33:I[196793,["/_next/static/chunks/3j31_2qapmh60.js?dpl=nogit-1792405568","/_next/static/chunks/313vt9_f08mnn.js?dpl=nogit-1792405568","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792405568","/_next/static/chunks/09bcgz1lr6dab.js?dpl=nogit-1792405568","/_next/static/chunks/0tzbwbomzrnqw.js?dpl=nogit-1792405568","/_next/static/chunks/14trqrdc9kqx-.js?dpl=nogit-1792405568","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792405568","/_next/static/chunks/0zsno9tz6vkpw.js?dpl=nogit-1792405568","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792405568","/_next/static/chunks/11md10s8bxc41.js?dpl=nogit-1792405568","/_next/static/chunks/0oqj6s0nlv2xz.js?dpl=nogit-1792405568","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792405568","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792405568","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792405568","/_next/static/chunks/3z1ylg82-_yrl.js?dpl=nogit-1792405568","/_next/static/chunks/230359bhbwn7h.js?dpl=nogit-1792405568","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792405568","/_next/static/chunks/2_v9g5x0fx29m.js?dpl=nogit-1792405568","/_next/static/chunks/0r0qwy8stn_uo.js?dpl=nogit-1792405568","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792405568","/_next/static/chunks/10j6l_dhgof6p.js?dpl=nogit-1792405568","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792405568","/_next/static/chunks/3fr9hkx77wvja.js?dpl=nogit-1792405568","/_next/static/chunks/3adfywl6-8_k3.js?dpl=nogit-1792405568","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792405568","/_next/static/chunks/1vqil3_m7bft1.js?dpl=nogit-1792405568","/_next/static/chunks/40cqw2xvsic28.js?dpl=nogit-1792405568","/_next/static/chunks/3j5g1ks7ljj92.js?dpl=nogit-1792405568","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792405568","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792405568","/_next/static/chunks/2p5xv3ji4tpih.js?dpl=nogit-1792405568","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792405568","/_next/static/chunks/2a2ryz-j6u084.js?dpl=nogit-1792405568","/_next/static/chunks/2o9c7zxab747y.js?dpl=nogit-1792405568","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792405568","/_next/static/chunks/2zjfedap7f8sc.js?dpl=nogit-1792405568","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1792405568","/_next/static/chunks/3nnmps7-hhzc_.js?dpl=nogit-1792405568","/_next/static/chunks/0wg6314tous77.js?dpl=nogit-1792405568","/_next/static/chunks/1u3eptdza7lch.js?dpl=nogit-1792405568","/_next/static/chunks/2mo1yi9s3sppb.js?dpl=nogit-1792405568","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792405568","/_next/static/chunks/0_95ok1we81_2.js?dpl=nogit-1792405568","/_next/static/chunks/1gz0d2bu8bcyn.js?dpl=nogit-1792405568","/_next/static/chunks/00k10pkv-tsak.js?dpl=nogit-1792405568","/_next/static/chunks/12rdo7yi_xy67.js?dpl=nogit-1792405568","/_next/static/chunks/29add9pj_xa2g.js?dpl=nogit-1792405568","/_next/static/chunks/2xx3y5ow--aem.js?dpl=nogit-1792405568","/_next/static/chunks/42ur87elx-dz3.js?dpl=nogit-1792405568","/_next/static/chunks/3ibs-m0_upaf_.js?dpl=nogit-1792405568","/_next/static/chunks/3nfomzjk3zoe-.js?dpl=nogit-1792405568","/_next/static/chunks/0horbp43y5c_k.js?dpl=nogit-1792405568","/_next/static/chunks/3ttet-2qytkke.js?dpl=nogit-1792405568","/_next/static/chunks/2-1ciy1x4js7n.js?dpl=nogit-1792405568","/_next/static/chunks/32s6q_rspd3hd.js?dpl=nogit-1792405568","/_next/static/chunks/0g1nv87sz_mxq.js?dpl=nogit-1792405568","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792405568","/_next/static/chunks/1vjwc_dfj8v1b.js?dpl=nogit-1792405568","/_next/static/chunks/30u3fsn115_1j.js?dpl=nogit-1792405568","/_next/static/chunks/2wrktd_v4ahrw.js?dpl=nogit-1792405568","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792405568"],"Gingerbread"] 35:I[647935,["/_next/static/chunks/3j31_2qapmh60.js?dpl=nogit-1792405568","/_next/static/chunks/313vt9_f08mnn.js?dpl=nogit-1792405568","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792405568","/_next/static/chunks/09bcgz1lr6dab.js?dpl=nogit-1792405568","/_next/static/chunks/0tzbwbomzrnqw.js?dpl=nogit-1792405568","/_next/static/chunks/14trqrdc9kqx-.js?dpl=nogit-1792405568","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792405568","/_next/static/chunks/0zsno9tz6vkpw.js?dpl=nogit-1792405568","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792405568","/_next/static/chunks/11md10s8bxc41.js?dpl=nogit-1792405568","/_next/static/chunks/0oqj6s0nlv2xz.js?dpl=nogit-1792405568","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792405568","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792405568","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792405568","/_next/static/chunks/3z1ylg82-_yrl.js?dpl=nogit-1792405568","/_next/static/chunks/230359bhbwn7h.js?dpl=nogit-1792405568","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792405568","/_next/static/chunks/2_v9g5x0fx29m.js?dpl=nogit-1792405568","/_next/static/chunks/0r0qwy8stn_uo.js?dpl=nogit-1792405568","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792405568","/_next/static/chunks/10j6l_dhgof6p.js?dpl=nogit-1792405568","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792405568","/_next/static/chunks/3fr9hkx77wvja.js?dpl=nogit-1792405568","/_next/static/chunks/3adfywl6-8_k3.js?dpl=nogit-1792405568","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792405568","/_next/static/chunks/1vqil3_m7bft1.js?dpl=nogit-1792405568","/_next/static/chunks/40cqw2xvsic28.js?dpl=nogit-1792405568","/_next/static/chunks/3j5g1ks7ljj92.js?dpl=nogit-1792405568","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792405568","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792405568","/_next/static/chunks/2p5xv3ji4tpih.js?dpl=nogit-1792405568","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792405568","/_next/static/chunks/2a2ryz-j6u084.js?dpl=nogit-1792405568","/_next/static/chunks/2o9c7zxab