ - 4sin³ $\theta $) = sin$\theta $ + 2sin $\theta $ cos $\theta $

$ \Rightarrow $ 2(3 - 4sin² $\theta $) = 1 + 2cos $\theta $

$ \Rightarrow $ 2(3 - 4 + 4cos² $\theta $) = 1 + 2cos $\theta $

$ \Rightarrow $ 2(4cos² $\theta $ - 1) = 1 + 2cos $\theta $

$ \Rightarrow $ 8 cos² $\theta $ – 2 cos $\theta $ – 3 = 0

$ \Rightarrow $ 8 cos² $\theta $ – 6 cos $\theta $ + 4 cos $\theta $ – 3 = 0

$ \Rightarrow $ (2 cos $\theta $ + 1) ( 4 cos $\theta $ - 3) = 0

$ \Rightarrow $ cos $\theta $ = ${3 \over 4}$ or cos $\theta $ = $-{1 \over 2}$

But cos $\theta $ = $-{1 \over 2}$ is not possible as $\angle C$ is acute angle.

Now we have to find,

a : b : c

$ \Rightarrow $ sin A : sin B : sin C [ from sin rule]

$ \Rightarrow $ sin $\theta $ : sin ($\pi $ - 3$\theta $) : sin 2$\theta $

$ \Rightarrow $ sin $\theta $ : sin 3$\theta $ : sin 2$\theta $

$ \Rightarrow $ sin $\theta $ : (3sin $\theta $ - 4sin³ $\theta $) : 2sin $\theta $ cos $\theta $

$ \Rightarrow $ 1 : (3 - 4sin² $\theta $) : 2 cos $\theta $

$ \Rightarrow $ 1 : (3 - 4 + 4cos² $\theta $) : 2 cos $\theta $

$ \Rightarrow $ 1 : (4cos² $\theta $ - 1) : 2 cos $\theta $

$ \Rightarrow $ 1 : (4${\left( {{3 \over 4}} \right)^2}$ - 1) : 2 $ \times $ ${{3 \over 4}}$

$ \Rightarrow $ 1 : ${5 \over 4}$ : ${{3 \over 2}}$

$ \Rightarrow $ 4 : 5 : 6

b + c = 11$\lambda $, c + a = 12$\lambda $, a + b = 13$\lambda $

$ \Rightarrow $  a = 7$\lambda $, b = 6$\lambda $, c = 5$\lambda $

(using cosine formula)

cosA = ${1 \over 5},$ cosB = ${19 \over 35},$ cosC = ${5 \over 7},$

$\alpha $ : $\beta $ : $\gamma $ $ \Rightarrow $  7 : 19 : 25

Given a + b = x and ab = y

If x² $-$ c² = y $ \Rightarrow $ (a + b)² $-$ c² = ab

$ \Rightarrow $  a² + b² $-$ c² = $-$ ab

$ \Rightarrow $   ${{{a^2} + {b^2} - {c^2}} \over {2ab}} = - {1 \over 2}$

$ \Rightarrow \cos C = - {1 \over 2}$

$ \Rightarrow \angle C = {{2\pi } \over 3}$

$R = {c \over {2\sin C}} = {c \over {\sqrt 3 }}$

A + B = 120^o


$\tan {{A - B} \over 2} = {{a - b} \over {a + b}}\cot \left( {{c \over 2}} \right)$

$ = {{\sqrt 3 + 1 - \sqrt 3 + 1} \over {2\left( {\sqrt 3 } \right)}}\cot \left( {{{30}^o}} \right) = {1 \over {\sqrt 3 }}.\sqrt 3 = 1$

${{A - B} \over 2} = {45^o}$

$ \Rightarrow A - B = {90^o}$

$ \ A + B = {120^o}$

$2A = {210^o}$

$A = {105^o}$

$B = {15^o}$

$ \therefore $ $\angle A:\angle B,$ = 7 : 1

In a triangle, orthocentre, centroid and circumcenter are collinear and centroid divides orthocenter and circumcenter in 2 : 1 ratio.



Here AB = 2 BC

$AB = \sqrt {{{\left( {3 + 3} \right)}^2} + {{\left( {3 - 5} \right)}^2}} $

$ = \sqrt {36 + 4} $

$ = \sqrt {40} $

$ = 2\sqrt {10} $

$\therefore\,\,\,\,$ BC $ = \sqrt {10} $

$\therefore\,\,\,\,$ AB + BC

$ = 2\sqrt {10} + \sqrt {10} $

$ = 3\sqrt {10} $

AC is the diameter of the circle.

