36:I[651451,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"UserProgressProvider"] 37:I[196793,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"Gingerbread"] 39:I[716297,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"TopicRecentlyViewed"] 3a:I[647935,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"RecordResourceResume"] 3b:I[426249,["/_next/static/chunks/2msykuvi-vydm.js?dpl=nogit-1790362613","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362613","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362613","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362613","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362613","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362613","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362613","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362613","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362613","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362613","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362613","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362613","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362613","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362613","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362613","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362613","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362613","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362613","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362613","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362613","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362613","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362613","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362613","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362613","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362613","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362613","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362613","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362613","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362613","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362613","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362613","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362613","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362613","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362613","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362613","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362613","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362613","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362613","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362613","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362613","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362613","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362613","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362613","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362613","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362613","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362613","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362613","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362613","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362613","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362613","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362613","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362613","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362613","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362613","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362613","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362613","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362613","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362613","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362613","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362613","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362613","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362613","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362613","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362613","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362613","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362613","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362613","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362613","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362613","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362613"],"TopicQuestionBank"] 24:["$","$L36",null,{"bundle":{"topicIds":["23c24b47-7b3c-4108-bb70-3fe34105de38"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L37",null,{"feature":"question-bank"}],"$L38",["$","$L39",null,{"subjectCode":"a_level_maths__wjec","subjectName":"Maths","topicSlug":"proof-6","topicTitle":"3.1.1 Proof (A-level only)","sectionTitle":null,"activity":"worksheet","resourceTitle":"3.1.1 Proof (A-level only) Questions","resourceHref":"/a-level/maths/wjec/proof-6/questions/q1","saveTrail":false}],["$","$L3a",null,{"href":"/a-level/maths/wjec/proof-6/questions/q1","label":"3.1.1 Proof (A-level