34:I[651451,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"UserProgressProvider"] 35:I[196793,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"Gingerbread"] 37:I[716297,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"TopicRecentlyViewed"] 38:I[647935,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"RecordResourceResume"] 39:I[426249,["/_next/static/chunks/3j194e876w45f.js?dpl=nogit-1790227106","/_next/static/chunks/060-i_62-vse1.js?dpl=nogit-1790227106","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790227106","/_next/static/chunks/2yhf9vq09dqoa.js?dpl=nogit-1790227106","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790227106","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790227106","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790227106","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790227106","/_next/static/chunks/316m85d5s15pp.js?dpl=nogit-1790227106","/_next/static/chunks/2_itvjgqsumf8.js?dpl=nogit-1790227106","/_next/static/chunks/0be1c60akah5d.js?dpl=nogit-1790227106","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790227106","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790227106","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790227106","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790227106","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790227106","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790227106","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790227106","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790227106","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790227106","/_next/static/chunks/1bbv7o2-m-0ni.js?dpl=nogit-1790227106","/_next/static/chunks/097slu7xaeast.js?dpl=nogit-1790227106","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790227106","/_next/static/chunks/117yqm55h4y_p.js?dpl=nogit-1790227106","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790227106","/_next/static/chunks/2eoustw-nfu98.js?dpl=nogit-1790227106","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790227106","/_next/static/chunks/11ryphth5_v5b.js?dpl=nogit-1790227106","/_next/static/chunks/0gvj8_2lj8s8m.js?dpl=nogit-1790227106","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790227106","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790227106","/_next/static/chunks/2ydqdr10ndcei.js?dpl=nogit-1790227106","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790227106","/_next/static/chunks/1br6ag7ig1vnn.js?dpl=nogit-1790227106","/_next/static/chunks/1bnld2zwzjlzx.js?dpl=nogit-1790227106","/_next/static/chunks/09_ag440ieydu.js?dpl=nogit-1790227106","/_next/static/chunks/3_o3al2nkyqic.js?dpl=nogit-1790227106","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790227106","/_next/static/chunks/2m_zh5-tarne0.js?dpl=nogit-1790227106","/_next/static/chunks/3kvhppczyvgp2.js?dpl=nogit-1790227106","/_next/static/chunks/2fk-2o4srnsgz.js?dpl=nogit-1790227106","/_next/static/chunks/00ol_16j3a2oa.js?dpl=nogit-1790227106","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790227106","/_next/static/chunks/2srfni4gkx9g6.js?dpl=nogit-1790227106","/_next/static/chunks/21sij0j_ujna2.js?dpl=nogit-1790227106","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790227106","/_next/static/chunks/2hccrcvndngge.js?dpl=nogit-1790227106","/_next/static/chunks/0ix6b88ybjhi_.js?dpl=nogit-1790227106","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790227106","/_next/static/chunks/1c50trxtlnlob.js?dpl=nogit-1790227106","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790227106","/_next/static/chunks/1jgxupn92tn-_.js?dpl=nogit-1790227106","/_next/static/chunks/2v0ixm9u4yz3r.js?dpl=nogit-1790227106","/_next/static/chunks/3ghbpnibq2fva.js?dpl=nogit-1790227106","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790227106","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790227106","/_next/static/chunks/1y5xmyjgu8grb.js?dpl=nogit-1790227106","/_next/static/chunks/1nnw6yom6-qhh.js?dpl=nogit-1790227106","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790227106","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790227106","/_next/static/chunks/12_xkdi18mnhe.js?dpl=nogit-1790227106","/_next/static/chunks/2wp4esku6l29e.js?dpl=nogit-1790227106","/_next/static/chunks/044q84i54lw83.js?dpl=nogit-1790227106","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790227106","/_next/static/chunks/2e0e0lyynppq_.js?dpl=nogit-1790227106","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790227106","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790227106","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790227106","/_next/static/chunks/2qrlyi_b3nfc0.js?dpl=nogit-1790227106","/_next/static/chunks/2s1hkstzlmj2o.js?dpl=nogit-1790227106"],"TopicQuestionBank"] 24:["$","$L34",null,{"bundle":{"topicIds":["30ac4c05-8d09-4ec4-ac0a-b9371d6b1f6e","8d894817-36cb-40f0-8968-d1d415ffb98f","683a920c-73f7-4a01-8dfc-64a655f29aa8","d44aee81-cb1e-48fe-948d-5396f0f1f365","938e6248-950b-489c-9a96-7f4a26e31850","9ff7b8ea-0937-484d-9c06-fa7aeb59ebaf","05578944-695e-4821-a312-9c0f01f63afc","85990def-e028-4a2c-9cd7-e4ebfcebec36","ba2745c7-ed70-423d-9c1d-83e9bb578508"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L35",null,{"feature":"question-bank"}],"$L36",["$","$L37",null,{"subjectCode":"a_level_maths__wjec","subjectName":"Maths","topicSlug":"trigonometry-7","topicTitle":"3.5 Trigonometry (A-level only)","sectionTitle":null,"activity":"worksheet","resourceTitle":"3.5 Trigonometry (A-level only) Questions","resourceHref":"/a-level/maths/wjec/trigonometry-7/questions/q1","saveTrail":false}],["$","$L38",null,{"href":"/a-level/maths/wjec/trigonometry-7/questions/q1","label":"3.5 Trigonometry (A-level only)","subjectCode":"a_level_maths__wjec","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 210 – 3.5 Trigonometry (A-level only) - Questions\",\"description\":\"Practise 3.5 Trigonometry (A-level only) A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"3.5 Trigonometry (A-level only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/wjec/trigonometry-7/questions/q210\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L39",null,{"backHref":"/a-level/maths/wjec/questions","basePath":"/a-level/maths/wjec/trigonometry-7/questions","formulaUrl":"https://www.wjec.co.uk/media/4jfdytcz/ss-w27-18-e.pdf","initialIndex":209,"initialQuestionWindow":[{"id":"ec6a07be-f4c6-49e6-9f51-db095a2d048e","revisionId":"45c531a2-bbe8-4dac-bed4-37b593bccc87","questionNumber":null,"questionType":"LA","specification":"A mechanical linkage tracks an angular displacement \$\\theta\$ such that its extension \$E\$ is modelled by the function \\[E(\\theta) = -2 \\csc(\\theta + \\pi)\\] for \$0 \\le \\theta \\le 2\\pi\$.","difficulty":5,"difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"7c57a61b-7714-4bbc-8aca-2afb3b1fa751","label":null,"content":"Sketch the graph of \$E\$ against \$\\theta\$, stating the equations of any vertical asymptotes and the coordinates of the local turning points.","markscheme":"- Utilises the identity $\\csc(\\theta + \\pi) = -\\csc \\theta$ (or via expansion of $\\sin(\\theta + \\pi)$) to simplify the function to $E(\\theta) = 2 \\csc \\theta$ M1\n- Identifies and states the equations of the vertical asymptotes as $\\theta = 0, \\theta = \\pi,$ and $\\theta = 2\\pi$ B1\n- Sketches two correct 'U-shaped' reciprocal curves: one opening upwards in the interval $0 < \\theta < \\pi$ and one opening downwards in the interval $\\pi < \\theta < 2\\pi$ M1\n- Correctly identifies the local minimum at $(\\frac{\\pi}{2}, 2)$ and the local maximum at $(\\frac{3\\pi}{2}, -2)$ A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"trig_simplification","label":"Trigonometric Identity Simplification","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sin(\\theta + \\pi) = -\\sin\\theta$"},{"latex":"$$E(\\theta) = -2 \\csc(\\theta + \\pi) = \\htmlData{anchor=step1_orig}{\\color{@sky}{-2 \\left(\\frac{1}{\\sin(\\theta + \\pi)}\\right)}}$"},{"latex":"$$E(\\theta) = -2 \\left(\\frac{1}{\\htmlData{anchor=step1_sub}{\\color{@sky}{-\\sin\\theta}}}\\right) = \\htmlData{anchor=step1_final}{\\color{@amber}{2 \\csc\\theta}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"step1_orig"},{"color":"sky","style":"box","anchor":"step1_sub"},{"color":"amber","style":"text-color","anchor":"step1_final"}],"connections":[{"to":"step1_sub","from":"step1_orig","color":"sky","label":"Substitute identity","style":"solid"}]},"order":0,"title":"Simplify the Function","description":"We begin by simplifying $E(\\theta) = -2 \\csc(\\theta + \\pi)$ using the trigonometric identity $\\sin(\\theta + \\pi) = -\\sin\\theta$, which means $\\csc(\\theta + \\pi) = -\\csc\\theta$. This earns the first mark M1."},{"type":"text","order":1,"title":"Find the Vertical Asymptotes","content":"The simplified function is:\n\n$$E(\\theta) = \\frac{2}{\\sin\\theta}$$\n\nVertical asymptotes occur where the denominator is equal to zero, i.e., where $\\sin\\theta = 0$. Within the domain $0 \\le \\theta \\le 2\\pi$, this occurs at:\n\n$$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$\n\nStating these three vertical asymptotes correctly earns B1."},{"type":"text","order":2,"title":"Determine the Turning Points","content":"To find the coordinates of the turning points, we evaluate $E(\\theta) = 2\\csc\\theta$ at critical values:\n\n- On the interval $0 < \\theta < \\pi$, $\\sin\\theta > 0$. The maximum value of $\\sin\\theta$ is $1$ at $\\theta = \\frac{\\pi}{2}$, which corresponds to the minimum value of the reciprocal function:\n $$E\\left(\\frac{\\pi}{2}\\right) = \\frac{2}{\\sin\\left(\\frac{\\pi}{2}\\right)} = 2$$\n Thus, we have a local minimum at $\\left(\\frac{\\pi}{2}, 2\\right)$.\n\n- On the interval $\\pi < \\theta < 2\\pi$, $\\sin\\theta < 0$. The minimum value of $\\sin\\theta$ is $-1$ at $\\theta = \\frac{3\\pi}{2}$, which corresponds to the maximum value of the reciprocal function:\n $$E\\left(\\frac{3\\pi}{2}\\right) = \\frac{2}{\\sin\\left(\\frac{3\\pi}{2}\\right)} = -2$$\n Thus, we have a local maximum at $\\left(\\frac{3\\pi}{2}, -2\\right)$.\n\nCorrectly identifying both turning points earns A1."},{"type":"text","order":3,"title":"Sketch the Graph","content":"Now we sketch the curve $E(\\theta) = 2 \\csc \\theta$ on the domain $[0, 2\\pi]$:\n\n1. Draw dashed vertical lines at the asymptotes $\\theta = 0$, $\\theta = \\pi$, and $\\theta = 2\\pi$.\n2. Plot the local minimum at $\\left(\\frac{\\pi}{2}, 2\\right)$ and sketch a 'U-shaped' curve opening upwards on the interval $0 < \\theta < \\pi$.\n3. Plot the local maximum at $\\left(\\frac{3\\pi}{2}, -2\\right)$ and sketch an inverted 'U-shaped' curve opening downwards on the interval $\\pi < \\theta < 2\\pi$.\n\nProducing a sketch with these correct characteristics earns M1.","calculator":[{"gdc":{"expression":"2/sin(x)","instructions":"1. Go to the Graph application.\n2. Enter the function f1(x) = 2 / sin(x).\n3. Set the viewing window domain from 0 to 2π and range from -10 to 10.\n4. Use the Analyze Graph tool to find the minimum and maximum coordinates."},"ti84":{"expression":"2/sin(x)","instructions":"1. Press [Y=] and enter Y1 = 2 / sin(X).\n2. Press [WINDOW] and set Xmin = 0, Xmax = 2*π (approx 6.28), Xscl = π/2 (approx 1.57), Ymin = -10, Ymax = 10, Yscl = 1.\n3. Press [GRAPH].\n4. To find the minimum, press [2nd] [TRACE] and select '3: minimum', then set bounds around x = π/2."},"desmos":{"expression":"y = 2 \\csc(x)","instructions":"1. Type `y = 2 \\csc(x)` in the first expression line.\n2. Click the wrench icon to set the x-axis limits from `0` to `2\\pi` and step size to `\\pi/2`.\n3. Click on the curve to display and highlight the coordinates of the peak at (1.571, 2) and trough at (4.712, -2)."},"description":"Visualize the graph and check the asymptotes and turning points on a graphing calculator."