True False[1 Marks ]

Question 11 Mark

Is it true that the distance travelled by a circular wheel of diameter d cm in one nrevolution is $2\pi\text{d cm}?$ Why?

Answer

False:
Distance travelled by wheel in one revolution is equal to the circumference of wheel $=2\pi\text{r}=\pi(2\text{r})=\pi\text{d}.$
Hence, the given statement is false.

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Question 21 Mark

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

Answer

False: Consider two circles of radii, $r _1, r _2$ of are length, $I _1$ and $I _2$, and their correponding angles of sectors $\theta_1, \theta_2$ respectively.
$\text{l}_1=\frac{2\pi\text{r}_1\theta_1}{360^\circ},\text{l}_2\frac{2\pi\text{r}_2\theta_2}{360^\circ}$
Now, $l_1 = 1_2$ [Given]
$\Rightarrow\ \ \frac{2\pi\text{r}_1\theta_1}{360^\circ}=\frac{2\pi\text{r}_2\theta_2}{360^\circ}$
$\Rightarrow\ \ \text{r}_1\theta_1=\text{r}_2\theta_2=\text{x}\ \ [\text{say}]$
Area of sectors $A_1$ and $A_2$ are given by
$\text{A}_1=\frac{\pi\text{r}_1^2\theta_1}{360^\circ},\ \text{A}_2=\frac{\pi\text{r}_2^2\theta_2}{360^\circ}$
$\therefore\ \ \frac{\text{A}_1}{\text{A}_2}=\frac{\frac{\pi\text{r}_1\theta_1\text{r}_1}{360^\circ}}{\frac{\pi\text{r}_2\theta_2\text{r}_2}{360^\circ}}=\frac{\text{xr}_1}{\text{xr}_2}=\frac{\text{r}_1}{\text{r}_2}$
$\Rightarrow\ \ \frac{\text{A}_1}{\text{A}_2}=\frac{\text{r}_1}{\text{r}_2}$
Area of sector can be equal when $\frac{\text{r}_1}{\text{r}_2}=1$
i.e., equal radii. So, the area of sectors of two circles of same arcs length are not equal.
Hence, the given statement false.

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Question 31 Mark

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?

Answer

True.
False: Consider two circles $C _1$ and $C _2$ of radii 2 r respectively.
Let the lengths of two arcs be $I _1$ and $I _2$.
$\text{l}_1=\text{AB of C}_1=\frac{2\pi\text{r}\theta_1}{360^\circ}$
$\text{l}_2=\text{CD of C}_2=\frac{2\pi\text{r}'\theta_2}{360^\circ}=\frac{2\pi.2\text{r}\theta_2}{360^\circ}$
According to question,
$l_1 = l_2$
$\Rightarrow\ \ \frac{2\pi\text{r}\theta_1}{360^\circ}=\frac{2\pi.2\pi\theta_2}{360^\circ}$
$\Rightarrow\ \ \theta_1=2\theta_2$
i.e., Angle of sector of the Ist circle is twice the angle of the sector of the other circle.
Hence, the given statement false.

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Question 41 Mark

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

Answer

False: Using usual abbreviations for sectors
$\text{A}_1=\text{A}_2$
$\Rightarrow\ \ \frac{\pi\text{r}^2_1\theta_1}{360^\circ}=\frac{\pi\text{r}_2^2\theta_2}{360^\circ}$
$\Rightarrow\ \ \text{r}^2_1\theta_1=\text{r}^2_2\theta_2$
$\Rightarrow\ \ \frac{\theta_1}{\theta_2}==\frac{\text{r}_2^2}{\text{r}_1^2}$
Now, $\frac{\text{l}_1}{\text{l}_2}=\frac{2\pi\text{r}_1\theta_1}{2\pi\text{r}_2\theta_2}$ $=\frac{\text{r}_1}{\text{r}_2}\times\frac{\text{r}^2_2}{\text{r}_1^2}=\frac{\text{r}_2}{\text{r}_1}$
$\Rightarrow\ \frac{\text{l}_1}{\text{l}_2}=\frac{\text{r}_2}{\text{r}_1}$
Hence, arcs lenght can be equal if $\frac{\text{r}_2}{\text{r}_1}=1$
i.e., $r_1 = r_2 = r.$
Hence, the given statement is false.

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Question 51 Mark

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

Answer

True:
Distance covered by a cicular wheel in n revolution $=2\pi\text{rn}$
where n = number of revolutions
$\therefore\ \text{s}=2\pi\text{rn}$ or $\text{n}=\frac{\text{s}}{2\pi\text{r}}.$
Hence, the given statement true.

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Question 61 Mark

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

Answer

True:$\because\ \ 2\pi\text{r}_1=2\pi\text{r}_2$ [Given]
$\Rightarrow\ \ \text{r}_1=\text{r}_2=\text{r}$ [say]
Now, $\frac{\text{A}_1}{\text{A}_2}=\frac{\pi\text{r}^2_1}{\pi\text{r}^2_2}=\frac{\text{r}^2}{\text{r}^2}=1$
$\therefore A_1 = A_2$ i.e., their areas are equal.
Hence, the given statement is true.