only)","subjectCode":"a_level_maths__wjec","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 3 – 3.1.1 Proof (A-level only) - Maths Questions\",\"description\":\"Practise 3.1.1 Proof (A-level only) A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"3.1.1 Proof (A-level only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/wjec/proof-6/questions/q3\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/wjec/questions","basePath":"/a-level/maths/wjec/proof-6/questions","formulaUrl":"https://www.wjec.co.uk/media/4jfdytcz/ss-w27-18-e.pdf","initialIndex":2,"initialQuestionWindow":[{"id":"0f28af03-b719-4ce4-8b8a-9e1d7a79b4a7","revisionId":"324c7352-b543-4e17-9dd1-df01c5209be9","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"4d9bbc87-f857-47e1-b785-70d816b38969","label":"a","content":"Prove by contradiction that the difference between a rational number and an irrational number is irrational.","markscheme":"- Assume the difference between a rational number and an irrational number is rational B1\n- Let rational = $\\frac{a}{b}$, irrational = $c$, and difference = $\\frac{d}{e}$ (where $a, b, d, e$ are integers) M1\n- Set up equation: $\\frac{a}{b} - c = \\frac{d}{e}$ M1\n- Rearrange for $c$: $c = \\frac{a}{b} - \\frac{d}{e}$ M1\n- Combine fractions: $c = \\frac{ae - bd}{be}$ A1\n- State that $c$ has been written as a fraction of integers, which contradicts the fact that $c$ is irrational. A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Assume the negation (the difference is rational).","code":"B"},{"id":"p2","marks":1,"description":"Define the rational number and the assumed result as fractions of integers (e.g., $a/b$, $d/e$).","code":"M"},{"id":"p3","marks":1,"description":"Formulate the equation representing the difference.","code":"M"},{"id":"p4","marks":1,"description":"Correctly rearrange the equation to isolate the irrational number $c$.","code":"M"},{"id":"p5","marks":1,"description":"Simplify the expression into a single fraction of integers.","code":"A"},{"id":"p6","marks":1,"description":"Conclude that $c$ must be rational, providing a contradiction.","code":"A"}],"finalAnswers":["The difference is irrational"],"generatedAt":"2026-05-06T14:04:45.685Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"fraction_definition","label":"Expressing Rationals as Fractions","steps":[{"type":"text","order":0,"title":"State the counter-assumption","content":"To prove the statement by contradiction, we begin by assuming the opposite of what we want to prove.\n\nWe assume that the difference between a rational number and an irrational number is **rational**.\n\nThis is a crucial first step that earns the first mark **(B1)**."},{"type":"text","order":1,"title":"Define the variables","content":"Let's define our numbers formally:\n- Let the rational number be $r = \\frac{a}{b}$, where $a$ and $b$ are integers and $b \\neq 0$.\n- Let the irrational number be $c$.\n- Let their difference be another rational number, $d = \\frac{p}{q}$, where $p$ and $q$ are integers and $q \\neq 0$.\n\nDefining these terms algebraically using fractions of integers earns the next mark **(M1)**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=difference-eq}{\\frac{a}{b} - c = \\frac{p}{q}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"difference-eq"}],"connections":[]},"order":2,"title":"Set up the equation","description":"We now set up an equation representing their difference:\n\n$$\\frac{a}{b} - c = \\frac{p}{q}$$\n\nThis expression earns the method mark **(M1)**."},{"type":"text","order":3,"title":"Isolate the irrational number","content":"Next, we rearrange the equation to isolate the irrational number $c$ on one side:\n\n$$\\begin{aligned} \\frac{a}{b} - \\frac{p}{q} &= c \\\\ c &= \\frac{a}{b} - \\frac{p}{q} \\end{aligned}$$\n\nThis step earns a rearrangement mark **(M1)**."},{"type":"text","order":4,"title":"Combine into a single fraction","content":"We can write the right-hand side as a single fraction by finding a common denominator:\n\n$$c = \\frac{aq - bp}{bq}$$\n\nSince $a, b, p,$ and $q$ are all integers, it follows that:\n- The numerator $aq - bp$ is an integer (since the product and difference of integers are also integers).\n- The denominator $bq$ is a non-zero integer (since $b \\neq 0$ and $q \\neq 0$).\n\nWriting $c$ as a single fraction of integers earns the accuracy mark **(A1)**."},{"type":"text","order":5,"title":"Conclude the proof","content":"Because $c = \\frac{aq - bp}{bq}$ is the ratio of two integers, $c$ must be a **rational** number.\n\nHowever, this directly contradicts our initial definition that $c$ is **irrational**.\n\nTherefore, our original assumption must be false, proving that the difference between a rational number and an irrational number is indeed **irrational** **(A1)**."}]},{"id":"closure_properties","label":"Using Closure Properties of Rationals","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove this statement by contradiction, we begin by assuming the exact opposite of what we want to show. \n\nTherefore, we assume that the difference between a rational number and an irrational number is **rational**."},{"type":"text","order":1,"title":"Define variables algebraically","content":"Let's translate our assumption into algebraic terms:\n- Let the rational number be $\\frac{a}{b}$, where $a$ and $b$ are integers and $b \\neq 0$.