Question 1: Palette 1/31

JEE Advanced 2021 MSQ

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?

A

$\cos P \ge 1 - {{{p^2}} \over {2qr}}$

B

$\cos R \ge \left( {{{q - r} \over {p + q}}} \right)\cos P + \left( {{{p - r} \over {p + q}}} \right)\cos Q$

C

${{q + r} \over p} < 2{{\sqrt {\sin q\sin R} } \over {\sin P}}$

D

If p < q and p < r, then $\cos Q > {p \over r}$ and $\cos R > {p \over q}$

Question 2: Palette 2/31

JEE Advanced 2020 MSQ

Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y, and Z, respectively. If

$\tan {X \over 2} + \tan {Z \over 2} = {{2y} \over {x + y + z}}$, then which of the following statements is/are TRUE?

A

2Y = X + Z

B

Y = X + Z

C

$\tan {X \over 2}$ = ${x \over {y + z}}$

D

x² + z² $-$ y² = xz

Question 3: Palette 3/31

JEE Advanced 2019 MSQ

In a non-right-angled triangle $\Delta $PQR, let p, q, r denote the lengths of the sides opposite to the angles At P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = ${\sqrt 3 }$, q = 1, and the radius of the circumcircle of the $\Delta $PQR equals 1, then which of the following options is/are correct?

A

Length of OE = ${1 \over 6}$

B

Length of RS = ${{\sqrt 7 } \over 2}$

C

Area of $\Delta $SOE = ${{\sqrt 3 } \over {12}}$

D

Radius of incircle of $\Delta $PQR = ${{\sqrt 3 } \over {2}}$(${2 - \sqrt 3 }$)

Question 4: Palette 4/31

JEE Advanced 2016 MSQ

In a triangle $\Delta $$XYZ$, let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$ respectively, and $2s = x + y + z$.
If ${{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}$ and area of incircle of the triangle $XYZ$ is ${{8\pi } \over 3}$, then

A

area of the triangle $XYZ$ is $6\sqrt 6 $

B

the radius of circumcircle of the triangle $XYZ$ is ${{35} \over 6}\sqrt 6 $

C

$\sin {X \over 2}\sin {Y \over 2}\sin {Z \over 2} = {4 \over {35}}$

D

${\sin ^2}\left( {{{X + Y} \over 2}} \right) = {3 \over 5}$

Question 5: Palette 5/31

JEE Advanced 2015 MCQ

Match the following :

	Column I		Column I
(A)	$\begin{array}{l}\text { In a triangle } \Delta X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 2\left(a^2-b^2\right)=c^2 \\\text { and } \lambda=\frac{\sin (X-Y)}{\sin Z} \text {, then possible values of } n \text { for which } \cos (n \lambda) \\=0 \text { is (are) }\end{array}$	(P)	1
(B)	$\begin{array}{l}\text { In a triangle } \triangle X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 1+\cos 2 X-2 \\\cos 2 Y=2 \sin X \sin Y \text {, then possible value(s) of } \frac{a}{b} \text { is (are) }\end{array}$	(Q)	2
(C)	$\begin{array}{l}\text { In } \mathbb{R}^2 \text {, let } \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j} \text { and } \beta \hat{i}+(1-\beta) \hat{j} \text { be the position } \\\text { vectors of } X, Y \text { and } Z \text { with respect of the origin } \mathrm{O} \text {, respectively. If } \\\text { the distance of } \mathrm{Z} \text { from the bisector of the acute angle of } \overrightarrow{\mathrm{OX}} \text { with } \\\overrightarrow{\mathrm{OY}} \text { is } \frac{3}{\sqrt{2}} \text {, then possible value(s) of }|\beta| \text { is (are) }\end{array}$	(R)	3
(D)	$\begin{array}{l}\text { Suppose that } F(\alpha) \text { denotes the area of the region bounded by } \\x=0, x=2, y^2=4 x \text { and } y=|\alpha x-1|+|\alpha x-2|+\alpha x \text {, } \\\text { where, } \alpha \in\{0,1\} \text {. Then the value(s) of } F(\alpha)+\frac{8}{2} \sqrt{2} \text {, when } \alpha=0 \\\text { and } \alpha=1 \text {, is (are) }\end{array}$	(S)	5
		(T)	6

A

$\left( A \right) \to P,R;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P,Q;\,\,\left( D \right) \to S,T$

B

$\left( A \right) \to P,R,S;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P,Q;\,\,\left( D \right) \to S,T$

C

$\left( A \right) \to P,R,S;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P;\,\,\left( D \right) \to S,T$

D

$\left( A \right) \to S;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P;\,\,\left( D \right) \to S,T$

Question 6: Palette 6/31

JEE Advanced 2014 MCQ

In a triangle the sum of two sides is $x$ and the product of the same sides is $y$. If ${x^2} - {c^2} = y$, where $c$ is the third side of the triangle, then the ratio of the in radius to the circum-radius of the triangle is