}]}]},{"id":"graph_transformations","label":"Transforming the Parent Cosecant Function","steps":[{"type":"text","order":0,"title":"Understand the Parent Function","content":"To sketch the graph of $E(\\theta) = -2 \\csc(\\theta + \\pi)$ using transformations, we start with the standard parent cosecant function, $y = \\csc \\theta$.\n\nRecall that the parent function $y = \\csc \\theta$ is the reciprocal of $\\sin \\theta$. On the interval $[-\\pi, 3\\pi]$, it has:\n\n- **Vertical Asymptotes** where $\\sin \\theta = 0$:\n $$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$\n\n- **Local Turning Points**:\n - A local minimum at $\\left(\\frac{\\pi}{2}, 1\\right)$ (where the U-shaped branch opens upwards)\n - A local maximum at $\\left(\\frac{3\\pi}{2}, -1\\right)$ (where the U-shaped branch opens downwards)\n - Another local minimum at $\\left(\\frac{5\\pi}{2}, 1\\right)$"},{"type":"text","order":1,"title":"Apply Horizontal Phase Shift","content":"The $+\\pi$ inside the argument of $E(\\theta) = -2 \\csc(\\theta + \\pi)$ represents a horizontal translation of $\\pi$ units to the **left**.\n\nLet's apply this shift to our parent features by subtracting $\\pi$ from each $\\theta$-coordinate:\n\n1. **Vertical Asymptotes**:\n $$\\begin{aligned} \\theta = 0 &\\;\\to\\; \\theta = -\\pi \\\\ \\theta = \\pi &\\;\\to\\; \\theta = 0 \\\\ \\theta = 2\\pi &\\;\\to\\; \\theta = \\pi \\\\ \\theta = 3\\pi &\\;\\to\\; \\theta = 2\\pi \\end{aligned}$$\n Since we are restricted to the domain $0 \\le \\theta \\le 2\\pi$, our relevant vertical asymptotes inside this interval are:\n $$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$\n\n2. **Turning Points**:\n - The local maximum $\\left(\\frac{3\\pi}{2}, -1\\right)$ shifts left to:\n $$\\left(\\frac{3\\pi}{2} - \\pi, -1\\right) = \\left(\\frac{\\pi}{2}, -1\\right)$$\n - The local minimum $\\left(\\frac{5\\pi}{2}, 1\\right)$ shifts left to:\n $$\\left(\\frac{5\\pi}{2} - \\pi, 1\\right) = \\left(\\frac{3\\pi}{2}, 1\\right)$$"},{"type":"text","order":2,"title":"Apply Vertical Transformations","content":"The multiplier of $-2$ in $E(\\theta) = -2 \\csc(\\theta + \\pi)$ represents two vertical transformations:\n\n1. A **vertical stretch** by a scale factor of $2$.\n2. A **reflection** in the $\\theta$-axis due to the negative sign.\n\nTo apply this, we multiply the $y$-coordinates of our shifted turning points by $-2$:\n\n- **For the point at $\\theta = \\frac{\\pi}{2}$**:\n $$y = -1 \\times (-2) = 2 \\implies \\left(\\frac{\\pi}{2}, 2\\right)$$\n Because of the reflection, the branch that originally opened downwards now opens upwards. Thus, this becomes a **local minimum**.\n\n- **For the point at $\\theta = \\frac{3\\pi}{2}$**:\n $$y = 1 \\times (-2) = -2 \\implies \\left(\\frac{3\\pi}{2}, -2\\right)$$\n Because of the reflection, the branch that originally opened upwards now opens downwards. Thus, this becomes a **local maximum**."},{"type":"text","order":3,"title":"Sketch and Marks","content":"Now we can sketch the graph of $E(\\theta)$ against $\\theta$ for $0 \\le \\theta \\le 2\\pi$ using the features we derived:\n\n1. Draw dashed vertical lines for the asymptotes at $\\theta = 0$, $\\theta = \\pi$, and $\\theta = 2\\pi$.\n2. In the interval $0 < \\theta < \\pi$, draw an upward-opening U-shape with its vertex (local minimum) at $\\left(\\frac{\\pi}{2}, 2\\right)$.\n3. In the interval $\\pi < \\theta < 2\\pi$, draw a downward-opening U-shape with its vertex (local maximum) at $\\left(\\frac{3\\pi}{2}, -2\\right)$.\n\n### Mark Scheme Breakdown\n- **M1**: Recognizing the correct transformations/simplification leading to $E(\\theta) = 2\\csc\\theta$.\n- **B1**: Correctly identifying the vertical asymptotes: $\\theta = 0, \\theta = \\pi, \\theta = 2\\pi$.\n- **M1**: Sketching two correct U-shaped reciprocal curves in the correct intervals.\n- **A1**: Labeling the local minimum at $\\left(\\frac{\\pi}{2}, 2\\right)$ and the local maximum at $\\left(\\frac{3\\pi}{2}, -2\\right)$.","calculator":[{"gdc":{"expression":"y1 = -2 / sin(x + \\pi)","instructions":"1. Set the calculator angle mode to 'Radians'.\n2. Open the Graphing menu and enter \$y1 = -2 / \\sin(x + \\pi)\$.\n3. Set the viewing window (V-Window) for \$x\$ from \$0\$ to \$2\\pi\$ and \$y\$ from \$-5\$ to \$5\$.\n4. Draw the graph.\n5. Use 'G-Solve' (F5) then select 'MAX' or 'MIN' to verify the coordinates of the turning points."},"ti84":{"expression":"Y_1 = -2 / sin(X + \\pi)","instructions":"1. Press [MODE] and ensure 'RADIAN' is highlighted.\n2. Press [Y=] and enter \$Y_1 = -2 / \\sin(X + \\pi)\$.\n3. Press [WINDOW] and set: \$Xmin = 0\$, \$Xmax = 2\\pi\$ (approx 6.28), \$Xscl = \\pi/2\$, \$Ymin = -5\$, \$Ymax = 5\$, \$Yscl = 1\$.\n4. Press [GRAPH].\n5. To find the minimum point, press [2nd] [TRACE] (CALC), select 'minimum', and set left/right bounds around \$X = \\pi/2\$."},"desmos":{"expression":"y = -2/\\sin(x + \\pi)","instructions":"1. Click on the wrench icon (Settings) and select 'Radians'.\n2. Type the function: \$E(x) = -2\\csc(x + \\pi)\$ or \$y = -2/\\sin(x + \\pi)\$.\n3. Set the x-axis limits from \$0\$ to \$2\\pi\$.\n4. Click on the curves to reveal the gray dots representing the local minimum at \$(\\pi/2, 2)\$ and local maximum at \$(3\\pi/2, -2)\$."},"description":"To verify your sketch, graph the function in radian mode. Since most calculators do not have a dedicated cosecant button, enter the function using the identity $\\csc(x) = \\frac{1}{\\sin(x)}$."}]}]},{"id":"reciprocal_sine","label":"Reciprocal of Sine Curve","steps":[{"type":"text","order":0,"title":"Define the reciprocal relationship","content":"To sketch the graph of the cosecant function \n\n$$E(\\theta) = -2 \\csc(\\theta + \\pi)$$\n\nwe first rewrite it using its reciprocal relationship with the sine function. Recall that $\\csc(x) = \\frac{1}{\\sin(x)}$. Therefore, our function can be written as:\n\n$$E(\\theta) = \\frac{-2}{\\sin(\\theta + \\pi)}$$\n\nTo sketch this curve using the **Reciprocal of Sine Curve** method, we will first sketch the auxiliary sine function in the denominator:\n\n$$y = -2 \\sin(\\theta + \\pi)$$\n\nThen, we will find its reciprocal to obtain the final graph.","highlights":[{"text":"E(\\theta) = -2 \\csc(\\theta + \\pi)","location":"specification"}]},{"type":"text","order":1,"title":"Simplify the auxiliary sine function","content":"Before sketching the auxiliary sine function, we can simplify it using trigonometric identities. Using the angle addition formula or our knowledge of phase shifts:\n\n$$\\sin(\\theta + \\pi) = -\\sin\\theta$$\n\nSubstituting this back into our auxiliary function gives:\n\n$$y = -2(-\\sin\\theta) = 2\\sin\\theta$$\n\nThus, our auxiliary curve is simply a standard sine wave with an amplitude of $2$:\n\n$$y = 2\\sin\\theta$$\n\nConsequently, the cosecant function simplifies to:\n\n$$E(\\theta) = 2\\csc\\theta$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\htmlData{anchor=amp-val}{\\color{@sky}{2}}\\sin\\theta$"},{"latex":"$$\\text{Roots where } \\sin\\theta = 0 \\implies \\theta = \\htmlData{anchor=root-zero}{\\color{@amber}{0}}, \\; \\theta = \\htmlData{anchor=root-pi}{\\color{@amber}{\\pi}}, \\; \\theta = \\htmlData{anchor=root-2pi}{\\color{@amber}{2\\pi}}$"},{"latex":"$$\\text{Turning points: Max at } \\left(\\frac{\\pi}{2}, \\htmlData{anchor=max-y}{\\color{@purple}{2}}\\right) \\text{ and Min at } \\left(\\frac{3\\pi}{2}, \\htmlData{anchor=min-y}{\\color{-purple}{-2}}\\right)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"amp-val"},{"color":"amber","style":"underline","anchor":"root-zero"},{"color":"amber","style":"underline","anchor":"root-pi"},{"color":"amber","style":"underline","anchor":"root-2pi"}],"connections":[{"to":"max-y","from":"amp-val","color":"sky","label":"amplitude","style":"dashed"}]},"order":2,"title":"Analyze the auxiliary curve","description":"We sketch the auxiliary sine graph $y = 2\\sin\\theta$ over the interval $0 \\le \\theta \\le 2\\pi$ to locate key geometric features for the reciprocal sketch."},{"type":"text","order":3,"title":"Identify vertical asymptotes","content":"The vertical asymptotes of the reciprocal cosecant function $E(\\theta) = \\frac{2}{\\sin\\theta}$ occur where the denominator is zero.\n\nSince the roots of $y = 2\\sin\\theta$ occur at $\\theta = 0, \\pi,$ and $2\\pi$, these values make the denominator of $E(\\theta)$ zero, causing the function to be undefined.\n\nTherefore, the equations of the vertical asymptotes are:\n\n$$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$"},{"type":"text","order":4,"title":"Determine local turning points","content":"Taking the reciprocal of the auxiliary curve $y = 2\\sin\\theta$ swaps the nature of its peak and trough:\n\n* At the peak of the sine wave $\\left(\\frac{\\pi}{2}, 2\\right)$, the reciprocal function reaches a local **minimum** value of:\n $$\\frac{1}{2} \\times 2 = 2 \\implies \\left(\\frac{\\pi}{2}, 2\\right)$$\n* At the trough of the sine wave $\\left(\\frac{3\\pi}{2}, -2\\right)$, the reciprocal function reaches a local **maximum** value of:\n $$\\frac{1}{-2} \\times (-2) = -2 \\implies \\left(\\frac{3\\pi}{2}, -2\\right)$$\n\nAs $\\theta$ approaches the asymptotes, the magnitude of the sine curve decreases toward $0$, meaning the reciprocal curve $E(\\theta)$ shoots off towards $\\pm\\infty$."},{"type":"text","order":5,"title":"Sketch the final graph","content":"To complete your sketch:\n\n1. Draw the coordinate axes for $0 \\le \\theta \\le 2\\pi$.\n2. Mark the vertical asymptotes at $\\theta = 0$, $\\theta = \\pi$, and $\\theta = 2\\pi$ using dashed vertical lines.\n3. Plot the turning points at $\\left(\\frac{\\pi}{2}, 2\\right)$ and $\\left(\\frac{3\\pi}{2}, -2\\right)$.\n4. Sketch a U-shaped curve opening upwards in the region $0 < \\theta < \\pi$, with its lowest point at $\\left(\\frac{\\pi}{2}, 2\\right)$, asymptotic to $\\theta = 0$ and $\\theta = \\pi$.\n5. Sketch an inverted U-shaped curve opening downwards in the region $\\pi < \\theta < 2\\pi$, with its highest point at $\\left(\\frac{3\\pi}{2}, -2\\right)$, asymptotic to $\\theta = \\pi$ and $\\theta = 2\\pi$.","calculator":[{"gdc":{"expression":"y = -2/sin(x + π)","instructions":"1. Open your graphing application.\n2. Enter the auxiliary function \\\$f1(x) = 2\\sin(x)\\\$.\n3. Enter the target function \\\$f2(x) = -2\\csc(x + \\pi)\\\$ or \\\$f2(x) = 2/\\sin(x)\\\$.\n4. Set your horizontal viewing window from 0 to 2π."},"ti84":{"expression":"Y_2 = -2 / sin(X + π)","instructions":"1. Press [MODE] and ensure your calculator is in RADIAN mode.\n2. Press [Y=].\n3. Enter \\\$Y_1 = 2\\sin(X)\\\$.\n4. Enter \\\$Y_2 = -2 / \\sin(X + \\pi)\\\$.\n5. Press [WINDOW] and set Xmin = 0, Xmax = 2π, Ymin = -5, Ymax = 5.\n6. Press [GRAPH] to view the curves."},"desmos":{"expression":"y = -2\\csc(x + \\pi)","instructions":"1. Type `y = 2\\sin(x)` in the first expression bar to plot the auxiliary curve.\n2. Type `y = -2\\csc(x + \\pi)` in the second bar.\n3. Click on the wrench icon (Graph Settings) and set the x-axis limits from 0 to 2π with a step of π/2."},"description":"You can visualize both the auxiliary sine curve and the cosecant curve on your graphing calculator to verify your sketch."}]}]}],"generatedAt":"2026-06-11T03:46:31.099Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"7c57a61b-7714-4bbc-8aca-2afb3b1fa751:trig_simplification","label":"Trigonometric Identity Simplification","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sin(\\theta + \\pi) = -\\sin\\theta$"},{"latex":"$$E(\\theta) = -2 \\csc(\\theta + \\pi) = \\htmlData{anchor=step1_orig}{\\color{@sky}{-2 \\left(\\frac{1}{\\sin(\\theta + \\pi)}\\right)}}$"},{"latex":"$$E(\\theta) = -2 \\left(\\frac{1}{\\htmlData{anchor=step1_sub}{\\color{@sky}{-\\sin\\theta}}}\\right) = \\htmlData{anchor=step1_final}{\\color{@amber}{2 \\csc\\theta}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"step1_orig"},{"color":"sky","style":"box","anchor":"step1_sub"},{"color":"amber","style":"text-color","anchor":"step1_final"}],"connections":[{"to":"step1_sub","from":"step1_orig","color":"sky","label":"Substitute identity","style":"solid"}]},"order":0,"title":"Simplify the Function","description":"We begin by simplifying $E(\\theta) = -2 \\csc(\\theta + \\pi)$ using the trigonometric identity $\\sin(\\theta + \\pi) = -\\sin\\theta$, which means $\\csc(\\theta + \\pi) = -\\csc\\theta$. This earns the first mark M1."