\n- Let the irrational number be $c$.\n- Since we assume their difference is a rational number, we can write the difference as $\\frac{d}{e}$, where $d$ and $e$ are integers and $e \\neq 0$.\n\nThis gives us our initial equation:\n$$\\frac{a}{b} - c = \\frac{d}{e}$$"},{"type":"text","order":2,"title":"Isolate the irrational number","content":"Next, we rearrange the equation to isolate the irrational number $c$ on one side:\n\n$$\\begin{aligned} \\frac{a}{b} - c &= \\frac{d}{e} \\\\ c &= \\frac{a}{b} - \\frac{d}{e} \\end{aligned}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$c = \\frac{\\htmlData{anchor=num}{\\color{@sky}{ae - bd}}}{\\htmlData{anchor=den}{\\color{@purple}{be}}}$"},{"text":"Since $a, b, d, e$ are integers, both the numerator and denominator are integers."}],"highlights":[{"color":"sky","style":"box","anchor":"num"},{"color":"purple","style":"box","anchor":"den"}],"connections":[]},"order":3,"title":"Apply closure properties","description":"We combine the fractions on the right-hand side using a common denominator to express $c$ as a single quotient."},{"type":"text","order":4,"title":"Identify the contradiction","content":"Let's analyze the fraction we just found:\n1. The numerator, $ae - bd$, is an integer because integers are closed under multiplication and subtraction.\n2. The denominator, $be$, is a non-zero integer since neither $b$ nor $e$ is zero.\n\nBecause $c$ can be written as the ratio of two integers, $c$ must be a **rational** number. \n\nHowever, this directly contradicts our given information that $c$ is an **irrational** number. Since our assumption leads to a contradiction, the assumption must be false.\n\nThus, the difference between a rational number and an irrational number is indeed **irrational**."}]}],"generatedAt":"2026-06-09T03:24:41.908Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4d9bbc87-f857-47e1-b785-70d816b38969:fraction_definition","label":"Expressing Rationals as Fractions","steps":[{"type":"text","order":0,"title":"State the counter-assumption","content":"To prove the statement by contradiction, we begin by assuming the opposite of what we want to prove.\n\nWe assume that the difference between a rational number and an irrational number is **rational**.\n\nThis is a crucial first step that earns the first mark **(B1)**."},{"type":"text","order":1,"title":"Define the variables","content":"Let's define our numbers formally:\n- Let the rational number be $r = \\frac{a}{b}$, where $a$ and $b$ are integers and $b \\neq 0$.\n- Let the irrational number be $c$.\n- Let their difference be another rational number, $d = \\frac{p}{q}$, where $p$ and $q$ are integers and $q \\neq 0$.\n\nDefining these terms algebraically using fractions of integers earns the next mark **(M1)**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=difference-eq}{\\frac{a}{b} - c = \\frac{p}{q}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"difference-eq"}],"connections":[]},"order":2,"title":"Set up the equation","description":"We now set up an equation representing their difference:\n\n$$\\frac{a}{b} - c = \\frac{p}{q}$$\n\nThis expression earns the method mark **(M1)**."},{"type":"text","order":3,"title":"Isolate the irrational number","content":"Next, we rearrange the equation to isolate the irrational number $c$ on one side:\n\n$$\\begin{aligned} \\frac{a}{b} - \\frac{p}{q} &= c \\\\ c &= \\frac{a}{b} - \\frac{p}{q} \\end{aligned}$$\n\nThis step earns a rearrangement mark **(M1)**."},{"type":"text","order":4,"title":"Combine into a single fraction","content":"We can write the right-hand side as a single fraction by finding a common denominator:\n\n$$c = \\frac{aq - bp}{bq}$$\n\nSince $a, b, p,$ and $q$ are all integers, it follows that:\n- The numerator $aq - bp$ is an integer (since the product and difference of integers are also integers).\n- The denominator $bq$ is a non-zero integer (since $b \\neq 0$ and $q \\neq 0$).\n\nWriting $c$ as a single fraction of integers earns the accuracy mark **(A1)**."},{"type":"text","order":5,"title":"Conclude the proof","content":"Because $c = \\frac{aq - bp}{bq}$ is the ratio of two integers, $c$ must be a **rational** number.\n\nHowever, this directly contradicts our initial definition that $c$ is **irrational**.\n\nTherefore, our original assumption must be false, proving that the difference between a rational number and an irrational number is indeed **irrational** **(A1)**."}]},{"id":"4d9bbc87-f857-47e1-b785-70d816b38969:closure_properties","label":"Using Closure Properties of Rationals","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove this statement by contradiction, we begin by assuming the exact opposite of what we want to show. \n\nTherefore, we assume that the difference between a rational number and an irrational number is **rational**."},{"type":"text","order":1,"title":"Define variables algebraically","content":"Let's translate our assumption into algebraic terms:\n- Let the rational number be $\\frac{a}{b}$, where $a$ and $b$ are integers and $b \\neq 0$.\n- Let the irrational number be $c$.\n- Since we assume their difference is a rational number, we can write the difference as $\\frac{d}{e}$, where $d$ and $e$ are integers and $e \\neq 0$.