747y.js?dpl=nogit-1792405568","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792405568","/_next/static/chunks/2zjfedap7f8sc.js?dpl=nogit-1792405568","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1792405568","/_next/static/chunks/3nnmps7-hhzc_.js?dpl=nogit-1792405568","/_next/static/chunks/0wg6314tous77.js?dpl=nogit-1792405568","/_next/static/chunks/1u3eptdza7lch.js?dpl=nogit-1792405568","/_next/static/chunks/2mo1yi9s3sppb.js?dpl=nogit-1792405568","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792405568","/_next/static/chunks/0_95ok1we81_2.js?dpl=nogit-1792405568","/_next/static/chunks/1gz0d2bu8bcyn.js?dpl=nogit-1792405568","/_next/static/chunks/00k10pkv-tsak.js?dpl=nogit-1792405568","/_next/static/chunks/12rdo7yi_xy67.js?dpl=nogit-1792405568","/_next/static/chunks/29add9pj_xa2g.js?dpl=nogit-1792405568","/_next/static/chunks/2xx3y5ow--aem.js?dpl=nogit-1792405568","/_next/static/chunks/42ur87elx-dz3.js?dpl=nogit-1792405568","/_next/static/chunks/3ibs-m0_upaf_.js?dpl=nogit-1792405568","/_next/static/chunks/3nfomzjk3zoe-.js?dpl=nogit-1792405568","/_next/static/chunks/0horbp43y5c_k.js?dpl=nogit-1792405568","/_next/static/chunks/3ttet-2qytkke.js?dpl=nogit-1792405568","/_next/static/chunks/2-1ciy1x4js7n.js?dpl=nogit-1792405568","/_next/static/chunks/32s6q_rspd3hd.js?dpl=nogit-1792405568","/_next/static/chunks/0g1nv87sz_mxq.js?dpl=nogit-1792405568","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792405568","/_next/static/chunks/1vjwc_dfj8v1b.js?dpl=nogit-1792405568","/_next/static/chunks/30u3fsn115_1j.js?dpl=nogit-1792405568","/_next/static/chunks/2wrktd_v4ahrw.js?dpl=nogit-1792405568","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792405568"],"RecordResourceResume"] 36:I[473028,["/_next/static/chunks/3j31_2qapmh60.js?dpl=nogit-1792405568","/_next/static/chunks/313vt9_f08mnn.js?dpl=nogit-1792405568","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792405568","/_next/static/chunks/09bcgz1lr6dab.js?dpl=nogit-1792405568","/_next/static/chunks/0tzbwbomzrnqw.js?dpl=nogit-1792405568","/_next/static/chunks/14trqrdc9kqx-.js?dpl=nogit-1792405568","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792405568","/_next/static/chunks/0zsno9tz6vkpw.js?dpl=nogit-1792405568","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792405568","/_next/static/chunks/11md10s8bxc41.js?dpl=nogit-1792405568","/_next/static/chunks/0oqj6s0nlv2xz.js?dpl=nogit-1792405568","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792405568","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792405568","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792405568","/_next/static/chunks/3z1ylg82-_yrl.js?dpl=nogit-1792405568","/_next/static/chunks/230359bhbwn7h.js?dpl=nogit-1792405568","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792405568","/_next/static/chunks/2_v9g5x0fx29m.js?dpl=nogit-1792405568","/_next/static/chunks/0r0qwy8stn_uo.js?dpl=nogit-1792405568","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792405568","/_next/static/chunks/10j6l_dhgof6p.js?dpl=nogit-1792405568","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792405568","/_next/static/chunks/3fr9hkx77wvja.js?dpl=nogit-1792405568","/_next/static/chunks/3adfywl6-8_k3.js?dpl=nogit-1792405568","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792405568","/_next/static/chunks/1vqil3_m7bft1.js?dpl=nogit-1792405568","/_next/static/chunks/40cqw2xvsic28.js?dpl=nogit-1792405568","/_next/static/chunks/3j5g1ks7ljj92.js?dpl=nogit-1792405568","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792405568","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792405568","/_next/static/chunks/2p5xv3ji4tpih.js?dpl=nogit-1792405568","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792405568","/_next/static/chunks/2a2ryz-j6u084.js?dpl=nogit-1792405568","/_next/static/chunks/2o9c7zxab747y.js?dpl=nogit-1792405568","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792405568","/_next/static/chunks/2zjfedap7f8sc.js?dpl=nogit-1792405568","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1792405568","/_next/static/chunks/3nnmps7-hhzc_.js?dpl=nogit-1792405568","/_next/static/chunks/0wg6314tous77.js?dpl=nogit-1792405568","/_next/static/chunks/1u3eptdza7lch.js?dpl=nogit-1792405568","/_next/static/chunks/2mo1yi9s3sppb.js?dpl=nogit-1792405568","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792405568","/_next/static/chunks/0_95ok1we81_2.js?dpl=nogit-1792405568","/_next/static/chunks/1gz0d2bu8bcyn.js?dpl=nogit-1792405568","/_next/static/chunks/00k10pkv-tsak.js?dpl=nogit-1792405568","/_next/static/chunks/12rdo7yi_xy67.js?dpl=nogit-1792405568","/_next/static/chunks/29add9pj_xa2g.js?dpl=nogit-1792405568","/_next/static/chunks/2xx3y5ow--aem.js?dpl=nogit-1792405568","/_next/static/chunks/42ur87elx-dz3.js?dpl=nogit-1792405568","/_next/static/chunks/3ibs-m0_upaf_.js?dpl=nogit-1792405568","/_next/static/chunks/3nfomzjk3zoe-.js?dpl=nogit-1792405568","/_next/static/chunks/0horbp43y5c_k.js?dpl=nogit-1792405568","/_next/static/chunks/3ttet-2qytkke.js?dpl=nogit-1792405568","/_next/static/chunks/2-1ciy1x4js7n.js?dpl=nogit-1792405568","/_next/static/chunks/32s6q_rspd3hd.js?dpl=nogit-1792405568","/_next/static/chunks/0g1nv87sz_mxq.js?dpl=nogit-1792405568","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792405568","/_next/static/chunks/1vjwc_dfj8v1b.js?dpl=nogit-1792405568","/_next/static/chunks/30u3fsn115_1j.js?dpl=nogit-1792405568","/_next/static/chunks/2wrktd_v4ahrw.js?dpl=nogit-1792405568","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792405568"],"RecordTopicView"] 