Column I Column II (A) In a triangle ∆ XYZ let a, b and c be the lengths of the angles X, Y and Z , respectively. If 2( a ^ 2 - b ^ 2 ) = c ^ 2 and λ = ( (X - Y))/( Z) then possible values of n for which (nπλ) = 0 is

Correct answer: A. a-p,r,s;b-p;c-p,q;d-s,t; (A) λ = ( (X - Y))/(sin Z) ( X Y - X Y)/( Z) (acosY - bcosX)/(c) a( a ^ 2 + c ^ 2 - b ^ 2 2ac ) - b( b ^ 2 + c ^ 2 - a ^ 2 2bc ) c = λ a ^ 2 - b ^ 2 c ^ 2 = λ As 2( a ^ 2 - b ^ 2 ) = c ^ 2 ∴ λ = (1)/(2) ∴ ((nπ)/(2)) = 0 ∴ n = 1, 3, 5 (B) 1 + 2X - 2 2Y = 2sin Xsin Y Solving 4 ^ 2 Y - 2 ^ 2 X = 2 X Y 4 b ^ 2 - 2 a ^ 2 = 2ab 4 - 2 ((a)/(b)) ^ 2 = (2a)/(b) (a)/(b) = X X ^ 2 + X - 2 = 0 X = 1 (a)/(b) = 1 x = - 2 (a)/(b) = - 2 (rejected) (C) OY ≡ y = 3 x OX ≡ y = 1 3 x ∴ Equation of bisector of OX, OY y = x ∴ | x - y 2 | = 3 2 β - (1 - β) 2 = ± 3 2 2β - 1 = ± 3 β = 2 β = - 1 |β| = 1, 2 ( D ) ∴ Area = 6 - ∫ _ 0 ^ 2 2 x dx = 6 - 8 2 3 ∴ Required value = 6 - 8 2 3 + 8 2 3 = 6 Area= Shaded region - area under parabola. = 5 - 8 2 3 ∴ Required value = 5

JEE Advanced 2015Paper 1

Exam: JEE Advanced
Year: 2015
Paper: Paper 1
Subject: Mathematics
Chapter: Properties of Triangles

JEE Advanced 2015 · Paper 1 · Mathematics · Properties of Triangles

Column I Column II (A) In a triangle ∆ XYZ let a, b and c be the...

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

Options

Aa-p,r,s;b-p;c-p,q;d-s,t;
Ba-q,r,s,t;b-q,r,s,t;c-r,s,t;d-q,r,s;
Ca-q;b-r;c-r,s;d-r,s;
Da-q,r,s,t;b-t;c-r,s,t;d-q,r,s,t;

Correct answer

A. a-p,r,s;b-p;c-p,q;d-s,t;

Step-by-step solution

\left(A\right) \lambda = \dfrac{\sin \left(X - Y\right)}{sin Z} $ $ \dfrac{\sin X \cos Y - \cos X \sin Y}{\sin Z} $ $ \dfrac{acosY - bcosX}{c} $ $ \dfrac{a\left(\dfrac{{a}^{2} + {c}^{2} - {b}^{2}}{2ac}\right) - b\left(\dfrac{{b}^{2} + {c}^{2} - {a}^{2}}{2bc}\right)}{c} = \lambda $ $ \dfrac{{a}^{2} - {b}^{2}}{{c}^{2}} = \lambda $ As$ 2\left({a}^{2} - {b}^{2}\right) = {c}^{2} $ $ ∴ \lambda = \dfrac{1}{2} $ $ ∴ \cos \left(\dfrac{n\pi}{2}\right) = 0 $ $ ∴ n = 1, 3, 5 $ \left(B\right) 1 + \cos 2X - 2\cos 2Y = 2sin Xsin Y$
Solving
$4\sin^{2}Y - 2\sin^{2}X = 2\sin X \sin Y$
$4{b}^{2} - 2{a}^{2} = 2ab$
$4 - 2{\left(\dfrac{a}{b}\right)}^{2} = \dfrac{2a}{b}$

Column I Column II (A) In a triangle ∆ XYZ let a, b and c be the lengths of the angles X, Y and Z , respectively. If 2(
$\dfrac{a}{b} = X$

${X}^{2} + X - 2 = 0$

$X = 1$

$\dfrac{a}{b} = 1$

$x = - 2$

$\dfrac{a}{b} = - 2$ (rejected)

\left(C\right) OY ≡ y = \sqrt{3}x $$ OX ≡ y = \dfrac{1}{\sqrt{3}}x $$ ∴ $Equation of bisector of$ OX, OY $$ y = x $$ ∴ \left|\dfrac{x - y}{\sqrt{2}}\right| = \dfrac{3}{\sqrt{2}} $$ \dfrac{\beta - \left(1 - \beta\right)}{\sqrt{2}} = \pm \dfrac{3}{\sqrt{2}} $$ 2\beta - 1 = \pm 3 $$ \beta = 2 $$ \beta = - 1 $\left|\beta\right| = 1, 2$

$( D )$
Column I Column II (A) In a triangle ∆ XYZ let a, b and c be the lengths of the angles X, Y and Z , respectively. If 2(

$∴$ Area $= 6 - {∫}_{0}^{2}2\sqrt{x}dx$

$= 6 - \dfrac{8\sqrt{2}}{3}$

$∴$ Required value $= 6 - \dfrac{8\sqrt{2}}{3} + \dfrac{8\sqrt{2}}{3} = 6$

Column I Column II (A) In a triangle ∆ XYZ let a, b and c be the lengths of the angles X, Y and Z , respectively. If 2(
Area= Shaded region $-$ area under parabola.

$= 5 - \dfrac{8\sqrt{2}}{3}$

$∴$ Required value $= 5$

Practice Properties of Triangles on Quantrex Academy →

More from Properties of Triangles

In a triangle ABC, BC=7, AC=8, AB=α ∈ N and A=(2)/(3) . If 49 (3 C)+42= m n , where (m… JEE Main 2024 Consider an obtuse angled triangle ABC in which the difference between the largest and… JEE Advanced 2023 Consider an obtuse angled triangle ABC in which the difference between the largest and… JEE Advanced 2023 For a triangle ABC , the value of 2A + 2B + 2C is least. If its inradius is 3 and… JEE Main 2023 Let ABC be the triangle with AB = 1, AC = 3 and ∠ BAC = (π)/(2) . If a circle of radius r… JEE Advanced 2022 Let a, b and c be the length of sides of a triangle ABC such that (a + b)/(7) = (b +… JEE Main 2022 Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q… JEE Advanced 2021 In a triangle ABC , let AB = 23 , BC = 3 and CA = 4 . Then the value of (cotA +… JEE Advanced 2021 Let (sinA)/(sinB) = ( ( A - C ))/( ( C - B )), where A, B, C are angles of a triangle ABC… JEE Main 2021 Let in a right angled triangle, the smallest angle be θ . If a triangle formed by taking… JEE Main 2021 If in a triangle ABC, AB = 5 units, ∠ B = ^ - 1 ((3)/(5)) and radius of circumcircle of ∆… JEE Main 2021 In Δ ABC, the lengths of sides AC and AB are 12cm and 5cm, respectively. If the area of ∆… JEE Main 2021 Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides… JEE Advanced 2020 Suppose that the side lengths of a triangle are three consecutive integers and one of the… NTA Abhyas JEE Main 2020 The inradius of the triangle having sides 26, 28, 30 units is NTA Abhyas JEE Main 2020 The lengths of two adjacent sides of a cyclic quadrilateral are 2 units and 5 units and… NTA Abhyas JEE Main 2020 Full Properties of Triangles list Properties of Triangles JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs