Binomial Expansion MCQ Quiz - Objective Question with Answer for Binomial Expansion - Download Free PDF
Last updated on Apr 17, 2025
Latest Binomial Expansion MCQ Objective Questions
Binomial Expansion Question 1:
If the constant term in the expansion of
Answer (Detailed Solution Below)
Binomial Expansion Question 1 Detailed Solution
Concept:
The general term of the binomial (a + b)n is given by Tr+1 = nCran-rbr.
Calculation:
Given,
∴ General term,
⇒
For constant term, power of x = 0
⇒ 2r - 12 = 0
⇒ r = 6
∴
⇒ α =
⇒ 25α = 9 × 11 × 7 = 693.
∴ The value of 25α is equal to 693.
The correct value is Option 3.
Binomial Expansion Question 2:
In a binomial distribution, if the mean is 6 and the standard deviation is √2, then what are the values of the parameters n and p respectively?
Answer (Detailed Solution Below)
Binomial Expansion Question 2 Detailed Solution
Explanation:
Given:
⇒ Mean x = np = 6....(i)
⇒ SD =
Dividing (ii) by (i), we get
⇒
⇒ q= 1/3
Now
⇒ p = 1 – q = 1- 1\3 = 2/3
From (i)
⇒
⇒n =9
∴ Option (d) is correct.
Binomial Expansion Question 3:
Comprehension:
Direction : Consider the following for the items that follow :
Let (8 + 3√7) 20 = U + V and (8 - 3√7) 20 = W, where U is an integer and 0
What is the value of (U + V)W?
Answer (Detailed Solution Below)
Binomial Expansion Question 3 Detailed Solution
Explanation:
Given:
(8 + 3√7) 20 = U + V...(i)
(8 - 3√7) 20 = W...(ii)
⇒ (U + V)W =] (8 + 3√7) 20][(8 - 3√7) 20]
= (64 – 63)20 = 120 = 1
∴ Option (b) is correct.
Binomial Expansion Question 4:
Comprehension:
Direction : Consider the following for the items that follow :
Let (8 + 3√7) 20 = U + V and (8 - 3√7) 20 = W, where U is an integer and 0
What is V + W equal to?
Answer (Detailed Solution Below)
Binomial Expansion Question 4 Detailed Solution
Explanation:
Given:
(8 + 3√7) 20 = U + V...(i)
(8 - 3√7) 20 = W...(ii)
Here, 0
Now, adding Eqs. (i) and (ii), we get
U + V +W = (8 + 3√7) 20 + (8 - 3√7) 20
= 2[20C0820+ 20C2818. (3√7 )2+........+(3√7)20
⇒ It is an even number.
Also, 0
Thus, V + W is an integer
V + W = 1
∴ Option (d) is correct.
Binomial Expansion Question 5:
What is the remainder when 7n - 6n is divided by 36 for n = 100?
Answer (Detailed Solution Below)
Binomial Expansion Question 5 Detailed Solution
Explanation:
Let p(n) = 7n – 6n
= (6+1)n – 6n
= 1+6n +nC262 + nC363+.......... 6n -6n)
= 1+ 62(nC2+ nC3 × 6 +....... 6n-2)
= 1+ 36(100C2 + 100C3 × 6 +........698).....(n =100)
= 1 + 36 k
where k = (100C2 + 100C3 × 6 +........698)
Thus When 7n – 6n is divided by 36, the remainder will be 1
∴ Option (b) is correct
Top Binomial Expansion MCQ Objective Questions
What is C(n, 1) + C(n, 2) + _ _ _ _ _ + C(n, n) equal to
Answer (Detailed Solution Below)
Binomial Expansion Question 6 Detailed Solution
Download Solution PDFConcept:
(1 + x)n = nC0 × 1(n-0) × x 0+ nC1 × 1(n-1) × x 1 + nC2 × 1(n-2) × x2 + …. + nCn × 1(n-n) × xn
nth term of the G.P. is an = arn−1
Sum of n terms = s =
Sum of n terms = s =
Calculation:
C(n, 1) + C(n, 2) + _ _ _ _ _ + C(n, n)
⇒ nC1 + nC2 + ... + nCn
⇒ nC0 + nC1 + nC2 + ... + nCn - nC0
⇒ (1 + 1)n - nCo
⇒ 2n - 1 =
Comparing it with a G.P sum = a ×
∴ 2n - 1 = 1 + 2 + 22 + ... +2n-1 which will give us n terms in total.
What is the sum of the coefficients of first and last terms in the expansion of (1 + x)2n, where n is a natural number?
Answer (Detailed Solution Below)
Binomial Expansion Question 7 Detailed Solution
Download Solution PDFConcept:
(1 + x)n = nC0 × 1(n-0) × x 0+ nC1 × 1(n-1) × x 1 + nC2 × 1(n-2) × x2 + …. + nCn × 1(n-n) × xn
Calculation:
Given expansion is (1 + x)2n
⇒ 2nC0 ×1(2n-0) × x0 + 2nC1 ×1(2n-1) × x1 + ... + 2nC2n ×1(2n-2n) × x2n
First term = 2nC0 ×1 × 1 = 1
Last term = 2nC2n ×1 × x2n = 1 × x2n = x2n
⇒ Sum = 1 + x2n
Coefficient of 1 = 1, coefficient of x2n = 1
∴ sum of the coefficients = 1 + 1 = 2.