37:I[830476,["/_next/static/chunks/3j31_2qapmh60.js?dpl=nogit-1792405568","/_next/static/chunks/313vt9_f08mnn.js?dpl=nogit-1792405568","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792405568","/_next/static/chunks/09bcgz1lr6dab.js?dpl=nogit-1792405568","/_next/static/chunks/0tzbwbomzrnqw.js?dpl=nogit-1792405568","/_next/static/chunks/14trqrdc9kqx-.js?dpl=nogit-1792405568","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792405568","/_next/static/chunks/0zsno9tz6vkpw.js?dpl=nogit-1792405568","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792405568","/_next/static/chunks/11md10s8bxc41.js?dpl=nogit-1792405568","/_next/static/chunks/0oqj6s0nlv2xz.js?dpl=nogit-1792405568","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792405568","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792405568","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792405568","/_next/static/chunks/3z1ylg82-_yrl.js?dpl=nogit-1792405568","/_next/static/chunks/230359bhbwn7h.js?dpl=nogit-1792405568","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792405568","/_next/static/chunks/2_v9g5x0fx29m.js?dpl=nogit-1792405568","/_next/static/chunks/0r0qwy8stn_uo.js?dpl=nogit-1792405568","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792405568","/_next/static/chunks/10j6l_dhgof6p.js?dpl=nogit-1792405568","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792405568","/_next/static/chunks/3fr9hkx77wvja.js?dpl=nogit-1792405568","/_next/static/chunks/3adfywl6-8_k3.js?dpl=nogit-1792405568","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792405568","/_next/static/chunks/1vqil3_m7bft1.js?dpl=nogit-1792405568","/_next/static/chunks/40cqw2xvsic28.js?dpl=nogit-1792405568","/_next/static/chunks/3j5g1ks7ljj92.js?dpl=nogit-1792405568","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792405568","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792405568","/_next/static/chunks/2p5xv3ji4tpih.js?dpl=nogit-1792405568","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792405568","/_next/static/chunks/2a2ryz-j6u084.js?dpl=nogit-1792405568","/_next/static/chunks/2o9c7zxab747y.js?dpl=nogit-1792405568","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792405568","/_next/static/chunks/2zjfedap7f8sc.js?dpl=nogit-1792405568","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1792405568","/_next/static/chunks/3nnmps7-hhzc_.js?dpl=nogit-1792405568","/_next/static/chunks/0wg6314tous77.js?dpl=nogit-1792405568","/_next/static/chunks/1u3eptdza7lch.js?dpl=nogit-1792405568","/_next/static/chunks/2mo1yi9s3sppb.js?dpl=nogit-1792405568","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792405568","/_next/static/chunks/0_95ok1we81_2.js?dpl=nogit-1792405568","/_next/static/chunks/1gz0d2bu8bcyn.js?dpl=nogit-1792405568","/_next/static/chunks/00k10pkv-tsak.js?dpl=nogit-1792405568","/_next/static/chunks/12rdo7yi_xy67.js?dpl=nogit-1792405568","/_next/static/chunks/29add9pj_xa2g.js?dpl=nogit-1792405568","/_next/static/chunks/2xx3y5ow--aem.js?dpl=nogit-1792405568","/_next/static/chunks/42ur87elx-dz3.js?dpl=nogit-1792405568","/_next/static/chunks/3ibs-m0_upaf_.js?dpl=nogit-1792405568","/_next/static/chunks/3nfomzjk3zoe-.js?dpl=nogit-1792405568","/_next/static/chunks/0horbp43y5c_k.js?dpl=nogit-1792405568","/_next/static/chunks/3ttet-2qytkke.js?dpl=nogit-1792405568","/_next/static/chunks/2-1ciy1x4js7n.js?dpl=nogit-1792405568","/_next/static/chunks/32s6q_rspd3hd.js?dpl=nogit-1792405568","/_next/static/chunks/0g1nv87sz_mxq.js?dpl=nogit-1792405568","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792405568","/_next/static/chunks/1vjwc_dfj8v1b.js?dpl=nogit-1792405568","/_next/static/chunks/30u3fsn115_1j.js?dpl=nogit-1792405568","/_next/static/chunks/2wrktd_v4ahrw.js?dpl=nogit-1792405568","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792405568"],"FlashcardAccessBranch"] 38:I[315438,["/_next/static/chunks/3j31_2qapmh60.js?dpl=nogit-1792405568","/_next/static/chunks/313vt9_f08mnn.js?dpl=nogit-1792405568","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792405568","/_next/static/chunks/09bcgz1lr6dab.js?dpl=nogit-1792405568","/_next/static/chunks/0tzbwbomzrnqw.js?dpl=nogit-1792405568","/_next/static/chunks/14trqrdc9kqx-.js?dpl=nogit-1792405568","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792405568","/_next/static/chunks/0zsno9tz6vkpw.js?dpl=nogit-1792405568","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792405568","/_next/static/chunks/11md10s8bxc41.js?dpl=nogit-1792405568","/_next/static/chunks/0oqj6s0nlv2xz.js?dpl=nogit-1792405568","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792405568","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792405568","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792405568","/_next/static/chunks/3z1ylg82-_yrl.js?dpl=nogit-1792405568","/_next/static/chunks/230359bhbwn7h.js?dpl=nogit-1792405568","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792405568","/_next/static/chunks/2_v9g5x0fx29m.js?dpl=nogit-1792405568","/_next/static/chunks/0r0qwy8stn_uo.js?dpl=nogit-1792405568","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792405568","/_next/static/chunks/10j6l_dhgof6p.js?dpl=nogit-1792405568","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792405568","/_next/static/chunks/3fr9hkx77wvja.js?dpl=nogit-1792405568","/_next/static/chunks/3adfywl6-8_k3.js?dpl=nogit-1792405568","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792405568","/_next/static/chunks/1vqil3_m7bft1.js?dpl=nogit-1792405568","/_next/static/chunks/40cqw2xvsic28.js?dpl=nogit-1792405568","/_next/static/chunks/3j5g1ks7ljj92.js?dpl=nogit-1792405568","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792405568","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792405568","/_next/static/chunks/2p5xv3ji4tpih.js?dpl=nogit-1792405568","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792405568","/_next/static/chunks/2a2ryz-j6u084.js?dpl=nogit-1792405568","/_next/static/chunks/2o9c7zxab747y.js?dpl=nogit-1792405568","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792405568","/_next/static/chunks/2zjfedap7f8sc.js?dpl=nogit-1792405568","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1792405568","/_next/static/chunks/3nnmps7-hhzc_.js?dpl=nogit-1792405568","/_next/static/chunks/0wg6314tous77.js?dpl=nogit-1792405568","/_next/static/chunks/1u