(1) The domain of the function \( f(x)=\frac{1}{\sqrt{x-|x|}} \) is \( \qquad \) (A) \( \mathbb{R} \) (C) \( (-\infty, 0) \) (B) \( (0, \infty) \) (D) \( f \) is not defined for any \( x \in \mathbb{R} \) (2) What is the fourth term of a geometric sequence whose third term is 1 and whose eighth term is \( \frac{1}{32} \) ? (A) 4 (C) \( \frac{1}{8} \) (B) \( \frac{1}{2} \) (D) \( \frac{1}{16} \) (3) In a random arrangement of the word "SOCIETY" what is the total number of words in which all the three vowels come together? (A) 120 (C) 420 (B) 720 (D) 840 (4) Consider the experiment of rolling a die whose sample space is \( \{1,2,3,4,5,6\} \). If two dice are rolled simultaneously, what is the probability that a prime number turns up on one of the dice and a composite number on the other? (A) \( \frac{1}{3} \) (B) \( \frac{2}{3} \) (C) \( \frac{1}{6} \) (D) \( \frac{5}{6} \) (5) A three digited library identification card is to be printed from the numbers \( 0,1,2,3,4 \) and 5 in such a way that the first is non-zero and no number is to be repeated, how many such cards can be printed? (A) 100 (B) 120 (C) 180 (D) 150

Let's solve each problem step by step: --- ### **Problem 1** **Find the domain of the function** \( f(x) = \frac{1}{\sqrt{x - |x|}} \). **Options:** - (A) \( \mathbb{R} \) - (B) \( (0, \infty) \) - (C) \( (-\infty, 0) \) - (D) \( f \) is not defined for any \( x \in \mathbb{R} \) **Solution:** 1. **Condition for the square root to be defined and non-zero:** \[ x - |x| > 0 \] 2. **Consider two cases for \( x \):** - **Case 1:** \( x \geq 0 \) \[ |x| = x \Rightarrow x - x = 0 \not> 0 \] - **Case 2:** \( x < 0 \) \[ |x| = -x \Rightarrow x - (-x) = 2x > 0 \Rightarrow x > 0 \] But \( x < 0 \) contradicts \( x > 0 \). No solution in this case. 3. **Conclusion:** There is **no real number** \( x \) that satisfies the condition. Therefore, the function is **not defined** for any real \( x \). **Answer:** **(D) \( f \) is not defined for any \( x \in \mathbb{R} \)** --- ### **Problem 2** **Find the fourth term of a geometric sequence whose third term is 1 and whose eighth term is** \( \frac{1}{32} \). **Options:** - (A) 4 - (B) \( \frac{1}{2} \) - (C) \( \frac{1}{8} \) - (D) \( \frac{1}{16} \) **Solution:** 1. **Let the first term be** \( a \) **and the common ratio be** \( r \). 2. **Given:** \[ \text{3rd term} = ar^2 = 1 \] \[ \text{8th term} = ar^7 = \frac{1}{32} \] 3. **Divide the two equations to solve for \( r \):** \[ \frac{ar^7}{ar^2} = r^5 = \frac{1}{32} \Rightarrow r^5 = \frac{1}{32} = \left(\frac{1}{2}\right)^5 \Rightarrow r = \frac{1}{2} \] 4. **Find \( a \) using the 3rd term:** \[ ar^2 = 1 \Rightarrow a \left(\frac{1}{2}\right)^2 = 1 \Rightarrow a \cdot \frac{1}{4} = 1 \Rightarrow a = 4 \] 5. **Find the 4th term:** \[ ar^3 = 4 \left(\frac{1}{2}\right)^3 = 4 \times \frac{1}{8} = \frac{1}{2} \] **Answer:** **(B) \( \frac{1}{2} \)** --- ### **Problem 3** **In a random arrangement of the word "SOCIETY", what is the total number of words in which all the three vowels come together?** **Options:** - (A) 120 - (B) 720 - (C) 420 - (D) 840 **Solution:** 1. **Identify vowels and consonants in "SOCIETY":** - **Vowels:** O, I, E (3 vowels) - **Consonants:** S, C, T, Y (4 consonants) 2. **Treat all vowels as a single unit to ensure they come together.** - Total units to arrange: 4 consonants + 1 vowel group = 5 units. 3. **Number of ways to arrange these 5 units:** \[ 5! = 120 \] 4. **Number of ways to arrange the 3 vowels within their group:** \[ 3! = 6 \] 5. **Total number of desired arrangements:** \[ 5! \times 3! = 120 \times 6 = 720 \] **Answer:** **(B) 720** --- ### **Problem 4** **Rolling two dice simultaneously, what is the probability that a prime number turns up on one die and a composite number on the other?** **Options:** - (A) \( \frac{1}{3} \) - (B) \( \frac{2}{3} \) - (C) \( \frac{1}{6} \) - (D) \( \frac{5}{6} \) **Solution:** 1. **Possible outcomes when rolling two dice:** \( 6 \times 6 = 36 \) 2. **Identify prime and composite numbers on a die:** - **Prime numbers:** 2, 3, 5 (3 numbers) - **Composite numbers:** 4, 6 (2 numbers) - **Neither:** 1 (excluded) 3. **Favorable outcomes:** - **First die is prime and second die is composite:** \[ 3 \text{ (primes)} \times 2 \text{ (composites)} = 6 \] - **First die is composite and second die is prime:** \[ 2 \text{ (composites)} \times 3 \text{ (primes)} = 6 \] - **Total favorable outcomes:** \( 6 + 6 = 12 \) 4. **Probability:** \[ \frac{12}{36} = \frac{1}{3} \] **Answer:** **(A) \( \frac{1}{3} \)** --- ### **Problem 5** **A three-digit library identification card is to be printed from the numbers 0,1,2,3,4,5 in such a way that the first digit is non-zero and no number is repeated. How many such cards can be printed?** **Options:** - (A) 100 - (B) 120 - (C) 180 - (D) 150 **Solution:** 1. **Digits available:** 0, 1, 2, 3, 4, 5 (6 digits) 2. **Conditions:** - **First digit (Hundreds place):** Cannot be 0 → 5 choices (1,2,3,4,5) - **Second digit (Tens place):** Any of the remaining 5 digits (including 0) - **Third digit (Units place):** Any of the remaining 4 digits 3. **Total number of possible cards:** \[ 5 \times 5 \times 4 = 100 \] **Answer:** **(A) 100** ---