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(i) Square number

(ii) A multiple of 7.

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The box contains cards numbered from 11 to 123. So, we first need to calculate the number of cards in the box. To calculate the probability of drawing a square number, we must first calculate the number of square numbers present between 11 to 123 and then divide this by the total number of cards. To calculate the probability of drawing a multiple of 7, we must first calculate the number of cards between 11 to 123 that are multiples of 7 and then divide this by the total number of cards.

We know that there are cards numbered from 11 to 123. Let us begin by calculating the number of cards in the box. The number of cards in the box \[ = 123 - 10 = 113\]. Here, we subtract one less than the number from which the cards are numbered.

(i) Now, we need to determine the number of square numbers present between 11 to 123. The square numbers between 11 to 123 are\[{4^2} = 16,{5^2} = 25,{6^2} = 36,{7^2} = 49,{8^2} = 64,{9^2} = 81,{10^2} = 100,{11^2} = 121.\]

Number of square numbers between 11 to 123 is 8.

Probability of drawing a card that has a square number\[ = \dfrac{8}{{{\rm{total number of cards}}}} = \dfrac{8}{{113}}\]

(ii) The multiples of 7 between 11 to 123 are\[7 \times 2 = 14,7 \times 3 = 21,7 \times 4 = 28,7 \times 5 = 35,...,7 \times 17 = 119\].

Number of multiples of 7 between 11 to 123 is 16.

Probability of drawing a card that is a multiple of 7\[ = \dfrac{{16}}{{{\rm{total number of cards}}}}\]

\[ = \dfrac{{16}}{{113}}\]

To solve this question, you must know how to calculate the probability of drawing cards. You must be able to determine all square numbers in a given range, here it was 11 to 123. You must also be able to determine multiples of a given number in the provided range.