3eptdza7lch.js?dpl=nogit-1792405568","/_next/static/chunks/2mo1yi9s3sppb.js?dpl=nogit-1792405568","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792405568","/_next/static/chunks/0_95ok1we81_2.js?dpl=nogit-1792405568","/_next/static/chunks/1gz0d2bu8bcyn.js?dpl=nogit-1792405568","/_next/static/chunks/00k10pkv-tsak.js?dpl=nogit-1792405568","/_next/static/chunks/12rdo7yi_xy67.js?dpl=nogit-1792405568","/_next/static/chunks/29add9pj_xa2g.js?dpl=nogit-1792405568","/_next/static/chunks/2xx3y5ow--aem.js?dpl=nogit-1792405568","/_next/static/chunks/42ur87elx-dz3.js?dpl=nogit-1792405568","/_next/static/chunks/3ibs-m0_upaf_.js?dpl=nogit-1792405568","/_next/static/chunks/3nfomzjk3zoe-.js?dpl=nogit-1792405568","/_next/static/chunks/0horbp43y5c_k.js?dpl=nogit-1792405568","/_next/static/chunks/3ttet-2qytkke.js?dpl=nogit-1792405568","/_next/static/chunks/2-1ciy1x4js7n.js?dpl=nogit-1792405568","/_next/static/chunks/32s6q_rspd3hd.js?dpl=nogit-1792405568","/_next/static/chunks/0g1nv87sz_mxq.js?dpl=nogit-1792405568","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792405568","/_next/static/chunks/1vjwc_dfj8v1b.js?dpl=nogit-1792405568","/_next/static/chunks/30u3fsn115_1j.js?dpl=nogit-1792405568","/_next/static/chunks/2wrktd_v4ahrw.js?dpl=nogit-1792405568","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792405568"],"TopicFlashcardRunner"] 24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/gcse/maths/ccea/grade-7-7-10-iteration/flashcards","label":"Iteration","subjectCode":"gcse_maths__ccea","surface":"flashcards"}],["$","$L36",null,{"kind":"flashcards","topicId":"e68e0b60-6c56-66ce-8368-e4f51ff3825a"}],["$","$L37",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L38",null,{"decks":[{"id":"5fb9971d-2c16-70c2-8d43-9c5869eb8a5d","title":"Iteration Flashcards","topicId":"e68e0b60-6c56-66ce-8368-e4f51ff3825a","topicSlug":"grade-7-7-10-iteration","cards":[{"id":"351c8761-e1bf-977a-b786-722d06732e25","cardType":"cloze","points":null,"front":"Iteration means [ ].","back":"Iteration means repeatedly applying the same rule.","mcqOptions":[{"text":"repeatedly applying the same rule","correct":true,"explanation":"Iteration generates successive values by applying the same rule repeatedly, with each result normally becoming the input for the next application."},{"text":"applying several different rules once","correct":false,"explanation":"Using different rules once is not iteration. Iteration means repeatedly applying the same rule, usually feeding each calculated value back into that rule."},{"text":"reversing the same rule repeatedly","correct":false,"explanation":"Iteration does not require a rule to be reversed. It means repeatedly applying the same rule to generate a sequence of successive values."},{"text":"repeatedly choosing a different rule","correct":false,"explanation":"Changing the rule at every step contradicts the definition of iteration. The right description is repeatedly applying the same rule."}]},{"id":"0ac6b34c-21c3-7739-8a44-1d093b9d80c1","cardType":"basic","points":1,"front":"In \$x_3\$, what does the subscript 3 tell you?","back":"The term with subscript 3.","mcqOptions":[{"text":"The term with subscript 3.","correct":true,"explanation":"A subscript identifies a term’s position or label in a sequence, so \$x_3\$ denotes the term with subscript 3."},{"text":"The term raised to power 3.","correct":false,"explanation":"This confuses a subscript with an exponent. The 3 labels the term with subscript 3; a power would be written \$x^3\$."},{"text":"The term multiplied by 3.","correct":false,"explanation":"A subscript does not indicate multiplication. In \$x_3\$, the 3 identifies the term with subscript 3, rather than meaning \$3x\$."},{"text":"The term divided by 3.","correct":false,"explanation":"A subscript labels a term and does not mean division. The notation \$x_3\$ identifies the term with subscript 3."}]},{"id":"6e99650b-22e4-e768-116f-e99cce1bb8b6","cardType":"cloze","points":null,"front":"In \$x_3\$, the 3 is a [ ], not a [ ].","back":"In \$x_3\$, the 3 is a subscript, not a power.","mcqOptions":[{"text":"In \$x_3\$, the 3 is a subscript, not a power.","correct":true,"explanation":"The 3 is written below the variable, so it is a subscript labelling a term. An exponent or power would be written above the variable."},{"text":"In \$x_3\$, the 3 is a power, not a subscript.","correct":false,"explanation":"This reverses the meanings of the two positions. In \$x_3\$, the 3 is a subscript; a power of 3 would be written \$x^3\$."},{"text":"In \$x_3\$, the 3 is a coefficient, not a power.","correct":false,"explanation":"A coefficient is placed before a variable, as in \$3x\$. Here the 3 is below the variable, making it a subscript rather than a power."},{"text":"In \$x_3\$, the 3 is a subscript, not a coefficient.","correct":false,"explanation":"Although 3 is a subscript, the intended contrast is with a power, a common notation error. In \$x_3\$, it is a subscript, not a power."}]},{"id":"bbcd8052-61dc-0ced-3eac-11faf8eea34a","cardType":"cloze","points":null,"front":"A starting value, such as \$x_0\$, is used [ ] the recurrence formula.","back":"A starting value, such as \$x_0\$, is used before the recurrence formula.","mcqOptions":[{"text":"before","correct":true,"explanation":"The starting value must be known before the recurrence formula can first be used, because it provides the input needed to calculate the next term."},{"text":"after","correct":false,"explanation":"Placing the starting value after the recurrence formula reverses the process. The starting value is used before the formula to calculate the first new term."},{"text":"instead of","correct":false,"explanation":"The starting value does not replace the recurrence formula. It is used before the formula as the initial input from which later terms are generated."