$\therefore\,\,\,$ Radius $ = {1 \over 2}\,\,$ AC

$ = {1 \over 2} \times 3\sqrt {10} $

$ = 3\sqrt {{5 \over 2}} $

Given $3$ $\sin \,P + 4\cos Q = 6$ $\,\,\,\,\,\,\,\,...\left( i \right)$

$4\sin Q + 3\cos P = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$

Squaring and adding $(i)$ & $(ii)$ we get

$9\,{\sin ^2}P + 16{\cos ^2}Q + 24\sin P\cos Q$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 16\,{\sin ^2}Q + 9{\cos ^2}P + 24\sin Q\cos P$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $ = 36 + 1 = 37$

$ \Rightarrow 9\left( {{{\sin }^2}p + {{\cos }^2}P} \right) + 16\left( {{{\sin }^2}Q + {{\cos }^2}q} \right)$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 24\left( {\sin P\cos Q + \cos P\sin Q} \right) = 37$

$ \Rightarrow 9 + 16 + 24\sin \left( {P + Q} \right) = 37$

[ As ${\sin ^2}\theta + {\cos ^2}\theta = 1$ and

$\sin A\cos B + \cos A\sin B$ $ = \sin \left( {A + B} \right)$ ]

$ \Rightarrow \sin \left( {P + Q} \right) = {1 \over 2}$

$ \Rightarrow P + Q = {\pi \over 6}$ or ${{5\pi } \over 6}$

$ \Rightarrow R = {{5\pi } \over 6}$ or ${\pi \over 6}$

(as $P + Q + R = \pi $ )

If $R = {{5\pi } \over 6}$ then $0 < P,Q < {\pi \over 6}$

$ \Rightarrow \cos Q < 1$ and $\sin P < {1 \over 2}$

$ \Rightarrow 3\sin P + 4\cos Q < {{11} \over 2}$ which is not true.

So $R = {\pi \over 6}$

If $O$ is center of polygon and

$AB$ is one of the side, then by figure

$\cos {\pi \over n} = {r \over R}$

$ \Rightarrow {r \over R} = {1 \over 2},{1 \over {\sqrt 2 }},{{\sqrt 3 } \over 2}\,\,for$

$n = 3,4,6$ respectively.

We know by sin c rule

${c \over {\sin C}} = 2R \Rightarrow c = 2R\sin C$

$ \Rightarrow c = 2R$

$\left( \, \right.$as $\,\,\,\angle C = {90^ \circ }$ $\left. \, \right)$

Also $\tan {C \over 2} = {r \over {s - c}}$

$ \Rightarrow \tan {\pi \over 4} = {r \over {s - c}}$

$\left( \, \right.$as $\,\,\,\angle C = {90^ \circ }$ $\left. \, \right)$

$ \Rightarrow r = s - c = {{a + b - c} \over 2}$

$ \Rightarrow 2r + c = a + b$

$ \Rightarrow 2r + 2R = a + b$

(using $c=2R$)

$\Delta = {1 \over 2}{p_1}a = {1 \over 2}{p_2}b = {1 \over 2}{p_3}b$

${p_1},{p_2},{p_3},$ are in $H.P.$

$ \Rightarrow {{2\Delta } \over a},{{2\Delta } \over b},{{2\Delta } \over c}$ are in $H.P.$

$ \Rightarrow {1 \over a},{1 \over b},{1 \over c},$ are in $H.P.$

$ \Rightarrow a,b,c$ are in $A.P.$

$ \Rightarrow $ $K\sin A,K\sin B,K\sin C$ are in $A.P.$

$ \Rightarrow $ $\sin A,\sin B,\sin C$ are in $A.P.$

Let $a = \sin \alpha ,b = \cos \alpha $

and $c = \sqrt {1 + \sin \alpha \cos \alpha } $

Clearly $a$ and $b < 1$ but $c > 1$

as $\,\,\,\sin \alpha > 0$ and $\cos \alpha > 0$

$\therefore$ $c$ is the greatest side and greatest angle is $C$

$\therefore$ $\cos C = {{{a^2} + {b^2} - {c^2}} \over {2ab}}$

$ = {{{{\sin }^2}\alpha + {{\cos }^2}\alpha - 1 - \sin \alpha \cos \alpha } \over {2\,\sin \alpha \cos \alpha }}$

$ = - {1 \over 2}$

$\therefore$ $C = {120^ \circ }$

If $a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2}$

$a\left[ {\cos C + 1} \right] + c\left[ {\cos A + 1} \right] = 3b$

$\left( {a + c} \right) + \left( {a\cos C + c\cos \,B} \right) = 3b$

$a + c + b = 3b$ or $a + c = 2b$

or $a,b,c$ are in $A.P.$

$AP = {2 \over 3}AD = {8 \over 3};\,\,PD = {4 \over 3};\,\,$

Let $PB=x$

$\tan {60^ \circ } = {{8/3} \over x}$

or $x = {8 \over {3\sqrt 3 }}$

Area of $\Delta ABD$

$ = {1 \over 2} \times 4 \times {8 \over {3\sqrt 3 }} = {{16} \over {3\sqrt 3 }}$

$\therefore$ Area of $\Delta ABC$

$ = 2 \times {{16} \over {3\sqrt 3 }} = {{32} \over {3\sqrt 3 }}$

$\left[ \, \right.$ As median of a $\Delta $ divides it into two $\Delta 's$ of equal area. $\left. \, \right]$

$\tan \left( {{\pi \over n}} \right) = {a \over {2r}};\,\,\sin \left( {{\pi \over n}} \right) = {a \over {2R}}$

$r + R = {a \over 2}\left[ {\cot {\pi \over n} + \cos ec{\pi \over n}} \right]$


$ = {a \over 2}\left[ {{{\cos {\pi \over n} + 1} \over {\sin {\pi \over n}}}} \right]$

$ = {a \over 2}\left[ {{{2{{\cos }^2}{\pi \over {2n}}} \over {2\sin {\pi \over {2n}}\cos {\pi \over {2n}}}}} \right]$

$ = {a \over 2}\cot {\pi \over {2\pi }}$

Let $\,\,\,\,a = 3x + 4y,b = 4x + 3y$

and $c = 5x + 5y$

as $\,\,\,\,x,y > 0,c = 5x + 5y$ is the largest side

$\therefore$ $C$ is the largest angle. Now

$\cos \,C = {{{{\left( {3x + 4y} \right)}^2} + {{\left( {4x + 3y} \right)}^3} - {{\left( {5x + 5y} \right)}^2}} \over {2\left( {3x + 4y} \right)\left( {4x + 3y} \right)}}$

$ = {{ - 2xy} \over {2\left( {3x + 4y} \right)\left( {4x + 3y} \right)}} < 0$

$\therefore$ $C$ is obtuse angle $ \Rightarrow \Delta ABC$ is obtuse angled

${r_1} > {r_2} > {r_3}$

$ \Rightarrow {\Delta \over {s - a}} > {\Delta \over {s - b}} > {\Delta \over {s - c}};$

$ \Rightarrow s - a < s - b < s - c$

$ \Rightarrow - a < - b < - c$

$ \Rightarrow a > b > c$

Option (A):

$ \begin{aligned} & 2\left(a^2-b^2\right)=c^2 \quad\quad(1)\\\\ & \lambda=\frac{\sin (x-y)}{\sin z} \quad\quad(2)\\\\ & \cos (n \pi \lambda)=0 \quad\quad(3)\\\\ & \Rightarrow n \lambda=\frac{(2 m+1)}{2}\quad\quad(4) \end{aligned} $

From Eq. (2), we have

$ \lambda=\frac{\sin x(0) y-\cos x \sin y}{\sin z} $

$ \begin{aligned} & \Rightarrow \lambda=\frac{a \cos y-b \cos x}{c} \\\\ & \Rightarrow \lambda=\frac{a\left(\frac{a^2+c^2-b^2}{2 a c}\right)-b\left(\frac{b^2+c^2-a^2}{2 b c}\right)}{2 c} \quad \text { (by sine formula) } \\\\ & \Rightarrow \lambda=\frac{2\left(a^2-b^2\right)}{2 c^2}=\frac{1}{2}\quad\quad(5) \end{aligned} $

Therefore, from Eqs. (4) and (5), we get

$ \frac{x}{2}=\frac{2 m+1}{2} \Rightarrow x=(2 m+1) $

Hence, (A)→(P), (R), (S).

