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Is it true to say that area of a square inscribed in a circle of diameter p cm is p2cm2? Why?

False

In the figure we see that the diameter of circle is equal to diagonal of square

Hence, diagonal of square = p cm

Let side of square = a cm            Using Pythagoras theorem we get

\\p^{2}=a^{2}+a^{2}\\ p^{2}=2a^{2}\\ \frac{p^{2}}{2}=a^{2}\\ a=\frac{p}{\sqrt{2}}

Area of square = side × side

                  =\frac{p}{\sqrt{2}} \times \frac{p}{\sqrt{2}}=\frac{p^{2}}{2}cm^{2}

Here we found that area of square is not equal to p2cm2.                

Hence the given statement is False

           

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Areas of two circles are equal. Is it necessary that their circumferences are equal? Why?

True

Solution

Use area of circle \pi r^{2}

Let two circles having radius r1  and r2

Here it is given that their areas are equal

\pi r_{1}^{2}=\pi r_{2}^{2}\\ \\ r_{1}^{2}= r_{2}^{2}\\ \\ r_{1}= r_{2}\\

We know that circumference of circle =2\pi r

Circumference of circle with radius r1 = 2πr1

Circumference of circle with radius r2 = 2πr2    …..(1)

Put r2 = r1 in (1) we get

2πr1 = 2πr2

Hence the circumference of given circle are also equal because two circles with equal radii will also have equal circumference.                               

Therefore, the given statement is True.

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Circumferences of two circles are equal. Is it necessary that their areas be equal? Why?

True

Solution

Use circumference of circle =2 \pi r

Let two circles having radius r1  and r2 

Here it is given that their circumferences are equal

2 \pi r_{1}=2 \pi r_{2}\\ \Rightarrow r_{1}= r_{2}

We know that area of circle = \pi r^{2}

Area of circle with radius r1 = πr12

Area of circle with radius r2 = πr22     ……..(1)

Put r2 = r1   in (1) we get

πr22 = πr12

Hence the area of given circles are also equal because two circles with equal radii will also have equal areas.

Hence the given statement is True.

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Is the area of the largest circle that can be drawn inside a rectangle of length acm and breadth b cm (a >b) is \pib2 cm2?Why?

False

Solution

            

Diameter of circle = b

Radius =\frac{b}{2}

Area =\pi r^{2}=\pi \left (\frac{b}{2} \right )^{2}=\frac{1}{4}\pi b^{2}cm^{2}

Here we found that the area of the largest circle is not equal πb2cm2.                                                                                                     

Hence the given statement is False

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The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?

True

Solution

Let the radius of first circle is r1  and of other is r2

Let the arc length of both circles are same.

Let the arc length is a.                                 

length of arc (a)= 2\pi r \times \frac{\theta }{360^{\circ}}

Area of sector of first circle = a \times \frac{r_{1} }{2}       

(because area of sector = \pi r^{2}\times \frac{\theta }{360^{\circ}}=\left [ 2\pi r \times \frac{\theta }{360^{\circ}} \right ] \times \frac{r}{2}=\frac{r \times a}{2} )

Area of sector of other circle = a \times \frac{r_{2} }{2}

Here we found that both areas are equal in the case of when r1 = r2

Hence the area of two sectors of two different circles would be equal only in case of  both the circles have equal radii and equal corresponding arc length.

Hence it is necessary that their corresponding arcs lengths are equal.

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The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why?

False

Solution.

Let the radius of first circle is r1 and of other is r2

The length of arcs of both circles is same.

Let the arc length = a.                                

length of arc (a)=2\pi r \times \frac{\theta }{360^{\circ}}

Area of sector of first circle = a \times \frac{r_{1} }{2}     (because area of sector = \pi r^{2}\times \frac{\theta }{360^{\circ}}=\left [ 2\pi r \times \frac{\theta }{360^{\circ}} \right ] \times \frac{r}{2} )

Area of sector of second circle = a \times \frac{r_{2} }{2}

Here we found that the area of sector is depending on radius of circles.

When the circle is same then radius is also same then the given statement is true.

But in case of different circles then the radius is also different

Hence the given statement is False.

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If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why?

True

Solution

\because  Formula of length of arc= \frac{2 \pi r\theta }{360}

Let Radius of first circle = r

Length of arc =\frac{2 \pi r\theta_{1} }{360}                         ….. (1)                                                 {\theta _{1} is the angle of first circle}

Radius of second circle = 2r

Length of arc= \frac{2 \pi (2r)\theta_{2} }{360}

                   =\frac{4\pi r\theta_{2} }{360}                       …..(2) {\theta _{2}  is the angle of second circle}

According to question

\\\frac{2 \pi r\theta_{1} }{360}=\frac{4\pi r\theta_{2} }{360}\\\\ \theta_{1} =2\theta_{2}

No, this statement is True

 

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The numerical value of the area of a circle is greater than the numerical value of its circumference. Is this statement true? Why?

False

Solution

Area of circle =\pi r^{2}

Circumference of circle =2\pi r

Case 1:

Let r = 1

Area of circle =\pi r^{2}=\pi(1)^{2}=\pi

Circumference of circle = 2\pi r= 2\pi (1)=2 \pi

Case 2:

Let r = 3

Area of circle = πr2 = π(3)2 = 9π

Circumference of circle = 2πr = 2π(3) = 6π

Conclusion:- In case (1) we found that the area is less than the circumference but in case (2) we found that the area is greater than the circumference.

So, from conclusion we observe that it depend on the value of radius of the circle.

Hence the given statement false.

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In covering a distance s metres, a circular wheel of radius r m makes \frac{s}{2 \pi r} revolution. Is this statement true? Why?

[True]

Solution

\because Circumference of circle =2 \pi r

Radius of circular wheel = r m

Circumference of wheel = 2 \pi r

Distance covered in One revolution= circumference of wheel  =2 \pi r

In covering a distance of s number of revolution required =\frac{s}{2 \pi r}    

Hence the given statement is True

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Is it true that the distance travelled by a circular wheel of diameter d cm in one revolution is 2\pid cm?  Why?

False

Solution

Circumference of circle =2\pir

Diameter = d

Radius =\frac{d}{2}

Circumference =2\pir

 = 2 \pi \frac{d}{2}=\pi d

Here we found that the distance travelled by a circular wheel of diameter d cm in one revolution is πd which is not equal to 2πd.

Hence the given statement is False.

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