},{"text":"independently of","correct":false,"explanation":"The starting value and recurrence formula are not independent. The starting value is used before the formula and supplies its first input."}]},{"id":"3f3d3860-5405-d63d-1a13-3a2063be0979","cardType":"basic","points":1,"front":"When using \$x_{n+1}=f(x_n)\$, which value is substituted to find the next term?","back":"The current term \$x_n\$.","mcqOptions":[{"text":"The current term \$x_n\$.","correct":true,"explanation":"The recurrence formula applies \$f\$ to the current term \$x_n\$, producing the next term \$x_{n+1}\$."},{"text":"The starting term \$x_0\$.","correct":false,"explanation":"The starting term is used only to begin the iteration. At each later step, the current term \$x_n\$ is substituted to obtain \$x_{n+1}\$."},{"text":"The next term \$x_{n+1}\$.","correct":false,"explanation":"Substituting the next term would use the unknown value being calculated. The current term \$x_n\$ is substituted into \$f\$ to find \$x_{n+1}\$."},{"text":"The term number \$n\$.","correct":false,"explanation":"The index \$n\$ labels the term but is not the value substituted here. The recurrence uses the current term \$x_n\$ to calculate the next one."}]},{"id":"5d8d9138-0f1d-8f93-a88a-b614cabb98f0","cardType":"cloze","points":null,"front":"In \$x_{n+1}=2+\\frac{6}{x_n^2}\$, \$x_n\$ is the [ ] value and \$x_{n+1}\$ is the [ ] value.","back":"In \$x_{n+1}=2+\\frac{6}{x_n^2}\$, \$x_n\$ is the current value and \$x_{n+1}\$ is the next value.","mcqOptions":[{"text":"In \$x_{n+1}=2+\\frac{6}{x_n^2}\$, \$x_n\$ is the current value and \$x_{n+1}\$ is the next value.","correct":true,"explanation":"The formula uses the current value \$x_n\$ as its input and produces the next value \$x_{n+1}\$, advancing the sequence by one term."},{"text":"In \$x_{n+1}=2+\\frac{6}{x_n^2}\$, \$x_n\$ is the previous value and \$x_{n+1}\$ is the current value.","correct":false,"explanation":"This labels the terms from the following step rather than the calculation shown. Here \$x_n\$ is the current value and \$x_{n+1}\$ is the next value."},{"text":"In \$x_{n+1}=2+\\frac{6}{x_n^2}\$, \$x_n\$ is the next value and \$x_{n+1}\$ is the current value.","correct":false,"explanation":"This reverses the order of the sequence. The input \$x_n\$ is the current value, while the output \$x_{n+1}\$ is the next value."},{"text":"In \$x_{n+1}=2+\\frac{6}{x_n^2}\$, \$x_n\$ is the starting value and \$x_{n+1}\$ is the final value.","correct":false,"explanation":"The indices do not necessarily represent the beginning and end of the iteration. They identify consecutive terms: \$x_n\$ is current and \$x_{n+1}\$ is next."}]},{"id":"ccadfd82-0645-92a2-fa72-b9c09f5075a0","cardType":"cloze","points":null,"front":"Starting from \$x_0\$, after one use you get [ ]; after three uses you get [ ].","back":"Starting from \$x_0\$, after one use you get \$x_1\$; after three uses you get \$x_3\$.","mcqOptions":[{"text":"Starting from \$x_0\$, after one use you get \$x_1\$; after three uses you get \$x_3\$.","correct":true,"explanation":"Each use of the recurrence increases the subscript by one. Starting at \$x_0\$, one use reaches \$x_1\$ and three uses reach \$x_3\$."},{"text":"Starting from \$x_0\$, after one use you get \$x_0\$; after three uses you get \$x_2\$.","correct":false,"explanation":"This fails to count the first application as a step. Each use increases the subscript, so one use gives \$x_1\$ and three uses give \$x_3\$."},{"text":"Starting from \$x_0\$, after one use you get \$x_1\$; after three uses you get \$x_4\$.","correct":false,"explanation":"The first result is counted correctly, but the third is one step too far. Three applications from \$x_0\$ produce \$x_1\$, \$x_2\$, then \$x_3\$."},{"text":"Starting from \$x_0\$, after one use you get \$x_2\$; after three uses you get \$x_3\$.","correct":false,"explanation":"This skips \$x_1\$ on the first application. A recurrence advances one subscript at a time, so one use gives \$x_1\$, while three uses give \$x_3\$."}]},{"id":"6cce4b6f-a41a-6f6e-c4d5-46a73d3e3336","cardType":"basic","points":1,"front":"For \$P_{t+1}=1.10(P_t-12)\$, which value is used to calculate \$P_2\$?","back":"\$P_1\$, not \$P_0\$.","mcqOptions":[{"text":"\$P_1\$, not \$P_0\$.","correct":true,"explanation":"To calculate \$P_2\$, set \$t=1\$, giving \$P_2=1.10(P_1-12)\$. The recurrence therefore uses the immediately preceding value \$P_1\$."},{"text":"\$P_0\$, not \$P_1\$.","correct":false,"explanation":"Using \$P_0\$ would calculate \$P_1\$, not \$P_2\$. To find \$P_2\$, the current value must be \$P_1\$."},{"text":"\$P_2\$, not \$P_1\$.","correct":false,"explanation":"The recurrence cannot use the unknown value \$P_2\$ to calculate itself. It uses the preceding value \$P_1\$ to produce \$P_2\$."},{"text":"\$P_3\$, not \$P_2\$.","correct":false,"explanation":"This uses a later term rather than the preceding one. Calculating \$P_2\$ requires setting \$t=1\$, so the substituted value is \$P_1\$."}]},{"id":"f97e5997-9b1a-3389-f6fa-6e2e3168549e","cardType":"basic","points":1,"front":"Given \$P_0=180\$ and \$P_{t+1}=1.10(P_t-12)\$, find \$P_1\$.","back":"\$184.8\$","mcqOptions":[{"text":"\$184.8\$","correct":true,"explanation":"Substituting \$P_0=180\$ into the rule gives \$P_1=1.10(180-12)=1.10\\times168=184.8\$."},{"text":"\$184.68\$","correct":false,"explanation":"This comes from a small arithmetic error after subtracting 12. The correct calculation is \$1.10(180-12)=1.10\\times168=184.8\$."},{"text":"\$198\$","correct":false,"explanation":"This multiplies 180 by 1.10 without first subtracting 12. The iteration rule requires \$1.10(180-12)\$, which gives \$184.8\$."},{"text":"\$172.8\$","correct":false,"explanation":"This subtracts 12 after applying the multiplier, rather than inside the brackets. The required order is \$1.10(180-12)=184.8\$."