## MP Board Class 11th Maths Important Questions Chapter 8 Binomial Theorem

### Binomial Theorem Important Questions

**Binomial Theorem Objective Type Questions**

(A) Choose the correct option :

Question 1.

The total number of terms in the expansion of

(a) 7

(b) 12

(c) 13

(d) 6.

Answer:

(c) 13

Question 2.

If y = 3x + 6x^{2} + 10x^{3} + …………… ∞, then the correct relation will be :

(a) x = 1 – (1 + y)^{–\(\frac { 1 }{ 3 }\)}

(b) x = (1 + y)^{–\(\frac { 1 }{ 3 }\)}

(c) y = 1 – (1 – x)^{-3}

(d) x = 1 + (1 + y)^{–\(\frac { 1 }{ 3 }\)}

Answer:

(a) x = 1 – (1 + y)^{–\(\frac { 1 }{ 3 }\)}

Question 3.

The total number of terms in the expansion of (1+ x)^{-1} will be :

(a) 0

(b) ∞

(c) 2

(d) It can not be expand

Answer:

(b) ∞

Question 4.

The mid – term in the expansion of (x – \(\frac { 1 }{ x }\))^{10} will be :

(a) – ^{10}C_{5}

(b) ^{10}C_{5}

(c) 251

(d) 252

Answer:

(a) – ^{10}C_{5}

An online remainder theorem calculator allows you to determine the remainder of given polynomial expressions by remainder theorem.

Question 5.

For all positive integer of n, n(n – 1) is :

(a) Integer

(b) Natural number

(c) Even positive integer

(d) Odd positive integer.

Answer:

(c) Even positive integer

Question 6.

Expansion of (a + x)^{n} is :

(a) a^{n} + ^{n}C_{1}a^{n-1}x + ^{n}C_{2}a^{n-2}x^{2} + ……….. + ^{n}C_{r}a^{n-r}x^{r} + ………… + a^{n}

(b) x^{n} + ^{n}C_{1}x^{n-1}a +^{n}C_{2}x^{n-2}a^{2} + ……….. + ^{n}C_{r}x^{n-r}a^{r} + ………… + a^{n}

(c) a^{n} – ^{n}C_{1}a^{n-1}x + ^{n}C_{2}a^{n-2}x^{2} + ……….. + (-1)^{r}^{n}C_{r}a^{n-r}x^{r} + ………… + (-1)^{n}a^{n}

(d) x^{n} – ^{n}C_{1}a^{n-1}a + ^{n}C_{2}a^{n-2}a^{2} + ……….. + (-1)^{r}^{n}C_{r}a^{n-r}x^{r} + ………… + (-1)^{n}x^{n}

Answer:

(a) a^{n} + ^{n}C_{1}a^{n-1}x +^{n}C_{2}a^{n-2}x^{2} + ……….. + ^{n}C_{r}a^{n-r}x^{r} + ………… + a^{n}

Question 7.

The fifth term in the expansion of ( x – \(\frac { 1 }{ x }\) )^{10} from the end, will be :

(a) \(\frac { ^{ 10 }{ C }_{ 6 } }{ x }\)

(b) \(\frac { 105 }{ 32{ x }^{ 2 } }\)

(c) \(\frac { ^{ 10 }{ C }_{ 6 } }{ { x }^{ 2 } }\)

(d) \(\frac { ^{ 10 }{ C }_{ 6 } }{ { x }^{ 10 } }\)

Answer:

(a) \(\frac { ^{ 10 }{ C }_{ 6 } }{ x }\)

[Hint: The total number of terms be 11 in the expansion of it.

∵ The 5^{th} term from end = (11 – 5)^{th} = 7^{th} term from the beginning.]

Question 8.

The number of mid – terms in the expansion of ( x – \(\frac { 1 }{ x }\) )^{10}

(a) 1

(b) 2

(c) – \(\frac { ^{ 13 }{ C }_{ 7 } }{ x }\)

(d) 1716x

Answer:

(b) 2

Question 9.

The value of ^{n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + …………….. + ^{n}C_{n} :

(a) 2^{n} + 1

(b) 2^{n – 1}

(c) 2^{n} – 1

(d) 2^{n}

Answer:

(d) 2^{n}

Question 10.

The value of ^{n}C_{0} + ^{n}C_{2} + ^{n}C_{4} + …………….. = ^{n}C_{1} + ^{n}C_{3} + ……………. will be :

(a) 2^{n} + 1

(b) 2^{n – 1}

(c) 2^{n} – 1

(d) 2^{n}

Answer:

(b) 2^{n – 1}

The Chebyshev’s Theorem calculator, above, will allow you to enter any value of k greater than 1.

Question 11.

