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Properties of Triangle question

2015 · Shift 1 · Q36
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  1. PYQ Library
  2. /JEE Advanced
  3. /Mathematics
  4. /Properties of Triangle
  5. /2015 · Shift 1 · Q36

Properties of Triangle question

2015 · Shift 1 · Q36

JEE AdvancedMathematicsProperties of TriangleMCQ+4 / −1
Match the following :

Column I Column I
(A)  In a triangle ΔXYZ, let a,b and c be the lengths of the sides  opposite to the angles X,Y and Z, respectively. If 2(a2−b2)=c2 and λ=sin⁡(X−Y)sin⁡Z, then possible values of n for which cos⁡(nλ)=0 is (are) \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} In a triangle ΔXYZ, let a,b and c be the lengths of the sides  opposite to the angles X,Y and Z, respectively. If 2(a2−b2)=c2 and λ=sinZsin(X−Y)​, then possible values of n for which cos(nλ)=0 is (are) ​ (P) 1
(B)  In a triangle △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−2cos⁡2Y=2sin⁡Xsin⁡Y, then possible value(s) of ab is (are) \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} In a triangle △XYZ, let a,b and c be the lengths of the sides  opposite to the angles X,Y and Z, respectively. If 1+cos2X−2cos2Y=2sinXsinY, then possible value(s) of ba​ is (are) ​ (Q) 2
(C)  In R2, let 3i^+j^,i^+3j^ and βi^+(1−β)j^ be the position  vectors of X,Y and Z with respect of the origin O, respectively. If  the distance of Z from the bisector of the acute angle of OX→ with OY→ is 32, then possible value(s) of ∣β∣ is (are) \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} In R2, let 3​i^+j^​,i^+3​j^​ and βi^+(1−β)j^​ be the position  vectors of X,Y and Z with respect of the origin O, respectively. If  the distance of Z from the bisector of the acute angle of OX with OY is 2​3​, then possible value(s) of ∣β∣ is (are) ​ (R) 3
(D)  Suppose that F(α) denotes the area of the region bounded by x=0,x=2,y2=4x and y=∣αx−1∣+∣αx−2∣+αx,  where, α∈{0,1}. Then the value(s) of F(α)+822, when α=0 and α=1, is (are) \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} Suppose that F(α) denotes the area of the region bounded by x=0,x=2,y2=4x and y=∣αx−1∣+∣αx−2∣+αx,  where, α∈{0,1}. Then the value(s) of F(α)+28​2​, when α=0 and α=1, is (are) ​ (S) 5
(T) 6
  1. A
    (A)→P,R;  (B)→P;  (C)→P,Q;  (D)→S,T\left( A \right) \to P,R;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P,Q;\,\,\left( D \right) \to S,T(A)→P,R;(B)→P;(C)→P,Q;(D)→S,T
  2. B
    (A)→P,R,S;  (B)→P;  (C)→P,Q;  (D)→S,T\left( A \right) \to P,R,S;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P,Q;\,\,\left( D \right) \to S,T(A)→P,R,S;(B)→P;(C)→P,Q;(D)→S,T
  3. C
    (A)→P,R,S;  (B)→P;  (C)→P;  (D)→S,T\left( A \right) \to P,R,S;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P;\,\,\left( D \right) \to S,T(A)→P,R,S;(B)→P;(C)→P;(D)→S,T
  4. D
    (A)→S;  (B)→P;  (C)→P;  (D)→S,T\left( A \right) \to S;\,\,\left( B \right) \to P;\,\,\left( C \right) \to P;\,\,\left( D \right) \to S,T(A)→S;(B)→P;(C)→P;(D)→S,T
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