A

${{3y} \over {2x\left( {x + c} \right)}}$

B

${{3y} \over {2c\left( {x + c} \right)}}$

C

${{3y} \over {4x\left( {x + c} \right)}}$

D

${{3y} \over {4c\left( {x + c} \right)}}$

Question 7: Palette 7/31

JEE Advanced 2013 MSQ

In a triangle $PQR$, $P$ is the largest angle and $\cos P = {1 \over 3}$. Further the incircle of the triangle touches the sides $PQ$, $QR$ and $RP$ at $N,L$ and $M$ respectively, such that the lengths of $PN, QL$ and $RM$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

A

$16$

B

$18$

C

$24$

D

$22$

Question 8: Palette 8/31

JEE Advanced 2012 MCQ

Let $PQR$ be a triangle of area $\Delta $ with $a=2$, $b = {7 \over 2}$ and $c = {5 \over 2}$; where $a, b,$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P.Q$ and $R$ respectively. Then ${{2\sin P - \sin 2P} \over {2\sin P + \sin 2P}}$ equals.

A

${3 \over {4\Delta }}$

B

${45 \over {4\Delta }}$

C

${\left( {{3 \over {4\Delta }}} \right)^2}$

D

${\left( {{45 \over {4\Delta }}} \right)^2}$

Question 9: Palette 9/31

JEE Advanced 2010 MCQ

Let $ABC$ be a triangle such that $\angle ACB = {\pi \over 6}$ and let $a, b$ and $c$ denote the lengths of the sides opposite to $A$, $B$ and $C$ respectively. The value(s) of $x$ for which $a = {x^2} + x + 1,\,\,\,b = {x^2} - 1\,\,\,$ and $c = 2x + 1$ is (are)

A

$ - \left( {2 + \sqrt 3 } \right)$

B

${1 + \sqrt 3 }$

C

${2 + \sqrt 3 }$

D

${4 \sqrt 3 }$

Question 10: Palette 10/31

JEE Advanced 2010 MCQ

If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression ${a \over c}\sin 2C + {c \over a}\sin 2A$ is

A

${1 \over 2}$

B

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

C

$1$

D

${\sqrt 3 }$

Question 11: Palette 11/31

JEE Advanced 2007 MCQ

Let $ABCD$ be a quadrilateral with area $18$, with side $AB$ parallel to the side $CD$ and $2AB=CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides, then its radius is

A

$3$

B

$2$

C

${3 \over 2}$

D

$1$

Question 12: Palette 12/31

JEE Advanced 2006 MCQ

Given an isosceles triangle, whose one angle is $120^{\circ}$ and radius of its incircle $=\sqrt{3}$. Then the area of the triangle in sq. units is

A

$7+12 \sqrt{3}$

B

$12-7 \sqrt{3}$

C

$12+7 \sqrt{3}$

D

$4 \pi$

Question 13: Palette 13/31

JEE Advanced 2006 MSQ

Internal bisector of $\angle A$ of triangle $A B C$ meets side BC at D . A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F . If $a, b, c$ represent sides of $\triangle \mathrm{ABC}$ then

A

AE is HM of $b$ and $c$

B

$\mathrm{AD}=\frac{2 b c}{b+c} \cos \frac{\mathrm{~A}}{2}$

C

$\mathrm{EF}=\frac{4 b c}{b+c} \sin \frac{\mathrm{~A}}{2}$

D

the triangle AEF is isosceles

Question 14: Palette 14/31

JEE Advanced 2005 MCQ

In a triangle $ABC$, $a,b,c$ are the lengths of its sides and $A,B,C$ are the angles of triangle $ABC$. The correct relation is given by

A

$\left( {b - c} \right)\sin \left( {{{B - C} \over 2}} \right) = a\cos {A \over 2}$

B

$\left( {b - c} \right)cos\left( {{A \over 2}} \right) = a\,sin{{B - C} \over 2}$

C

$\left( {b + c} \right)\sin \left( {{{B + C} \over 2}} \right) = a\cos {A \over 2}$

D

$\left( {b - c} \right)cos\left( {{A \over 2}} \right) = 2a\,sin{{B + C} \over 2}$

Question 15: Palette 15/31

JEE Advanced 2005 MCQ

In an equilateral triangle, $3$ coins of radii $1$ unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

A

$4 + 2\sqrt 3 $

B

$6 + 4\sqrt 3 $

C

$12 + {{7\sqrt 3 } \over 4}$

D

$3 + {{7\sqrt 3 } \over 4}$

Question 16: Palette 16/31

JEE Advanced 2004 MCQ

The sides of a triangle are in the ratio $1:\sqrt 3 :2$, then the angles of the triangle are in the ratio

A

$1:3:5$

B

$2:3:4$

C

$3:2:1$

D

$1:2:3$

Question 17: Palette 17/31

JEE Advanced 2003 MCQ

If the angles of a triangle are in the ratio $4:1:1$, then the ratio of the longest side to the perimeter is

A

$\sqrt 3 :\left( {2 + \sqrt 3 } \right)$

B

$1:6$

C

$1:2 + \sqrt 3 $

D

$2:3$

Question 18: Palette 18/31

JEE Advanced 2002 MCQ

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $ABC$ ($R$ being the radius of the circumcircle)?

