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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Heights and Distances

NDA 2017 · Mathematics · Heights and Distances

The angles of elevation of the top of a tower from the top and foot...

The angles of elevation of the top of a tower from the top and foot of a pole are respectively 30∘30^{{}^{\circ}} and 45∘45^{{}^{\circ}}. If hT\mathrm{h}_{\mathrm{T}} is the height of the tower and hph_p is the height of the pole, then which of the following are correct? 1. \frac{2 \mathrm{ h}_{\mathrm{P}} \mathrm{h}_{\mathrm{T}}}{3+\sqrt{3}}=\mathrm{h}_{\mathrm{P}^2 2. \frac{\mathrm{h}_{\mathrm{T}}-\mathrm{h}_{\mathrm{P}}}{\sqrt{3}+1}=\frac{\mathrm{h}_{\mathrm{P}}}{2} 3. \frac{2\left(\mathrm{ h}_{\mathrm{P}}+\mathrm{h}_{\mathrm{T}}\right)}{\mathrm{h}_{\mathrm{P}}}=4+\sqrt{3} Select the correct answer using the code given below.

Options

  1. A11 and 33 only
  2. B22 and 33 only
  3. C11 and 22 only
  4. D1,21,2 and 33

Correct answer

C. 11 and 22 only

Step-by-step solution

Let the distance between tower and pole be ' xx '.
tan⁡45∘=htx⇒htx=1⇒ht=x.\begin{aligned} & \tan 45^{{}^{\circ}}=\frac{\mathrm{h}_{\mathrm{t}}}{\mathrm{x}} \Rightarrow \frac{\mathrm{h}_{\mathrm{t}}}{\mathrm{x}}=1 \\ & \Rightarrow \mathrm{ h}_{\mathrm{t}}=\mathrm{x} .\end{aligned} Now, tan⁡30∘=ht−hpx⇒13=ht−hpx\tan 30^{{}^{\circ}}=\frac{h_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}}{\mathrm{x}} \Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{h}_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}}{\mathrm{x}}⇒13=ht−hpht⇒13−1=ht−hpht−(ht−hp)\begin{aligned} & \Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{h}_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}}{\mathrm{h}_{\mathrm{t}}} \\ & \Rightarrow \frac{1}{\sqrt{3}-1}=\frac{\mathrm{h}_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}}{\mathrm{h}_{\mathrm{t}}-\left(\mathrm{h}_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}\right)}\end{aligned}⇒3+12=ht−hphp⇒ht−hp3+1=hp2\Rightarrow \frac{\sqrt{3}+1}{2}=\frac{\mathrm{h}_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}}{\mathrm{h}_{\mathrm{p}}} \Rightarrow \frac{\mathrm{h}_{\mathrm{t}}-\mathrm{h}_{\mathrm{p}}}{\sqrt{3}+1}=\frac{\mathrm{h}_{\mathrm{p}}}{2}\ldots(1) So, statement ' 2 ' is correct. Now, ht−hphp=3+12⇒ht−hp+2hphp=3+1+42⇒ht+hphp=5+32\frac{h_t-h_p}{h_p}=\frac{\sqrt{3}+1}{2} \Rightarrow \frac{h_t-h_p+2 h_p}{h_p}=\frac{\sqrt{3}+1+4}{2}\Rightarrow \frac{\mathrm{h}_{\mathrm{t}}+\mathrm{h}_{\mathrm{p}}}{\mathrm{h}_{\mathrm{p}}}=\frac{5+\sqrt{3}}{2}\ldots(2) So, statement ‘3’ is incorrect.
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