},{"type":"text","order":1,"title":"Find the Vertical Asymptotes","content":"The simplified function is:\n\n$$E(\\theta) = \\frac{2}{\\sin\\theta}$$\n\nVertical asymptotes occur where the denominator is equal to zero, i.e., where $\\sin\\theta = 0$. Within the domain $0 \\le \\theta \\le 2\\pi$, this occurs at:\n\n$$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$\n\nStating these three vertical asymptotes correctly earns B1."},{"type":"text","order":2,"title":"Determine the Turning Points","content":"To find the coordinates of the turning points, we evaluate $E(\\theta) = 2\\csc\\theta$ at critical values:\n\n- On the interval $0 < \\theta < \\pi$, $\\sin\\theta > 0$. The maximum value of $\\sin\\theta$ is $1$ at $\\theta = \\frac{\\pi}{2}$, which corresponds to the minimum value of the reciprocal function:\n $$E\\left(\\frac{\\pi}{2}\\right) = \\frac{2}{\\sin\\left(\\frac{\\pi}{2}\\right)} = 2$$\n Thus, we have a local minimum at $\\left(\\frac{\\pi}{2}, 2\\right)$.\n\n- On the interval $\\pi < \\theta < 2\\pi$, $\\sin\\theta < 0$. The minimum value of $\\sin\\theta$ is $-1$ at $\\theta = \\frac{3\\pi}{2}$, which corresponds to the maximum value of the reciprocal function:\n $$E\\left(\\frac{3\\pi}{2}\\right) = \\frac{2}{\\sin\\left(\\frac{3\\pi}{2}\\right)} = -2$$\n Thus, we have a local maximum at $\\left(\\frac{3\\pi}{2}, -2\\right)$.\n\nCorrectly identifying both turning points earns A1."},{"type":"text","order":3,"title":"Sketch the Graph","content":"Now we sketch the curve $E(\\theta) = 2 \\csc \\theta$ on the domain $[0, 2\\pi]$:\n\n1. Draw dashed vertical lines at the asymptotes $\\theta = 0$, $\\theta = \\pi$, and $\\theta = 2\\pi$.\n2. Plot the local minimum at $\\left(\\frac{\\pi}{2}, 2\\right)$ and sketch a 'U-shaped' curve opening upwards on the interval $0 < \\theta < \\pi$.\n3. Plot the local maximum at $\\left(\\frac{3\\pi}{2}, -2\\right)$ and sketch an inverted 'U-shaped' curve opening downwards on the interval $\\pi < \\theta < 2\\pi$.\n\nProducing a sketch with these correct characteristics earns M1.","calculator":[{"gdc":{"expression":"2/sin(x)","instructions":"1. Go to the Graph application.\n2. Enter the function f1(x) = 2 / sin(x).\n3. Set the viewing window domain from 0 to 2π and range from -10 to 10.\n4. Use the Analyze Graph tool to find the minimum and maximum coordinates."},"ti84":{"expression":"2/sin(x)","instructions":"1. Press [Y=] and enter Y1 = 2 / sin(X).\n2. Press [WINDOW] and set Xmin = 0, Xmax = 2*π (approx 6.28), Xscl = π/2 (approx 1.57), Ymin = -10, Ymax = 10, Yscl = 1.\n3. Press [GRAPH].\n4. To find the minimum, press [2nd] [TRACE] and select '3: minimum', then set bounds around x = π/2."},"desmos":{"expression":"y = 2 \\csc(x)","instructions":"1. Type `y = 2 \\csc(x)` in the first expression line.\n2. Click the wrench icon to set the x-axis limits from `0` to `2\\pi` and step size to `\\pi/2`.\n3. Click on the curve to display and highlight the coordinates of the peak at (1.571, 2) and trough at (4.712, -2)."},"description":"Visualize the graph and check the asymptotes and turning points on a graphing calculator."}]}]},{"id":"7c57a61b-7714-4bbc-8aca-2afb3b1fa751:graph_transformations","label":"Transforming the Parent Cosecant Function","steps":[{"type":"text","order":0,"title":"Understand the Parent Function","content":"To sketch the graph of $E(\\theta) = -2 \\csc(\\theta + \\pi)$ using transformations, we start with the standard parent cosecant function, $y = \\csc \\theta$.\n\nRecall that the parent function $y = \\csc \\theta$ is the reciprocal of $\\sin \\theta$. On the interval $[-\\pi, 3\\pi]$, it has:\n\n- **Vertical Asymptotes** where $\\sin \\theta = 0$:\n $$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$\n\n- **Local Turning Points**:\n - A local minimum at $\\left(\\frac{\\pi}{2}, 1\\right)$ (where the U-shaped branch opens upwards)\n - A local maximum at $\\left(\\frac{3\\pi}{2}, -1\\right)$ (where the U-shaped branch opens downwards)\n - Another local minimum at $\\left(\\frac{5\\pi}{2}, 1\\right)$"},{"type":"text","order":1,"title":"Apply Horizontal Phase Shift","content":"The $+\\pi$ inside the argument of $E(\\theta) = -2 \\csc(\\theta + \\pi)$ represents a horizontal translation of $\\pi$ units to the **left**.\n\nLet's apply this shift to our parent features by subtracting $\\pi$ from each $\\theta$-coordinate:\n\n1. **Vertical Asymptotes**:\n $$\\begin{aligned} \\theta = 0 &\\;\\to\\; \\theta = -\\pi \\\\ \\theta = \\pi &\\;\\to\\; \\theta = 0 \\\\ \\theta = 2\\pi &\\;\\to\\; \\theta = \\pi \\\\ \\theta = 3\\pi &\\;\\to\\; \\theta = 2\\pi \\end{aligned}$$\n Since we are restricted to the domain $0 \\le \\theta \\le 2\\pi$, our relevant vertical asymptotes inside this interval are:\n $$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$\n\n2. **Turning Points**:\n - The local maximum $\\left(\\frac{3\\pi}{2}, -1\\right)$ shifts left to:\n $$\\left(\\frac{3\\pi}{2} - \\pi, -1\\right) = \\left(\\frac{\\pi}{2}, -1\\right)$$\n - The local minimum $\\left(\\frac{5\\pi}{2}, 1\\right)$ shifts left to:\n $$\\left(\\frac{5\\pi}{2} - \\pi, 1\\right) = \\left(\\frac{3\\pi}{2}, 1\\right)$$"},{"type":"text","order":2,"title":"Apply Vertical Transformations","content":"The multiplier of $-2$ in $E(\\theta) = -2 \\csc(\\theta + \\pi)$ represents two vertical transformations:\n\n1. A **vertical stretch** by a scale factor of $2$.\n2. A **reflection** in the $\\theta$-axis due to the negative sign.\n\nTo apply this, we multiply the $y$-coordinates of our shifted turning points by $-2$:\n\n- **For the point at $\\theta = \\frac{\\pi}{2}$**:\n $$y = -1 \\times (-2) = 2 \\implies \\left(\\frac{\\pi}{2}, 2\\right)$$\n Because of the reflection, the branch that originally opened downwards now opens upwards. Thus, this becomes a **local minimum**.\n\n- **For the point at $\\theta = \\frac{3\\pi}{2}$**:\n $$y = 1 \\times (-2) = -2 \\implies \\left(\\frac{3\\pi}{2}, -2\\right)$$\n Because of the reflection, the branch that originally opened upwards now opens downwards. Thus, this becomes a **local maximum**."},{"type":"text","order":3,"title":"Sketch and Marks","content":"Now we can sketch the graph of $E(\\theta)$ against $\\theta$ for $0 \\le \\theta \\le 2\\pi$ using the features we derived:\n\n1. Draw dashed vertical lines for the asymptotes at $\\theta = 0$, $\\theta = \\pi$, and $\\theta = 2\\pi$.\n2. In the interval $0 < \\theta < \\pi$, draw an upward-opening U-shape with its vertex (local minimum) at $\\left(\\frac{\\pi}{2}, 2\\right)$.\n3. In the interval $\\pi < \\theta < 2\\pi$, draw a downward-opening U-shape with its vertex (local maximum) at $\\left(\\frac{3\\pi}{2}, -2\\right)$.\n\n### Mark Scheme Breakdown\n- **M1**: Recognizing the correct transformations/simplification leading to $E(\\theta) = 2\\csc\\theta$.\n- **B1**: Correctly identifying the vertical asymptotes: $\\theta = 0, \\theta = \\pi, \\theta = 2\\pi$.\n- **M1**: Sketching two correct U-shaped reciprocal curves in the correct intervals.\n- **A1**: Labeling the local minimum at $\\left(\\frac{\\pi}{2}, 2\\right)$ and the local maximum at $\\left(\\frac{3\\pi}{2}, -2\\right)$.","calculator":[{"gdc":{"expression":"y1 = -2 / sin(x + \\pi)","instructions":"1. Set the calculator angle mode to 'Radians'.\n2. Open the Graphing menu and enter \$y1 = -2 / \\sin(x + \\pi)\$.\n3. Set the viewing window (V-Window) for \$x\$ from \$0\$ to \$2\\pi\$ and \$y\$ from \$-5\$ to \$5\$.\n4. Draw the graph.\n5. Use 'G-Solve' (F5) then select 'MAX' or 'MIN' to verify the coordinates of the turning points."},"ti84":{"expression":"Y_1 = -2 / sin(X + \\pi)","instructions":"1. Press [MODE] and ensure 'RADIAN' is highlighted.\n2. Press [Y=] and enter \$Y_1 = -2 / \\sin(X + \\pi)\$.\n3. Press [WINDOW] and set: \$Xmin = 0\$, \$Xmax = 2\\pi\$ (approx 6.28), \$Xscl = \\pi/2\$, \$Ymin = -5\$, \$Ymax = 5\$, \$Yscl = 1\$.\n4. Press [GRAPH].\n5. To find the minimum point, press [2nd] [TRACE] (CALC), select 'minimum', and set left/right bounds around \$X = \\pi/2\$."},"desmos":{"expression":"y = -2/\\sin(x + \\pi)","instructions":"1. Click on the wrench icon (Settings) and select 'Radians'.\n2. Type the function: \$E(x) = -2\\csc(x + \\pi)\$ or \$y = -2/\\sin(x + \\pi)\$.\n3. Set the x-axis limits from \$0\$ to \$2\\pi\$.\n4. Click on the curves to reveal the gray dots representing the local minimum at \$(\\pi/2, 2)\$ and local maximum at \$(3\\pi/2, -2)\$."},"description":"To verify your sketch, graph the function in radian mode. Since most calculators do not have a dedicated cosecant button, enter the function using the identity $\\csc(x) = \\frac{1}{\\sin(x)}$."}]}]},{"id":"7c57a61b-7714-4bbc-8aca-2afb3b1fa751:reciprocal_sine","label":"Reciprocal of Sine Curve","steps":[{"type":"text","order":0,"title":"Define the reciprocal relationship","content":"To sketch the graph of the cosecant function \n\n$$E(\\theta) = -2 \\csc(\\theta + \\pi)$$\n\nwe first rewrite it using its reciprocal relationship with the sine function. Recall that $\\csc(x) = \\frac{1}{\\sin(x)}$. Therefore, our function can be written as:\n\n$$E(\\theta) = \\frac{-2}{\\sin(\\theta + \\pi)}$$\n\nTo sketch this curve using the **Reciprocal of Sine Curve** method, we will first sketch the auxiliary sine function in the denominator:\n\n$$y = -2 \\sin(\\theta + \\pi)$$\n\nThen, we will find its reciprocal to obtain the final graph.","highlights":[{"text":"E(\\theta) = -2 \\csc(\\theta + \\pi)","location":"specification"}]},{"type":"text","order":1,"title":"Simplify the auxiliary sine function","content":"Before sketching the auxiliary sine function, we can simplify it using trigonometric identities. Using the angle addition formula or our knowledge of phase shifts:\n\n$$\\sin(\\theta + \\pi) = -\\sin\\theta$$\n\nSubstituting this back into our auxiliary function gives:\n\n$$y = -2(-\\sin\\theta) = 2\\sin\\theta$$\n\nThus, our auxiliary curve is simply a standard sine wave with an amplitude of $2$:\n\n$$y = 2\\sin\\theta$$\n\nConsequently, the cosecant function simplifies to:\n\n$$E(\\theta) = 2\\csc\\theta$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\htmlData{anchor=amp-val}{\\color{@sky}{2}}\\sin\\theta$"},{"latex":"$$\\text{Roots where } \\sin\\theta = 0 \\implies \\theta = \\htmlData{anchor=root-zero}{\\color{@amber}{0}}, \\; \\theta = \\htmlData{anchor=root-pi}{\\color{@amber}{\\pi}}, \\; \\theta = \\htmlData{anchor=root-2pi}{\\color{@amber}{2\\pi}}$"},{"latex":"$$\\text{Turning points: Max at } \\left(\\frac{\\pi}{2}, \\htmlData{anchor=max-y}{\\color{@purple}{2}}\\right) \\text{ and Min at } \\left(\\frac{3\\pi}{2}, \\htmlData{anchor=min-y}{\\color{-purple}{-2}}\\right)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"amp-val"},{"color":"amber","style":"underline","anchor":"root-zero"},{"color":"amber","style":"underline","anchor":"root-pi"},{"color":"amber","style":"underline","anchor":"root-2pi"}],"connections":[{"to":"max-y","from":"amp-val","color":"sky","label":"amplitude","style":"dashed"}]},"order":2,"title":"Analyze the auxiliary curve","description":"We sketch the auxiliary sine graph $y = 2\\sin\\theta$ over the interval $0 \\le \\theta \\le 2\\pi$ to locate key geometric features for the reciprocal sketch."},{"type":"text","order":3,"title":"Identify vertical asymptotes","content":"The vertical asymptotes of the reciprocal cosecant function $E(\\theta) = \\frac{2}{\\sin\\theta}$ occur where the denominator is zero.\n\nSince the roots of $y = 2\\sin\\theta$ occur at $\\theta = 0, \\pi,$ and $2\\pi$, these values make the denominator of $E(\\theta)$ zero, causing the function to be undefined.