Binomial expansions and their validity Test

{"questions":[{"content":"Enter the coefficient of $x^3$ in $(1+x)^2(1+2x)$. [[input-binomial-expansions-test-1]]","widgets":{"input-binomial-expansions-test-1":{"type":"input","answer":"2"}},"hints":[],"explanation":"Only x² from the first factor and 2x from the second contribute to x³; their product is 2x³.","id":"binomial-expansions-test-1","coverage_topics":["aa-hl-binomial-expansion"],"correct_feedback":"Only x² from the first factor and 2x from the second contribute to x³; their product is 2x³.","incorrect_feedback":"Find every pairing of powers whose exponents add to 3.","evidence_kind":"objective-concept-test"},{"content":"Enter the coefficient of x² in $(3-2x)^4$. [[input-binomial-expansions-test-2]]","widgets":{"input-binomial-expansions-test-2":{"type":"input","answer":"216"}},"hints":[],"explanation":"C(4,2)3²(−2)² = 6 × 9 × 4.","id":"binomial-expansions-test-2","coverage_topics":["aa-hl-binomial-expansion"],"correct_feedback":"C(4,2)3²(−2)² = 6 × 9 × 4.","incorrect_feedback":"C(4,2)3²(−2)² = 6 × 9 × 4.","evidence_kind":"objective-concept-test"},{"content":"Enter the constant term in $(x+2/x)^4$ for x nonzero. [[input-binomial-expansions-test-3]]","widgets":{"input-binomial-expansions-test-3":{"type":"input","answer":"24"}},"hints":[],"explanation":"The general power is x^(4−2r), so r = 2; C(4,2)2² = 24.","id":"binomial-expansions-test-3","coverage_topics":["aa-hl-binomial-expansion"],"correct_feedback":"The general power is x^(4−2r), so r = 2; C(4,2)2² = 24.","incorrect_feedback":"The general power is x^(4−2r), so r = 2; C(4,2)2² = 24.","evidence_kind":"objective-concept-test"},{"content":"The coefficient of x in $(1+kx)^5$ is 30. Enter k. [[input-binomial-expansions-test-4]]","widgets":{"input-binomial-expansions-test-4":{"type":"input","answer":"6"}},"hints":[],"explanation":"The linear coefficient is 5k.","id":"binomial-expansions-test-4","coverage_topics":["aa-hl-binomial-expansion"],"correct_feedback":"The linear coefficient is 5k.","incorrect_feedback":"The linear coefficient is 5k.","evidence_kind":"objective-concept-test"},{"content":"Enter the coefficient of x² in $(1-3x)^{-2}$. [[input-binomial-expansions-test-5]]","widgets":{"input-binomial-expansions-test-5":{"type":"input","answer":"27"}},"hints":[],"explanation":"The generalized coefficient is 3, multiplied by (−3)².","id":"binomial-expansions-test-5","coverage_topics":["aa-hl-generalized-binomial-expansion"],"correct_feedback":"The generalized coefficient is 3, multiplied by (−3)².","incorrect_feedback":"The generalized coefficient is 3, multiplied by (−3)².","evidence_kind":"objective-concept-test"},{"content":"Which are the first three terms of $(1+x)^{1/3}$?\u003Cbr\u003E[[choice-binomial-expansions-test-6]]","widgets":{"choice-binomial-expansions-test-6":{"type":"choice","options":["$1+x/3-2x^2/9$","$1+x/3-x^2/9$","$1+x/3+x^2/9$"],"answer":[1],"choice_feedback":["The quadratic product must be divided by 2!.","The quadratic coefficient is (1/3)(−2/3)/2 = −1/9.","The factor α − 1 is negative."]}},"hints":[],"explanation":"The quadratic coefficient is (1/3)(−2/3)/2 = −1/9.","id":"binomial-expansions-test-6","coverage_topics":["aa-hl-generalized-binomial-expansion"],"correct_feedback":"The quadratic coefficient is (1/3)(−2/3)/2 = −1/9.","incorrect_feedback":"The quadratic coefficient is (1/3)(−2/3)/2 = −1/9.","evidence_kind":"objective-concept-test"},{"content":"State the open validity interval for expanding $(4+x)^{-1}$ about x=0.\u003Cbr\u003E[[choice-binomial-expansions-test-7]]","widgets":{"choice-binomial-expansions-test-7":{"type":"choice","options":["$-4\\lt x\\lt 4$","$-1\\lt x\\lt 1$","$x\\ne-4$"],"answer":[0],"choice_feedback":["Factor out 4, leaving input x/4; require |x/4| \u0026lt; 1.","That unnecessarily treats x itself as the series input.","That is the function domain, not the convergence interval."]}},"hints":[],"explanation":"Factor out 4, leaving input x/4; require |x/4| \u0026lt; 1.","id":"binomial-expansions-test-7","coverage_topics":["aa-hl-generalized-binomial-expansion"],"correct_feedback":"Factor out 4, leaving input x/4; require |x/4| \u0026lt; 1.","incorrect_feedback":"Factor out 4, leaving input x/4; require |x/4| \u0026lt; 1.","evidence_kind":"objective-concept-test"},{"content":"For small positive x, $\\sqrt{1+x}\\approx1+x/2-x^2/8$. Compared with the approximation $1+x/2$, what does the quadratic correction tell us?\u003Cbr\u003E[[choice-binomial-expansions-test-8]]","widgets":{"choice-binomial-expansions-test-8":{"type":"choice","options":["The linear approximation is too small","The two approximations are equal for every valid x","The linear approximation is too large near x = 0"],"answer":[2],"choice_feedback":["The quadratic term is subtracted, so it moves the estimate down.","The quadratic correction is nonzero whenever x is nonzero.","The next nonzero correction is negative. For sufficiently small positive x it lowers the estimate; later terms are smaller-order corrections."]}},"hints":[],"explanation":"The next nonzero correction is negative. For sufficiently small positive x it lowers the estimate; later terms are smaller-order corrections.","id":"binomial-expansions-test-8","coverage_topics":["aa-hl-generalized-binomial-expansion"],"correct_feedback":"The next nonzero correction is negative. For sufficiently small positive x it lowers the estimate; later terms are smaller-order corrections.","incorrect_feedback":"The next nonzero correction is negative. For sufficiently small positive x it lowers the estimate; later terms are smaller-order corrections.","evidence_kind":"objective-concept-test"}],"mix":0,"tinkumo_module_mode":"test"}