}]},{"id":"a631baf1-533e-8057-a041-514114c4e3d0","cardType":"basic","points":1,"front":"Given \$P_2=190.08\$ and \$P_{t+1}=1.10(P_t-12)\$, find \$P_3\$.","back":"\$195.888\$","mcqOptions":[{"text":"\$195.888\$","correct":true,"explanation":"Using \$P_2=190.08\$, the rule gives \$P_3=1.10(190.08-12)=1.10\\times178.08=195.888\$."},{"text":"\$207.088\$","correct":false,"explanation":"This adds the multiplier effect to the original value without subtracting 12 first. Applying the brackets correctly gives \$1.10(190.08-12)=195.888\$."},{"text":"\$196.08\$","correct":false,"explanation":"This effectively adds 6% to \$190.08\$, rather than applying the 10% increase to the value after subtracting 12. The correct result is \$195.888\$."},{"text":"\$185.688\$","correct":false,"explanation":"This subtracts 12 from the result after multiplying, reversing the intended order of operations. The rule gives \$1.10(190.08-12)=195.888\$."}]},{"id":"86536c41-d3da-6a94-f071-60e3cbb2826a","cardType":"basic","points":1,"front":"With \$x_0=2\$ and \$x_{n+1}=2+\\frac{6}{x_n^2}\$, find \$x_1\$.","back":"\$3.5\$","mcqOptions":[{"text":"\$3.5\$","correct":true,"explanation":"Substitute \$x_0=2\$ into the rule: \$x_1=2+\\frac{6}{2^2}=2+\\frac{6}{4}=3.5\$."},{"text":"\$5\$","correct":false,"explanation":"This treats the denominator as \$x_0\$ instead of \$x_0^2\$, giving \$2+6\\div2=5\$. Squaring 2 first gives the correct value, \$3.5\$."},{"text":"\$2.75\$","correct":false,"explanation":"This halves the fraction incorrectly: \$6\\div2^2\$ is \$6\\div4=1.5\$, not \$0.75\$. Therefore \$x_1=2+1.5=3.5\$."},{"text":"\$4\$","correct":false,"explanation":"This uses \$6\\div2^2=2\$, confusing the numerator and denominator calculation. Since \$6\\div4=1.5\$, the iteration gives \$x_1=3.5\$."}]},{"id":"bb973384-52b1-8d0e-4f05-7eb6dc446571","cardType":"basic","points":1,"front":"Given \$x_1=3.5\$ and \$x_{n+1}=2+\\frac{6}{x_n^2}\$, find \$x_2\$ to 3 d.p.","back":"\$2.490\$","mcqOptions":[{"text":"\$2.490\$","correct":true,"explanation":"Substituting \$x_1=3.5\$ gives \$x_2=2+\\frac{6}{3.5^2}=2+\\frac{6}{12.25}=2.489795\\ldots\$, which rounds to \$2.490\$ to 3 d.p."},{"text":"\$2.489\$","correct":false,"explanation":"The unrounded value is \$2.489795\\ldots\$, so the fourth decimal place is 7 and the third decimal place must round up. The correct rounded value is \$2.490\$."},{"text":"\$2.857\$","correct":false,"explanation":"This results from using \$3.5\$ rather than squaring it in the denominator. The rule requires \$2+6\\div(3.5^2)=2.489795\\ldots\$, giving \$2.490\$."},{"text":"\$3.714\$","correct":false,"explanation":"This adds \$6\\div3.5\$ instead of \$6\\div3.5^2\$, omitting the square on the denominator. Squaring first gives the correct result, \$2.490\$."}]},{"id":"7caac00b-2fb6-9e7c-310d-5f888e9b2815","cardType":"basic","points":1,"front":"Given \$x_2=2.489795918\\ldots\$ and \$x_{n+1}=2+\\frac{6}{x_n^2}\$, find \$x_3\$ to 3 d.p.","back":"\$2.968\$","mcqOptions":[{"text":"\$2.968\$","correct":true,"explanation":"Using \$x_2=2.489795918\\ldots\$, the rule gives \$x_3=2+\\frac{6}{x_2^2}=2.9677\\ldots\$, which rounds to \$2.968\$ to 3 d.p."},{"text":"\$2.968\\ldots\$","correct":false,"explanation":"The value is approximately \$2.9677\\ldots\$, but the question requires 3 decimal places. Rounding to that precision gives \$2.968\$, not an unrounded recurring value."},{"text":"\$2.410\$","correct":false,"explanation":"This uses a rough or inaccurate square for \$x_2\$, making the fraction too small. Calculating \$2+6/(2.489795918\\ldots)^2\$ gives \$2.9677\\ldots\$, hence \$2.968\$."},{"text":"\$3.506\$","correct":false,"explanation":"This effectively uses \$6\\div x_2\$ rather than \$6\\div x_2^2\$, producing too large an added fraction. Squaring the denominator gives the correct rounded value \$2.968\$."}]},{"id":"bf3fd58c-ee1f-ade5-50ff-95404b40a781","cardType":"cloze","points":null,"front":"If the terms get closer and closer to one value, the sequence [ ].","back":"If the terms get closer and closer to one value, the sequence converges.","mcqOptions":[{"text":"converges","correct":true,"explanation":"A sequence converges when its terms approach one fixed value more and more closely, even if the limiting value is never reached exactly."},{"text":"diverges","correct":false,"explanation":"Divergence means the terms do not approach one limiting value. The described behaviour is convergence, so the sequence converges."},{"text":"terminates","correct":false,"explanation":"A terminating sequence stops after a finite number of terms, whereas getting closer to one value describes convergence. The correct term is converges."},{"text":"cycles","correct":false,"explanation":"A cycling sequence repeatedly returns to a pattern of values rather than approaching one value. Terms getting closer to one value means the sequence converges."}]},{"id":"0a65b06e-fa27-3403-eab6-83c3b26bd0ef","cardType":"cloze","points":null,"front":"At a fixed point, applying the iteration rule leaves the value [ ]; the current and next values are [ ].","back":"At a fixed point, applying the iteration rule leaves the value unchanged; the current and next values are equal.","mcqOptions":[{"text":"At a fixed point, applying the iteration rule leaves the value unchanged; the current and next values are equal.","correct":true,"explanation":"At a fixed point, the iteration rule maps the value to itself. Therefore the value remains unchanged, and the current and next terms are equal."},{"text":"At a fixed point, applying the iteration rule leaves the value changed; the current and next values are equal.","correct":false,"explanation":"A fixed point is not changed by applying the rule, although the current and next values are equal. The correct description is unchanged and equal."},{"text":"At a fixed point, applying the iteration rule leaves the value unchanged; the current and next values are different.","correct":false,"explanation":"If the value is unchanged by the rule, the next value must be the same as the current value. Thus the correct pair is unchanged and equal."