\n\nThis gives us our initial equation:\n$$\\frac{a}{b} - c = \\frac{d}{e}$$"},{"type":"text","order":2,"title":"Isolate the irrational number","content":"Next, we rearrange the equation to isolate the irrational number $c$ on one side:\n\n$$\\begin{aligned} \\frac{a}{b} - c &= \\frac{d}{e} \\\\ c &= \\frac{a}{b} - \\frac{d}{e} \\end{aligned}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$c = \\frac{\\htmlData{anchor=num}{\\color{@sky}{ae - bd}}}{\\htmlData{anchor=den}{\\color{@purple}{be}}}$"},{"text":"Since $a, b, d, e$ are integers, both the numerator and denominator are integers."}],"highlights":[{"color":"sky","style":"box","anchor":"num"},{"color":"purple","style":"box","anchor":"den"}],"connections":[]},"order":3,"title":"Apply closure properties","description":"We combine the fractions on the right-hand side using a common denominator to express $c$ as a single quotient."},{"type":"text","order":4,"title":"Identify the contradiction","content":"Let's analyze the fraction we just found:\n1. The numerator, $ae - bd$, is an integer because integers are closed under multiplication and subtraction.\n2. The denominator, $be$, is a non-zero integer since neither $b$ nor $e$ is zero.\n\nBecause $c$ can be written as the ratio of two integers, $c$ must be a **rational** number. \n\nHowever, this directly contradicts our given information that $c$ is an **irrational** number. Since our assumption leads to a contradiction, the assumption must be false.\n\nThus, the difference between a rational number and an irrational number is indeed **irrational**."}]}],"generatedAt":"2026-06-09T03:24:53.986Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":41},"walkthroughVideo":null},{"id":"135c4fa2-edaf-426c-aff5-6dff03228d23","revisionId":"274f1618-3d69-42c6-8f04-a0dcdf160b51","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"0f2c61c9-2637-4bf9-863f-bbf2faba715e","label":"a","content":"Prove by contradiction that $\\sqrt{3}$ is irrational.","markscheme":"- Assume $\\sqrt{3} = \\frac{a}{b}$ where $a$ and $b$ are integers with no common factors except 1 B1\n- Square both sides: $3 = \\frac{a^2}{b^2} \\Rightarrow 3b^2 = a^2$ M1\n- State $a^2$ (and $a$) must be a multiple of 3 A1\n- Substitute $a = 3n$: $3b^2 = (3n)^2 = 9n^2 \\Rightarrow b^2 = 3n^2$ M1\n- State $b^2$ (and $b$) must also be a multiple of 3 A1\n- Conclude that $a$ and $b$ have a common factor of 3, contradicting the assumption that they have no common factors. A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Assume rationality in simplest form with coprime integers.","code":"B"},{"id":"p2","marks":1,"description":"Square and rearrange to relate a^2 and b^2.","code":"M"},{"id":"p3","marks":1,"description":"Deduce that a^2 and a are multiples of 3.","code":"A"},{"id":"p4","marks":1,"description":"Substitute a=3n and simplify for b^2.","code":"M"},{"id":"p5","marks":1,"description":"Deduce that b^2 and b are multiples of 3.","code":"A"},{"id":"p6","marks":1,"description":"Conclude that both share a factor, leading to a contradiction.","code":"A"}],"finalAnswers":["Hence sqrt(3) is irrational"],"generatedAt":"2026-05-06T14:04:27.672Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"coprime_integers","label":"Standard Coprime Assumption","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming the opposite: that $\\sqrt{3}$ is a rational number.\n\nBy definition, any rational number can be written as a fraction in its simplest form:\n\n$$\\text{Assume } \\sqrt{3} = \\frac{a}{b}$$\n\nwhere $a$ and $b$ are integers with no common factors other than $1$ (meaning they are coprime), and $b \\neq 0$.\n\n*Award **1 mark (B1)** for stating this initial assumption clearly.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sqrt{3} = \\frac{\\htmlData{anchor=a-term}{a}}{\\htmlData{anchor=b-term}{b}}$"},{"latex":"$$3 = \\frac{\\htmlData{anchor=asq-term}{a^2}}{\\htmlData{anchor=bsq-term}{b^2}}$"},{"latex":"$$\\htmlData{anchor=lhs}{3b^2} = \\htmlData{anchor=rhs}{a^2}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"b-term"}],"connections":[{"to":"asq-term","from":"a-term","color":"sky","label":"square","style":"solid"},{"to":"bsq-term","from":"b-term","color":"amber","label":"square","style":"solid"}]},"order":1,"title":"Square and rearrange","description":"Square both sides of the equation to eliminate the square root, then multiply both sides by $b^2$ to clear the fraction. This gives us $3b^2 = a^2$, earning the **M1** mark."},{"type":"text","order":2,"title":"Analyze $a^2$ and $a$","content":"Look at the rearranged equation:\n\n$$a^2 = 3b^2$$\n\nSince $b$ is an integer, $b^2$ is also an integer. This means $a^2$ is equal to $3$ times an integer, so $a^2$ must be a multiple of $3$.\n\nBecause $3$ is a prime number, if the square of an integer is a multiple of $3$, then the integer itself must also be a multiple of $3$. \n\nTherefore, $a$ is a multiple of $3$.\n\n*Award **1 mark (A1)** for concluding that $a^2$ (and $a$) is a multiple of $3$.