A) 17
B) 23
C) 22
D) 19
Correct Answer: C
Solution :
Let \[P=(a,\,b,\,c)\] \[M\to \,(a,\,b,\,0),\,N\to \,(0,\,b,\,c)\] DR?s of \[OP=\,(a,\,b,\,c)\] DR?s of perpendicular vector to xy-plane is (0, 0, 1). \[\sin \,{{30}^{o}}\,=\,\frac{a\times 0+b\times 0+c\times 1}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\,\sqrt{{{0}^{2}}+{{0}^{2}}+{{1}^{2}}}}\] \[\sin \,{{30}^{o}}\,=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\] ?(i) and similarly, \[\sin \,{{45}^{o}}\,=\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\] ?(ii) and \[\sin \,{{60}^{o}}\,=\,\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\] ?(iii) Plane OMN passes through (0, 0, 0), (a, b, 0) and (0, b, c). \[A\,(x-0)\,+B\,(y-0)+C\,(z-0)=0\] \[Aa+Bb=0\] \[\Rightarrow \] \[\frac{A}{B}=-\frac{b}{a}\] and \[Bb+Cc=0\] \[\Rightarrow \] \[\frac{B}{C}=-\frac{c}{b}\] Let \[A=-bk,\,B=ak,\] \[C=\frac{abk}{-c}\] \[\Rightarrow \] \[-bx+ay-\frac{abz}{c}=0\] \[\Rightarrow \] \[\frac{x}{a}-\frac{y}{b}+\frac{z}{c}=0\] DR?s of perpendicular vector \[=\,(-bc,\,ac,\,-ab)\] \[\therefore \] \[\sin \,\theta \,=\frac{a(-bc)+b(ac)-a\,(bc)}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\,\sqrt{{{b}^{2}}{{c}^{2}}+{{a}^{2}}{{c}^{2}}+{{a}^{2}}{{b}^{2}}}}}\] \[\Rightarrow \] \[{{\sin }^{2}}\theta \,=\,\frac{{{a}^{2}}{{b}^{2}}{{c}^{2}}}{({{a}^{2}}{{b}^{2}}+{{b}^{2}}{{c}^{2}}+{{c}^{2}}{{a}^{2}})\,({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\] \[\Rightarrow \] \[\cos e{{c}^{2}}\theta =\frac{({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}{{{a}^{2}}{{b}^{2}}{{c}^{2}}}\] ?(iv) Taking reciprocal of Eqs. (i), (ii) and (iii) \[\frac{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}{c}=\cos ec\,{{30}^{o}}\] \[\frac{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}{b}=\cos ec\,{{45}^{o}}\] \[\frac{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}{b}=\cos ec\,{{60}^{o}}\] \[\Rightarrow \] \[\cos e{{c}^{2}}\,{{30}^{o}}+\cos e{{c}^{2}}\,{{45}^{o}}\,+\cos e{{c}^{2}}\,{{60}^{o}}\] \[=({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\,\left( \frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}} \right)\] \[\Rightarrow \] \[4+2+\frac{4}{3}=\,\cos e{{c}^{2}}\theta \] [from Eq. (iv)] \[\Rightarrow \] \[3\,\cos e{{c}^{2}}\theta =22\]
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