(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

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Let's solve each problem step by step: --- **1. Domain of $$ f(x) = \frac{1}{\sqrt{x - |x|}} $$** To find the domain, the expression inside the square root must be positive: $$ x - |x| > 0 $$ Consider two cases: - **Case 1:** $$ x \geq 0 $$ $$ x - |x| = x - x = 0 \quad (\text{Not greater than } 0) $$ No solutions in this case. - **Case 2:** $$ x < 0 $$ $$ x - |x| = x - (-x) = 2x > 0 \implies x > 0 $$ This contradicts $$ x < 0 $$, so no solutions here either. **Conclusion:** There is no real number $$ x $$ for which the function is defined. **Answer:** **(D) $$ f $$ is not defined for any $$ x \in \mathbb{R} $$** --- **2. Fourth Term of the Geometric Sequence** Given: $$ t_3 = 1 \quad \text{and} \quad t_8 = \frac{1}{32} $$ For a geometric sequence: $$ t_n = t_1 \cdot r^{n-1} $$ Thus: $$ t_3 = t_1 \cdot r^2 = 1 \quad \text{and} \quad t_8 = t_1 \cdot r^7 = \frac{1}{32} $$ Divide the two equations: $$ \frac{t_8}{t_3} = r^5 = \frac{1}{32} \implies r = \frac{1}{2} $$ Find $$ t_1 $$: $$ t_1 \cdot \left(\frac{1}{2}\right)^2 = 1 \implies t_1 = 4 $$ The fourth term: $$ t_4 = t_1 \cdot r^3 = 4 \cdot \left(\frac{1}{2}\right)^3 = \frac{1}{2} $$ **Answer:** **(B) $$ \frac{1}{2} $$** --- **3. Arrangements of "SOCIETY" with All Vowels Together** The word "SOCIETY" has 7 letters: S, O, C, I, E, T, Y. The vowels are O, I, E. - Treat the vowels as a single unit: [OIE] - Total units to arrange: [OIE], S, C, T, Y → 5 units - Arrangements of these units: $$ 5! = 120 $$ - Arrangements of vowels within the unit: $$ 3! = 6 $$ - Total arrangements: $$ 120 \times 6 = 720 $$ **Answer:** **(B) 720** --- **4. Probability with Two Dice** Numbers on a die: 1, 2, 3, 4, 5, 6 - **Prime numbers:** 2, 3, 5 (3 numbers) - **Composite numbers:** 4, 6 (2 numbers) - **Neither:** 1 We want one die to show a prime number and the other to show a composite number. - **Possible favorable outcomes:** - Die 1: Prime (3 choices) × Die 2: Composite (2 choices) = $$ 3 \times 2 = 6 $$ - Die 1: Composite (2 choices) × Die 2: Prime (3 choices) = $$ 2 \times 3 = 6 $$ - **Total favorable outcomes:** $$ 6 + 6 = 12 $$ - **Total possible outcomes:** $$ 6 \times 6 = 36 $$ - **Probability:** $$ \frac{12}{36} = \frac{1}{3} $$ **Answer:** **(A) $$ \frac{1}{3} $$** --- **5. Number of Possible Three-Digit Library Cards** Digits available: 0, 1, 2, 3, 4, 5 Conditions: - The first digit cannot be 0. - No digit is repeated. - **First digit:** 5 choices (1, 2, 3, 4, 5) - **Second digit:** 5 choices (remaining 5 digits, including 0) - **Third digit:** 4 choices (remaining 4 digits) - **Total possible cards:** $$ 5 \times 5 \times 4 = 100 $$ **Answer:** **(A) 100** --- **Summary of Answers:** 1. **D** 2. **B** 3. **B** 4. **A** 5. **A** **Answers:** 1. D 2. B 3. B 4. A 5. A

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