Since all elements in S are distinct prime numbers, an element x = p_i S divides y = _p T p if and only if p_i is one of the prime factors of y, which means p_i T. Therefore, for a fixed prime x = p_i, the number of valid elements y A such that x divides y equals the number of non-empty subsets T S that contain p_i.

Since all elements in S are distinct prime numbers, an element x = p_i S divides y = _p T p if and only if p_i is one of the prime factors of y, which means p_i T. Therefore, for a fixed prime x = p_i, the number of valid elements y A such that x divides y equals the number of non-empty subsets T S that contain p_i.

1. Let $S = \{p_1, p_2, \dots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$ , where $P$ is the set of all possible products of distinct elements of $S$ . Then the number of all ordered pairs (x, y) in $S \times Y$ in $A$ such that $x$ divides $y$ is 2. Let y(x) be the solution of the differential equation $2\cos(x) \frac{dy}{dx} - \sin(2x)y - \sin(x) = x$ in $\left(0, \frac{\pi}{2}\right)$ . If $y\left(\frac{\pi}{3}\right) = 0$ , then $y\left(\frac{\pi}{4}\right) \cdot y'\left(\frac{\pi}{4}\right)$ is equal to 3. Find the number of different signals that can be generated by arranging at least 2 flags in order one below the other on a vertical staff, if five different flags are available. 4. The marks distribution of 30 students in a mathematics examination are given in the table below. Find the mode of this data: $\begin{array}{|l|c|c|} \hline \text{Class interval} & \text{Number of students } (f_i) & \text{Class mark } (x_i) \hline 10\text{--}25 & 2 & 17.5 25\text{--}40 & 3 & 32.5 40\text{--}55 & 7 & 47.5 55\text{--}70 & 6 & 62.5 70\text{--}85 & 6 & 77.5 85\text{--}100 & 6 & 92.5 \hline \text{Total} & 30 & \hline \end{array}$ 5. (a) The product of the digits of a 2-digit number is 18. When 27 is subtracted from the number, the digits interchange their places. Find the number. OR (b) Two numbers are in the ratio $5:6$ . If 8 is subtracted from each of the numbers, the ratio becomes $4:5$ . Find the numbers. 6. Let $A = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} \mid \alpha - 1 \le 4 \text{ and } \beta - 5 \le 6\}$ and $B = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} \mid 16(\alpha - 2)^2 - 9(\beta - 6)^2 \le 144\}$ , then

Answer: 5120

Step-by-step solution

Step 1: Define the sets $S$ , $P$ , and $A$

Here, $S$ contains 10 distinct prime numbers. The set $P$ consists of all products of 2 or more distinct elements of $S$ . Since the single-element products correspond to the elements of $S$ itself, the union $A = S \cup P$ represents the products of all non-empty subsets $T \subseteq S$ . Thus, each element $y \in A$ uniquely corresponds to a non-empty subset $T \subseteq S$ such that $y = \prod_{p \in T} p$ .

Step 2: Establish the divisibility condition for pairs (x, y)

Since all elements in $S$ are distinct prime numbers, an element $x = p_i \in S$ divides $y = \prod_{p \in T} p$ if and only if $p_i$ is one of the prime factors of $y$ , which means $p_i \in T$ . Therefore, for a fixed prime $x = p_i$ , the number of valid elements $y \in A$ such that $x$ divides $y$ equals the number of non-empty subsets $T \subseteq S$ that contain $p_i$ .

Step 3: Count the number of multiples for each prime element

To form a subset $T \subseteq S$ containing a fixed prime $p_i$ , the element $p_i$ must be included, and for each of the remaining 9 primes in $S \setminus \{p_i\}$ , we have 2 choices: either include it or exclude it. This gives $2^9 = 512$ non-empty subsets containing $p_i$ . Hence, for each $x \in S$ , there are exactly 512 choices of $y \in A$ such that $x$ divides $y$ .

Step 4: Calculate the total number of ordered pairs

Since there are 10 distinct primes in $S$ , and each prime $x$ divides exactly 512 elements in $A$ , the total number of ordered pairs (x, y) such that $x \in S$ , $y \in A$ , and $x$ divides $y$ is $10 \times 512 = 5120$ .

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Step-by-step solution

Step 1: Define the sets SSS, PPP, and AAA

Step 2: Establish the divisibility condition for pairs (x, y)

Step 3: Count the number of multiples for each prime element

Step 4: Calculate the total number of ordered pairs

Step 1: Define the sets $S$ , $P$ , and $A$