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Binomial Theorem is a type of algebraic expansion that is expressed in terms of (a+b) for a positive integral exponent n.
- Binomial Theorem is used to solve expressions that are lengthy and tedious.
- It is also known as binomial expansion.
- The theorem is used in the field of algebra and probability.
- The value of the exponent used in the expression can be negative or fraction.
- It can easily solve expansion that involves polynomial function.
- The coefficient of binomial expansion can be applied to Pascal’s Triangle.
- The total number of terms used in the binomial theorem is n +1.
- The binomial theorem formula is as follows:
(x + y)n = axuyc
Some of the important properties of binomial theorem are as follows:
- C0 + C1 + C2 +…+ Cn = 2n
- C0–C1+C2 –…+(–1)nCn = 0
- C0 + C2 + C4 +…= C1 + C3 + C5 +…= 2n–1
- nCr = nCn–r
- r(nCr)=nn-1 Cr–1
- nCr/r+1 = (n+1)Cr+1/(n+1)
- nCr + nCr–1 = (n+1)Cr
- Where n ∈N, r ∈ W and r ≤ n
Binomial Theorem MCQs
Ques: Find the number of terms in (1 + 14x +49x2)60
- 101
- 22
- 121
- 300
Click here for the answer
Ans: (c) 121
Explanation: (1 + 14x + 49x2)60
= [(1 + 7x)2]60
= (1 + 7x)120
∴ The number of terms = (120 + 1) = 121
Ques: Find the middle term of (1 −6x + 9x2 )2n
- (4n/2) + 1
- (4n) + 1
- (4n/2) + 2
- (4n/2)
Click here for the answer
Ans: (a) (4n/2) + 1
Explanation: (1 − 6x + 9x2 )2n
= [(1 − 3x)2]2n
= (1 − 3x)4n
∴ Middle Term = [(4n/2) + 1] term
Ques: Find the independent term of x in (x+1/x)6
- 17
- 10
- 2
- 20
Click here for the answer
Ans: (d) 20
Explanation: r = [6(1)/1+1] = 3
∴ The independent term is 6C3 = 20
Ques: Find the 3th term in (2x - y)7.
- -16x5y2
- -126xy2
- -126x5y2
- -136x5y2
Click here for the answer
Ans: (c)-126x5y2
Explanation: Now, to determine the 3th term,
⇒ we can consider r = 2, a = 2x, b = -y and n = 7
Therefore, T3 = T2+1
= 7C2(3x)7-2.(-y)2
= -126x5y2
Ques: Determine the expansion of (x + 6)2 using the binomial theorem formula?
- x2 + 11x + 36
- x2 + x + 6
- x2 + x + 16
- x2 + 12x + 36
Click here for the answer
Ans: (d) x2 + 12x + 36
Explanation: (x + 6)2
= x2 + 12x + 36
Ques: The coefficient of the middle term in the expansion of (1+3x)4 is:
- 54
- 6
- 216
- 8!
Click here for the answer
Ans: (a) 54
Explanation: If the exponent of the expression is n, then the total number of terms is n+1.
⇒ The total number of terms is 4+1 = 5.
⇒ The middle term is the 3rd term.
⇒ We know that general term of (x+a)n is Tr+1 =nCr xn-r ar
∵ Here, n=4, r=2
Therefore, T3 = 4C2.(1)2.(3x)2
⇒ T3 = (6).(1).(9x2)
⇒ T3 = 54x2.
∴ the coefficient of the middle term is 54.
Ques: The value of (217)1/3 up to three decimal places is
- 6.011
- 5.012
- 6.013
- 6.009
Click here for the answer
Ans: (d) 6.009
Explanation: (217)⅓ can also be written as the cube root of 217.
Hence, (217)⅓ is approximately equal to 6.009.
Ques: Consider a binomial expansion where n is even in the expansion of (a+b)n, the middle term is:
- nth term
- [(n/2)+1]th term
- [(n/2)-1]th term
- [(n/2)]th term
Click here for the answer
Ans: (b) [(n/2)+1]th term
Explanation: Since “n” is the even in the expansion of (a+b)n, then the number of terms will be odd. (i.e) n+1.
Hence, the middle term of the expansion (a+b)n is [(n/2)+1]th term.
Ques: The largest coefficient in the expansion of (1+x)12 is:
- 10! / (5!)2
- 13!/(7!)(6!).
- 13!/(7!)(8!).
- 10! / (5!×4!)
Click here for the answer
Ans: (b) 13!/(7!)(6!).
Explanation: Given: (1+x)12
∵ The greatest coefficient will always occur in the middle term.
⇒ The total number of terms in an expansion is 11. (i.e. 12+1 = 13)
⇒ Middle term = [(12/2) + 1] = 6+1 = 7th term.
⇒ We know that general term of (x+a)n is Tr+1 =nCr xn-r ar
⇒ n=13, r=7
∴ T7 = 13C6.x6
Therefore, the coefficient of the greatest term = 13C7 = 13!/(7!)(6!).
Ques: Find the number of terms in (1 + 16x +64x2)60
- 121
- 220
- 1210
- 300
Click here for the answer
Ans: (a) 121
Explanation: (1 + 16x + 64x2)60
⇒ [(1 + 8x)2]60 = (1 + 8x)120
∴ The number of terms = (120 + 1) = 121
Ques: Find the middle term of (1 −8x + 16x2 )2n
- (4n/2) + 10
- (4n)
- (4n/2) + 1
- (4n/2) + 5
Click here for the answer
Ans: (c) (4n/2) + 1
Explanation: (1 − 8x + 16x2 )2n
⇒ [(1 − 4x)2]2n = (1 − 4x)4n
∴ Middle Term = [(4n/2) + 1] term
Ques: Find the independent term of x in (x+1/x)10
- 17
- 100
- 288
- 252
Click here for the answer
Ans: (d) 252
Explanation: r = [10(1)/1+1] = 5
The independent term is 10C5 = 252
Ques: The value of (1002)1/3 up to three decimal places is
- 6.011
- 5.012
- 10.0066
- 6.009
Click here for the answer
Ans: (c) 10.0066
Explanation: (1002)⅓ can also be written as the cube root of 1002.
∴ (1002)⅓ is approximately equal to 10.0066.
Ques: Determine the expansion of (x + 8)2 using the binomial theorem formula?
- x2 + 16x + 64
- x2 + x + 6
- x2 + x + 16
- x2 + 12x + 36
Click here for the answer
Ans: (a) x2 + 16x + 64
Explanation: (x + 8)2
= x2 + 16x + 64
Ques: What is the formula of a binomial theorem?
- (x + y)n
- xy
- x
- y
Click here for the answer
Ans: (a) (x + y)n
Explanation: The binomial theorem formula is as follows:
(x + y)n = axuyc
Ques: The value of (347)1/3 up to three decimal places is
- 5.011
- 7.022
- 6.013
- 7.009
Click here for the answer
Ans: (b) 7.022
Explanation: (347)⅓ can also be written as the cube root of 347.
Hence, (347)⅓ is approximately equal to 7.022
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