(1)

We have,
circumference of a circle = 176 cm

$$\begin{gathered} 2\pi r=176\Rightarrow 2\times \frac{22}{7}\times r=176 \\ \Rightarrow r=\frac{176\times 7}{44}=28 \mathrm{cm}\Rightarrow r=28 \mathrm{cm} \\ \mathrm{Both} \mathrm{Assertion} \left(\mathrm{A}\right) \mathrm{and} \mathrm{Reason} \left(\mathrm{R}\right) \mathrm{are} \mathrm{true} \mathrm{and} \mathrm{Reason} \left(\mathrm{R}\right) \mathrm{is} \mathrm{the} \mathrm{correct} \mathrm{explanation} \mathrm{of} \mathrm{Assertion} \left(\mathrm{A}\right). \end{gathered}$$

(2)

We have radius of circle = 28  cm, therefore length of wire = circumference of circle

$\Rightarrow \text{Length of wire}=2\pi r=2\times \frac{22}{7}\times 28=176\text{ cm}$

Since, this wire is bent to form a square

$\therefore \text{Perimeter of square}=\text{length of wire}$

$$\begin{gathered} \Rightarrow 4x=176 \\ \Rightarrow x=44\text{ cm} \left(\text{Where }x\text{ is the length of the side of the square}\right) \end{gathered}$$

$\therefore \text{Area of square}=x^2=(44{)}^2=1936{\text{ cm}}^2$

Also, Angle described by a minute hand in a minute $=6°$

Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(1)

Let two circles having centres $O_1,O_{2 }and radii r_1,r_2$ touch internally at point P .

$$\begin{gathered} \therefore Sum of their areas=116\pi c{m}^2 \\ \pi r_1^2+\pi r_2^2=116\pi \\ \Rightarrow r_1^2+r_2^2=116...\left(\mathrm{i}\right) \\ Also,r_1-r_2=6...\left(\mathrm{ii}\right) \\ The values r_1=10 \mathrm{cm} and r_2=4 \mathrm{cm} satisfy both equations\left(i\right)and\left(ii\right). \end{gathered}$$

∴ Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(2)

We have the length of the minute hand = 7 cm
∴ Radius of the sector of the circle = 7 cm 

$$\begin{gathered} and angle made by minute hand in 5 minutes = \theta = 30° \\ \therefore Area swept by the minute hand in 5 minutes \\ =\frac{30°}{360°}\times \pi \times (7{)}^2=\frac{1}{12}\times \frac{22}{7}\times 7\times 7=\frac{77}{6}{ \mathrm{cm}}^2 \\ Also, the length of an arc of a sector of angle \theta and radius r is given by \\ l=2\pi r\times \frac{\theta }{360°} \\ \therefore Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A). \end{gathered}$$

(2)

We have

 $$\begin{gathered} r=6 \mathrm{cm} \mathrm{and} \mathrm{central} \mathrm{angle} \theta =60° \\ \therefore Area of sector \\ =\frac{\theta }{360°}\times \pi r^2=\frac{60°}{360°}\times \frac{22}{7}\times (6{)}^2=\frac{1}{6}\times \frac{22}{7}\times 36 \\ =\frac{132}{7}{ \mathrm{cm}}^2=18\frac{6}{7}{ \mathrm{cm}}^2 \\ Also, circumference of the circle with radius r is 2\pi r \\ Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A). \end{gathered}$$

(3)

Area of minor segmen

t $$\begin{gathered} =\mathrm{Area} \mathrm{of} \mathrm{sector}-\mathrm{Area} \mathrm{of} ∆OAB \\ =\frac{90°}{360°}\times \pi \times (14{)}^2 -\frac{1}{2}\times 14\times 14 \end{gathered}$$

$$\begin{gathered} =\frac{1}{4}\times \frac{22}{7}\times 14\times 14- 7\times 14 \\ =154-98=56{ \mathrm{cm}}^2 \\ \therefore Assertion \left(A\right) is true. \end{gathered}$$

$Angle described by the minute hand in 5 minutes = 5 \times 6° = 30° So Reason \left(R\right) is false.$

(1)

From the figure, 

$$\begin{gathered} \angle DCB+\angle CDA+\angle MAB+\angle ABC=360°\left(\mathrm{Sum} \mathrm{of} \mathrm{angles} \mathrm{of} \mathrm{a} \mathrm{quadrilateral}\right) \\ \Rightarrow 90°+90°+\angle MAB+\angle ABC=360° \\ \Rightarrow \angle MAB+\angle ABC=180° \\ \Rightarrow \angle MAN=180°-\angle ABC=120° \\ \therefore \mathrm{Area} \mathrm{of} \mathrm{the} \mathrm{shaded} \mathrm{region}=\frac{120°}{360°}\times \frac{22}{7}\times 7\times 7=\frac{154}{3}{\mathrm{cm}}^2 \end{gathered}$$

Thus, Assertion (A) and Reason (R) both are true and Reason (R) is correct explanation of Assertion (A).

(1)

Let R and r be the radii of outer and inner circle respectively. 
Diameter of inner circle = 2a

$$\begin{gathered} \therefore r=a \\ \Rightarrow R=\sqrt{a^2+a^2}=\sqrt{2}a \\ Area enclosed between two circles \\ =\mathrm{Area} \mathrm{of} \mathrm{outer} \mathrm{circle}-\mathrm{Area} \mathrm{of} \mathrm{inner} \mathrm{circle} \\ ={\mathrm{\pi R}}^2-{\mathrm{\pi r}}^2=\mathrm{\pi }(\sqrt{2}\mathrm{a}{)}^2-{\mathrm{\pi a}}^2 \\ =2{\mathrm{\pi a}}^2-{\mathrm{\pi a}}^2={\mathrm{\pi a}}^2 \\ =\mathrm{Area} \mathrm{of} \mathrm{smaller} \mathrm{circle} \end{gathered}$$

Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).

(2)

Here, we can say,
Area of shaded region 
=Area of quadrilateral ABCD-Area of smaller semi-circle-Area of larger semi-circle

$$\begin{gathered} Also,if R=r,then only AB\parallel CD \\ But here R=r+2 \\ \therefore AB \mathrm{is} \mathrm{not} \mathrm{parallel} \mathrm{to} CD \end{gathered}$$

Thus, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).