the statement sec __ nt is wrong. Roll No statement be SCHOOL GRADE 11TH FIRST SEMESTER MATHS TEST-2 C3.The simplified domain Prh(x)=metex is correct and false if f frrm of r (2) 2x-15^2, fxin ard false ir (h) if false ir (h __ unction is never crossis horizontal asymptote. 4. The staph of (x^2-6x+9)/(x^2)-x-6div (2x-6)/(x+2)=(1)/(2),xneq -2& xneq 3 ationer from the given 6. The solution set of the rations!equation of (2)/(x+2)-(1)/(x-2)=(-4)/(x^2)-4 8. C. 2 D. 13 tet (f(x))/(Ehoose) Then f has a hole at x x=1 __ The simplified form of the rational expression of 7. Which of the following is the partial fraction of the decomposition of rational expression A. -2,2 B. -2 A. (3)/(x+2)+(2)/(x-1) C. (-2)/(x+2)+(3)/(x-1) A. x-3 __ f(x)=(x^2-x-6)/(x^2)-4 is a rational function.then which one of the following is not true? D. (3)/(x-2)+(2)/(x+1) (5x-4)/(x^2)-x-2 A. The domain of f is Rvert pm 2 B. (2)/(x-2)+(3)/(x+1) (x^3-27)/(x^2)-9times (x+3)/(x^2)+3x+9,pm 3 C. x=2 is V.A and y=1 is H.A __ clean the whole school rooms, if they working together? rooms in 6hours and Genet can do the same job in 12hours How long will it take asxarrow pm infty ,f(x)arrow 1 B. The graph has an x-intercept at x=2 8 x=-3 III. Show the necessary steps clearly and briefly 11. Write the behaviors of the given rational function of A. 2hrs B. 3hrs f(x)=(x-1)/(x+2) and sketch the graph of the function C. 4hrs D. 6hrs

Let's address each part of the provided content:<br /><br />1. **Simplified Rational Expression**:<br /> - The expression given is \(\frac{x^2 - 6x + 9}{x^2 - x - 6} \div \frac{2x - 6}{x + 2}\).<br /> - Simplify it step-by-step:<br /> - Factorize: <br /> - \(x^2 - 6x + 9 = (x-3)^2\)<br /> - \(x^2 - x - 6 = (x-3)(x+2)\)<br /> - \(2x - 6 = 2(x-3)\)<br /> - Rewrite the division as multiplication by the reciprocal:<br /> \[<br /> \frac{(x-3)^2}{(x-3)(x+2)} \times \frac{x+2}{2(x-3)}<br /> \]<br /> - Cancel common factors:<br /> \[<br /> \frac{x-3}{2}<br /> \]<br /> - Therefore, the simplified form is \(\frac{x-3}{2}\), and the condition \(x \neq -2\) & \(x \neq 3\) ensures no division by zero.<br /><br />2. **Solution Set of Rational Equation**:<br /> - Solve \(\frac{2}{x+2} - \frac{1}{x-2} = \frac{-4}{x^2-4}\).<br /> - Recognize that \(x^2 - 4 = (x+2)(x-2)\).<br /> - Combine terms over a common denominator:<br /> \[<br /> \frac{2(x-2) - 1(x+2)}{(x+2)(x-2)} = \frac{-4}{(x+2)(x-2)}<br /> \]<br /> \[<br /> \frac{2x - 4 - x - 2}{(x+2)(x-2)} = \frac{-4}{(x+2)(x-2)}<br /> \]<br /> \[<br /> \frac{x - 6}{(x+2)(x-2)} = \frac{-4}{(x+2)(x-2)}<br /> \]<br /> - Equate numerators since denominators are equal:<br /> \[<br /> x - 6 = -4<br /> \]<br /> \[<br /> x = 2<br /> \]<br /> - Check if \(x = 2\) is valid in the original equation. Since \(x = 2\) makes the denominator zero, it is not in the domain.<br /> - Thus, the solution set is empty.<br /><br />3. **Partial Fraction Decomposition**:<br /> - For the expression \(\frac{5x-4}{x^2-x-2}\), factor the denominator:<br /> - \(x^2 - x - 2 = (x-2)(x+1)\).<br /> - The partial fraction decomposition is:<br /> \[<br /> \frac{A}{x-2} + \frac{B}{x+1}<br /> \]<br /> - Solving for \(A\) and \(B\):<br /> \[<br /> 5x - 4 = A(x+1) + B(x-2)<br /> \]<br /> - Expand and equate coefficients to solve for \(A\) and \(B\).<br /><br />4. **Work Problem**:<br /> - Eden's rate: \(\frac{1}{6}\) rooms per hour.<br /> - Genet's rate: \(\frac{1}{12}\) rooms per hour.<br /> - Combined rate: \(\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}\).<br /> - Time taken together: \(\frac{1}{\frac{1}{4}} = 4\) hours.<br /><br />5. **Behavior of Rational Function**:<br /> - For \(f(x) = \frac{x-1}{x+2}\):<br /> - Vertical asymptote at \(x = -2\).<br /> - Horizontal asymptote as \(x \to \pm \infty\) is \(y = 1\).<br /> - X-intercept at \(x = 1\).<br /><br />These solutions cover the mathematical problems presented in your text. If you have any specific questions or need further clarification on any part, feel free to ask!