Rational functions and asymptotes Test

{"questions":[{"content":"For f(x) = (x + 2)/(x − 4), identify the asymptotes.<br>[[choice-rat-q1]]","widgets":{"choice-rat-q1":{"type":"choice","options":["x = 4 and y = 1","x = −2 and y = 4","x = 1 and y = 4","x = 4 only"],"answer":[0],"explanations":["[M1] x − 4 = 0. [A1] Equal degrees give y = 1.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Use the denominator zero and leading coefficient ratio."],"explanation":"[M1] x − 4 = 0. [A1] Equal degrees give y = 1.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"The hole in (x² − 9)/(x − 3) occurs at x = [[input-rat-q2]]","widgets":{"input-rat-q2":{"type":"input","answer":"3"}},"hints":["Factor the difference of squares."],"explanation":"[M1] Cancel x − 3 while retaining the original restriction. [A1] x = 3.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"Divide (x² + 1)/(x − 1). Which form is correct?<br>[[choice-rat-q3]]","widgets":{"choice-rat-q3":{"type":"choice","options":["x + 1 + 2/(x − 1)","x − 1","x + 2/(x − 1)","1 + x/(x − 1)"],"answer":[0],"explanations":["[M1] Polynomial division. [A1] Quotient x + 1, remainder 2.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Multiply (x − 1)(x + 1)."],"explanation":"[M1] Polynomial division. [A1] Quotient x + 1, remainder 2.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"A GDC graph of 1/(x − 100) looks almost flat near the origin. Which response is best?<br>[[choice-rat-q4]]","widgets":{"choice-rat-q4":{"type":"choice","options":["Change the window and retain the algebraic asymptote x = 100","Conclude there is no asymptote","Replace 100 by 0","Use regression"],"answer":[0],"explanations":["[R1] A display window cannot override the algebra; choose a window containing x = 100.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["The important feature lies outside the current window."],"explanation":"[R1] A display window cannot override the algebra; choose a window containing x = 100.","assessment":{"paper_mode":"paper_2","calculator_mode":"recommended","gdc_skill":"graph","marks":["R1"],"follow_through":false}},{"content":"For f(x) = 3 + 2/(x + 1), which point is the center of the reciprocal transformation?<br>[[choice-rat-q5]]","widgets":{"choice-rat-q5":{"type":"choice","options":["(−1, 3)","(1, 3)","(−1, 2)","(3, −1)"],"answer":[0],"explanations":["[M1] Find x = −1 and y = 3. [A1] Their intersection is (−1, 3).","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Use the vertical and horizontal asymptotes."],"explanation":"[M1] Find x = −1 and y = 3. [A1] Their intersection is (−1, 3).","assessment":{"paper_mode":"paper_2","calculator_mode":"optional","gdc_skill":"graph","marks":["M1","A1"],"follow_through":false}}],"mix":1}