If the third term in the binomial expansion of (1 + x)m is (-1/8)x² then the rational value of m is
Answer (Detailed Solution Below)
Binomial Expansion Question 8 Detailed Solution
Download Solution PDFConcept:
Expansion of (1 + x)n:
Calculation:
Given: the third term in the binomial expansion of (1 + x)m is (-1/8)x²
So, the third term in the binomial expansion of (1 + x)m is
⇒
⇒ 4m2 - 4m + 1 = 0
⇒ (2m - 1)2 = 0
⇒ 2m - 1 = 0
∴ m =
The coefficient of x2 in the expansion of
Answer (Detailed Solution Below)
Binomial Expansion Question 9 Detailed Solution
Download Solution PDFConcept:
General term: General term in the expansion of (x + y)n is given by
Expansion of (1 + x)n:
Calculation:
To Find: coefficient of x2 in the expansion of
Now, the coefficient of x2 in the expansion =
What is
Answer (Detailed Solution Below)
Binomial Expansion Question 10 Detailed Solution
Download Solution PDFConcept:
Binomial expansion of (x + y)n is given by
(x + y)n = nC0xn + nC1xn-1y + nC2xn-2y2+.....+ nCn-1xyn-1 + nCnyn
Calculation:
Given,
Expanding the expression,
⇒ nC020 + nC1 21 + nC222+.....+ nCn-12n-1 + nCn2n
⇒ nC01n 20 + nC1 1n-1 21 + nC21n-2 22+.....+ nCn-1 2n-1 + nCn2n
Comparing with binomial expansion x = 1 and y = 2
⇒
⇒
∴ The correct option is (2).
The number of terms in the expansion of (x + y + z)10 is
Answer (Detailed Solution Below)
Binomial Expansion Question 11 Detailed Solution
Download Solution PDFConcept:
- Number of terms in the expansion of (x1 + x2 + x3 +......+ xr)n is =
, Where n = exponent of the term to be expanded, r = number of terms to be expanded -
Explanation:
Number of terms in (x + y + z)10 =
⇒
Since,
If 4× nC5 = 9 × n-1C5 then the value of n will be?
Answer (Detailed Solution Below)
Binomial Expansion Question 12 Detailed Solution
Download Solution PDFFormula used:
n! = n × (n - 1) × (n - 2).......3 × 2 × 1
Calculation:
Given that,
4 × nC5 = 9 × n-1C5
Using the above formula
⇒
⇒
⇒ 9n - 45 = 49
⇒ 5n = 4n
⇒ n = 9
The 9th term from the end in (x – 1/x) 12 is
Answer (Detailed Solution Below)
Binomial Expansion Question 13 Detailed Solution
Download Solution PDFConcept:
We have (x + y) n = nC0 xn + nC1 xn-1 . y + nC2 xn-2. y2 + …. + nCn yn
- General term: General term in the expansion of (x + y) n is given by
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th term.
Note:
- In the binomial expansion of (x + y)n, the rth term from end = In the binomial expansion of (y + x)n, the rth term from the start.
- If we interchange the term x → y, it will give rth term from the beginning.
Calculation:
We have to find 9th term from the end in (x – 1/x) 12
We know that rth term from end means (n – r + 2)th term from start.
So. 9th term from the end = [12 – 9 + 2]th term from start = 5th term from start
General term:
= 12C4 x4
The coefficient of x12 in the expansion of
Answer (Detailed Solution Below)
Binomial Expansion Question 14 Detailed Solution
Download Solution PDFConcept:
The expansion of (x + y)n is given by the binomial expansion. The expansion will be
(x + y)n = nC0 xn + nC1 xn-1 y + nC2 xn-2 y2 + ... + nCr xn-r yr + ... + nCn yn;
The general term is given by
Tr = nCr xn-r yr ;
Calculation:
Given expansion is
Let the rth term has x12 and the coefficient be A;
⇒ Tr = A x12;
The general rth term of the given expansion will be
Tr =
⇒ Tr = 30Cr 330-r × x60-2r × x-r = 30Cr 330-r × x60-3r
Equating the power of x with 12,
⇒ 60 - 3r = 12 ⇒ r = 16
Now the coefficient will be
A = 30Cr 330-r = 30C16 314;
The coefficient of x6 in the expansion of (1 + x + x2)-3, is:
Answer (Detailed Solution Below)
Binomial Expansion Question 15 Detailed Solution
Download Solution PDFConcept:
Binomial Expansion:
(a + b)n = C0 an b0 + C1 an-1 b1 + C2 an-2 b2 + … + Cr an-r br + … + Cn-1 a1 bn-1 + Cn a0 bn, where C0, C1, …, Cn are the Binomial Coefficients defined as Cr = nCr =
(1 - x)-n = 1 + nC1 x + n+1C2 x2 + n+2C3 x3 + ....
Algebraic Identities:
a3 - b3 = (a - b)(a2 + ab + b2).
Calculation:
We note that 13 - x3 = (1 - x)(1 + x + x2).
⇒ 1 + x + x2 =
Now, (1 + x + x2)-3 can be written as:
=
= (1 - x)3 × (1 - x3)-3
= (1 - 3x + 3x2 - x3)(1 + 3C1 x3 + 4C2 x6 + ... higher powers of x)
x6 will be obtained by multiplying (1 and 4C2 x6) and (-x3 and 3C1 x3)
∴ Coefficient of x6 will be: (1 × 4C2) + (-1 × 3C1) = 6 - 3 = 3.