CBSE 10th Maths Question Paper 2023 (Set 3)

A number ends with zero if and only if its prime factorization contains the pair $(2 \times 5)$. Choice (B) is $(2 \times 5)^n = 10^n$, which always ends in zero for any natural number $n$. Other choices do not contain both 2 and 5 as factors.

Total number of outcomes = Number of prizes + Number of blanks = $5 + 20 = 25$. Number of favorable outcomes = 5. Probability = $\frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{5}{25} = \frac{1}{5}$. Wait, checking the options provided: $5/25 = 1/5$. Since 1/5 is option D, let's re-read. Oh, the standard probability is $5/25 = 1/5$. Given the choices provided, D is the mathematical result. Checking choice A: $1/4$ would imply $5/20$, which is incorrect. The probability is $5/(5+20) = 5/25 = 1/5$. Answer D is correct.

Given equations: (1) $2x + 3y = 15$ and (2) $3x + 2y = 25$. Subtracting (2) from (1): $(2x - 3x) + (3y - 2y) = 15 - 25 \implies -x + y = -10$. Multiplying by -1, we get $x - y = 10$.

In $\triangle ABP$ and $\triangle CDP$, $\angle A = \angle C$ and $\angle APB = \angle CPD$ (vertically opposite). By AA similarity, $\triangle ABP \sim \triangle CDP$. Therefore, $\frac{AB}{CD} = \frac{AP}{CP} \implies \frac{6}{CD} = \frac{12}{4} \implies \frac{6}{CD} = 3 \implies CD = \frac{6}{3} = 2$ cm. Wait, calculating again: $CD = (6 \times 4) / 12 = 24 / 12 = 2$. Option A is correct.

From similarity $\triangle ABP \sim \triangle CDP$, we have $\frac{AB}{CD} = \frac{AP}{CP}$. Substituting the values: $\frac{6}{CD} = \frac{12}{4} \implies \frac{6}{CD} = 3 \implies CD = 2$ cm.