\n\nTherefore, the equations of the vertical asymptotes are:\n\n$$\\theta = 0, \\quad \\theta = \\pi, \\quad \\theta = 2\\pi$$"},{"type":"text","order":4,"title":"Determine local turning points","content":"Taking the reciprocal of the auxiliary curve $y = 2\\sin\\theta$ swaps the nature of its peak and trough:\n\n* At the peak of the sine wave $\\left(\\frac{\\pi}{2}, 2\\right)$, the reciprocal function reaches a local **minimum** value of:\n $$\\frac{1}{2} \\times 2 = 2 \\implies \\left(\\frac{\\pi}{2}, 2\\right)$$\n* At the trough of the sine wave $\\left(\\frac{3\\pi}{2}, -2\\right)$, the reciprocal function reaches a local **maximum** value of:\n $$\\frac{1}{-2} \\times (-2) = -2 \\implies \\left(\\frac{3\\pi}{2}, -2\\right)$$\n\nAs $\\theta$ approaches the asymptotes, the magnitude of the sine curve decreases toward $0$, meaning the reciprocal curve $E(\\theta)$ shoots off towards $\\pm\\infty$."},{"type":"text","order":5,"title":"Sketch the final graph","content":"To complete your sketch:\n\n1. Draw the coordinate axes for $0 \\le \\theta \\le 2\\pi$.\n2. Mark the vertical asymptotes at $\\theta = 0$, $\\theta = \\pi$, and $\\theta = 2\\pi$ using dashed vertical lines.\n3. Plot the turning points at $\\left(\\frac{\\pi}{2}, 2\\right)$ and $\\left(\\frac{3\\pi}{2}, -2\\right)$.\n4. Sketch a U-shaped curve opening upwards in the region $0 < \\theta < \\pi$, with its lowest point at $\\left(\\frac{\\pi}{2}, 2\\right)$, asymptotic to $\\theta = 0$ and $\\theta = \\pi$.\n5. Sketch an inverted U-shaped curve opening downwards in the region $\\pi < \\theta < 2\\pi$, with its highest point at $\\left(\\frac{3\\pi}{2}, -2\\right)$, asymptotic to $\\theta = \\pi$ and $\\theta = 2\\pi$.","calculator":[{"gdc":{"expression":"y = -2/sin(x + π)","instructions":"1. Open your graphing application.\n2. Enter the auxiliary function \\\$f1(x) = 2\\sin(x)\\\$.\n3. Enter the target function \\\$f2(x) = -2\\csc(x + \\pi)\\\$ or \\\$f2(x) = 2/\\sin(x)\\\$.\n4. Set your horizontal viewing window from 0 to 2π."},"ti84":{"expression":"Y_2 = -2 / sin(X + π)","instructions":"1. Press [MODE] and ensure your calculator is in RADIAN mode.\n2. Press [Y=].\n3. Enter \\\$Y_1 = 2\\sin(X)\\\$.\n4. Enter \\\$Y_2 = -2 / \\sin(X + \\pi)\\\$.\n5. Press [WINDOW] and set Xmin = 0, Xmax = 2π, Ymin = -5, Ymax = 5.\n6. Press [GRAPH] to view the curves."},"desmos":{"expression":"y = -2\\csc(x + \\pi)","instructions":"1. Type `y = 2\\sin(x)` in the first expression bar to plot the auxiliary curve.\n2. Type `y = -2\\csc(x + \\pi)` in the second bar.\n3. Click on the wrench icon (Graph Settings) and set the x-axis limits from 0 to 2π with a step of π/2."},"description":"You can visualize both the auxiliary sine curve and the cosecant curve on your graphing calculator to verify your sketch."}]}]}],"generatedAt":"2026-06-11T03:46:33.598Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"eca855ec-c608-464f-a630-573257686af5","revisionId":"9a1bd390-1ee9-4010-9e36-feaaabc48f60","questionNumber":null,"questionType":"LA","specification":"Solve, for $0 \\leq x \\leq 2\\pi$, the equation\n\n$2\\cot x = 3\\sec x$","difficulty":null,"difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"fa222a84-1e44-43c8-951e-429ae423c5e2","label":null,"content":"Give your answers in terms of $\\pi$.","markscheme":"- Writes as $\\frac{2\\cos x}{\\sin x} = \\frac{3}{\\cos x}$ M1\n- $2\\cos^2 x = 3\\sin x$ A1\n- Uses $\\cos^2 x \\equiv 1 - \\sin^2 x$: $2\\sin^2 x + 3\\sin x - 2 = 0$ M1\n- $(2\\sin x - 1)(\\sin x + 2) = 0$, and $\\sin x = -2$ is rejected A1\n- $\\sin x = \\frac{1}{2}$, so $x = \\frac{\\pi}{6}$ and $x = \\frac{5\\pi}{6}$ A1\n\nBoth $\\cot x$ and $\\sec x$ require $\\sin x \\neq 0$ and $\\cos x \\neq 0$; both solutions satisfy this.","marks":5,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"ee916fe5-848f-4b1e-8c34-ab8d8ef5c64e","revisionId":"068c133f-6d33-496c-8073-90f6f1db9c82","questionNumber":null,"questionType":"LA","specification":"An amphitheatre stage is designed as a composite of two circular sectors sharing a common centre $O$. The outer part of the stage is a minor sector $POQ$ with radius $OP$ and arc $PQ$. The inner part is a reflex (major) sector $ROS$ with radius $OR$ and arc $RTS$. It is given that points $R$ and $S$ lie on the radii $OP$ and $OQ$ respectively, such that $O, R, P$ and $O, S, Q$ are collinear.\n\nThe layout specifications are as follows:\n- The central angle $\\angle POQ$ of the minor sector is $0.8$ radians.\n- The length of the outer arc $PQ$ is $20\\text{ m}$.\n- The ratio of the lengths $PR : RO = 1 : 4$.","difficulty":4,"difficultyBand":"medium","tier":null,"totalMarks":14,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"0948dddc-b857-4f0e-a940-a03794089e36","label":"a","content":"Show that $RO = 20\\text{ m}$.","markscheme":"- Uses $s = r\\theta$ for the outer sector $POQ$: $20 = OP \\times 0.8$ M1\n- Calculates the outer radius $OP = \\frac{20}{0.8} = 25\\text{ m}$ A1\n- Uses the ratio $1 : 4$ to find the length of $RO$: $RO = \\frac{4}{1+4} \\times 25$ M1\n- Obtains $RO = 20\\text{ m}$ as required A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"sequential_calculation","label":"Sequential Calculation","steps":[{"type":"text","order":0,"title":"State arc length formula","content":"The arc length of a circular sector is given by the formula $s = r\\theta$, where $s$ is the arc length, $r$ is the radius, and $\\theta$ is the central angle in radians."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=s-val}{\\color{@sky}{20}} \\text{ m}, \\quad \\htmlData{anchor=theta-val}{\\color{@amber}{0.8}} \\text{ rad}$"},{"latex":"$$$\\htmlData{anchor=s-eq}{\\color{@sky}{20}} = OP \\times \\htmlData{anchor=theta-eq}{\\color{@amber}{0.8}}$$"},{"latex":"$$$OP = 25 \\text{ m}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"s-val"},{"color":"amber","style":"text-color","anchor":"theta-val"}],"connections":[{"to":"s-eq","from":"s-val","color":"sky","label":"Substitute s","style":"solid"},{"to":"theta-eq","from":"theta-val","color":"amber","label":"Substitute theta","style":"solid"}]},"order":1,"title":"Calculate outer radius","description":"Substitute the given values $s = 20\\text{ m}$ and $\\theta = 0.8\\text{ rad}$ into the formula to find the outer radius $OP$."},{"type":"text","order":2,"title":"Understand the ratio","content":"Since points $O, R, P$ are collinear, the total radius is $OP = PR + RO$. The ratio $PR : RO = 1 : 4$ means the segment $RO$ is $\\frac{4}{1+4} = \\frac{4}{5}$ of the total radius $OP$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=op-val}{\\color{@purple}{OP = 25\\text{ m}}}, \\quad \\htmlData{anchor=frac-val}{\\color{@teal}{\\frac{4}{5}}}$"},{"latex":"$$$RO = \\htmlData{anchor=frac-sub}{\\color{@teal}{\\frac{4}{5}}} \\times \\htmlData{anchor=op-sub}{\\color{@purple}{25}}$$"},{"latex":"$$$RO = 20\\text{ m}$$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"op-val"},{"color":"teal","style":"text-color","anchor":"frac-val"}],"connections":[{"to":"op-sub","from":"op-val","color":"purple","label":"Substitute OP","style":"solid"},{"to":"frac-sub","from":"frac-val","color":"teal","label":"Apply ratio fraction","style":"solid"}]},"order":3,"title":"Calculate length RO","description":"Multiply the fraction by the total radius length of $25\\text{ m}$ to show that $RO = 20\\text{ m}$."}]},{"id":"algebraic_expression","label":"Algebraic Expression","steps":[{"type":"text","order":0,"title":"Express OP algebraically","content":"Since $O, R, P$ are collinear, $OP = RO + PR$. Given the ratio $PR : RO = 1 : 4$, we have $PR = 0.25 RO$, which gives $OP = RO + 0.25 RO = 1.25 RO$."},{"type":"text","order":1,"title":"Set up equation","content":"Using the arc length formula $s = r\\theta$ for the outer sector $POQ$ with arc length $s = 20\\text{ m}$ and angle $\\theta = 0.8\\text{ rad}$, we get the equation $20 = OP \\times 0.8$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{We have: } \\htmlData{anchor=op-expr}{\\color{@sky}{OP = 1.25 RO}}$"},{"latex":"$$$20 = \\htmlData{anchor=op-sub}{\\color{@sky}{OP}} \\times 0.8$$"},{"latex":"$$$20 = (\\htmlData{anchor=ro-sub}{\\color{@sky}{1.25 RO}}) \\times 0.8$$"},{"latex":"$$$20 = 1.0 RO \\implies RO = 20\\text{ m}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"op-expr"},{"color":"sky","style":"text-color","anchor":"ro-sub"}],"connections":[{"to":"ro-sub","from":"op-expr","color":"sky","label":"Substitute OP","style":"solid"}]},"order":2,"title":"Substitute and solve","description":"Substitute the algebraic expression $OP = 1.25 RO$ into the arc length equation and solve for $RO$."},{"type":"text","order":3,"title":"Verify with markscheme","content":"Our steps align perfectly with the markscheme: we used the arc length formula $20 = OP \\times 0.8$, calculated $OP = 25\\text{ m}$, applied the ratio $RO = \\frac{4}{5} \\times 25$, and obtained $RO = 20\\text{ m}$."}]}],"generatedAt":"2026-06-09T10:16:07.940Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"58cbfa34-b649-44a3-9f6d-804ade54eb24","label":"b","content":"Calculate the total perimeter of the amphitheatre stage, giving your answer in m to 3 significant figures.","markscheme":"- Identifies the components of the perimeter: arc $PQ$, segment $PR$, segment $QS$, and major arc $RTS$ M1\n- Calculates the length of the straight segments: $PR = QS = 25 - 20 = 5\\text{ m}$ B1\n- Calculates the length of major arc $RTS$ using $r\\theta_{major} = 20(2\\pi - 0.8)$ M1\n- Sums all components: $P = 20 + 5 + 5 + 20(2\\pi - 0.8)$ M1\n- Evaluates $P = 30 + 40\\pi - 16 \\approx 139.66...$ and rounds to $140\\text{ m}$ A1","marks":5,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"reflex_angle_method","label":"Direct Reflex Angle Approach","steps":[{"type":"text","order":0,"title":"Identify perimeter components","content":"To find the total perimeter of the amphitheatre stage, we need to identify all boundary segments that make up its outline.\n\nThe stage consists of:\n1. The outer arc $PQ$\n2. The straight segment $PR$\n3. The straight segment $QS$\n4. The major (reflex) arc $RTS$\n\nThus, the perimeter formula is:\n$$\\text{Perimeter} = \\text{arc } PQ + PR + QS + \\text{arc } RTS$$"},{"type":"text","order":1,"title":"Calculate outer radius","content":"We are given that the length of the outer arc $PQ$ is $20\\text{ m}$ and the central angle of the minor sector is $\\angle POQ = 0.8\\text{ rad}$.\n\nUsing the arc length formula:\n$$s = r \\theta$$\n\nLetting $r_{\\text{outer}} = OP$, we have:\n$$20 = OP \\times 0.8$$\n\nSolving for $OP$:\n$$OP = \\frac{20}{0.8} = 25\\text{ m}$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Since $O, R, P$ are collinear:"},{"latex":"$$\\htmlData{anchor=op-total}{OP} = \\htmlData{anchor=ro-part}{RO} + \\htmlData{anchor=pr-part}{PR}$"},{"text":"With the given ratio:"},{"latex":"$$\\htmlData{anchor=ratio-spec}{PR : RO = 1 : 4}$"},{"latex":"$$\\htmlData{anchor=ro-eq}{RO} = \\frac{4}{1+4} \\times \\htmlData{anchor=op-val}{25} = 20\\text{ m}$"},{"latex":"$$\\htmlData{anchor=pr-eq}{PR} = 25 - 20 = 5\\text{ m}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"ratio-spec"},{"color":"sky","style":"underline","anchor":"ro-eq"},{"color":"green","style":"box","anchor":"pr-eq"}],"connections":[{"to":"ro-eq","from":"op-total","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Find inner dimensions","description":"We use the given ratio to split the outer radius into the inner radius and the boundary segment length."},{"type":"text","order":3,"title":"Calculate major arc length","content":"Since $ROS$ is a major (reflex) sector, the major angle $\\theta_{\\text{major}}$ is the reflex of the central angle $\\theta = 0.8\\text{ rad}$.