1. **Problem:** Which of the following numbers is not a real number? Options: (a) $\frac{22}{7}$, (b) $\sqrt{25}$, (c) $\sqrt{-25}$, (d) 25 Step 1: Recall that the square root of a negative number is not a real number. Step 2: $\sqrt{-25}$ involves the square root of a negative number, which is imaginary. Step 3: All other options are real numbers. **Answer:** (c) $\sqrt{-25}$ is not a real number. 2. **Problem:** If $x < \sqrt{51} < x + 1$, where $x \in \mathbb{Z}$, find $2x$. Step 1: Calculate $\sqrt{51} \approx 7.1414$. Step 2: Find integer $x$ such that $x < 7.1414 < x + 1$. Step 3: $x = 7$ satisfies $7 < 7.1414 < 8$. Step 4: Calculate $2x = 2 \times 7 = 14$. **Answer:** (d) 14 3. **Problem:** If $a \sqrt{3} + 3 \sqrt{3} = 9 \sqrt{3}$, find $a$. Step 1: Factor out $\sqrt{3}$: $(a + 3) \sqrt{3} = 9 \sqrt{3}$. Step 2: Divide both sides by $\sqrt{3}$: $a + 3 = 9$. Step 3: Solve for $a$: $a = 9 - 3 = 6$. **Answer:** (b) 6 4. **Problem:** Calculate $\left(\frac{\sqrt{3}}{\sqrt{3}}\right)^6$. Step 1: Simplify inside the parentheses: $\frac{\sqrt{3}}{\sqrt{3}} = 1$. Step 2: Raise to the 6th power: $1^6 = 1$. Step 3: Among options, closest is 3 (c), but actual value is 1, which is not listed. Step 4: Re-examine options: (a) $\sqrt{3}$, (b) $3 \sqrt{3}$, (c) 3, (d) 9. Step 5: Since $1$ is not listed, the expression equals 1, which is not among options; possibly a trick or typo. **Answer:** None of the options exactly equal 1; the expression equals 1. 5. **Problem:** Find the interval expressed by $]1,4[ \cup \{1\}$. Step 1: $]1,4[$ means the open interval $(1,4)$ excluding 1 and 4. Step 2: $\{1\}$ is the singleton set containing 1. Step 3: Union adds 1 to the open interval, making it $[1,4[$. **Answer:** (d) $[1,4[$ 6. **Problem:** Express $6x^2 + 18y$ as $6x(\ldots)$. Step 1: Factor out $6x$ from $6x^2 + 18y$. Step 2: $6x^2 = 6x \times x$. Step 3: $18y = 6x \times 3y$ only if $x$ divides $18y$, but $x$ is variable. Step 4: Since $18y$ does not contain $x$, factoring $6x$ is not straightforward. Step 5: Possibly a typo; if $18y$ is $18xy$, then $6x^2 + 18xy = 6x(x + 3y)$. Step 6: Assuming typo, answer is (c) $x + 3y$. **Answer:** (c) $x + 3y$ 7. **Problem:** Find $N \cap [1,4]$ where $N$ is natural numbers. Step 1: Natural numbers $N = \{1,2,3,4,5,\ldots\}$. Step 2: Intersection with $[1,4]$ includes natural numbers from 1 to 4. Step 3: So, $N \cap [1,4] = \{1,2,3,4\}$. **Answer:** (b) $\{1,2,3,4\}$ 8. **Problem:** If $\frac{x}{\sqrt{2}} = \frac{y}{\sqrt{5}} = 1$, find $x^2 + y^2$. Step 1: From $\frac{x}{\sqrt{2}} = 1$, multiply both sides by $\sqrt{2}$: $x = \sqrt{2}$. Step 2: From $\frac{y}{\sqrt{5}} = 1$, multiply both sides by $\sqrt{5}$: $y = \sqrt{5}$. Step 3: Calculate $x^2 + y^2 = (\sqrt{2})^2 + (\sqrt{5})^2 = 2 + 5 = 7$. **Answer:** (a) 7 9. **Problem:** Find the solution set of $x^2 + 25x = 0$ in $\mathbb{R}$. Step 1: Factor the equation: $x(x + 25) = 0$. Step 2: Set each factor to zero: $x = 0$ or $x + 25 = 0 \Rightarrow x = -25$. Step 3: Solution set is $\{0, -25\}$. **Answer:** (c) $\{0, -25\}$