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1a:Te64,受験勉強お疲れ様です。
結論から述べると、目的と段階によりますが、基礎問題精講と1対1シリーズの2冊で、基礎から標準レベルの、数学Ⅲの一通りの解法パターンは習得できる(※ただ基礎問題精講だけでは解法パターンは網羅できていないかな)と思いますし、一旦まず数Ⅲを一通り学習するというのであればそれで十分ですが、東工大を目指す上では別の観点から、多少馬力不足な気がします。また数Ⅲを学習する際は、数Ⅲの性格をよく知った上でやるのが効率も良いですし、得策かと思われます。僕の受験体験から数Ⅲに関して2点特徴を挙げます。
1点目、入試問題の数Ⅲは、おおよそ、傍用問題集に乗るような基礎的標準的な問題から、誰も完答できないような難問奇問まで多岐に渡ります。数1A2Bの場合には難問奇問はあまり出ません。ですが、最難関大を狙う学生たちはやはりレベルが高いため、難度の高い問題(過去問で言うレベルCやD)でも部分点ぐらいは狙ってきます。ですので、標準問題を反復するだけでは足りません(もちろん標準問題の反復は大事ですが)。もしAkiさんが一通り数Ⅲの標準問題を解けるようになったのなら、少し難度の高い問題や思考が必要な問題にも触れる必要があります。
2点目、数Ⅲは1A2Bに比べて計算量が著しく多いです。特に東工大は工業大学であるがゆえ、数Ⅲの出題では極限と微積が大部分を占めており、計算量も日本のどの大学に比べても類を見ないほどです。その一方で、基礎問題精講や1対1、チャートなどは解法パターンを習得することに主眼を置いているため本物の入試数学(特に東工大の数学)とは少々趣が異なります。つまり計算は軽めです。
以上2点からアドバイスを述べますと、基礎問題精講と1対1で解法パターンの習得は十分です。チャート式などに手を出す必要はあまりないと思われます。それよりかは、東工大レベルの息の長い計算力と思考力を少しでも鍛えるためにも、上記2冊で解法パターンの習得が済んだのならば少し上のレベルの問題を解く方が良いです。おすすめとしては、それこそ東工大の過去問に触れてみるのが一番手っ取り早いです。もちろん受験生の身としては過去問は残しておきたいのも分かりますが、結局は過去問は直近の5〜10年ぐらいをやれば十分ですので、直近10年だけ残しておいてそれより前の過去問を問題集のように解くのが得策です。他には上級問題精講・やさしい理系数学(←簡単ではない)、理系プラチカ数Ⅲなども東工大等の最難関大受験生にはおすすめです。これらの少し上のレベルの問題を解くことで負担の大きい計算力と息の長い思考力を鍛えましょう。
ちなみに1A2Bは難度の高い問題ばかりを解くよりは標準的な問題をこなす方が良いです。
長ったらしく拙い文章で申し訳ありませんが、僕のアドバイスが少しでもためになれば幸いです。
東工大の数学はやはり難しいです。ですが、そのレベルの高さにめげずに、むしろ数学極めてやるぐらいの勢いで、晴れて合格を掴み取って欲しいです。頑張ってください！1d:Tcb2,計画としてはそれで良いと思います。

すべての科目に置いて、学習は基礎から発展へと段階を踏むのが基本です。
質問者さんの計画はこの基本原則に則っており、
無駄も少なく十分実現可能でしょう。
例えば1日10題ずつ進めていけば、問題精講1冊がおよそ1ヶ月で終わりますので、
時間的にも十分です。

質問者さんの数学の理解度にもよりますが、
入門問題精講は解説が充実している分、
問題数は少なめで、難易度もかなり低いです。
基礎的な部分がわかっているのであれば、
基礎問題精講から入っても良いのではないでしょうか？
そこまで問題のレベルも高くありませんし。
もちろん、基礎問題精講が解けない！となれば入門から始めるのが良いでしょう。
(すでに問題集を手にとって入門からやらねば！となっているのであればすみません。)

チャートか問題精講かということですが、
これは質問者さんがどのような学習を望んでいるのかによります。
チャート系列は問題数が多く、演習を積みたいという方には非常に有用です。
ただし解説は蛋白で、全くの初心者がスラスラ進めることは難しいです。
一方問題精講はその名の通り厳選された問題のみを掲載しているので、
解法のパターンは大体つかめますが、演習量は確保できません。
解説は非常に豊かでわかりやすいと思います。

基礎を素早く固めたいならば問題精講、
演習量を増やし、基礎を盤石なものにしたいならチャートを選ぶのが良いでしょう。
質問者さんはなるべく早く基礎を固めたいということですので、
問題精講を使う計画が適切かと思います。

ーーー
ここからは得点戦略の話になります。
本来の質問内容とは少しずれるので読み飛ばしても構いません。

数学が苦手ということですが、どれほど苦手なのでしょうか。
学校の内容についていけない、またはギリギリついて行っている、
というレベルなのであれば、数学を得点源にすることは難しいです。
質問者さんはなぜ数学を得点源にしたいのでしょうか？

理系だからといって数学を得意になる必要はありません。
実際私も上でいろいろ偉そうなことを言っていますが、
数学が得意というわけでもなく、本番では半分も取れませんでした。

苦手教科で大切なのは高得点を取ることではなく、
点数の底割れを避けることです。
その分得意教科を伸ばしてカバーするほうがよっぽど簡単だからです。

もう一度、自分の得意不得意を見直して、
どの教科で何点取るのか、戦略を立てるのが良いと思います。
ーーー

以上、参考になれば幸いです。
高2の時点から基礎の重要性をよく理解できているのは素晴らしいことです。
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