\n\nA full revolution is $2\\pi$ radians, so:\n$$\\theta_{\\text{major}} = 2\\pi - 0.8$$\n\nNow, calculate the length of the major arc $RTS$ using the inner radius $r_{\\text{inner}} = RO = 20\\text{ m}$:\n$$\\text{arc } RTS = r_{\\text{inner}} \\theta_{\\text{major}} = 20(2\\pi - 0.8) = 40\\pi - 16\\text{ m}$$"},{"type":"text","order":4,"title":"Calculate total perimeter","content":"Now we sum all the segments to find the total perimeter:\n$$\\text{Perimeter} = \\text{arc } PQ + PR + QS + \\text{arc } RTS$$\n$$\\text{Perimeter} = 20 + 5 + 5 + (40\\pi - 16) = 14 + 40\\pi\\text{ m}$$\n\nUsing a calculator to find the decimal value:\n$$\\text{Perimeter} \\approx 14 + 125.664 = 139.664\\text{ m}$$\n\nRounding to 3 significant figures gives **$140\\text{ m}$**.","calculator":[{"gdc":{"expression":"14 + 40 * π","instructions":"Enter `14 + 40 * π` into the main calculation screen."},"ti84":{"expression":"14 + 40 * π","instructions":"Type `14 + 40 * π` and press `ENTER`."},"desmos":{"expression":"14 + 40\\pi","instructions":"Type `14 + 40\\pi` into the expression line."},"description":"Evaluate the exact expression on your calculator to get the decimal approximation."}]}]},{"id":"circumference_subtraction_method","label":"Circumference Subtraction Approach","steps":[{"type":"text","order":0,"title":"Calculate the outer radius","content":"To begin, we need to find the radius of the outer sector, $OP$. The formula for arc length is:\n$$s = r\\theta$$\n\nHere, the outer arc length is $PQ = 20\\text{ m}$ and the central angle $\\angle POQ = 0.8\\text{ rad}$. Substituting these into the formula:\n$$20 = OP \\times 0.8$$\n$$OP = \\frac{20}{0.8} = 25\\text{ m}$$"},{"type":"text","order":1,"title":"Determine the segment lengths","content":"We are given that the points $O, R, P$ are collinear and the ratio of the lengths is $PR : RO = 1 : 4$. This means the total length $OP$ is divided into $1 + 4 = 5$ parts:\n$$PR = \\frac{1}{5} \\times OP = \\frac{1}{5} \\times 25 = 5\\text{ m}$$\n$$RO = \\frac{4}{5} \\times OP = \\frac{4}{5} \\times 25 = 20\\text{ m}$$\n\nBy symmetry, the other straight border of the stage is $QS = PR = 5\\text{ m}$, and the inner radius is $RO = SO = 20\\text{ m}$."},{"type":"text","order":2,"title":"Find the major arc length","content":"Using the **Circumference Subtraction Approach**, we can find the major arc length $RTS$ by calculating the total circumference of the inner circle (with radius $r = 20\\text{ m}$) and subtracting the unshared minor arc $RS$.\n\n1. **Full Circumference of the inner circle**:\n $$C = 2\\pi r = 2\\pi(20) = 40\\pi\\text{ m}$$\n\n2. **Minor arc $RS$ length**:\n $$\\text{Arc } RS = r\\theta = 20 \\times 0.8 = 16\\text{ m}$$\n\n3. **Major arc $RTS$ length**:\n $$\\text{Arc } RTS = C - \\text{Arc } RS = 40\\pi - 16\\text{ m}$$"},{"type":"text","order":3,"title":"Calculate total perimeter","content":"The total perimeter of the stage is the sum of the outer arc $PQ$, the two straight side segments $PR$ and $QS$, and the major arc $RTS$:\n$$\\text{Perimeter} = PQ + PR + QS + \\text{Arc } RTS$$\n\nSubstituting the values we have found:\n$$\\text{Perimeter} = 20 + 5 + 5 + (40\\pi - 16) = 14 + 40\\pi\\text{ m}$$"},{"type":"text","order":4,"title":"Evaluate and round","content":"Now we evaluate this expression numerically and round to $3$ significant figures:\n$$\\text{Perimeter} = 14 + 40\\pi \\approx 139.66\\text{ m}$$\n\nRounding to $3$ significant figures gives:\n$$\\text{Perimeter} \\approx 140\\text{ m}$$","calculator":[{"gdc":{"expression":"14+40*π","instructions":"Type '14 + 40 * π' into the main calculation screen and execute."},"ti84":{"expression":"14+40*π","instructions":"Enter '14 + 40 * π' on the home screen and press ENTER."},"desmos":{"expression":"14+40\\pi","instructions":"Type '14 + 40\\pi' in the input line to see the decimal result."},"description":"Calculate the perimeter value using a scientific or graphing calculator."}]}]}],"generatedAt":"2026-06-09T10:18:37.890Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"81df3f2c-8fc6-4b0d-b93c-e0c9d41df905","label":"c","content":"Calculate the total surface area of the stage, giving your answer in $\\text{m}^2$ to the nearest integer.","markscheme":"- Calculates the area of the minor sector $POQ = \\frac{1}{2} \\times 25^2 \\times 0.8 = 250\\text{ m}^2$ M1\n- Identifies the angle of the major sector $ROS$ as $2\\pi - 0.8$ M1\n- Calculates the area of the major sector $ROS = \\frac{1}{2} \\times 20^2 \\times (2\\pi - 0.8) = 200(2\\pi - 0.8)$ A1\n- Sums the areas: $250 + 1096.63... = 1346.63...$ M1\n- Rounds to $1347\\text{ m}^2$ A1","marks":5,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_reflex_angle","label":"Direct Sector Summation with Reflex Angle","steps":[{"type":"text","order":0,"title":"Find outer radius $OP$","content":"The outer part of the stage is the minor sector $POQ$ with central angle $\\theta = 0.8$ radians and arc length $PQ = 20\\text{ m}$.\n\nUsing the radian formula for arc length from our data booklet:\n\n$$s = r\\theta$$\n\nWe can substitute our values to find the radius $OP$ (let this be $R_{\\text{outer}}$):\n\n$$\\begin{aligned} 20 &= R_{\\text{outer}} \\times 0.8 \\\\ R_{\\text{outer}} &= \\frac{20}{0.8} = 25\\text{ m} \\end{aligned}$$\n\nSo, the radius of the minor sector $OP = 25\\text{ m}$.","highlights":[{"text":"0.8 radians","location":"specification"},{"text":"The length of the outer arc PQ is 20 m.","location":"specification"}]},{"type":"text","order":1,"title":"Find inner radius $OR$","content":"We are given the ratio of the lengths along the radius $PR : RO = 1 : 4$. \n\nSince $O, R, P$ are collinear, the total length is $OP = RO + PR$. Under this ratio, we can write:\n\n$$RO = \\frac{4}{1+4} OP = \\frac{4}{5} OP$$\n\nSubstituting $OP = 25\\text{ m}$:\n\n$$OR = \\frac{4}{5} \\times 25 = 20\\text{ m}$$\n\nThus, the radius of the inner reflex sector $ROS$ is $20\\text{ m}$.","highlights":[{"text":"The ratio of the lengths PR : RO = 1 : 4.","location":"specification"}]},{"type":"text","order":2,"title":"Area of minor sector","content":"The outer part of the stage is the minor sector $POQ$. \n\nUsing the sector area formula from our data booklet:\n\n$$A = \\frac{1}{2}r^2\\theta$$\n\nSubstituting $r = 25\\text{ m}$ and $\\theta = 0.8$ radians:\n\n$$\\begin{aligned} \\text{Area}_{POQ} &= \\frac{1}{2} \\times 25^2 \\times 0.8 \\\\ &= 0.4 \\times 625 \\\\ &= 250\\text{ m}^2 \\end{aligned}$$\n\nThis awards the first mark M1."},{"type":"text","order":3,"title":"Area of reflex sector","content":"The inner part of the stage is a reflex (major) sector $ROS$ with radius $OR = 20\\text{ m}$.\n\nSince the minor angle is $\\angle POQ = 0.8$ radians, the reflex angle for sector $ROS$ is:\n\n$$\\theta_{\\text{reflex}} = 2\\pi - 0.8 \\approx 5.483\\text{ rad}$$\n\nThis awards the mark M1.\n\nNow, we calculate the area of this major sector directly using the sector area formula:\n\n$$\\text{Area}_{ROS} = \\frac{1}{2} \\times 20^2 \\times (2\\pi - 0.8) = 200(2\\pi - 0.8) \\approx 1096.64\\text{ m}^2$$\n\nThis awards the mark A1."},{"type":"text","order":4,"title":"Calculate total area","content":"The total surface area of the stage is the sum of the two non-overlapping areas we just calculated:\n\n$$\\begin{aligned} \\text{Total Area} &= \\text{Area}_{POQ} + \\text{Area}_{ROS} \\\\ &= 250 + 200(2\\pi - 0.8) \\\\ &\\approx 250 + 1096.64 \\\\ &= 1346.64\\text{ m}^2 \\end{aligned}$$\n\nThis sum awards the mark M1.\n\nRounding to the nearest integer as requested, we obtain:\n\n$$\\text{Total Area} \\approx 1347\\text{ m}^2$$\n\nThis final answer awards the mark A1.","highlights":[{"text":"to the nearest integer.","location":"specification"}]}]},{"id":"complementary_area","label":"Complementary Area Approach","steps":[{"type":"text","order":0,"title":"Determine the radius OP","content":"To calculate the area of the outer minor sector $POQ$, we first need to find its radius $OP$.\n\nRecall the arc length formula from your radian measure studies:\n\n$$s = r\\theta$$\n\nHere, the outer arc length $s = 20\\text{ m}$ and the angle $\\theta = 0.8\\text{ rad}$. Letting $r = OP$, we have:\n\n$$20 = OP \\times 0.8$$\n\n$$OP = \\frac{20}{0.8} = 25\\text{ m}$$\n\nThis gives us a radius of $25\\text{ m}$ for the larger sector.","highlights":[{"text":"The length of the outer arc PQ is 20 m.","location":"specification"},{"text":"0.8 radians","location":"specification"}]},{"type":"text","order":1,"title":"Calculate area of POQ","content":"Now we calculate the area of the outer minor sector $POQ$ using the sector area formula:\n\n$$\\text{Area} = \\frac{1}{2} r^2 \\theta$$\n\nSubstituting $r = 25\\text{ m}$ and $\\theta = 0.8\\text{ rad}$:\n\n$$\\text{Area}_{POQ} = \\frac{1}{2} \\times 25^2 \\times 0.8 = 250\\text{ m}^2$$"},{"type":"text","order":2,"title":"Find the radius OR","content":"Next, we find the radius of the inner part, $OR$.\n\nWe are given the ratio of lengths $PR : RO = 1 : 4$. Since $O, R,$ and $P$ are collinear, the total length $OP$ is divided into $1 + 4 = 5$ equal parts, where $RO$ represents $\\frac{4}{5}$ of the total length $OP$.\n\n$$\\begin{aligned} RO &= \\frac{4}{5} \\times OP \\\\ RO &= \\frac{4}{5} \\times 25 \\\\ RO &= 20\\text{ m} \\end{aligned}$$\n\nSo, the radius of the inner circle $OR$ is $20\\text{ m}$.","highlights":[{"text":"The ratio of the lengths PR : RO = 1 : 4.","location":"specification"}]},{"type":"text","order":3,"title":"Apply Complementary Area Approach","content":"Using the **Complementary Area Approach**, we can find the area of the major sector $ROS$ by subtracting the area of the inner minor sector with angle $0.8\\text{ rad}$ from the area of the full inner circle of radius $OR = 20\\text{ m}$.\n\n- **Area of full inner circle:**\n$$\\text{Area}_{\\text{circle}} = \\pi r^2 = \\pi \\times 20^2 = 400\\pi\\text{ m}^2$$\n\n- **Area of inner minor sector (angle $0.8$):**\n$$\\text{Area}_{\\text{minor}} = \\frac{1}{2} r^2 \\theta = \\frac{1}{2} \\times 20^2 \\times 0.8 = 160\\text{ m}^2$$\n\n- **Area of major sector $ROS$:**\n$$\\begin{aligned} \\text{Area}_{ROS} &= \\text{Area}_{\\text{circle}} - \\text{Area}_{\\text{minor}} \\\\ &= 400\\pi - 160 \\\\ &\\approx 1256.64 - 160 \\\\ &= 1096.64\\text{ m}^2 \\end{aligned}$$"},{"type":"text","order":4,"title":"Sum the areas","content":"The total surface area of the stage is the sum of the area of the outer minor sector $POQ$ and the area of the major sector $ROS$:\n\n$$\\begin{aligned} \\text{Total Area} &= \\text{Area}_{POQ} + \\text{Area}_{ROS} \\\\ &= 250 + (400\\pi - 160) \\\\ &= 90 + 400\\pi \\\\ &\\approx 1346.64\\text{ m}^2 \\end{aligned}$$\n\nRounding to the nearest integer, we get:\n\n$$\\text{Total Area} \\approx 1347\\text{ m}^2$$"}]}],"generatedAt":"2026-06-09T10:21:30.267Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0948dddc-b857-4f0e-a940-a03794089e36:sequential_calculation","label":"(a) Sequential Calculation","steps":[{"type":"text","order":0,"title":"State arc length formula","content":"The arc length of a circular sector is given by the formula $s = r\\theta$, where $s$ is the arc length, $r$ is the radius, and $\\theta$ is the central angle in radians."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=s-val}{\\color{@sky}{20}} \\text{ m}, \\quad \\htmlData{anchor=theta-val}{\\color{@amber}{0.8}} \\text{ rad}$"},{"latex":"$$$\\htmlData{anchor=s-eq}{\\color{@sky}{20}} = OP \\times \\htmlData{anchor=theta-eq}{\\color{@amber}{0.8}}$$"},{"latex":"$$$OP = 25 \\text{ m}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"s-val"},{"color":"amber","style":"text-color","anchor":"theta-val"}],"connections":[{"to":"s-eq","from":"s-val","color":"sky","label":"Substitute s","style":"solid"},{"to":"theta-eq","from":"theta-val","color":"amber","label":"Substitute theta","style":"solid"}]},"order":1,"title":"Calculate outer radius","description":"Substitute the given values $s = 20\\text{ m}$ and $\\theta = 0.8\\text{ rad}$ into the formula to find the outer radius $OP$."