The total number of terms in the expansion of (a + b + c + d)^{n} will be :

(a) \(\frac { (n+1)(n+2) }{ 2 }\)

(b) \(\frac { n(n+1) }{ 2 }\)

(c) \(\frac { (n+1)(n+2)(n+3) }{ 6 }\)

(d) \(\frac { (n+1)(n+2) }{ 6 }\)

Answer:

(c) \(\frac { (n+1)(n+2)(n+3) }{ 6 }\)

Question 12.

The necessary condition for expansion of (1 + x)^{-1} is :

(a) | x | < 1

(b) | x | > 1

(c) | x | = 1

(d) | x | = – 1.

Answer:

(a) | x | < 1

Question 13.

The general term in the expansion of (x + a)^{n} will be :

(a) r^{th}

(b) (r+1)^{th}term

(c) (r-1)^{th}

(d) (r+2)^{th}term

Answer:

(b) (r+1)^{th}term

Question 14.

In the expansion of ( 2x + \(\frac { 1 }{ { 3x }^{ 2 } }\) )^{9}, then term independent of x will be :

(a) \(\frac { 8 }{ 127 }\)

(b) \(\frac { 124 }{ 81 }\)

(c) \(\frac { 1792 }{ 9 }\)

(d) \(\frac { 256 }{ 243 }\)

Answer:

(c) \(\frac { 1792 }{ 9 }\)

Question 15.

The coefficient of x3 in the expansion of ( x – \(\frac { 1 }{ x }\) )^{15} is :

(a) 14

(b) 21

(c) 28

(d) 35

Answer:

(b) 21

Question 16.

The independent term in the expansion of (x^{2} – \(\frac { 2 }{ { x }^{ 3 } }\))^{15} is :

(a) 5^{th}

(b) 6^{th}

(c) 7^{th}

(d) 8^{th}

Answer:

(c) 7^{th}

Question 17.

The value of ^{n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + …………….. = ^{n}C_{n} the expansion of (l + x)^{n} is :

(a) 2^{n} – 1

(b) 2^{n} – 2

(c) 2^{n}

(d) 2^{n-1}

Answer:

(c) 2^{n}

Question 18.

The value of ^{15}C_{0} + ^{15}C_{2} + ^{15}C_{4} + ^{15}C_{6} + …………….. = ^{15}C_{14} is :

(a) 2^{14}

(b) 2^{15}

(c) 2^{15} – 1

(d) None of these

Answer:

(a) 2^{14}

(B) Match the following :

Answer:

- (d)
- (a)
- (e)
- (c)
- (b)
- (g)
- (f)
- (i)
- (b)

(C) Fill in the blanks :

- By binomial theorem the value of (102)
^{4}is …………….. - The value of second term in the expansion of (1 – x)
^{-3/2}is …………….. - The 5
^{th}term from the end in the expansion of (x – \(\frac { 1 }{ 2x }\) )^{10}is …………….. - The constant term will be …………….. in the expansion of ( x
^{2}– 2 + \(\frac { 1 }{ { x }^{ 2 } }\) )^{6} - The value of C
_{1}+ 2C_{2}+ 3C_{3}+ ………….. + nC_{n}will be …………….. - The value of
^{n}C_{0}–^{n}C_{1}+^{n}C_{2}–^{n}C_{3}+ ………….. will be …………….. - The value of is ……………..
- The value of is ……………..
- The value of is ……………..
- (2x + 3y)
^{5}= …………….. up to three terms. - The coefficient of x
^{7}in the expansion of (x^{2}+ \(\frac { 1 }{ x }\) )^{11}will be ……………. - In the expansion of (1 – x)
^{10}the value of middle term is …………….. - Third (3
^{rd}) term in the expansion of e^{-3x}will be …………….. - If n is odd, in the expansion of (x + a)
^{n}, then number of middle terms are …………….. - The middle term in the expansion of (\(\frac { x }{ a }\) + \(\frac { a }{ x }\) )
^{10} - The coefficient of x
^{n}in the expansion of (1 + x) (1 – x)^{n}will be ……………..