Option (B):


$ \begin{aligned} & 1+\cos 2 x-2 \cos 2 y=2 \sin x \sin y \\\\ \Rightarrow & 2 \cos ^2 x-2\left(2 \cos ^2 y-1\right)=2 \sin x \sin y \\\\ \Rightarrow & 2 \cos ^2 x-4 \cos ^2 y+2=2 \sin x \sin y \\\\ \Rightarrow & 2 \sin ^2 y-2 \sin x \sin y+\sin x \sin y-\sin ^2 x=0 \\\\ \Rightarrow & 2 \sin y(\sin y-\sin x)+\sin x(\sin y-\sin x)=0 \\\\ \Rightarrow & (\sin y-\sin x)(2 \sin y+\sin x)=0 \\\\ \Rightarrow & b=a \text { or } 2 b=-a(\text { which is impossible }) \\\\ \Rightarrow & \frac{a}{b}=1 \end{aligned} $

Hence, $(\mathrm{B}) \rightarrow(\mathrm{P})$

Option (C):


Vector along the bisector of acute angle between $\overrightarrow{O X}$ and $\overrightarrow{O Y}$ is

$ \frac{\sqrt{3} \hat{i}+\hat{j}}{2}+\frac{\hat{i}+\sqrt{3} \hat{j}}{2}=\frac{(\sqrt{3}+1)}{2}(\hat{i}+\hat{j}) $

Slope of $\overrightarrow{O B}=\tan \left(\frac{\pi}{4}\right)=1$

Equation of $O B$ is $y=x$.

Since

$ \begin{aligned} Z L & =\frac{3}{\sqrt{3}} \Rightarrow \frac{|\beta-(1-\beta)|}{\sqrt{2}}=\frac{3}{\sqrt{2}} \\\\ & \Rightarrow|2 \beta-1|=3 \\\\ & \Rightarrow(2 \beta-1)= \pm 3 \\\\ & \Rightarrow \beta=2 \text { or } \beta=-1 \\\\ & \Rightarrow|\beta|=1 \text { or } 2 \end{aligned} $

$(\mathrm{C}) \rightarrow(\mathrm{P}),(\mathrm{Q})$

Option (D):


$ \Rightarrow y=|\alpha x-1|+|\alpha x-2|+\alpha x ; \alpha \in\{0,1\} $

Case (I): For $\alpha=0, y=3$

Case (II): For $\alpha=1, y=|x-1|+|x-2|+x$

$\Rightarrow y=\left\{\begin{array}{cc}3-x ; & x \leq 1 \\ x+1 ; & 1 < x < 2 \\ 3 x-3 ; & x \geq 2\end{array}\right.$

Hence,

$ \begin{aligned} F(0) & =\int_0^2(3-2 \sqrt{x}) d x=\left[3 x-\frac{4}{3} x^{3 / 2}\right]_0^2 \\\\ & =\left[6-\frac{4}{3}(2 \sqrt{2})\right]=6-\frac{8}{3} \sqrt{2} \\\\ \Rightarrow & F(0)+\frac{8}{3} \sqrt{2}=6 \Rightarrow(T) \end{aligned} $

and

$ \begin{aligned} F(1) & =F(0)-\text { area of } \triangle A C D \\\\ & =\left(6-\frac{8}{3} \sqrt{2}\right)-\frac{1}{2}(2)(1)=5-\frac{8}{3} \sqrt{2} \\\\ & \Rightarrow F(1)+\frac{8}{3} \sqrt{2}=5 \Rightarrow(S) \end{aligned} $

Hence, $(\mathrm{D}) \rightarrow(\mathrm{T}),(\mathrm{S})$

Let $a, b, c$ are the sides of a triangle

And $\quad a+b=x, a b =y$

Given, $\quad x^2-c^2 =y$

$\begin{array}{lrl} \Rightarrow & (x+c)(x-c) & =y \\ \Rightarrow & (a+b+c)(a+b-c) & =y \\ \Rightarrow & 2 s(2 s-2 c) & =y \\ \Rightarrow & s(s-c) & =\frac{y}{4} \quad \text{... (i)}\\ \Rightarrow & \frac{s(s-c)}{y} & =\frac{1}{4} \end{array}$

$\begin{array}{ll} \Rightarrow & \frac{s(s-c)}{a b}=\frac{1}{4} \\ \Rightarrow & \cos ^2 \frac{C}{2}=\frac{1}{4} \\ \Rightarrow & \cos \frac{C}{2}= \pm \frac{1}{2} \\ \Rightarrow & \frac{C}{2}=\frac{\pi}{3}, \frac{2 \pi}{3} \\ \Rightarrow & \mathrm{C}=\frac{2 \pi}{3}, \frac{4 \pi}{3} \end{array}$