},{"type":"text","order":2,"title":"Understand the ratio","content":"Since points $O, R, P$ are collinear, the total radius is $OP = PR + RO$. The ratio $PR : RO = 1 : 4$ means the segment $RO$ is $\\frac{4}{1+4} = \\frac{4}{5}$ of the total radius $OP$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=op-val}{\\color{@purple}{OP = 25\\text{ m}}}, \\quad \\htmlData{anchor=frac-val}{\\color{@teal}{\\frac{4}{5}}}$"},{"latex":"$$$RO = \\htmlData{anchor=frac-sub}{\\color{@teal}{\\frac{4}{5}}} \\times \\htmlData{anchor=op-sub}{\\color{@purple}{25}}$$"},{"latex":"$$$RO = 20\\text{ m}$$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"op-val"},{"color":"teal","style":"text-color","anchor":"frac-val"}],"connections":[{"to":"op-sub","from":"op-val","color":"purple","label":"Substitute OP","style":"solid"},{"to":"frac-sub","from":"frac-val","color":"teal","label":"Apply ratio fraction","style":"solid"}]},"order":3,"title":"Calculate length RO","description":"Multiply the fraction by the total radius length of $25\\text{ m}$ to show that $RO = 20\\text{ m}$."}]},{"id":"0948dddc-b857-4f0e-a940-a03794089e36:algebraic_expression","label":"(a) Algebraic Expression","steps":[{"type":"text","order":0,"title":"Express OP algebraically","content":"Since $O, R, P$ are collinear, $OP = RO + PR$. Given the ratio $PR : RO = 1 : 4$, we have $PR = 0.25 RO$, which gives $OP = RO + 0.25 RO = 1.25 RO$."},{"type":"text","order":1,"title":"Set up equation","content":"Using the arc length formula $s = r\\theta$ for the outer sector $POQ$ with arc length $s = 20\\text{ m}$ and angle $\\theta = 0.8\\text{ rad}$, we get the equation $20 = OP \\times 0.8$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{We have: } \\htmlData{anchor=op-expr}{\\color{@sky}{OP = 1.25 RO}}$"},{"latex":"$$$20 = \\htmlData{anchor=op-sub}{\\color{@sky}{OP}} \\times 0.8$$"},{"latex":"$$$20 = (\\htmlData{anchor=ro-sub}{\\color{@sky}{1.25 RO}}) \\times 0.8$$"},{"latex":"$$$20 = 1.0 RO \\implies RO = 20\\text{ m}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"op-expr"},{"color":"sky","style":"text-color","anchor":"ro-sub"}],"connections":[{"to":"ro-sub","from":"op-expr","color":"sky","label":"Substitute OP","style":"solid"}]},"order":2,"title":"Substitute and solve","description":"Substitute the algebraic expression $OP = 1.25 RO$ into the arc length equation and solve for $RO$."},{"type":"text","order":3,"title":"Verify with markscheme","content":"Our steps align perfectly with the markscheme: we used the arc length formula $20 = OP \\times 0.8$, calculated $OP = 25\\text{ m}$, applied the ratio $RO = \\frac{4}{5} \\times 25$, and obtained $RO = 20\\text{ m}$."}]},{"id":"58cbfa34-b649-44a3-9f6d-804ade54eb24:reflex_angle_method","label":"(b) Direct Reflex Angle Approach","steps":[{"type":"text","order":0,"title":"Identify perimeter components","content":"To find the total perimeter of the amphitheatre stage, we need to identify all boundary segments that make up its outline.\n\nThe stage consists of:\n1. The outer arc $PQ$\n2. The straight segment $PR$\n3. The straight segment $QS$\n4. The major (reflex) arc $RTS$\n\nThus, the perimeter formula is:\n$$\\text{Perimeter} = \\text{arc } PQ + PR + QS + \\text{arc } RTS$$"},{"type":"text","order":1,"title":"Calculate outer radius","content":"We are given that the length of the outer arc $PQ$ is $20\\text{ m}$ and the central angle of the minor sector is $\\angle POQ = 0.8\\text{ rad}$.\n\nUsing the arc length formula:\n$$s = r \\theta$$\n\nLetting $r_{\\text{outer}} = OP$, we have:\n$$20 = OP \\times 0.8$$\n\nSolving for $OP$:\n$$OP = \\frac{20}{0.8} = 25\\text{ m}$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Since $O, R, P$ are collinear:"},{"latex":"$$\\htmlData{anchor=op-total}{OP} = \\htmlData{anchor=ro-part}{RO} + \\htmlData{anchor=pr-part}{PR}$"},{"text":"With the given ratio:"},{"latex":"$$\\htmlData{anchor=ratio-spec}{PR : RO = 1 : 4}$"},{"latex":"$$\\htmlData{anchor=ro-eq}{RO} = \\frac{4}{1+4} \\times \\htmlData{anchor=op-val}{25} = 20\\text{ m}$"},{"latex":"$$\\htmlData{anchor=pr-eq}{PR} = 25 - 20 = 5\\text{ m}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"ratio-spec"},{"color":"sky","style":"underline","anchor":"ro-eq"},{"color":"green","style":"box","anchor":"pr-eq"}],"connections":[{"to":"ro-eq","from":"op-total","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Find inner dimensions","description":"We use the given ratio to split the outer radius into the inner radius and the boundary segment length."},{"type":"text","order":3,"title":"Calculate major arc length","content":"Since $ROS$ is a major (reflex) sector, the major angle $\\theta_{\\text{major}}$ is the reflex of the central angle $\\theta = 0.8\\text{ rad}$.\n\nA full revolution is $2\\pi$ radians, so:\n$$\\theta_{\\text{major}} = 2\\pi - 0.8$$\n\nNow, calculate the length of the major arc $RTS$ using the inner radius $r_{\\text{inner}} = RO = 20\\text{ m}$:\n$$\\text{arc } RTS = r_{\\text{inner}} \\theta_{\\text{major}} = 20(2\\pi - 0.8) = 40\\pi - 16\\text{ m}$$"},{"type":"text","order":4,"title":"Calculate total perimeter","content":"Now we sum all the segments to find the total perimeter:\n$$\\text{Perimeter} = \\text{arc } PQ + PR + QS + \\text{arc } RTS$$\n$$\\text{Perimeter} = 20 + 5 + 5 + (40\\pi - 16) = 14 + 40\\pi\\text{ m}$$\n\nUsing a calculator to find the decimal value:\n$$\\text{Perimeter} \\approx 14 + 125.664 = 139.664\\text{ m}$$\n\nRounding to 3 significant figures gives **$140\\text{ m}$**.","calculator":[{"gdc":{"expression":"14 + 40 * π","instructions":"Enter `14 + 40 * π` into the main calculation screen."},"ti84":{"expression":"14 + 40 * π","instructions":"Type `14 + 40 * π` and press `ENTER`."},"desmos":{"expression":"14 + 40\\pi","instructions":"Type `14 + 40\\pi` into the expression line."},"description":"Evaluate the exact expression on your calculator to get the decimal approximation."}]}]},{"id":"58cbfa34-b649-44a3-9f6d-804ade54eb24:circumference_subtraction_method","label":"(b) Circumference Subtraction Approach","steps":[{"type":"text","order":0,"title":"Calculate the outer radius","content":"To begin, we need to find the radius of the outer sector, $OP$. The formula for arc length is:\n$$s = r\\theta$$\n\nHere, the outer arc length is $PQ = 20\\text{ m}$ and the central angle $\\angle POQ = 0.8\\text{ rad}$. Substituting these into the formula:\n$$20 = OP \\times 0.8$$\n$$OP = \\frac{20}{0.8} = 25\\text{ m}$$"},{"type":"text","order":1,"title":"Determine the segment lengths","content":"We are given that the points $O, R, P$ are collinear and the ratio of the lengths is $PR : RO = 1 : 4$. This means the total length $OP$ is divided into $1 + 4 = 5$ parts:\n$$PR = \\frac{1}{5} \\times OP = \\frac{1}{5} \\times 25 = 5\\text{ m}$$\n$$RO = \\frac{4}{5} \\times OP = \\frac{4}{5} \\times 25 = 20\\text{ m}$$\n\nBy symmetry, the other straight border of the stage is $QS = PR = 5\\text{ m}$, and the inner radius is $RO = SO = 20\\text{ m}$."},{"type":"text","order":2,"title":"Find the major arc length","content":"Using the **Circumference Subtraction Approach**, we can find the major arc length $RTS$ by calculating the total circumference of the inner circle (with radius $r = 20\\text{ m}$) and subtracting the unshared minor arc $RS$.\n\n1. **Full Circumference of the inner circle**:\n $$C = 2\\pi r = 2\\pi(20) = 40\\pi\\text{ m}$$\n\n2. **Minor arc $RS$ length**:\n $$\\text{Arc } RS = r\\theta = 20 \\times 0.8 = 16\\text{ m}$$\n\n3. **Major arc $RTS$ length**:\n $$\\text{Arc } RTS = C - \\text{Arc } RS = 40\\pi - 16\\text{ m}$$"},{"type":"text","order":3,"title":"Calculate total perimeter","content":"The total perimeter of the stage is the sum of the outer arc $PQ$, the two straight side segments $PR$ and $QS$, and the major arc $RTS$:\n$$\\text{Perimeter} = PQ + PR + QS + \\text{Arc } RTS$$\n\nSubstituting the values we have found:\n$$\\text{Perimeter} = 20 + 5 + 5 + (40\\pi - 16) = 14 + 40\\pi\\text{ m}$$"},{"type":"text","order":4,"title":"Evaluate and round","content":"Now we evaluate this expression numerically and round to $3$ significant figures:\n$$\\text{Perimeter} = 14 + 40\\pi \\approx 139.66\\text{ m}$$\n\nRounding to $3$ significant figures gives:\n$$\\text{Perimeter} \\approx 140\\text{ m}$$","calculator":[{"gdc":{"expression":"14+40*π","instructions":"Type '14 + 40 * π' into the main calculation screen and execute."},"ti84":{"expression":"14+40*π","instructions":"Enter '14 + 40 * π' on the home screen and press ENTER."},"desmos":{"expression":"14+40\\pi","instructions":"Type '14 + 40\\pi' in the input line to see the decimal result."},"description":"Calculate the perimeter value using a scientific or graphing calculator."}]}]},{"id":"81df3f2c-8fc6-4b0d-b93c-e0c9d41df905:direct_reflex_angle","label":"(c) Direct Sector Summation with Reflex Angle","steps":[{"type":"text","order":0,"title":"Find outer radius $OP$","content":"The outer part of the stage is the minor sector $POQ$ with central angle $\\theta = 0.8$ radians and arc length $PQ = 20\\text{ m}$.\n\nUsing the radian formula for arc length from our data booklet:\n\n$$s = r\\theta$$\n\nWe can substitute our values to find the radius $OP$ (let this be $R_{\\text{outer}}$):\n\n$$\\begin{aligned} 20 &= R_{\\text{outer}} \\times 0.8 \\\\ R_{\\text{outer}} &= \\frac{20}{0.8} = 25\\text{ m} \\end{aligned}$$\n\nSo, the radius of the minor sector $OP = 25\\text{ m}$.","highlights":[{"text":"0.8 radians","location":"specification"},{"text":"The length of the outer arc PQ is 20 m.","location":"specification"}]},{"type":"text","order":1,"title":"Find inner radius $OR$","content":"We are given the ratio of the lengths along the radius $PR : RO = 1 : 4$. \n\nSince $O, R, P$ are collinear, the total length is $OP = RO + PR$. Under this ratio, we can write:\n\n$$RO = \\frac{4}{1+4} OP = \\frac{4}{5} OP$$\n\nSubstituting $OP = 25\\text{ m}$:\n\n$$OR = \\frac{4}{5} \\times 25 = 20\\text{ m}$$\n\nThus, the radius of the inner reflex sector $ROS$ is $20\\text{ m}$.","highlights":[{"text":"The ratio of the lengths PR : RO = 1 : 4.","location":"specification"}]},{"type":"text","order":2,"title":"Area of minor sector","content":"The outer part of the stage is the minor sector $POQ$. \n\nUsing the sector area formula from our data booklet:\n\n$$A = \\frac{1}{2}r^2\\theta$$\n\nSubstituting $r = 25\\text{ m}$ and $\\theta = 0.8$ radians:\n\n$$\\begin{aligned} \\text{Area}_{POQ} &= \\frac{1}{2} \\times 25^2 \\times 0.8 \\\\ &= 0.4 \\times 625 \\\\ &= 250\\text{ m}^2 \\end{aligned}$$\n\nThis awards the first mark M1."},{"type":"text","order":3,"title":"Area of reflex sector","content":"The inner part of the stage is a reflex (major) sector $ROS$ with radius $OR = 20\\text{ m}$.\n\nSince the minor angle is $\\angle POQ = 0.8$ radians, the reflex angle for sector $ROS$ is:\n\n$$\\theta_{\\text{reflex}} = 2\\pi - 0.8 \\approx 5.483\\text{ rad}$$\n\nThis awards the mark M1.\n\nNow, we calculate the area of this major sector directly using the sector area formula:\n\n$$\\text{Area}_{ROS} = \\frac{1}{2} \\times 20^2 \\times (2\\pi - 0.8) = 200(2\\pi - 0.8) \\approx 1096.64\\text{ m}^2$$\n\nThis awards the mark A1."},{"type":"text","order":4,"title":"Calculate total area","content":"The total surface area of the stage is the sum of the two non-overlapping areas we just calculated:\n\n$$\\begin{aligned} \\text{Total Area} &= \\text{Area}_{POQ} + \\text{Area}_{ROS} \\\\ &= 250 + 200(2\\pi - 0.8) \\\\ &\\approx 250 + 1096.64 \\\\ &= 1346.64\\text{ m}^2 \\end{aligned}$$\n\nThis sum awards the mark M1.\n\nRounding to the nearest integer as requested, we obtain:\n\n$$\\text{Total Area} \\approx 1347\\text{ m}^2$$\n\nThis final answer awards the mark A1.","highlights":[{"text":"to the nearest integer.","location":"specification"}]}]},{"id":"81df3f2c-8fc6-4b0d-b93c-e0c9d41df905:complementary_area","label":"(c) Complementary Area Approach","steps":[{"type":"text","order":0,"title":"Determine the radius OP","content":"To calculate the area of the outer minor sector $POQ$, we first need to find its radius $OP$.