Binomial theorem for positive integral index - One Line Questions
1.
What is the 4th term in the expansion of (2x - 3y)^7? —
-26460 x^4 y^3
2.
Find the term independent of x in the expansion of (x^2 - 2/x)^9. —
-84 * 2^6
3.
In the expansion of (x + a)^n, if n is odd, what are the indices of the two middle terms (starting from r=0)? —
(n-1)/2 and (n+1)/2
4.
What is the value of C(n, 0)? —
1
5.
What is the value of C(n, n)? —
1
6.
What is the sum of the coefficients of the terms in the expansion of (3x - 2y)^4? —
1
7.
What is the coefficient of x^2 in the expansion of (1 + x)^5? —
10
8.
What is the coefficient of the 5th term in the expansion of (a + b)^9? —
252
9.
Find the term independent of x in the expansion of (x - 2/x)^6. —
192
10.
What is the sum of the coefficients in the binomial expansion of (x + y)^n? —
2^n
11.
What is the sum of the binomial coefficients C(n, 0) + C(n, 1) + ... + C(n, n)? —
2^n
12.
The sum of the coefficients of the expansion of (x + y)^n is obtained by setting x=1 and y=1. What is the result? —
2^n
13.
Find the term containing x^3 in the expansion of (x + 1/x)^7. —
35 x^3
14.
Find the middle term in the expansion of (x + 3y)^10. —
252 * 243 * x^5 y^5
15.
The (r+1)th term in the expansion of (x + a)^n is T_{r+1} = C(n, r) x^(n-r) a^r. What is the value of r for the term containing a^5 in the expansion of (x+a)^10? —
5
16.
What is the coefficient of x^3 in the expansion of (x + 2)^5? —
80
17.
What is the sign of the terms in the expansion of (x - a)^n when n is a positive integer? —
Alternating positive and negative
18.
What is the coefficient of x^7 y^3 in the expansion of (x + y)^10? —
C(10, 7) and C(10, 3)
19.
The binomial theorem for a positive integral index n states that (x + a)^n = sum_{r=0}^n C(n, r) x^(n-r) a^r. What is the first term (r=0)? —
C(n, 0) x^n a^0
20.
What is the coefficient of x^k in the expansion of (x + a)^n? —
C(n, n-k) * a^k
21.
What is the coefficient of x^k in the expansion of (x+a)^n, where k is the power of x? —
C(n, k) * a^(n-k)
22.
What is the coefficient of a^k in the expansion of (x + a)^n? —
C(n, k) * x^(n-k)
23.
What is the coefficient of the term with x^p in the expansion of (x+a)^n? —
C(n, p) * a^(n-p)
24.
In the expansion of (x + a)^n, the coefficient of x^r a^(n-r) is: —
C(n, r) and C(n, n-r)
25.
In the expansion of (x + a)^n, what is the last term (r=n)? —
C(n, n) x^0 a^n
26.
What is the value of the middle term in the expansion of (x + a)^n when n is even? —
C(n, n/2) * x^(n/2) * a^(n/2)
27.
What is the term independent of x in the expansion of (x + 1/x)^n? —
C(n, n/2) if n is even, otherwise 0
28.
In the expansion of (a + b)^n, the coefficient of the term containing a^(n-r) * b^r is: —
Both C(n, r) and C(n, n-r)
29.
Consider the expansion of (1 + x)^n. What is the coefficient of x^r? —
C(n, r)
30.
In the expansion of (x + a)^n, what is the coefficient of the term containing x^(n-r) a^r? —
Both C(n, r) and C(n, n-r)
31.
What is the (r+1)th term in the expansion of (a+x)^n? —
C(n, r) * a^(n-r) * x^r
32.
What is the value of C(n, r) + C(n, r-1)? —
C(n+1, r)
33.
What is the value of C(n, r) + C(n, r+1)? —
C(n+1, r+1)
34.
The expansion of (x + a)^n has terms T_1, T_2, ..., T_{n+1}. What is the index of the term that contains a^k? —
k+1
35.
When n is an odd positive integer, how many terms are there in the expansion of (x + a)^n? —
n + 1
36.
When n is an even positive integer, how many terms are there in the expansion of (x + a)^n? —
n + 1
37.
The expansion of (x + a)^n has (n+1) terms. What is the index of the last term (starting from index 0)? —
n
38.
In the expansion of (x + a)^n, the sum of the powers of x and a in each term is always: —
n
39.
What is the value of C(n, 1)? —
n
40.
What is the value of C(n, n-1)? —
n
41.
If the term T_{r+1} in the expansion of (x + a)^n is C(n, r) x^(n-r) a^r, then the term containing x^k will have r equal to: —
n-k
42.
In the expansion of (x + a)^n, if n is even, what is the index of the middle term (starting from r=0)? —
n/2
43.
In the binomial expansion of (x + a)^n, what is the (r+1)th term? —
nCr * x^(n-r) * a^r
44.
Which property of binomial coefficients is represented by C(n, r) = C(n, n-r)? —
Symmetry Property
45.
For the expansion of (x - a)^n, where n is a positive integer, what is the general term? —
T_{r+1} = C(n, r) * x^(n-r) * (-a)^r
46.
What is the general term in the binomial expansion of (x + a)^n, where n is a positive integer? —
T_{r+1} = C(n, r) * x^(n-r) * a^r
47.
The binomial expansion of (x + a)^n is given by the sum of terms from r=0 to n. What is the formula for the (r+1)th term? —
T_{r+1} = C(n, r) * x^(n-r) * a^r
48.
In the expansion of (x + a)^n, the term with the highest power of 'a' is: —
The last term
49.
Which term in the expansion of (x + a)^n is independent of x? —
The term where the power of x is 0
50.
To find the sum of the coefficients in the expansion of (ax + by)^n, we substitute: —
x=1, y=1