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let } \\htmlData{anchor=a-def}{\\color{@sky}{a = 3n}} \\text{ for some integer } n$"},{"latex":"$$3b^2 = (\\htmlData{anchor=a-sub}{\\color{@sky}{3n}})^2$"},{"latex":"$$3b^2 = \\htmlData{anchor=expanded}{9n^2}$"},{"latex":"$$\\htmlData{anchor=final}{b^2 = 3n^2}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-def"},{"color":"sky","style":"text-color","anchor":"a-sub"}],"connections":[{"to":"a-sub","from":"a-def","color":"sky","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute and simplify","description":"Since $a$ is a multiple of $3$, we can express it as $a = 3n$ for some integer $n$. Substitute this back into our equation $3b^2 = a^2$ and divide both sides by $3$ to simplify. This substitution and simplification step earns the **M1** mark."},{"type":"text","order":4,"title":"Analyze $b^2$ and $b$","content":"Now examine our simplified equation:\n\n$$b^2 = 3n^2$$\n\nSince $n$ is an integer, $n^2$ is also an integer. This tells us that $b^2$ is $3$ times an integer, which means $b^2$ must be a multiple of $3$.\n\nUsing the same prime number property as before, if $b^2$ is a multiple of $3$, then $b$ must also be a multiple of $3$.\n\n*Award **1 mark (A1)** for concluding that $b^2$ (and $b$) is a multiple of $3$.*"},{"type":"text","order":5,"title":"State the contradiction","content":"We have established that:\n1. $a$ is a multiple of $3$.\n2. $b$ is a multiple of $3$.\n\nThis means both $a$ and $b$ share a common factor of $3$. \n\nHowever, this directly contradicts our initial assumption that $a$ and $b$ are coprime and share no common factors other than $1$.\n\nBecause this contradiction has arisen, our initial assumption that $\\sqrt{3}$ is rational must be false. Therefore, $\\sqrt{3}$ is irrational.\n\n*Award the final **1 mark (A1)** for completing the proof with this logical conclusion.*"}]},{"id":"unique_factorization","label":"Unique Prime Factorization","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming the opposite: that $\\sqrt{3}$ is a rational number.\n\nBy definition, a rational number can be expressed as a fraction of two positive integers $a$ and $b$:\n\n$$\\sqrt{3} = \\frac{a}{b} \\quad \\text{where } a, b \\in \\mathbb{Z}^+$$"},{"type":"text","order":1,"title":"Rearrange and square","content":"Next, we square both sides of the equation to eliminate the square root, and then rearrange it to express it in terms of squares:\n\n$$\\begin{aligned} 3 &= \\frac{a^2}{b^2} \\\\ a^2 &= 3b^2 \\end{aligned}$$"},{"type":"text","order":2,"title":"Analyze prime factors","content":"According to the **Fundamental Theorem of Arithmetic**, every integer greater than $1$ has a unique prime factorization.\n\nIf we look at the prime factorization of any positive integer $x$:\n$$x = p_1^{e_1} p_2^{e_2} \\cdots p_k^{e_k}$$\n\nWhen we square it, the exponents of all its prime factors double:\n$$x^2 = p_1^{2e_1} p_2^{2e_2} \\cdots p_k^{2e_k}$$\n\nThis means that in the unique prime factorization of any perfect square, **the exponent of every prime factor (including $3$) must be an even integer** (possibly $0$)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a^2 = 3b^2$"},{"latex":"\\text{Power of 3 in } a^2 \\text{ is } \\htmlData{anchor=even-pow}{\\color{@sky}{2m}} \\quad \\text{where } m \\ge 0"},{"latex":"\\text{Power of 3 in } 3b^2 \\text{ is } \\htmlData{anchor=odd-pow}{\\color{@orange}{2n + 1}} \\quad \\text{where } n \\ge 0"}],"highlights":[{"color":"sky","style":"box","anchor":"even-pow"},{"color":"orange","style":"box","anchor":"odd-pow"}],"connections":[{"to":"odd-pow","from":"even-pow","color":"red","label":"Contradiction: Even ≠ Odd","style":"dashed"}]},"order":3,"title":"Compare exponents of 3","description":"Let $m$ be the power of the prime factor $3$ in $a$, and $n$ be the power of $3$ in $b$. We compare the total exponent of $3$ on both sides of the equation."},{"type":"text","order":4,"title":"State the contradiction","content":"Since $a^2 = 3b^2$, the unique prime factorization of both sides must be identical. Therefore, the exponent of the prime $3$ on both sides must be equal:\n\n$$2m = 2n + 1$$\n\nHowever, $2m$ is always an **even** integer, while $2n + 1$ is always an **odd** integer. An even integer can never equal an odd integer, which violates the uniqueness of prime factorization."},{"type":"text","order":5,"title":"Conclude the proof","content":"Our assumption that $\\sqrt{3}$ is rational has led to a fundamental contradiction.\n\nTherefore, the assumption must be false, and we conclude that **$\\sqrt{3}$ is irrational**."}]}],"generatedAt":"2026-06-09T03:20:02.535Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0f2c61c9-2637-4bf9-863f-bbf2faba715e:coprime_integers","label":"Standard Coprime Assumption","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming the opposite: that $\\sqrt{3}$ is a rational number.\n\nBy definition, any rational number can be written as a fraction in its simplest form:\n\n$$\\text{Assume } \\sqrt{3} = \\frac{a}{b}$$\n\nwhere $a$ and $b$ are integers with no common factors other than $1$ (meaning they are coprime), and $b \\neq 0$.\n\n*Award **1 mark (B1)** for stating this initial assumption clearly.