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MCQ Class 11 Mathematics Binomial Theorem with Answers - Set - 2
Question 1:
In the binomial expansion of (a + b)n, the coefficient of fourth and thirteenth terms are equal to each other, then the value of n is
(a) 10
(b) 15
(c) 20
(d) 25
Correct Answer – (B)
Given, in the binomial expansion of (a + b)n, the coefficient of fourth and thirteenth terms are equal to each other
⇒ nC3 = nC12
This is possible when n = 15
Because 15C13 = 15C12
Question 2 :
The greatest coefficient in the expansion of (1 + x)10 is
(a) 10!/(5!)
(b) 10!/(5!)²
(c) 10!/(5! × 4!)²
(d) 10!/(5! × 4!)
Correct Answer – (B)
The coefficient of xr in the expansion of (1 + x)10 is 10Cr and 10Cr is maximum for
r = 10/2 = 5
Hence, the greatest coefficient = 10C5
= 10!/(5!)²
Question 3 :
In the binomial expansion of (71/2 + 51/3)37, the number of integers are
(a) 2
(b) 4
(c) 6
(d) 8
Correct Answer – (C)
Given, (71/2 + 51/3)37
Now, general term of this binomial Tr+1 = 37Cr × (71/2)37-r × (51/3)r
⇒ Tr+1 = 37Cr × 7(37-r)/2 × (5)r/3
This General term will be an integer if 37Cr is an integer, 7(37-r)/2 is an integer and (5)r/3 is an integer.
Now, 37Cr will always be a positive integer.
Since 37Cr denotes the number of ways of selecting r things out of 37 things, it can not be a fraction.
So, 37Cr is an integer.
Again, 7(37-r)/2Cr will be an integer if (37 – r)/2 is an integer.
So, r = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37 …………. 1
And if (5)r/3 is an integer, then r/3 should be an integer.
So, r = 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 ………….2
Now, take intersection of 1 and 2, we get
r = 3, 9, 15, 21, 27, 33
So, total possible value of r is 6
Hence, there are 6 integers are in the binomial expansion of (71/2 + 51/3)37
Question 4 :
In the expansion of (a + b)n, if n is odd then the number of middle term is/are
(a) 0
(b) 1
(c) 2
(d) More than 2
Correct Answer – (C)
In the expansion of (a + b)n,
if n is odd then there are two middle terms which are
{(n + 1)/2}th term and {(n+1)/2 + 1}th term
Question 5 :
If n is a positive integer, then (√5+1)2n + 1 − (√5−1)2n + 1 is
(a) an odd positive integer
(b) not an integer
(c) none of these
(d) an even positive integer
Correct Answer – (B)
Since n is a positive integer, assume n = 1
(√5+1)² + 1 − (√5−1)² + 1
= (5 + 2√5 + 1) + 1 – (5 – 2√5 + 1) + 1 {since (x+y)² = x² + 2xy + y²}
= 4√5 + 2, which is not an integer
MCQ Class 11 Mathematics Binomial Theorem with Answers
Question 6 :
If the third term in the binomial expansion of (1 + x)m is (-1/8)x² then the rational value of m is
(a) 2
(b) 1/2
(c) 3
(d) 4
Correct Answer – (B)
(1 + x)m = 1 + mx + {m(m – 1)/2}x² + ……..
Now, {m(m – 1)/2}x² = (-1/8)x²
⇒ m(m – 1)/2 = -1/8
⇒ 4m² – 4m = -1
⇒ 4m² – 4m + 1 = 0
⇒ (2m – 1)² = 0
⇒ 2m – 1 = 0
⇒ m = 1/2
Question 7 :
The number of ordered triplets of positive integers which are solution of the equation x + y + z = 100 is
(a) 4815
(b) 4851
(c) 8451
(d) 8415
Correct Answer – (B)
Given, x + y + z = 100;
where x ≥ 1, y ≥ 1, z ≥ 1
Let u = x – 1, v = y – 1, w = z – 1
where u ≥ 0, v ≥ 0, w ≥ 0
Now, equation becomes
u + v + w = 97
So, the total number of solution = 97+3-1C3-1
= 99C2
= (99 × 98)/2
= 4851
Question 8 :
if n is a positive ineger then 23nn – 7n – 1 is divisible by
(a) 7
(b) 9
(c) 49
(d) 81
Correct Answer – (C)
Given, 23n – 7n – 1 = 23×n – 7n – 1
= 8n – 7n – 1
= (1 + 7)n – 7n – 1
= {nC0 + nC1 7 + nC2 7² + …….. + nCn 7n} – 7n – 1
= {1 + 7n + nC2 7² + …….. + nCn 7n} – 7n – 1
= nC2 7² + …….. + nCn 7n
= 49(nC2 + …….. + nCn 7n-2)
which is divisible by 49
So, 23n – 7n – 1 is divisible by 49
Question 9 :
In the expansion of (a + b)n, if n is even then the middle term is
(a) (n/2 + 1)th term
(b) (n/2)th term
(c) nth term
(d) (n/2 – 1)th term
Correct Answer – (A)
if n is even then the middle term is (n/2 + 1)th term
Question 10 :
The coefficient of xn in the expansion (1 + x + x² + …..)-n is
(a) 1
(b) (-1)n
(c) n
(d) n+1
Correct Answer – (B)
We know that
(1 + x + x² + …..)-n = (1 – x)-n
Now, the coefficient of x = (-1)n × nCn
= (-1)n
- NCERT Solutions Class 11 Mathematics Binomial Theorem with Answers : Exercise 8.1
- NCERT Solutions Class 11 Mathematics Binomial Theorem with Answers : Exercise 8.2
- NCERT Solutions Class 11 Mathematics Binomial Theorem with Answers : Exercise 8 Misc
- NCERT Solutions Class 11 Mathematics Textbook download