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Question 71 Mark

Is it true to say that area of a square inscribed in a circle of diameter $p cm$ is $p ^2 cm^2$ ? Why?

Answer

False: The diameter of the circle is p cm.
So, the diagonal of the square will be p cm.

Now, $AB^2 + BC^2 = AC^2$
$\Rightarrow AB^2 + AB^2 = p^2$
$\Rightarrow 2AB^2 = p^2$
$\Rightarrow\ \text{AB}=\frac{\text{p}}{\sqrt{2}}$
Now, Area of square $=\frac{\text{p}}{\sqrt{2}}\times\frac{\text{p}}{\sqrt{2}}=\frac{\text{p}^2}{2}\text{cm}^2$
Hence, the given statement is false.

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Question 81 Mark

Is the area of the largest circle that can be drawn inside a rectangle of length a cm and breadth $b cm(a>b)$ is $\pi b^2$ $cm ^2$ ? Why?

Answer

False: The diameter of circle that can be drawn inside the rectangle is equal to the breadth of rectangle.
The length of the rectangle = a cm
The breadth of the rectangle = b cm
$\therefore$ Diameter of circle = b cm
$\Rightarrow\ \ \text{r}=\frac{\text{b}}{2}\text{cm}$
$\therefore$ Area of circle $\text{A}=\pi\text{r}^2=\pi\Big(\frac{\text{b}}{2}\Big)^2=\frac{1}{4}\pi\text{b}^2\text{cm}^2$
Hence, the given statement is false.

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Question 91 Mark

Is it true to say that area of a segment of a circle is less than the area of its corresponding sector? Why?

Answer

False:

Area of major segment (ACB) is always greater than its correponding sector (QACB) and area of minor segment (ACB) is smaller the its corresponding minor sector (QACB).
Hence, the given statement is false.

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Question 101 Mark

Will it be true to say that the perimeter of a square circumscribing a circle of radius a cm is 8a cm? Give reasons for your answer.

Answer

True: Side of square = Diameter of circle

$\therefore$ AB = 2a So, the perimeter of square = 4 × AB = 4 × 2a = 8a cm Hence, the given statement is true.

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Question 111 Mark

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

Answer

True:
$\because\ \ \text{A}_1=\text{A}_2$ [Given]
$\Rightarrow\ \ \pi\text{r}_1^2=\pi\text{r}_2^2$
$\Rightarrow\ \ \text{r}_1^2=\text{r}_2^2$
$\Rightarrow\ \ \text{r}_1=\text{r}_2=\text{r}$ (say) [taking square root]
Now, $\frac{\text{C}_1}{\text{C}_2}=\frac{2\pi\text{r}_1}{2\pi\text{r}_2}=\frac{\text{r}_1}{\text{r}_2}=\frac{\text{r}}{\text{r}}=\frac{1}{1}$
Hence, $C_1 = C_2$
Hence, the given statement is true.

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Question 121 Mark

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

Answer

False:Let the radius of circle is r (0 < r < 2).
Then, the area of circle
$\text{A}=\pi\text{r}^2$ for $\text{r}=1.5,\text{ A}=\pi\times(1.5)^2$
$\therefore\ \text{A}=2.25\pi$
Circumference $(\text{C})=2\pi\text{r}=2\times\pi\times1.5$
$\Rightarrow\ \ \text{C}=3.0\pi$
$\therefore\ \ \text{C}>\text{A}$
So, the of a circle is not always greater than its circumference.
Hence, the given statement false.

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Question 131 Mark

In Fig. a square is inscribed in a circle of diameter d and another square is circumscribing the circle. Is the area of the outer square four times the area of the inner square? Give reasons for your answer.

Answer

False: Let the side of the smaller square is a units and that of bigger square is b units.

Diameter of circle = d
So, diagonal of squaras EFGH = d
Then, by Pythagoras theorem,
$\text{a}^2+\text{a}^2=\text{a}^2$
$\Rightarrow\ \ 2\text{a}^2=\text{d}^2$
$\Rightarrow\ \ \text{a}^2=\frac{\text{d}^2}{2}$
$\therefore$ Area of outer square $=\text{a}^2=\frac{\text{d}^2}{2}$
Side of outer square = b = Diameter of circle
$\therefore$ Area of outer square $= b^2 = d^2$
$=\frac{2}{2}\text{d}^2=2\times\frac{1}{2}\text{d}^2$
⇒ Area of larger square = 2 Area of smaller square
So, the given statement is false.

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Question 141 Mark

Is the area of the circle inscribed in a square of side $\text{a cm},\pi\text{a}^2\text{cm}^2?$ Give reasons for your answer.

Answer

False: The radius of the circle inscribed in a square of side a cm is $\text{r}=\frac{\text{a}}{2}$

$\therefore$ Area of circle $=\pi\text{r}^2$ $=\pi\Big(\frac{\text{a}}{2}\Big)^2=\frac{\pi\text{a}^2}{4}\text{cm}^2$ $\neq\pi\text{a}^2\text{cm}^2$ Hence, the given statement is false.

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MATHS True False[1 Marks ]

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