### Solution:

The radius of the plate with diameter 10 cm is $\n \n \n r\n 1\n \n =\n \n 10\n 2\n \n =5cm\n $ , and with diameter 24 cm is $\n \n \n r\n 2\n \n =\n \n 24\n 2\n \n =12cm\n $ .We know that the area of the circle is $\n \n A=\pi \n r\n 2\n \n \n $ .

Substitute $\n \n \n r\n 1\n \n =\n \n 10\n 2\n \n =5cm\n $ in the above formula, and we have

$\n \n \n A\n 1\n \n =\pi \n \n (5)\n 2\n \n =25\pi c\n m\n 2\n \n \n $

Substitute $\n \n \n r\n 2\n \n =\n \n 24\n 2\n \n =12cm\n $ in the above formula, and we have

$\n \n \n A\n 2\n \n =\pi \n \n (12)\n 2\n \n =144\pi c\n m\n 2\n \n \n $

Add both the areas, and we get

$\n \n \n \n A=(25\pi +144\pi )c\n m\n 2\n \n \n \n \n \n \n =169\pi c\n m\n 2\n \n \n \n \n \n \n $

Now, the area of the bigger plate is $\n \n 169\pi c\n m\n 2\n \n \n $ . To find the radius R, we have

$\n \n \n \n \pi \n R\n 2\n \n =\pi 169\n \n \n \n \n \n R\n 2\n \n =169\n \n \n \n \n R=\n \n 169\n \n \n \n \n \n \n R=13cm\n \n \n \n \n $

Therefore, the diameter of the bigger plate is $\n \n 13\xd72=26cm\n $ .

Hence, the correct option is 3.