},{"text":"At a fixed point, applying the iteration rule leaves the value changed; the current and next values are different.","correct":false,"explanation":"Both terms describe movement away from the defining property of a fixed point. A fixed point remains unchanged, so the current and next values are equal."}]},{"id":"12fc308e-ddc8-8dd0-d193-6c6b8d7988a9","cardType":"basic","points":1,"front":"With \$x_0=0\$ and \$x_{n+1}=\\frac{2}{x_n^2+4}\$, find \$x_1\$.","back":"\$0.5\$","mcqOptions":[{"text":"\$0.5\$","correct":true,"explanation":"Substituting \$x_0=0\$ gives \$x_1=\\frac{2}{0^2+4}=\\frac{2}{4}=0.5\$."},{"text":"\$2\$","correct":false,"explanation":"This ignores the denominator and uses the numerator alone. The rule requires \$2\\div(0^2+4)=2\\div4=0.5\$."},{"text":"\$0.25\$","correct":false,"explanation":"This squares the numerator or treats the fraction incorrectly. With \$x_0=0\$, the denominator is 4, so \$x_1=2\\div4=0.5\$."},{"text":"\$0\$","correct":false,"explanation":"The zero is squared in the denominator, not used as the whole result. Since the denominator is \$0^2+4=4\$, the next value is \$0.5\$."}]},{"id":"7e8a6cb4-64ba-5c57-6181-2932a30df476","cardType":"basic","points":1,"front":"Given \$x_2=0.470588235\\ldots\$ and \$x_{n+1}=\\frac{2}{x_n^2+4}\$, find \$x_3\$ to 3 d.p.","back":"\$0.474\$","mcqOptions":[{"text":"\$0.474\$","correct":true,"explanation":"Substituting \$x_2=0.470588235\\ldots\$ gives \$x_3=\\frac{2}{x_2^2+4}=0.47377\\ldots\$, which rounds to \$0.474\$ to 3 decimal places."},{"text":"\$0.471\$","correct":false,"explanation":"This is close to the previous iterate and comes from treating the sequence as nearly unchanged. The formula gives \$0.47377\\ldots\$, which rounds to \$0.474\$."},{"text":"\$0.476\$","correct":false,"explanation":"This results from an inaccurate squaring or division step. Substituting the given iterate produces \$x_3=0.47377\\ldots\$, so the correct 3 d.p. value is \$0.474\$."},{"text":"\$0.529\$","correct":false,"explanation":"This can result from using \$2/x_n^2+4\$ instead of placing the whole denominator below 2. The required calculation is \$2/(x_n^2+4)=0.47377\\ldots\$, giving \$0.474\$."}]},{"id":"03895efe-f58c-33c3-5c87-d2e4467fac71","cardType":"basic","points":1,"front":"What substitution links a convergent iteration to its equation?","back":"Set \$x_{n+1}=x_n=a\$.","mcqOptions":[{"text":"Set \$x_{n+1}=x_n=a\$.","correct":true,"explanation":"At convergence, consecutive iterates approach the same limiting value. Calling that value \$a\$ means setting both \$x_{n+1}\$ and \$x_n\$ equal to \$a\$."},{"text":"Set \$x_{n+1}=x_n+1=a\$.","correct":false,"explanation":"Adding 1 incorrectly describes a sequence increasing by a fixed amount, not a convergent sequence. Convergence requires consecutive iterates to become equal: \$x_{n+1}=x_n=a\$."},{"text":"Set \$x_{n+1}=x_n=0\$.","correct":false,"explanation":"Zero is not automatically the limiting value; it must satisfy the iteration equation. The general fixed-point substitution is \$x_{n+1}=x_n=a\$, with \$a\$ then determined algebraically."},{"text":"Set \$x_{n+1}=\\frac{1}{x_n}=a\$.","correct":false,"explanation":"Taking a reciprocal imposes an unrelated condition and does not express convergence. A convergent iteration has equal consecutive values, so the substitution is \$x_{n+1}=x_n=a\$."}]},{"id":"bc2e184e-07d3-d472-dd02-28e67b9e49e8","cardType":"basic","points":2,"front":"From \$a=\\frac{2}{a^2+4}\$, what equation follows after rearranging?","back":"\$a^3+4a-2=0\$","mcqOptions":[{"text":"\$a^3+4a-2=0\$","correct":true,"explanation":"Multiplying by \$a^2+4\$ gives \$a(a^2+4)=2\$. Expanding and moving all terms to one side produces \$a^3+4a-2=0\$."},{"text":"\$a^3+4a+2=0\$","correct":false,"explanation":"The constant term changes sign incorrectly when 2 is moved to the left. Rearranging gives \$a^3+4a-2=0\$, because the original right-hand side is positive 2."},{"text":"\$a^2+4a-2=0\$","correct":false,"explanation":"This misses the factor of \$a\$ multiplying the denominator. Multiplying through gives \$a(a^2+4)=a^3+4a\$, leading to \$a^3+4a-2=0\$."},{"text":"\$a^3+2a-4=0\$","correct":false,"explanation":"The coefficients have been rearranged incorrectly. Expanding \$a(a^2+4)\$ gives \$a^3+4a\$, and subtracting 2 yields \$a^3+4a-2=0\$."}]},{"id":"ac99498f-e718-0efe-b20c-5d31497b51b4","cardType":"cloze","points":null,"front":"A sign change means one output is [ ] and the other is [ ].","back":"A sign change means one output is negative and the other is positive.","mcqOptions":[{"text":"A sign change means one output is negative and the other is positive.","correct":true,"explanation":"A sign change occurs when two function outputs have opposite signs: one is negative and the other is positive. This supports the existence of a root between them for a continuous function."},{"text":"A sign change means one output is positive and the other is positive.","correct":false,"explanation":"Two positive outputs have the same sign, so no sign change has occurred. A sign change requires one negative output and one positive output."},{"text":"A sign change means one output is negative and the other is negative.","correct":false,"explanation":"Two negative outputs also have the same sign. The required condition is that one output is negative and the other is positive."},{"text":"A sign change means one output is zero and the other is positive.","correct":false,"explanation":"An output of zero means a root has already been found, rather than showing a sign change across an interval. A sign change requires opposite non-zero signs: negative and positive."}]},{"id":"a0928b69-054f-2ce9-8300-67f72e6b37a5","cardType":"basic","points":3,"front":"Show that \$x^3+5x-2=0\$ has a solution between \$0\$ and \$1\$.","back":"

\$f(0)=-2\$
\$f(1)=4\$
Opposite signs and continuous, so a root lies between.