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sqrt{3} = \\frac{\\htmlData{anchor=a-term}{a}}{\\htmlData{anchor=b-term}{b}}$"},{"latex":"$$3 = \\frac{\\htmlData{anchor=asq-term}{a^2}}{\\htmlData{anchor=bsq-term}{b^2}}$"},{"latex":"$$\\htmlData{anchor=lhs}{3b^2} = \\htmlData{anchor=rhs}{a^2}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-term"},{"color":"amber","style":"text-color","anchor":"b-term"}],"connections":[{"to":"asq-term","from":"a-term","color":"sky","label":"square","style":"solid"},{"to":"bsq-term","from":"b-term","color":"amber","label":"square","style":"solid"}]},"order":1,"title":"Square and rearrange","description":"Square both sides of the equation to eliminate the square root, then multiply both sides by $b^2$ to clear the fraction. This gives us $3b^2 = a^2$, earning the **M1** mark."},{"type":"text","order":2,"title":"Analyze $a^2$ and $a$","content":"Look at the rearranged equation:\n\n$$a^2 = 3b^2$$\n\nSince $b$ is an integer, $b^2$ is also an integer. This means $a^2$ is equal to $3$ times an integer, so $a^2$ must be a multiple of $3$.\n\nBecause $3$ is a prime number, if the square of an integer is a multiple of $3$, then the integer itself must also be a multiple of $3$. \n\nTherefore, $a$ is a multiple of $3$.\n\n*Award **1 mark (A1)** for concluding that $a^2$ (and $a$) is a multiple of $3$.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let } \\htmlData{anchor=a-def}{\\color{@sky}{a = 3n}} \\text{ for some integer } n$"},{"latex":"$$3b^2 = (\\htmlData{anchor=a-sub}{\\color{@sky}{3n}})^2$"},{"latex":"$$3b^2 = \\htmlData{anchor=expanded}{9n^2}$"},{"latex":"$$\\htmlData{anchor=final}{b^2 = 3n^2}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-def"},{"color":"sky","style":"text-color","anchor":"a-sub"}],"connections":[{"to":"a-sub","from":"a-def","color":"sky","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute and simplify","description":"Since $a$ is a multiple of $3$, we can express it as $a = 3n$ for some integer $n$. Substitute this back into our equation $3b^2 = a^2$ and divide both sides by $3$ to simplify. This substitution and simplification step earns the **M1** mark."},{"type":"text","order":4,"title":"Analyze $b^2$ and $b$","content":"Now examine our simplified equation:\n\n$$b^2 = 3n^2$$\n\nSince $n$ is an integer, $n^2$ is also an integer. This tells us that $b^2$ is $3$ times an integer, which means $b^2$ must be a multiple of $3$.\n\nUsing the same prime number property as before, if $b^2$ is a multiple of $3$, then $b$ must also be a multiple of $3$.\n\n*Award **1 mark (A1)** for concluding that $b^2$ (and $b$) is a multiple of $3$.*"},{"type":"text","order":5,"title":"State the contradiction","content":"We have established that:\n1. $a$ is a multiple of $3$.\n2. $b$ is a multiple of $3$.\n\nThis means both $a$ and $b$ share a common factor of $3$. \n\nHowever, this directly contradicts our initial assumption that $a$ and $b$ are coprime and share no common factors other than $1$.\n\nBecause this contradiction has arisen, our initial assumption that $\\sqrt{3}$ is rational must be false. Therefore, $\\sqrt{3}$ is irrational.\n\n*Award the final **1 mark (A1)** for completing the proof with this logical conclusion.*"}]},{"id":"0f2c61c9-2637-4bf9-863f-bbf2faba715e:unique_factorization","label":"Unique Prime Factorization","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming the opposite: that $\\sqrt{3}$ is a rational number.\n\nBy definition, a rational number can be expressed as a fraction of two positive integers $a$ and $b$:\n\n$$\\sqrt{3} = \\frac{a}{b} \\quad \\text{where } a, b \\in \\mathbb{Z}^+$$"},{"type":"text","order":1,"title":"Rearrange and square","content":"Next, we square both sides of the equation to eliminate the square root, and then rearrange it to express it in terms of squares:\n\n$$\\begin{aligned} 3 &= \\frac{a^2}{b^2} \\\\ a^2 &= 3b^2 \\end{aligned}$$"},{"type":"text","order":2,"title":"Analyze prime factors","content":"According to the **Fundamental Theorem of Arithmetic**, every integer greater than $1$ has a unique prime factorization.\n\nIf we look at the prime factorization of any positive integer $x$:\n$$x = p_1^{e_1} p_2^{e_2} \\cdots p_k^{e_k}$$\n\nWhen we square it, the exponents of all its prime factors double:\n$$x^2 = p_1^{2e_1} p_2^{2e_2} \\cdots p_k^{2e_k}$$\n\nThis means that in the unique prime factorization of any perfect square, **the exponent of every prime factor (including $3$) must be an even integer** (possibly $0$)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a^2 = 3b^2$"},{"latex":"\\text{Power of 3 in } a^2 \\text{ is } \\htmlData{anchor=even-pow}{\\color{@sky}{2m}} \\quad \\text{where } m \\ge 0"},{"latex":"\\text{Power of 3 in } 3b^2 \\text{ is } \\htmlData{anchor=odd-pow}{\\color{@orange}{2n + 1}} \\quad \\text{where } n \\ge 0"}],"highlights":[{"color":"sky","style":"box","anchor":"even-pow"},{"color":"orange","style":"box","anchor":"odd-pow"}],"connections":[{"to":"odd-pow","from":"even-pow","color":"red","label":"Contradiction: Even ≠ Odd","style":"dashed"}]},"order":3,"title":"Compare exponents of 3","description":"Let $m$ be the power of the prime factor $3$ in $a$, and $n$ be the power of $3$ in $b$. We compare the total exponent of $3$ on both sides of the equation."},{"type":"text","order":4,"title":"State the contradiction","content":"Since $a^2 = 3b^2$, the unique prime factorization of both sides must be identical. Therefore, the exponent of the prime $3$ on both sides must be equal:\n\n$$2m = 2n + 1$$\n\nHowever, $2m$ is always an **even** integer, while $2n + 1$ is always an **odd** integer. An even integer can never equal an odd integer, which violates the uniqueness of prime factorization."