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1

MCQ (Single Correct Answer)

English

Hindi

If the angles of elevation of the top of a tower from three collinear points $$A, B$$ and $$C,$$ on a line leading to the foot of the tower, are $${30^ \circ }$$, $${45^ \circ }$$ and $${60^ \circ }$$ respectively, then the ratio, $$AB:BC,$$ is :

A

$$1:\sqrt 3 $$

B

$$2:3$$

C

$$\sqrt 3 :1$$

D

$$\sqrt 3 :\sqrt 2 $$

As $$PB$$ bisects $$\angle APC,$$ therefore $$AB$$ $$:$$ $$BC$$ $$=PA:PC$$

Also in $$\Delta APQ,\sin {30^ \circ } = {h \over {PA}} \Rightarrow PA = 2h$$

and in $$\Delta CPQ,$$ $$\sin {60^ \circ } = {h \over {PC}} \Rightarrow PC = {{2h} \over {\sqrt 3 }}$$

$$\therefore$$ $$AB:BC = 2h:{{2h} \over {\sqrt 3 }} = \sqrt 3 :1$$

तीन संरेख बिंदुओं $$\mathrm{A, B}$$ तथा $$\mathrm{C}$$, एक ऐसी रेखा पर स्थित हैं जो एक मीनार के पाद की दिशा में ले जाती है, से एक मीनार के शिखर के उन्नयन कोण क्रमशः $$30^{\circ}, 45^{\circ}$$ तथा $$60^{\circ}$$ हैं, तो $$\mathrm{AB}: \mathrm{BC}$$ का अनुपात है :

A

$$1:\sqrt 3 $$

B

$$2:3$$

C

$$\sqrt 3 :1$$

D

$$\sqrt 3 :\sqrt 2 $$

2

MCQ (Single Correct Answer)

A bird is sitting on the top of a vertical pole $$20$$ m high and its elevation from a point $$O$$ on the ground is $${45^ \circ }$$. It files off horizontally straight away from the point $$O$$. After one second, the elevation of the bird from $$O$$ is reduced to $${30^ \circ }$$. Then the speed (in m/s) of the bird is

A

$$20\sqrt 2 $$

B

$$20\left( {\sqrt 3 - 1} \right)$$

C

$$40\left( {\sqrt 2 - 1} \right)$$

D

$$40\left( {\sqrt 3 - \sqrt 2 } \right)$$

Let the speed be $$y$$ $$m/sec$$.

Let $$AC$$ be the vertical pole of height $$20$$ $$m.$$

Let $$O$$ be the point on the ground such that $$\angle AOC = {45^ \circ }$$

Let $$OC = x$$

Time $$t=1$$ $$s$$

From $$\Delta AOC,\,\,\tan {45^ \circ } = {{20} \over x}\,\,\,\,\,\,\,.....\left( i \right)$$

and from $$\Delta BOD,\,\,\tan {30^ \circ } = {{20} \over {x + y}}...\left( {ii} \right)$$

From $$(i)$$ and $$(ii),$$ we have $$x=20$$

and $${1 \over {\sqrt 3 }} = {{20} \over {x + y}}$$

$$ \Rightarrow {1 \over {\sqrt 3 }} = {{20} \over {20 + y}}$$

$$ \Rightarrow 20 + y = 20\sqrt 3 $$

So, $$y = 20\left( {\sqrt 3 - 1} \right)\,\,i.e.,$$

speed $$ = 20\left( {\sqrt 3 - 1} \right)m/s$$

Let $$AC$$ be the vertical pole of height $$20$$ $$m.$$

Let $$O$$ be the point on the ground such that $$\angle AOC = {45^ \circ }$$

Let $$OC = x$$

Time $$t=1$$ $$s$$

From $$\Delta AOC,\,\,\tan {45^ \circ } = {{20} \over x}\,\,\,\,\,\,\,.....\left( i \right)$$

and from $$\Delta BOD,\,\,\tan {30^ \circ } = {{20} \over {x + y}}...\left( {ii} \right)$$

From $$(i)$$ and $$(ii),$$ we have $$x=20$$

and $${1 \over {\sqrt 3 }} = {{20} \over {x + y}}$$

$$ \Rightarrow {1 \over {\sqrt 3 }} = {{20} \over {20 + y}}$$

$$ \Rightarrow 20 + y = 20\sqrt 3 $$

So, $$y = 20\left( {\sqrt 3 - 1} \right)\,\,i.e.,$$

speed $$ = 20\left( {\sqrt 3 - 1} \right)m/s$$

3

MCQ (Single Correct Answer)

For a regular polygon, let $$r$$ and $$R$$ be the radii of the inscribed and the circumscribed circles. A $$false$$ statement among the following is

A

There is a regular polygon with $${r \over R} = {1 \over {\sqrt 2 }}$$

B

There is a regular polygon with $${r \over R} = {2 \over 3}$$

C

There is a regular polygon with $${r \over R} = {{\sqrt 3 } \over 2}$$

D

There is a regular polygon with $${r \over R} = {1 \over 2}$$

If $$O$$ is center of polygon and

$$AB$$ is one of the side, then by figure

$$\cos {\pi \over n} = {r \over R}$$

$$ \Rightarrow {r \over R} = {1 \over 2},{1 \over {\sqrt 2 }},{{\sqrt 3 } \over 2}\,\,for$$

$$n = 3,4,6$$ respectively.

4

MCQ (Single Correct Answer)

$$AB$$ is a vertical pole with $$B$$ at the ground level and $$A$$ at the top. $$A$$ man finds that the angle of elevation of the point $$A$$ from a certain point $$C$$ on the ground is $${60^ \circ }$$. He moves away from the pole along the line $$BC$$ to a point $$D$$ such that $$CD=7$$ m. From $$D$$ the angle of elevation of the point $$A$$ is $${45^ \circ }$$. Then the height of the pole is

A

$${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 - 1} }}m$$

B

$${{7\sqrt 3 } \over 2}\left( {\sqrt {3 + 1} } \right)m$$

C

$${{7\sqrt 3 } \over 2}\left( {\sqrt {3 - 1} } \right)m$$

D

$${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 + 1} }}m$$

In $$\Delta ABC$$

$${h \over x} = \tan {60^ \circ } = \sqrt 3 $$

$$ \Rightarrow x = {h \over {\sqrt 3 }}$$

In $$\Delta ABD{h \over {x + 7}}$$

$$ = \tan {45^ \circ } = 1$$

$$ \Rightarrow h = x + 7 \Rightarrow h - {h \over {\sqrt 3 }} = 7$$

$$ \Rightarrow h = {{7\sqrt 3 } \over {\sqrt 3 - 1}} \times {{\sqrt 3 + 1} \over {\sqrt 3 + 1}}$$

$$ \Rightarrow h = {{7\sqrt 3 } \over 2}\left( {\sqrt 3 + 1\,m} \right)$$

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