1 . 下列等式中, 正确的是( )
A.![]() | B.![]() |
C.![]() | D.![]() |
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2 . 若关于
,
的三项式
的展开式中各项系数之和为64,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
______ ;其中
项系数的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1eb936f10114007b7714efa72b7b946.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f29d5f376c75c41ae6af0c8a8565449.png)
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3 . 已知
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8017515e1f03c7b2043e6f1328b813a3.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fc1757e20cbbee0b00a0b72a5479cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8017515e1f03c7b2043e6f1328b813a3.png)
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4 . 在二项式
的展开式中,奇数项的二项式系数的和为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9216a0f9d6e65ea4937ab7bf102c5db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7182390a8171e716530156dc6040a2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
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5 . 在探究
的展开式的二项式系数性质时,我们把系数列成一张表,借助它发现了一些规律.在我国南宋数学家杨辉1261年所著的《详解九章算法》一书中,出现了这个表,我们称这个表为杨辉三角.杨辉三角是中国古代数学中十分精彩的篇章.杨辉三角如下图所示:
第0行 1
第1行 1 1
第2行 1 2 1
第3行 1 3 3 1
第4行 1 4 6 4 1
第5行 1 5 10 10 5 1
第6行 1 6 15 20 15 6 1
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed8181dab2251aad31eef4d43413cf8.png)
如上图,杨辉三角第6行的7个数依次为
,
,
…
,
.现将杨辉三角中第
行的第
个数乘以
,第0行的一个数为0,得到一个新的三角数阵如下图:
第0行 0
第1行 0 1
第2行 0 2 2
第3行 0 3 6 3
第4行 0 4 12 12 4
第5行 0 5 20 30 20 5
第6行 0 6 30 60 60 30 6
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed8181dab2251aad31eef4d43413cf8.png)
在这个新的三角数阵中,第10行的第3个数为________ ;从第一行开始的前
行的所有数的和为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9216a0f9d6e65ea4937ab7bf102c5db.png)
第0行 1
第1行 1 1
第2行 1 2 1
第3行 1 3 3 1
第4行 1 4 6 4 1
第5行 1 5 10 10 5 1
第6行 1 6 15 20 15 6 1
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed8181dab2251aad31eef4d43413cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed8181dab2251aad31eef4d43413cf8.png)
如上图,杨辉三角第6行的7个数依次为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c157f0f16f099230f1831cff5a3aae3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14ad7c0d051c4a14c35ba35bd8e6675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d88120a74c84c257915b5c060e503008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93dc96c400e9f6aa2f55f646c427e02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59cfd02b309dbdda35440c860bac311.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8a7c6d10fc680085289ed89f2b4878.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35994cf95c433ff61cdcc6345acc53f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db68583bc5ded5e0cf7028c0fd4297ab.png)
第0行 0
第1行 0 1
第2行 0 2 2
第3行 0 3 6 3
第4行 0 4 12 12 4
第5行 0 5 20 30 20 5
第6行 0 6 30 60 60 30 6
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed8181dab2251aad31eef4d43413cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed8181dab2251aad31eef4d43413cf8.png)
在这个新的三角数阵中,第10行的第3个数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b568d18fba797efb24d3baf3be98768b.png)
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6 .
的二项展开式中
的系数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac252a8b5b997a4f032c42b3aa1425f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
A.![]() | B.40 | C.![]() | D.80 |
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7 . 已知
的展开式中,含
项的系数为
,
.则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d023b359b3ac2db2e0a553fd0f793a.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55ff01ffe4598976aac530e869f798f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3336c8ed5361c10c37300e41e03f9f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c6b3f2836e67b34c280001e52841a8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d023b359b3ac2db2e0a553fd0f793a.png)
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8 . 已知
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfd2ef3bba74622edaebeecc5bc25af9.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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9 . 对于
,
,
不是10的整数倍,且
,则称
为
级十全十美数.已知数列
满足:
,
,
.
(1)若
为等比数列,求
;
(2)求在
,
,
,…,
中,3级十全十美数的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b1cfbfdf8e1b22aab9583e12e3449c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53f0e26992724eafcba06d163d9ff470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4217b1854fee34983372bf4f3a877d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdf53108bee755f5aa9a34ea4d163e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c2b5e218eb815213d8bc0ce9a06ca5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ac416116febcf793fee4ccc78a27b15.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0f62daf8552adeb241c9b54a57cd83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)求在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11075f2c574b6c59b97fb3038000e38.png)
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10 . 阅读知识卡片,结合所学知识完成以下问题:
知识卡片1:
一般地,如果两数
在区间
上的图象连续不断,用分点
将区间
等分成
个小区间,在每个小区间
上任取一点
(
,2,…,n),作和式![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13820d7479810174b76da70f2b0c91a.png)
(其中
为小区间长度),当
时,上述和式无限接近某个常数,这个常数叫做函数
在区间
上的定积分,记作
,即
.这里,
与
分别叫做积分下限与积分上限,区间
叫做积分区间,函数
叫做被积函数,x叫做积分变量,
叫做被积式.从几何上看,如果在区间
上函数
的图象连续不断且恒有
,那么定积分
表示由直线
,
,
和曲线
所围成的区域(称为曲边梯形)的面积.
知识卡片2:
一般地,如果
在区间
上的图象连续不断,并且
,那么
.这个结论叫做微积分基本定理,又叫做牛顿——莱布尼茨公式.例如,如图所示,对于函数
(
),从几何上看,定积分
的值为由直线
,
,
和曲线
所围成的区域即曲边梯形
的面积,根据微积分基本定理可得
.
①
;
②
;
③
;
④
.
(2)已知
,计算:
①
;
②![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8be82da06bf4bad8efa5c65a4d960a2.png)
(3)当
,
时,有如下表达式:
.计算:
知识卡片1:
一般地,如果两数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2779faf49c4c603fdb73ef6f03cc8d82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78d74528a8974c188d81391d4f158c28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96c9975b7cb327da8634aabab7856095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13820d7479810174b76da70f2b0c91a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eb09a0fb843fbc6959ab5688771ee01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50b037afdaa0872d5004fe3e8206ee89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19893fdb67307d35a9115ef4f3f1202a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4471a3b99ea6b431271b76b7931cf272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0112f1dbb9abb6bbffc3ccfb77ba03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b5f852626c9a09019c20fba7b388183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58196b9e63ec00aa1119052b6de6ae12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4471a3b99ea6b431271b76b7931cf272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b53b86bd516400d6fa7dabb3603f31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b334dafda377c3db77647c8cf1e95f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
知识卡片2:
一般地,如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4dd421af6cde8dad1ef435eb12cc45a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71da6f8d70a273a75779e1aa818d4f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60e712d5093491bf63ab4cdd34e8ca44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/644ba16341e356b57ea153e840555290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b53b86bd516400d6fa7dabb3603f31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b334dafda377c3db77647c8cf1e95f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe83f1ae7e5f05d8bed6bf6f42db0e7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e16dc5c9406c6cdadbca9136ef97b99.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6bfbb31a1766fea0bf95473044d34b.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bdb735861115bfe3412c93538c4a522.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea107a985b40ed680a43dfb1fb9c2535.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dfd1660244935cb37ca57a08a26b05b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2e58d66635ee79693d107c8f3c49c5a.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bed11655d89fb2e5363377fc38d53d7c.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8be82da06bf4bad8efa5c65a4d960a2.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71842069f1fda57351db2de2e23a3262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2079e9798470edc75b66126cf06da150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bdc90b519006d41102a40a47a9fd87.png)
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