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Directions:  Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below.

Assertion The area of the sector of a circle of radius 5 cm is \[9.75\text{ }c{{m}^{2}}\], if the corresponding arc length is 3.5 cm.

Reason Area of a sector of a circle of radius r and central angle \[\theta \] is \[\frac{\theta }{360{}^\circ }\pi {{r}^{2}}\].

A) A is true, R is true; R is a correct explanation for A.

B) A is true, R is true; R is not a correct explanation for A.

C) A is true; R is false.

D) A is false; R is true.

Correct Answer: D

Solution :

Let the central angle of the sector be \[\theta \]. Given that, radius of the sector of circle \[\left( r \right)=5cm\] and arc length (l) = 3.5 cm \[\therefore \] Central angle of the sector, \[\theta =\frac{arc\,length\,\left( l \right)}{radius}\] \[\Rightarrow \,\,\theta =\frac{3.5}{5}=0.7\,\,R\] \[\left[ \because \,\,\theta =\frac{l}{r} \right]\] \[\Rightarrow \,\,\theta ={{\left( 0.7\times \frac{180}{\pi } \right)}^{{}^\circ }}\]     \[\left[ \because \,1R=\frac{180{}^\circ }{\pi }D{}^\circ  \right]\] Now, area of sector with angle \[\theta =0.7\] \[=\frac{\pi {{r}^{2}}}{360{}^\circ }\times \left( 0.7 \right)\times \frac{180{}^\circ }{\pi }=\frac{{{\left( 5 \right)}^{2}}}{2}\times 0.7\] \[=\frac{25\times 7}{2\times 10}\times \frac{175}{20}=8.75c{{m}^{2}}\] Hence, required area of the sector of a circle is \[8.75\text{ }c{{m}^{2}}\].