},{"type":"text","order":5,"title":"Conclude the proof","content":"Our assumption that $\\sqrt{3}$ is rational has led to a fundamental contradiction.\n\nTherefore, the assumption must be false, and we conclude that **$\\sqrt{3}$ is irrational**."}]}],"generatedAt":"2026-06-09T03:20:14.719Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":62},"walkthroughVideo":null}],"subjectCode":"a_level_maths__wjec","summaries":[{"id":"fd7ef5e5-e672-484d-827d-216a0bf187eb","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"004f1715-7f20-44bd-b303-beb33cb8f946","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0f28af03-b719-4ce4-8b8a-9e1d7a79b4a7","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"135c4fa2-edaf-426c-aff5-6dff03228d23","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"26f0bc1d-bd36-470c-847c-ee92ef94d932","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"41bbfce2-2bcf-4f6a-b290-e3abcbb7a077","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"510dbdcb-d107-4e6d-ab01-cf07807fc6f9","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"6587ad34-dbe7-4eaf-93db-d0d216186419","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6b925623-c577-4449-a2dd-7a78f36a658f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"be93578c-b7d2-4df6-acd7-6ef98a572b2c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"e56fe79b-a1e4-4b6d-9dca-a4042e1cf70b","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f5006b00-71ec-4fe1-bfd6-a3689efa8c22","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"fe9a1d93-a8a9-4e0d-aca3-70ae3b951191","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b5b2f95d-72a6-49b2-9d75-4662197e54af","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"23ef56dd-b092-41bc-abe8-4a2e89a6592d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"47c36499-dd4e-4e1f-92ce-655791bcc3c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6c36ee4d-6b4b-4ff9-accd-9ae7dbccb7ec","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6c4ee647-e514-46f6-88b7-ec0733bbe1ed","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8dc028ae-d62c-450b-ace3-35c6f6f6a7f8","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"93607e6d-2a9b-443f-a9bb-40c9577b10e1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9e651ba3-893d-4d7b-a060-3cd50b14dc4b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"a3db02dd-cb03-45ff-bdb6-a8d761189719","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"aa3791df-7481-43a1-b0b9-cb8e3bcc9c44","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c7d9a82a-fa0f-4093-aeb3-1480c2a1ac5f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d4188522-4341-49ba-a1c1-b0d2cd27b056","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fe45c078-ebe7-4562-88d0-081c3c872bf2","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9b74fb76-e5d9-40d0-9ed3-b77c0f17ab33","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"13c9487f-fdfe-413a-bd71-4821d6b1f4c3","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2228aba7-e076-4386-af3a-881069ea3ae9","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3f64531b-355c-4f81-9254-e606a290abcd","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4470672a-f3bb-48d3-96e8-994a0a98913b","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"5198bcf0-d6eb-42d9-8112-78ad2e6eaba6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6c7892f5-02da-412a-8c42-cc99863a9e88","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8ca4ab45-dbd5-45b2-958e-815a8065d7c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9121389d-68f4-4fe8-b1ac-5609535df620","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"99118445-8ab0-44a7-80de-6bf301bd4ed0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"cd862080-eb92-428a-a568-4df58ad39380","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"cef11906-7830-41ee-a754-d85efa8f052e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fe7a6f3f-171d-45bc-92d9-1de3f6bfdcee","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9c62f3fe-64d0-47f3-b702-a3a3e8ff75ae","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"16d05447-9328-4f4f-8b04-cfecf208a89a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d071f313-7932-4ad0-aab3-9f797eda3853","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6bf375b8-ee1c-473b-81a2-7e09b5dab084","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e422d31a-b12d-4fc1-944e-8dde69e2113c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"89d385f6-788d-4b29-8eb9-98e58f6cdf0b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"16ae10ed-591f-4bc2-9325-d69489e15934","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"163a5d57-77a2-4ba5-953f-1b38f65f8c9e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"242514c0-179f-43d9-8f6f-6909d182f51f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"04be3b2f-ee91-4e8e-aee3-c7538271be15","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0facf6a1-ce05-494a-a17f-d1e24cc5518d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"110f33b5-c52d-4784-96fb-188f2dbcc3d7","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"18ada11b-472e-44dd-8867-c451904a517a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"221a8bcc-2438-4aa5-9af2-a39c38971592","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"24e423bc-3e