$\because \quad C=\frac{4 \pi}{3}$ is impossible in a triangle

$\therefore \quad C=\frac{2 \pi}{3}$

Now, $\frac{r}{\mathrm{R}}=\frac{(s-c) \tan \frac{\mathrm{C}}{2}}{\frac{c}{2 \sin \mathrm{C}}}$

$\begin{aligned} \Rightarrow \quad \frac{r}{\mathrm{R}} & =\frac{(s-c) \cdot \tan \frac{\pi}{3}}{\frac{c}{2 \sin \frac{2 \pi}{3}}} \\ \Rightarrow \quad \frac{r}{\mathrm{R}} & =\frac{\frac{y}{4 s} \cdot \sqrt{3}}{\frac{c}{2 \cdot \frac{\sqrt{3}}{2}}} \quad \left[(s-c)=\frac{y}{4 s} \text { from equation (i) }\right] \end{aligned}$

$\begin{aligned} & \Rightarrow \quad \frac{r}{\mathrm{R}}=\frac{3 y}{2 \cdot 2 s \cdot c} \\ & \Rightarrow \quad \frac{r}{\mathrm{R}}=\frac{3 y}{2(a+b+c) c} \\ & \Rightarrow \quad \frac{r}{\mathrm{R}}=\frac{3 y}{2(x+c) \cdot c} \end{aligned}$

Hint:

$\Rightarrow$ In a triangle $\triangle \mathrm{ABC}$

(i) $r=(\mathrm{s}-c) \tan \frac{\mathrm{C}}{2}, \mathrm{R}=\frac{c}{2 \sin \mathrm{C}}$

(ii) $\cos \frac{c}{2}=\sqrt{\frac{s(s-c)}{a b}}, s=\frac{a+b+c}{2}$

Given, $\Delta$ be the area of $\triangle \mathrm{PQR}$ of side length

$\begin{gathered} a=2, b=\frac{7}{2} \text { and } c=\frac{5}{2} \\ \Rightarrow s=\frac{a+b+c}{2}=\frac{2+\frac{7}{2}+\frac{5}{2}}{2}=4 \end{gathered}$

$\begin{aligned} \text { Now, } & \frac{2 \sin \mathrm{P}-\sin 2 \mathrm{P}}{2 \sin \mathrm{P}+\sin 2 \mathrm{P}} \\ &=\frac{2 \sin \mathrm{P}-2 \sin \mathrm{P} \cdot \cos \mathrm{P}}{2 \sin \mathrm{P}+2 \sin \mathrm{P} \cdot \cos \mathrm{P}} \\ & \quad(\sin 2 \theta=2 \sin \theta \cdot \cos \theta) \end{aligned}$

$\begin{aligned} & =\frac{1-\cos \mathrm{P}}{1+\cos \mathrm{P}} \\ & =\frac{1-\left(1-2 \sin ^2 \frac{\mathrm{P}}{2}\right)}{1+\left(2 \cos ^2 \frac{\mathrm{P}}{2}-1\right)} \\ & \quad\left(\cos 2 \theta=2 \cos ^2 \theta-1=1-2 \sin ^2 \theta\right) \\ & =\tan ^2 \frac{\mathrm{P}}{2} \\ & =\left(\sqrt{\left.\frac{(s-b)(s-c)}{s(s-a)}\right)^2}\right. \\ & =\left(\frac{(s-b)(s-c)}{\sqrt{s(s-a)(s-b)(s-c)}}\right)^2 \end{aligned}$

$\begin{aligned} & =\left(\frac{(s-b)(s-c)}{\Delta}\right)^2 \\ & \therefore \quad(\Delta=\sqrt{s(s-a)(s-b)(s-c)}) \\ & =\frac{\left(4-\frac{7}{2}\right)^2\left(4-\frac{5}{2}\right)^2}{\Delta^2} \\ & =\frac{9}{16 \Delta^2} \\ & =\left(\frac{3}{4 \Delta}\right)^2 \end{aligned}$

Using, $\cos C = {{{a^2} + {b^2} - {c^2}} \over {2ab}}$

$ \Rightarrow {{\sqrt 3 } \over 2} = {{{{({x^2} + x + 1)}^2} + {{({x^2} - 1)}^2} - {{(2x + 1)}^2}} \over {2({x^2} + x + 1)({x^2} - 1)}}$