\n\nRecall the arc length formula from your radian measure studies:\n\n$$s = r\\theta$$\n\nHere, the outer arc length $s = 20\\text{ m}$ and the angle $\\theta = 0.8\\text{ rad}$. Letting $r = OP$, we have:\n\n$$20 = OP \\times 0.8$$\n\n$$OP = \\frac{20}{0.8} = 25\\text{ m}$$\n\nThis gives us a radius of $25\\text{ m}$ for the larger sector.","highlights":[{"text":"The length of the outer arc PQ is 20 m.","location":"specification"},{"text":"0.8 radians","location":"specification"}]},{"type":"text","order":1,"title":"Calculate area of POQ","content":"Now we calculate the area of the outer minor sector $POQ$ using the sector area formula:\n\n$$\\text{Area} = \\frac{1}{2} r^2 \\theta$$\n\nSubstituting $r = 25\\text{ m}$ and $\\theta = 0.8\\text{ rad}$:\n\n$$\\text{Area}_{POQ} = \\frac{1}{2} \\times 25^2 \\times 0.8 = 250\\text{ m}^2$$"},{"type":"text","order":2,"title":"Find the radius OR","content":"Next, we find the radius of the inner part, $OR$.\n\nWe are given the ratio of lengths $PR : RO = 1 : 4$. Since $O, R,$ and $P$ are collinear, the total length $OP$ is divided into $1 + 4 = 5$ equal parts, where $RO$ represents $\\frac{4}{5}$ of the total length $OP$.\n\n$$\\begin{aligned} RO &= \\frac{4}{5} \\times OP \\\\ RO &= \\frac{4}{5} \\times 25 \\\\ RO &= 20\\text{ m} \\end{aligned}$$\n\nSo, the radius of the inner circle $OR$ is $20\\text{ m}$.","highlights":[{"text":"The ratio of the lengths PR : RO = 1 : 4.","location":"specification"}]},{"type":"text","order":3,"title":"Apply Complementary Area Approach","content":"Using the **Complementary Area Approach**, we can find the area of the major sector $ROS$ by subtracting the area of the inner minor sector with angle $0.8\\text{ rad}$ from the area of the full inner circle of radius $OR = 20\\text{ m}$.\n\n- **Area of full inner circle:**\n$$\\text{Area}_{\\text{circle}} = \\pi r^2 = \\pi \\times 20^2 = 400\\pi\\text{ m}^2$$\n\n- **Area of inner minor sector (angle $0.8$):**\n$$\\text{Area}_{\\text{minor}} = \\frac{1}{2} r^2 \\theta = \\frac{1}{2} \\times 20^2 \\times 0.8 = 160\\text{ m}^2$$\n\n- **Area of major sector $ROS$:**\n$$\\begin{aligned} \\text{Area}_{ROS} &= \\text{Area}_{\\text{circle}} - \\text{Area}_{\\text{minor}} \\\\ &= 400\\pi - 160 \\\\ &\\approx 1256.64 - 160 \\\\ &= 1096.64\\text{ m}^2 \\end{aligned}$$"},{"type":"text","order":4,"title":"Sum the areas","content":"The total surface area of the stage is the sum of the area of the outer minor sector $POQ$ and the area of the major sector $ROS$:\n\n$$\\begin{aligned} \\text{Total Area} &= \\text{Area}_{POQ} + \\text{Area}_{ROS} \\\\ &= 250 + (400\\pi - 160) \\\\ &= 90 + 400\\pi \\\\ &\\approx 1346.64\\text{ m}^2 \\end{aligned}$$\n\nRounding to the nearest integer, we get:\n\n$$\\text{Total Area} \\approx 1347\\text{ m}^2$$"}]}],"generatedAt":"2026-06-09T10:21:48.408Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__wjec","summaries":[{"id":"65b33456-ebd4-4af9-b7b3-c47294de42db","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"94057f61-abbc-4527-807f-573c6d2bb3ef","difficultyBand":"easy","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"e8ba8a76-b67a-41d7-97e1-02d637d03671","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"0aec2d07-eeb2-4854-99e5-78c5846bc99a","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"11a10575-0a46-4902-8659-9ef5676fb790","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"180fa6a6-e6f8-467f-8eee-2bde765116b2","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1c3539bd-2b7b-42e2-96cd-ad5ecc1a9b85","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"3155a703-639c-42e9-a2a0-065a726bb8c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"47a37e7b-a87b-4583-8b17-28f02146334b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"51aea9f2-3e1a-41e2-b43a-1b5737a1c0fc","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"71c4bd57-dd52-4e28-94ac-f95b2dc4dc4a","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"938f7bbf-f10d-4cd5-a0b0-f8e7ed8cfd61","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"953a2097-8503-4a13-8b3c-315e5e488039","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e016be92-1d7c-4924-b219-abd4a2c0b1d2","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"e0cd8580-436f-4f08-8dee-284b953c607a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"f2b54dcf-595e-401a-b92f-9a3bba892d0d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"f4923309-8d0b-4a75-bdf8-bf6885937510","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f72cfefd-990a-475c-8fde-0ec350de12d9","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"04df7d63-f5cb-4485-9727-ce5baaaf8bdc","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"acd6f9f4-de00-431c-a1f3-5c3fddbc8625","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b9b10d51-18c3-4a4b-9a9b-6f4296660855","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"009035ab-b95e-4a9f-9087-d0a651547233","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"048c3a3e-e04e-4be6-b422-8029fdb4fa0c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"0721a4ee-e4ab-467f-97e9-5fda7c60e630","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0e1af8ff-3856-482d-bf8c-2e012d25c122","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"14274a7a-233a-4bf1-bc53-389b53840114","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"327076de-b61b-400a-bde2-937a30472d46","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"4ed7094e-cf03-4e1e-8691-1d3d0175a970","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"719c5525-de5d-44ce-b58b-8c4a302df638","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"78a118f7-2344-4c79-b026-e5f3ada312c2","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"8714d6b0-9b11-45c2-b4a2-7e1eab0cf698","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a2f875e1-470c-47b6-a47c-8dd83e0d38e0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"c6241b26-4040-4dd8-8d9e-7a596676f9b1","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"c982acf5-de7a-460e-bb95-d152177fb4e7","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f1729ffe-13fc-4056-b25f-93caa75c4833","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"f47798aa-2c98-491a-bf4e-cf6360ff2b3b","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5d80e0c9-3c2f-403c-9e75-25fcd8ade4dd","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"99e5d4bc-3815-4df7-8a89-82f3b79d4fb4","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"e9e7cf6a-e376-4175-8d05-3602929c5bbf","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0de526f2-fbb9-48fa-9f0b-cff5155321f4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"101fe1be-f19c-4f8d-a77b-0b39eb54297c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"2a5f70b1-aeff-4a85-a7ab-fd13723ddd4e","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"536015ff-d008-4174-9c6c-839c46217f8e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"775582c4-55c4-4dc3-8c5c-c0a63c63dcaa","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"7f2acd9f-9bf7-4e62-8ded-4fdbd71007f4","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"822179c6-49f5-4f12-90be-93033b9b6a58","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8ce7a8c3-8dff-4b7d-9451-5de24e975831","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"934c88ce-aae4-46e7-8f4e-4af676f41791","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9f0a6c6b-f862-44ad-9eaf-ce962ede65df","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"c9d3770d-2da5-4cbf-aa58-8e182df190b6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"cef9c774-cc2f-4693-a79a-6434cadcb3f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d9709422-e2ce-44dc-8fb1-5939c3d61419","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"e35f11f1-d7d2-44dc-b364-ece4104acfa3","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"fecf3975-1519-4c4d-bf32-9ef833dcf206","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"2cb328b9-19c8-4983-a09d-31ba9ff0ece8","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5ec2a8ee-f256-4256-b67d-e8fe4c80c5ad","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"536ab148-9e3e-429a-8f23-80ed01ef8557","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"83595e3c-e503-4372-8f45-0c7c60173098","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a2eec4d2-89c8-415c-8c8b-6b1e523138f7","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"cd1dffd5-3636-4a2e-a756-86ef66e4090d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"003c0e82-2afd-46b0-94ea-86720d42988a","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"006483fe-fe7e-4cc4-864d-fab5f152bf33","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"01d13dfe-e432-4a4b-9945-ad584dc111e6","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"05b5fc43-7c93-48d2-9e9d-42a401b325e4","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"069fe3ee-7983-46dc-9454-a09eddc3b08a","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"072b940e-4992-409f-86ca-539b215200a5","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0a94e690-04c9-44b2-bfdf-c313fc95f5db","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"0cfa9b5c-c1d5-463a-8c91-3f1df59cba74","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"10528f69-997d-49e4-a12b-26c2732bcb8d","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"10cbd4c3-001c-4e83-99ba-8867b8bf41f2","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"119e9362-1f46-4aa5-97be-625f6ea78c35","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"11e7f755-bfdb-461a-bc4f-9c1680670249","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"15406844-792d-4523-9ea8-df8567630da2","difficultyBand":"hard","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"162bd4d0-a91c-4e9f-a578-46fb86c6548b","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1676c933-f8e3-41c6-a567-c91286856ce8","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"16f3f335-0f29-4f60-a113-da528d5b88c8","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"17b13754-20c6-4ec6-be6a-1581e260a33b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1d2f7ecc-4de6-41a2-a31b-81618dd9b855","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"1eaf9e01-73bc-4fe0-bd8a-300d25705a0e","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1ef17704-2998-4146-acd6-4b9d6ee38279","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1f894d1f-8fb3-44c2-bc84-4abf0227d342","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1fde01cc-33ca-4271-a99a-2994b71cce0f","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"201338d3-e238-4cc6-a3b4-6bf932edbfbc","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"20d13abb-d90f-4fa4-b45b-c42509dfbd67","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"21390a73-0ded-4be9-9fc7-4562b609cfb0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"21c02aed-c565-49a6-88c1-520b965b5fcd","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"21e38944-86ad-4493-aa04-3accfc01da23","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"2423687b-ba39-452d-8863-34deb93dac6c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"25ce1081-c674-4460-88f1-f992c1e5ff79","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"27008d16-42d8-44c9-b92e-0ae0079ac1ef","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2ccafd98-a9eb-4e2f-a266-ccc5e7b024da","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"327eafad-b9b5-4ae5-a127-75e7b51b89bf","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"328a884a-2fd6-4e0f-b798-eb627a46f614","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"337be6c9-7358-491a-a890-000198ede2d4","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"351bcce9-ed5c-43c3-8908-ec8b06711f5e","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"38c05923-2889-40f5-996b-faab30fc7f8b","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"38e393d6-3902-4a11-a192-2773ae33d771","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"3a173901-91a2-4c54-9501-e0b0e25a9bf7","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3a4a20a7-52dd-4f31-947c-3096996cfb33","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3b182616-3abd-4431-ad20-5b06a37c1b84","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"3b398c94-c1d7-44b8-9b7d-260afd4d9297","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