1f-4e0f-a5c0-560cbaf277d9","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"28ff3fa0-a34c-47b1-a0da-e821e3dc1d38","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"2a409465-939f-416b-9cde-8ab37e855f68","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2f78ea22-8340-4230-ad26-c371057132cf","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"30867d72-5744-4bfc-8cb8-ad6997d311ea","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"32d377a6-7e21-4be7-b952-8d7abc33cb85","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"335f9cac-0fa2-4509-b809-57630ededfff","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"380d8433-3d23-41bc-9b56-96df9a7d3e2b","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"3989536d-520e-433b-a640-28d6948a0c5d","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3e0420d1-3854-4158-a1ca-3db7c0bb5383","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3f5dfa65-d35a-4dc6-b83c-24ec3a347c5d","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"3fc583bb-f920-4ef7-9d09-f8c7cf7a28c8","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"42297193-1333-43f2-aa61-ae3fa3307b39","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"4a91330d-930c-4f9b-ad01-662049f645a1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4b909e4c-24ab-43c5-9c0c-4120c12a3030","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"556163c1-5661-443f-9a3a-18d926fa39a0","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"55ce197c-d30d-4299-b46e-0fcccbaf663a","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5980d418-006d-4ce1-9f63-7b5328010c01","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"59fd7d5a-d4d9-461f-b440-b94339dd8d64","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"627d78fa-3b88-4bf9-b8c7-bce4d034d9ef","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"671e773e-16b4-4a30-a467-e17fc654b8f1","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"6783b9fd-dd5e-43af-80ba-899510b6566e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"6a6a3ade-c77a-4fc4-bc26-74bf76e58e6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7175b86e-9fff-43b4-a327-41703bba4199","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7507acb4-cc2e-4a16-9d97-2dd4b8c3c7b9","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7824cce4-cdc6-48e1-b186-a9dcdc172e79","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"78ece143-0564-451e-bf6c-2923f38164f3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"7ab76ecd-976a-485e-b4fa-57191d6ef0b3","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"7b4d24e7-ab8c-46d9-abf2-a96ab239f484","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7c3def0f-80a7-43dc-a681-5029075dc4ac","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7d969d3a-9fc7-4d26-ab86-c45c0b8ba0de","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"848766c4-ab44-48b4-8b99-de0761912209","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"8abefdd6-fbf2-4b4d-a23a-85262f20fe27","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8e8ef60f-c37c-474f-b8ca-4c68daeeb75f","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"999aa15f-87c9-4201-9c01-48914ab348df","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"9d70be20-7d2f-4ba4-abe4-81fe965b006e","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9fe290e1-2433-4a33-b885-81ce7d4d4972","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a30286d7-5984-4bd6-aa68-ad8db5cb5584","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"adaa56f4-551d-4bdc-9b73-45f3148bd551","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b67c9afd-5e2b-441d-8be2-91adbf81317c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b7f385f4-7aab-4565-8fdb-c7e461c7e4f3","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b7fbf628-3084-47f8-8f14-a603e293136e","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b9facce6-e77c-47a1-ae79-3c336b9eb192","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ca749a66-7845-4abd-9efb-f717cd52b067","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d5a1ffe9-fb01-4354-8faa-227ea2a944f7","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"d6b1d512-3484-4b9a-b965-7f69406f2fe0","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d9c2f77c-1079-4219-90d7-82ec2213012e","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e2a673a1-9266-4837-90f6-f29a690b2107","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ef88d367-8d71-41e2-9443-fc32c8140de7","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"f1574cc4-4197-4312-8406-33fa51e7701c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f6c128b2-2f8d-4f0b-ad82-f672fd179ac4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f77fc775-be8f-46a1-a32d-54e49082f630","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f8ecec2a-ee26-4f16-9936-e82039f53785","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fd93ba39-853b-4ec4-9b28-6f485059e32c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true}],"title":"3.1.1 Proof (A-level only)","topicId":"23c24b47-7b3c-4108-bb70-3fe34105de38","topicSidebar":"$L3c","topicSlug":"proof-6","topicTitle":"3.1.1 Proof (A-level only)"}]}],"$L3d","$L3e"]}]

3.1.1 Proof (A-level only)

Question 1

3.1.1 Proof (A-level only) Questions

3.1.1 Proof (A-level only)

Question 1

3.1.1 Proof (A-level only) Questions