$ \Rightarrow (x + 2)(x + 1)(x - 1)x + {({x^2} - 1)^2} = \sqrt 3 ({x^2} + x + 1)({x^2} - 1)$

$ \Rightarrow {x^2} + 2x + ({x^2} - 1) = \sqrt 3 ({x^2} + x + 1)$

$ \Rightarrow (2 - \sqrt 3 ){x^2} + (2 - \sqrt 3 )x - (\sqrt 3 + 1) = 0$

$ \Rightarrow x = - (2 + \sqrt 3 )$ and $x = 1 + \sqrt 3 $

But, $x = - (2 + \sqrt 3 ) \Rightarrow c$ is negative.

$\therefore$ $x = 1 + \sqrt 3 $ is the only solution.

Hence, (b) is the correct option.

Since, A, B, C are in AP

$\Rightarrow$ 2B = A + C i.e., $\angle$B = 60$^\circ$

$\therefore$ ${a \over c}$(2 sin C cos C) + ${c \over a}$ (2 sin A cos A)

= 2k (a cos C + c cos A)

[using, ${a \over {\sin A}} = {b \over {\sin B}} = {c \over {\sin C}} = {1 \over k}$]

= 2k (b)

= 2 sin B

[using, b = a cos C + c cos A]

= $\sqrt3$

$ \begin{aligned} \text { Given, } & \mathrm{AB} & =\mathrm{AC} \\ \text { and } & \angle \mathrm{A} & =120^{\circ} \end{aligned} $

Let, $\quad \mathrm{AB}=\mathrm{AC}=b$

and $\quad \mathrm{BC}=a$

Const: Draw $\mathrm{AD} \perp \mathrm{BC}$

Since, AD is also bisector of unequal side BC .

Therefore, area of $\triangle \mathrm{ABC}$

$ \begin{aligned} & =\frac{1}{2} \times \mathrm{BC} \times \mathrm{AD} \\ & =\frac{a}{2}(\mathrm{AD}) \end{aligned} $

To calculate AD, we use trigonometry In $\Delta \mathrm{ADB}$

$ \begin{aligned} \tan 30^{\circ} & =\frac{\mathrm{AD}}{\mathrm{BD}} \\ \Rightarrow \quad \frac{1}{\sqrt{3}} & =\frac{\mathrm{AD}}{\frac{a}{2}} \\ \Rightarrow \quad \mathrm{AD} & =\frac{a}{2 \sqrt{3}} \\ \text { Area of } \triangle \mathrm{ABC} & =\frac{a^2}{4 \sqrt{3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i) \end{aligned} $

We know that the relation.

Area of triangle $=$ semi-perimeter $\times$ in radius

Now, area of $\Delta \mathrm{ABC}=\left(\frac{a+2 b}{2}\right) \times \sqrt{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$

$ \begin{aligned} &\text { From (i) and (ii) we have. }\\ &\left(\frac{a+2 b}{2}\right) \sqrt{3}=\frac{a^2}{4 \sqrt{3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(iii) \end{aligned} $

Also in right $\triangle \mathrm{ABD}$

$ \begin{aligned} & & \cos 30^{\circ} & =\frac{\mathrm{BD}}{\mathrm{AB}} \\ & & \frac{\sqrt{3}}{2} & =\frac{a}{2} \\ \Rightarrow & & a & =\sqrt{3} b \end{aligned} $

Substitute $a=\sqrt{3} b$ in (iii), we have

$ \begin{aligned} & \left(\frac{\sqrt{3} b+2 b}{2}\right) \sqrt{3}=\frac{(\sqrt{3} b)^2}{4 \sqrt{3}} \\ & \Rightarrow \quad 3 b+2 \sqrt{3} b=\frac{3 b^2}{2 \sqrt{3}} \\ & \Rightarrow \quad b=\frac{6 \sqrt{3}+12}{3} \\ & \Rightarrow \quad b=(2 \sqrt{3}+4) \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\, a=\sqrt{3}(2 \sqrt{3}+4)=6+4 \sqrt{3} \\ & \text { Area, }=\frac{1}{2}\left(\frac{6+4 \sqrt{3}}{2}\right)(2 \sqrt{3}+4) \\ & =(3+2 \sqrt{3})(\sqrt{3}+2) \\ & =3 \sqrt{3}+6+6+4 \sqrt{3}=12+7 \sqrt{3} \end{aligned} $