"3d68b2b7-bc78-4fdc-baff-9fb42eeb6031","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"3db9f5cd-0c63-447a-a354-0c61624681b8","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"3dc00c69-a3ca-4873-a70b-6b54da9d9c4a","difficultyBand":"hard","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"44175577-9915-44de-b50e-9b1f830491dd","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":false},{"id":"4915fd73-abcd-48ff-973d-c8f1edf5fdd7","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"4a29e657-9041-498a-b6e4-05fbabfde19f","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"4c48d117-6f1c-484d-b752-5e73365de807","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"4f69c1e1-d803-45b8-a414-98fbde10e2bf","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"505bffff-22b1-4097-9302-fa5a280cb6e2","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"50e9b35b-5950-4c9f-9fce-dfdb9de941a3","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"512008b0-83d1-4934-a51b-f4b1e6545942","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"5395c452-63da-4b16-9af1-7a3316d2e542","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5cb67e34-cca9-4384-9798-b104408e1e12","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"5f889398-58a2-4bf9-8c25-500534324770","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"62979910-7ad4-48fb-a551-883523533268","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"6353b0eb-3a53-4d5d-b1e7-14fe34fe647c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"63ddc30c-c999-4f7a-b0df-ff1b572d4f32","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"65248db3-3871-42c2-9934-9d81bf9372dd","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"65b32b3b-6e53-436e-97e0-c601f95d031a","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"65f59c3a-65f7-4841-9244-7c205d69a370","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"68a4587e-20bd-414a-9532-c9dd05cb5862","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"690b556d-3755-415d-bf74-86c00c3f8a18","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6a9eac22-6c2f-4b41-b84c-b69e757a4d21","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"6db579f2-62a1-4497-8595-44d58cafac8e","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"70d01524-477a-415d-9f1c-20ee0779e3b9","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"70eb939c-299f-4169-a8ac-b10fa10673d6","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":false},{"id":"71f21cc3-3a3b-49ca-844d-13d8c0e6fbed","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"76b6db76-f0bd-452a-8c0c-f57b05d8ed9b","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"7728e5d1-8b02-4a82-80ab-406b9d9c9ca0","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"7a783d21-a95d-4b5d-9a49-84ac3e4e9b0b","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"7b7638d7-f416-4902-ad39-cb33f5962d33","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"84369d6f-3818-4ca6-ae1e-e24ee6a51011","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"8636a590-f43b-474c-ae55-4f692b880cff","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"887faea7-f861-43d0-bfff-4ae4fa3219bc","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"88bdb9fe-0922-4beb-aefe-d53283eff103","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"89d1b8be-a141-4c78-816c-1ad4f855475c","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"8a84717a-d6df-4795-be10-7b36e5151d09","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"8bf9b1f3-f203-46c8-a9a7-f757198310eb","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"8d69b973-4b4e-4685-80fd-d5a9df7745c3","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"8e1db89f-bb68-4109-9390-5447ba387392","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"92fb3515-c1ed-41c4-80fa-0557ace4c4b0","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"93c19269-dd2c-4499-b25d-a6ee663dc33c","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"93d4ecc1-84b7-43fc-a2d7-8d5edda38cde","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"964e79ae-abef-4b55-bacc-acd8c3f7a25c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"96a08353-fb4d-46d3-8988-dfa69d9bf244","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"97f6bbc6-9f6d-4654-81ca-bb83949a4c3d","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"98b058ce-baa8-4a28-b29f-fa8a3cafe6eb","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"99069f8b-e0cc-46ff-94f7-e8d305d9cdfe","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"9952e45d-1751-4c06-a231-a0b217e43cf0","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"9cbabc0a-acdf-44cf-8a6e-b4759b8ce9b3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"9d4fbc1e-8f13-46d1-b176-2a344627f87b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9e31fab4-d5f2-47ad-8280-d82571015111","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"a00a7fdb-5f80-4a62-bd04-f80355f610ca","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a07af439-46dd-4bf4-831d-465f0dc4e2b8","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a0d4e11e-efc6-44a3-967d-a9be224d2a32","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a140ab1b-1b42-4ccc-b64c-520972ec87a8","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"a1b76c1c-59c6-4065-8599-29c71c32b89a","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"a586d201-6153-4fc7-9282-d2915df83e75","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"a87cc961-1aff-4cfe-9248-3d996bce4fce","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a89aa596-b453-4a58-bb81-60f2d64d1490","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ab79c082-1eb6-4721-8714-72817b51f94c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ae197600-f96e-4c34-a37d-9d072b8921a0","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"af834080-9977-4a4e-8276-74b78b465578","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"b20a2474-5ae5-40ef-8d6c-fc4c3ad20f5f","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"b42521dd-95b5-4bac-a07d-33089fa9bbcc","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b5132760-3951-478d-bde0-e6c202172474","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"b5b31158-aef2-4f24-ac7d-799f93edb23a","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"b6e07195-371e-42d9-9ce8-93ff9bd7530f","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b7f0d6f4-9469-4bd8-8030-b2c0c1c54925","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"b9aa88ab-01c7-49e1-b05e-3d7e7e1ca9a3","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"ba2de761-6181-4a00-88d2-a60977bab86e","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"bb18ad5c-044c-4972-a77b-4a0710ba2fae","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bb3c6839-93fd-4a83-b464-90f3ccc32517","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"c0ad646c-6c2b-44fa-8293-307d3b169f18","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c18ec8b4-d2a5-4951-bbf0-e6f7e96c4d33","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c318f847-a741-4829-ba35-e95d34dac46a","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"c84b0c64-9305-4605-a47b-e9d229323b61","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"c92d4ddd-435d-44fc-95d4-53a12eded7a7","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ca54a56d-32a6-4645-94f1-fb125b781074","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"ca771393-9a00-4a16-8640-184911d04ec5","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ce19f4ad-3669-44cd-8b1b-017a0566f741","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"cf9fb6c7-9380-4eb1-8ecc-5065067b54b1","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"d0a7483a-cafe-4317-b34c-e09d367f0616","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d183dcb1-0e62-4f4c-acf3-7106d716d77c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d19f03a5-82ac-4f66-9e6b-2f7bcc8fc927","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"d22e18f9-9abb-41fe-9828-87f4b798ec7e","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d5e8195d-1438-450e-b754-5fb4f576ef40","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"d706b6df-3a05-4fed-85ca-58784e003009","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"d78325a1-250a-4250-9e59-8468872ba0e0","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"d8345c4d-4b5c-41fb-a14e-6831cbd07e5a","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d89d701f-be26-4e98-b823-fac1354c506f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"da2ab0cc-fbda-48dc-a64c-f2c2dd7902a5","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"da7aa08e-6de4-4c69-b513-2c3e117e9854","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"dab10df8-e044-4362-9948-06fb79223c97","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"dbc1b4c7-0738-4da6-b312-f44971b80142","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"dce918aa-bb1d-43f2-a21c-b04865be7cb1","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"dda0d545-7277-4446-812c-6903b50c9b2b","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"dff1a84a-52c5-4247-bcdf-bf51a5ff4ac4","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e0007dd7-e51e-4335-8fe7-94e46d39b9fb","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e1891da7-dfee-4d17-b5da-f9075ce2c6ca","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"e2161ede-fb25-47c2-9eb8-90b6c123bb33","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"e2d81f0f-46a9-4150-bdd7-3c3c1d96c18d","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"e38dd4ed-3a46-4c39-992f-658a38821b43","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"e3b512c5-7788-4f94-82c9-8cea5fbcf57e","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e7c25a3b-12c1-4ba6-aaa8-9e494bcb6ea5","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e888b7ed-23b7-468e-81ab-46cc13f17d40","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"eae09f26-df7e-4795-9214-7b124cc27302","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ec6a07be-f4c6-49e6-9f51-db095a2d048e","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"eca855ec-c608-464f-a630-573257686af5","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ecd1941d-d9ba-465d-9e13-2c6098913d89","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ee1b4b87-10ae-4865-8cbc-cbd314417399","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ee916fe5-848f-4b1e-8c34-ab8d8ef5c64e","difficultyBand":"medium","calculatorUse":null,"totalMarks":14,"questionType":"LA","isCalculation":true},{"id":"eeae3160-c5de-4ee7-8f3d-9f1d566ec4ed","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"ef5cf080-e1e7-4c90-84b6-fa095c03b881","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"efa95e12-c5fb-4eb2-a433-4ee366f2ae6e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f14a2f13-8463-417d-8cb2-5e2c4443e003","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f2054a5a-e003-46e0-8d27-63c056892162","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"f38a6057-ae29-4730-8cf4-b02cfc2faf35","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"f4b8f88e-1990-4d40-8572-ce92671d609e","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f6786a45-2ab2-4809-84e2-3b5f0bf4fd69","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f6de1d5e-522e-4ccf-a658-ef0c2f5a78dd","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fcd7b986-e532-410e-b16c-a93caf48f013","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"ff0e3c1f-dac2-4ebb-9b45-bbc0f0d11d77","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"ff449bd6-00dd-43ff-8b6d-82fec48b5223","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true}],"title":"3.5 Trigonometry (A-level only)","topicId":"30ac4c05-8d09-4ec4-ac0a-b9371d6b1f6e","topicSidebar":"$L3a","topicSlug":"trigonometry-7","topicTitle":"3.5 Trigonometry (A-level only)"}]}],"$L3b","$L3c"]}]

3.5 Trigonometry (A-level only)

Question 141

3.5 Trigonometry (A-level only) Questions

3.5 Trigonometry (A-level only)

Question 141

3.5 Trigonometry (A-level only) Questions