A

$a,\,\sin \,A,sin\,B$

B

$a,b,c$

C

$a,\,\sin \,B,R$

D

$a,\,\sin \,A,R$

Question 19: Palette 19/31

JEE Advanced 2001 MCQ

A man from the top of a $100$ metres high tower sees a car moving towards the tower at an angle of depression of ${30^ \circ }$. After some time,the angle of depression becomes ${60^ \circ }$. The distance (in metres) travelled by the car during this time is

A

$100\sqrt 3 $

B

$200\sqrt 3 /3$

C

$100\sqrt 3 /3$

D

$200\sqrt 3 $

Question 20: Palette 20/31

JEE Advanced 2000 MCQ

In a triangle $ABC$, let $\angle C = {\pi \over 2}$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then $2(r+R)$ is equal to

A

$a+b$

B

$b+c$

C

$c+a$

D

$a+b+c$

Question 21: Palette 21/31

JEE Advanced 2000 MCQ

A pole stands vertically inside a triangular park $\Delta ABC$. If the angle of elevation of the top of the pole from each corner of the park is same, then in $\Delta ABC$ the foot of the pole is at the

A

centroid

B

circumcentre

C

incentre

D

orthocentre

Question 22: Palette 22/31

JEE Advanced 2000 MCQ

In a triangle $ABC$, $2ac\,\sin {1 \over 2}\left( {A - B + C} \right) = $

A

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

B

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

C

${b^2} - {c^2} - {a^2}$

D

${c^2} - {a^2} - {b^2}$

Question 23: Palette 23/31

JEE Advanced 1998 MCQ

Let ${A_0}{A_1}{A_2}{A_3}{A_4}{A_5}$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments ${A_0}{A_1},{A_0}{A_2}$ and ${A_0}{A_4}$ is

A

${3 \over 4}$

B

$3\sqrt 3 $

C

$3$

D

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

Question 24: Palette 24/31

JEE Advanced 1998 MCQ

If in a triangle $PQR$, $\sin P,\sin Q,\sin R$ are in $A.P.,$ then

A

the altitudes are in $A.P.$

B

the altitudes are in $H.P.$

C

the medians are in $G.P.$

D

the medians are in $A.P$

Question 25: Palette 25/31

JEE Advanced 1995 MCQ

In a triangle $ABC$, $\angle B = {\pi \over 3}$ and $\angle C = {\pi \over 4}$. Let $D$ divide $BC$ internally in the ratio $1:3$ then ${{\sin \angle BAD} \over {\sin \angle CAD}}$ is equal to

A

${1 \over {\sqrt 6 }}$

B

${1 \over 3}$

C

${1 \over {\sqrt 3 }}$

D

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

Question 26: Palette 26/31

JEE Advanced 1994 MCQ

If the lengths of the sides of triangle are $3, 5, 7$ then the largest angle of the triangle is

A

${\pi \over 2}$

B

${5\pi \over 6}$

C

${2\pi \over 3}$

D

${3\pi \over 4}$

Question 27: Palette 27/31

JEE Advanced 1990 MCQ

In a triangle $ABC$, angle $A$ is greater than angle $B$. If the measures of angles $A$ and $B$ satify the equation $3{\mathop{\rm sinx}\nolimits} - 4si{n^3}x - k = 0,$ $0 < k < 1$, then the measure of angle $C$ is

A

${\pi \over 3}$

B

${\pi \over 2}$

C

${2\pi \over 3}$

D

${5\pi \over 6}$

Question 28: Palette 28/31

JEE Advanced 1987 MSQ

In a triangle, the lengths of the two larger sides are $10$ and $9$, respectively. If the angles are in $AP$. Then the length of the third side can be

A

$5 - \sqrt 6 $

B

$3\sqrt 3 $

C

$5$

D

$5 + \sqrt 6 $

Question 29: Palette 29/31

JEE Advanced 1986 MSQ

There exists a triangle $ABC$ satisfying the conditions

A

$b\sin A = a,A < \pi /2$

B

$b\sin A > a,A > \pi /2$

C

$b\sin A > a,A < \pi /2$

D

$b\sin A < a,A < \pi /2,b > a$

Question 30: Palette 30/31

JEE Advanced 1983 MCQ

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is ${15^ \circ }$. The distance of the boat from the foot of the light house is

A

$\left( {{{\sqrt 3 - 1} \over {\sqrt 3 + 1}}} \right)60$ metres

B

$\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right)60$ metres

C

${\left( {{{\sqrt 3 + 1} \over {\sqrt 3 - 1}}} \right)^2}$ metres

D

none of these

Question 31: Palette 31/31

JEE Advanced 1979 MCQ

If the bisector of the angle $P$ of a triangle $PQR$ meets $QR$ in $S$, then

A

$QS=SR$

B

$QS:SR$ $= PR:PQ$

C

$QS:SR=PQ:PR$

D

None of these