MCQ

MCQ 11 Mark

The angle between the minute and hour hands of a clock at $8 : 30$ is :

A
$80^\circ$
✓
$75^\circ$
C
$60^\circ$
D
$105^\circ$

Answer

Correct option: B.

$75^\circ$

We know that the hour of a clock completes one rotation in $12$ hours.
Angle traced by the hour hand in $12$ hours $= 360^\circ$
Now,
Angle traced by the hour hand in $8$ hours $30$ minutes, i.e.
We also know that the minute hand of a clock completes one rotation in $60$ minutes.
Angle traced by the minute hade in $30$ minutes $=\Big(\frac{360}{60}\times30^{\circ}\Big)=180^{\circ}$
Required angle between the two hands of the clock $= 255^\circ - 180^\circ = 75^\circ$

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MCQ 21 Mark

If the angles of a triangle are in $A.P$. then the measures of one of the angles in radians is :

A
$\frac{\pi}{6}$
✓
$\frac{\pi}{3}$
C
$\frac{\pi}{2}$
D
$\frac{2\pi}{3}$

Answer

Correct option: B.

$\frac{\pi}{3}$

Let the angles of the triangle be $(a - d), (a)$ and $(a + d)$.
Thus, we have:
$a - d + a + a + d = 180$
$\Rightarrow 3a = 180$
$\Rightarrow a = 60$

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MCQ 31 Mark

The radius of the circle whose arc of length $15\pi$ makes an angle of $\frac{3\pi}{4}$ radian at the centre is :

A
$10\text{ cm}$
✓
$20\text{ cm}$
C
$11\frac{1}{4}\text{ cm}$
D
$22\frac{1}{2}\text{ cm}$

Answer

Correct option: B.

$20\text{ cm}$

$\theta=\frac{\text{Arc}}{\text{Radius}}$
$\Rightarrow\frac{3\pi}{4}=\frac{15\pi}{\text{Radius}}$
$\Rightarrow\frac{60}{3}$
$\Rightarrow20\text{ cm}$

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MCQ 41 Mark

If the arac of the same langth in two circles subtend angles $65^\circ$ and $110^\circ$ at the center, the ratio of the circle is:

✓
$22 : 13$
B
$11 : 13$
C
$22 : 15$
D
$21 : 13$

Answer

Correct option: A.

$22 : 13$

Let the angle subtended at the by the arec and radii of the first second circle $\theta_{1}$ and $\text{r}_{1}$ and $\theta_{2}$ and $\text{r}_{2}.$
We have,
$\theta_{1}=65^{\circ}=\Big(65\times\frac{\pi}{180}\Big)\ \text{radian}$
$\theta_{2}=65^{\circ}=\Big(110\times\frac{\pi}{180}\Big)\ \text{radian}$
$\theta_{1}=\frac{1}{\text{r}_{1}}$
$\Rightarrow \text{r}_{1}=\frac{1}{\big(65\times\frac{\pi}{180}\big)}$
$\Rightarrow \text{r}_{2}=\frac{1}{\big(110\times\frac{\pi}{180}\big)}$
$\frac{\text{r}_{1}}{\text{r}_{2}}=\frac{\frac{l}{\big(65\times\frac{\pi}{180}\big)}}{\frac{i}{\big(110\times\frac{\pi}{180}\big)}} $
$=\frac{110}{65}=\frac{22}{13}$
$\text{r}_{1}:\text{r}_{2}=22:13$

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MCQ 51 Mark

If $OP$ makes $4$ revolutions in on second the angular velocity in radians per seconds is :

A
$\pi$
B
$2\pi$
C
$4\pi$
✓
$8\pi$

Answer

Correct option: D.

$8\pi$

$\text{Angular velocity}=\frac{\text{Distance}}{\text{Time}}$
$=\frac{4\times2\pi}{1}$
$=8\pi\ \text{radians}$

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MCQ 61 Mark

If $D, G$ and $R$ denote respectively the number of degrees, grades and radians in an angle, then :

A
$\frac{\text{D}}{100}=\frac{\text{G}}{90}=\frac{2\text{R}}{\pi}$
B
$\frac{\text{D}}{90}=\frac{\text{G}}{100}=\frac{\text{R}}{\pi}$
✓
$\frac{\text{D}}{100}=\frac{\text{G}}{100}=\frac{2\text{R}}{\pi}$
D
$\frac{\text{D}}{90}=\frac{\text{G}}{100}=\frac{\text{R}}{\pi}$

Answer

Correct option: C.

$\frac{\text{D}}{100}=\frac{\text{G}}{100}=\frac{2\text{R}}{\pi}$

$\frac{\text{D}}{100}=\frac{\text{G}}{100}=\frac{2\text{R}}{\pi}$
It is the relation between degree, grade and radian.

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MCQ 71 Mark

At $3 : 40,$ the hour and minute hands of a clock are inclined at :

A
$\frac{2\pi^{\text{c}}}{3}$
B
$\frac{7\pi^{\text{c}}}{12}$
✓
$\frac{13\pi^{\text{c}}}{18}$
D
$\frac{13\pi^{\text{c}}}{4}$

Answer

Correct option: C.

$\frac{13\pi^{\text{c}}}{18}$

We know that the hour of a clock completes one rotation in $12$ hours $= 360^\circ$
Angle traced by the hour hand in $12$ hours $= 360^\circ$
Now,
Angle traced by the hour hand in $8$ hours $30$ minutes, i.e.
We also know that the minute hand of a clock completes one rotation in $60$ minutes.
Angle traced by the minute hade in $30$ minutes $=\Big(\frac{360}{60}\times40^{\circ}\Big)=240^{\circ}$
Required angle between the two hands of the clock $= 240^\circ - 110^\circ = 130^\circ$
Value of the angle $($in radians$)$ between the two hands of the clock $\Big(130\times\frac{\pi}{180}\Big)^{\text{c}}=\frac{13\pi^{\text{c}}}{18}$

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MCQ 81 Mark

A circular wire of radius $7\ cm$ is cut and bent again into an arc of a circle of radius $12\ cm$. The angle subtended by the arc at the centre is :

A
$50^\circ$
✓
$210^\circ$
C
$100^\circ$
D
$60^\circ$

Answer

Correct option: B.

$210^\circ$

Length of the arc of radius $=$ Circumference of the circle of radium
Now, Angle subtended by the arc $=\frac{14\pi}{12}$

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