","mcqOptions":[{"text":"

\$f(0)=-2\$
\$f(1)=4\$
Opposite signs and continuous, so a root lies between.

","correct":true,"explanation":"Evaluating gives \$f(0)=-2\$ and \$f(1)=4\$, which have opposite signs. Since the polynomial is continuous, the Intermediate Value Theorem guarantees a root between 0 and 1."},{"text":"

\$f(0)=2\$
\$f(1)=4\$
Same signs and continuous, so a root lies between.

","correct":false,"explanation":"The constant term is \$-2\$, so \$f(0)=-2\$, not 2. The correct endpoint values are \$-2\$ and 4, giving opposite signs and therefore a root between them."},{"text":"

\$f(0)=-2\$
\$f(1)=2\$
Opposite signs and continuous, so a root lies between.

","correct":false,"explanation":"At 1, the value is \$1^3+5(1)-2=4\$, not 2. Although the sign conclusion is right, the endpoint calculation must be \$f(1)=4\$."},{"text":"

\$f(0)=-2\$
\$f(1)=4\$
Opposite signs but not continuous, so no conclusion follows.

","correct":false,"explanation":"A polynomial is continuous everywhere, so the opposite endpoint signs do guarantee a root in the interval. The mistake is wrongly denying continuity."}]},{"id":"b863ba9f-2990-e5f7-4a64-ff45c92e49bd","cardType":"basic","points":2,"front":"Rearrange \$x^3+5x=2\$ into the form \$x=\\cdots\$.","back":"\$x=\\frac{2}{5}-\\frac{x^3}{5}\$","mcqOptions":[{"text":"\$x=\\frac{2}{5}-\\frac{x^3}{5}\$","correct":true,"explanation":"Subtracting \$x^3\$ gives \$5x=2-x^3\$. Dividing every term by 5 produces \$x=\\frac{2}{5}-\\frac{x^3}{5}\$."},{"text":"\$x=\\frac{2}{5}+\\frac{x^3}{5}\$","correct":false,"explanation":"The cubic term must be subtracted when moved from the left side to the right. Thus \$5x=2-x^3\$, giving \$x=\\frac{2}{5}-\\frac{x^3}{5}\$."},{"text":"\$x=2-5x^3\$","correct":false,"explanation":"This reverses the effects of rearranging and dividing by 5. Dividing \$5x=2-x^3\$ by 5 gives \$x=\\frac{2}{5}-\\frac{x^3}{5}\$."},{"text":"\$x=\\frac{2-x^3}{5x}\$","correct":false,"explanation":"Dividing by \$5x\$ leaves the variable on the denominator and does not isolate \$x\$. Dividing the equation by 5 gives the required expression directly."}]},{"id":"9fbbac3d-3b74-1766-adac-228fa8fb90bd","cardType":"basic","points":1,"front":"Turn \$x=\\frac{2}{5}-\\frac{x^3}{5}\$ into an iteration formula.","back":"\$x_{n+1}=\\frac{2}{5}-\\frac{x_n^3}{5}\$","mcqOptions":[{"text":"\$x_{n+1}=\\frac{2}{5}-\\frac{x_n^3}{5}\$","correct":true,"explanation":"An iteration formula replaces the current value \$x\$ on the right-hand side with \$x_n\$, while defining the next value as \$x_{n+1}\$."},{"text":"\$x_{n+1}=\\frac{2}{5}-\\frac{x_{n+1}^3}{5}\$","correct":false,"explanation":"Using \$x_{n+1}\$ on both sides makes the formula implicit rather than an iteration from a known value. The right-hand side must use the previous iterate \$x_n\$."},{"text":"\$x_n=\\frac{2}{5}-\\frac{x_{n+1}^3}{5}\$","correct":false,"explanation":"This reverses the roles of the current and next terms. To calculate a new iterate from a known one, use \$x_{n+1}\$ on the left and \$x_n\$ on the right."},{"text":"\$x_{n+1}=\\frac{2}{5}-\\frac{x_n^2}{5}\$","correct":false,"explanation":"The original rearranged equation contains a cube, so changing it to a square changes the function being iterated. The power must remain \$x_n^3\$."}]},{"id":"f6ffaefb-9678-0539-8663-78df1a7ba9ea","cardType":"basic","points":2,"front":"Using \$x_{n+1}=\\frac{2}{5}-\\frac{x_n^3}{5}\$ with \$x_0=0\$, what is \$x_2\$?","back":"\$0.3872\$","mcqOptions":[{"text":"\$0.3872\$","correct":true,"explanation":"Starting with \$x_0=0\$, first find \$x_1=0.4\$. Then \$x_2=0.4-\\frac{0.4^3}{5}=0.4-0.0128=0.3872\$."},{"text":"\$0.3968\$","correct":false,"explanation":"This subtracts \$0.0032\$, which comes from mishandling the division by 5 after cubing. Since \$0.4^3/5=0.0128\$, the correct result is \$0.3872\$."},{"text":"\$0.3360\$","correct":false,"explanation":"This is obtained by subtracting \$0.064\$ rather than \$0.0128\$, forgetting to divide the cube by 5. The correct calculation gives \$x_2=0.3872\$."},{"text":"\$0.4000\$","correct":false,"explanation":"This uses the first iterate and stops before applying the formula a second time. Since \$x_1=0.4\$, substituting it gives \$x_2=0.3872\$."}]}]}],"examCountdown":null,"heading":"Iteration","title":"Iteration Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"e68e0b60-6c56-66ce-8368-e4f51ff3825a","walled":"$L3a"}],"$L3b","$L3c"]