In the expansion of the polynomial $2 ( x + a ) ^ { n }$, the coefficient of $x ^ { n - 1 }$ and the coefficient of $x ^ { n - 1 }$ in the expansion of the polynomial $( x - 1 ) ( x + a ) ^ { n }$ are equal. Find the maximum value of $a n$ for all ordered pairs $( a , n )$ satisfying this condition. (Here, $a$ is a natural number and $n$ is a natural number with $n \geqq 2$.) [4 points]

In the expansion of the polynomial $( x - a ) ^ { 5 }$, when the sum of the coefficient of $x$ and the constant term is 0, what is the value of the positive constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In the expansion of the polynomial $( 1 + x ) ^ { n }$, the coefficient of $x ^ { 2 }$ is 45. Find the natural number $n$. [3 points]

In the expansion of the polynomial $( x + a ) ^ { 7 }$, when the coefficient of $x ^ { 4 }$ is 280, what is the coefficient of $x ^ { 5 }$? (where $a$ is a constant) [3 points]
(1) 84
(2) 91
(3) 98
(4) 105
(5) 112

In the expansion of the polynomial $( x + a ) ^ { 6 }$, if the coefficient of $x ^ { 4 }$ is 60, what is the value of the positive number $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In the expansion of $( a + x ) ( 1 + x ) ^ { 4 }$, the sum of coefficients of odd-power terms of $x$ is 32. Then $a = $ $\_\_\_\_$ .

4. In the expansion of the binomial $( x + 1 ) ^ { n } \left( n \in N _ { + } \right)$, the coefficient of $x ^ { 2 }$ is 15. Then $n =$
A. 4
B. 5
C. 6
D. 7

Let $n > 1$, and let us arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the first three coefficients are in arithmetic progression. Then, the number of terms where $x$ appears with an integer power, is
(A) 3
(B) 2
(C) 1
(D) 0

The coefficients of three consecutive terms of $( 1 + x ) ^ { n + 5 }$ are in the ratio $5 : 10 : 14$. Then $n =$

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x ^ { 5 }$ in the expansion of $\left( a x ^ { 2 } + \frac { 70 } { 27 b x } \right) ^ { 4 }$ is equal to the coefficient of $x ^ { - 5 }$ in the expansion of $\left( a x - \frac { 1 } { b x ^ { 2 } } \right) ^ { 7 }$, then the value of $2 b$ is

In the binomial expansion of $( a - b ) ^ { n } , n \geq 5$, the sum of $5 ^ { \text {th } }$ and $6 ^ { \text {th } }$ terms is zero, then $\frac { a } { b }$ equals
(1) $\frac { 5 } { n - 4 }$
(2) $\frac { 6 } { n - 5 }$
(3) $\frac { n - 5 } { 6 }$
(4) $\frac { n - 4 } { 5 }$

If for positive integers $r > 1 , n > 2$, the coefficients of the $( 3r ) ^ { \text {th} }$ and $( r + 2 ) ^ { \text {th} }$ powers of $x$ in the expansion of $( 1 + x ) ^ { 2n }$ are equal, then $n$ is equal to:
(1) $2r + 1$
(2) $2r - 1$
(3) $3r$
(4) $r + 1$

If the coefficients of $x ^ { 3 }$ and $x ^ { 4 }$ in the expansion of $\left( 1 + a x + b x ^ { 2 } \right) ( 1 - 2 x ) ^ { 18 }$ in powers of $x$ are both zero, then $( a , b )$ is equal to
(1) $\left( 14 , \frac { 272 } { 3 } \right)$
(2) $\left( 16 , \frac { 272 } { 3 } \right)$
(3) $\left( 16 , \frac { 251 } { 3 } \right)$
(4) $\left( 14 , \frac { 251 } { 3 } \right)$

If $A$ and $B$ are coefficients of $x^n$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $\frac{A}{B}$ equals:
(1) $1$
(2) $2$
(3) $\frac{1}{2}$
(4) $\frac{1}{n}$

If the number of terms in the expansion of $\left(1 - \frac{2}{x} + \frac{4}{x^2}\right)^n$, $x \neq 0$, is 28, then the sum of the coefficients of all the terms in this expansion, is:
(1) 64
(2) 2187
(3) 243
(4) 729

The positive value of $\lambda$ for which the co-efficient of $x ^ { 2 }$ in the expansion $x ^ { 2 } \left( \sqrt { x } + \frac { \lambda } { x ^ { 2 } } \right) ^ { 10 }$ is 720, is
(1) $\sqrt { 5 }$
(2) 3
(3) 4
(4) $2 \sqrt { 2 }$

If the fourth term in the binomial expansion of $\sqrt { x ^ { \frac { 1 } { 1 + \log _ { 10 } x } } } + x ^ { \frac { 1 } { 12 } }$ is equal to 200 , and $x > 1$, then the value of $x$ is
(1) 100
(2) $10 ^ { 4 }$
(3) $10 ^ { 3 }$
(4) 10

If the fourth term in the Binomial expansion of $\left( \frac { 2 } { x } + x ^ { \log _ { 8 } x } \right) ^ { 6 } , ( x > 0 )$ is $20 \times 8 ^ { 7 }$, then a value of $x$ is
(1) $8 ^ { - 2 }$
(2) 8
(3) $8 ^ { 3 }$
(4) $8 ^ { 2 }$

If some three consecutive coefficients in the binomial expansion of $( x + 1 ) ^ { n }$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficients is:
(1) 227
(2) 964
(3) 625
(4) 232

If the sum of the coefficients of all even powers of $x$ in the product $\left(1 + x + x ^ { 2 } + \ldots + x ^ { 2n} \right) \left(1 - x + x ^ { 2 } - x ^ { 3 } + \ldots + x ^ { 2n } \right)$ is 61, then $n$ is equal to

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}}\right)^{10}$ is $10k$, then $k$ is equal to
(1) 336
(2) 352
(3) 84
(4) 176

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $( 1 + x ) ^ { n + 5 }$ are in the ratio $5 : 10 : 14$, then the largest coefficient in the expansion is:
(1) 462
(2) 330
(3) 792
(4) 252

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left( x ^ { m } + \frac { 1 } { x ^ { 2 } } \right) ^ { 22 }$ is 1540, is

If the constant term in the binomial expansion of $\left(\sqrt{\mathrm{x}}-\frac{\mathrm{k}}{\mathrm{x}^{2}}\right)^{10}$ is 405, then $|\mathrm{k}|$ equals:
(1) 9
(2) 1
(3) 3
(4) 2