Answer:

- 08243216
- \(\frac { { x }^{ 3 } }{ 16 }\)
- \(\frac { 105 }{ 32{ x }^{ 2 } }\)
- 924
- n.2
^{n-1} - 0
- \(\frac { n(n+1) }{ 2 }\)
- \(\frac { n(n+1)(2n+1) }{ 6 }\)
- [ \(\frac { n(n+1) }{ 2 }\) ]
^{2} - 32x
^{5}+ 240x^{4}y + 720x^{3}y^{2} - 462
- – 252 x
^{-5} - \(\frac { 1 }{ 2 }\)
- Two
- 252
- (- 1)
^{n}(1 – n)

(D) Write true / false :

- The expansion of (1 + x)
^{-3}is 1 – 3x + 6x^{2}– 10x^{3}+ ………….. + \(\frac { (- 1)(r + 1)(r +2) }{ 2! }\)x^{r}+ …………….. - The expansion of (1 – x)
^{-3}is 1 + 3x + 6x^{2}+ 10x^{3}+ ………….. + \(\frac { (- 1)^{r}(r + 1)(r +2) }{ 2! }\)x^{r}+ …………….. - The expansion of (1 – x)
^{-2}is 1 + 2x + 3x^{2}+ (r + 1) x^{r}+ ………….. + - The (r + 1 )
^{th}term in the expansion of (1 – x)^{-2}is (- 1)^{r}(r + 1) x^{r}+ ………….. + - The (r + 1 )
^{th}term in the expansion of (1 – x)^{n}will be x^{r} - The total number of terms in the expansion of (a + b + c)
^{n}is \(\frac { (n + 1)(n + 1) }{ 2 }\) - In the expansion of ( 3x – \(\frac { { x }^{ 3 } }{ 9 } \) )
^{9}, No . of terms is 9. - The number of term in the expansion of ( 3x – \(\frac { { x }^{ 3 } }{ 9 } \) )
^{9}is 8. - In the expansion of (x + a)
^{n}then sum of powers of x and α in any term is n. - The coefficient of x in the expansion of (1 – 2x)
^{-3}is 6. - The second term in the expansion of (2x + 3y)
^{5}is 240x^{4}y. - The value of second term in the expansion of (1 – x)
^{-3/2}is \(\frac { 3 }{ 2 }\)x.

Answer:

- True
- True
- True
- True
- False
- True
- False
- False
- True
- False
- True
- True.

(E) Write answer in one word / sentence :

- Find the value of 999
^{3}by the binomial theorem - Find the middle term in expansion of (x
^{2}– \(\frac { 1 }{ x }\))^{6} - General term in the expansion of (x + a)
^{n}will be. - If in the expansion of (1+x)
^{51}the coefficient of x^{r}and x^{r – 5}are equal, then the value of r will be. - Find the coefficient of x
^{n}in the expansion of (1 + x + x^{2}+ …………… ∞)^{2}, if | x | < 1. - In the expansion of (\(\frac { x }{ 3 }\) – \(\frac { 2 }{ { x }^{ 2 } }\))
^{10}, x^{4}comes in r^{th}term, then the value of r will be. - The 5
^{th}term in the expansion of (1 – 2x)^{– 1}will be.

Answer:

- 997002999
- – 20 x
^{5} ^{n}C_{r }x^{n – r }a^{r}- 28
- (n + 1)
- 3
- 16x
^{2}

**Binomial Theorem Long Answer Type Questions**

Question 1.

Expand : (\(\frac { 2 }{ x }\) – \(\frac { x }{ 2 }\))^{5} (NCERT)

Solution:

Question 2.

Expand : (2x – 3)^{6} (NCERT)

Solution:

Question 3.

Expand : (\(\frac { x }{ 3 }\) + \(\frac { 1 }{ x }\))^{5} (NCERT)

Solution:

Question 4.

Expand : (x + \(\frac { 1 }{ x }\))^{6}. (NCERT)

Solution:

Question 5.

find 13^{th} term in the expansion of ( 9x – \(\frac { 1 }{ 3\sqrt { x } }\) )^{18}. (NCERT)

Solution:

Question 6.

Find the middle term of (3 – \(\frac { { x }^{ 3 } }{ 6 }\))^{7}

Solution:

Question 7.

Find the middle term in the expansion of (\(\frac { x }{ 3 }\) + 9y)^{10}

Solution:

Here n =10

Total number of terms = n + 1 = 10 + 1 = 11 (odd)

Here, the term will be middle term.

Question 8.

If coefficient of x^{2} and x^{3} in the expansion of (3 + ax)^{9} are equal, the value of a.

Solution:

Question 9.

Find the coefficient of x^{5} in the expansion of (x + 3)^{8}

Solution:

Suppose x^{5} appears in (r + 1)^{th} term T_{r+1} = nC_{1}x^{n-r}a^{r}

Here n = 8, x = x, a = 3

T_{r+1} = ^{8}C_{r}(x)^{8 – r}(3)^{r}

For the coefficient of x^{5},

8 – r = 5

=> r = 3

T_{3+1} = ^{8}C_{3}(3)^{3}

= \(\frac { 8 × 7 × 6 }{ 3 × 2 × 1}\) × 3 × 3 × 3 × x^{5}

= 1512 × x^{5}

Hence coefficient of x^{5} is 1512.

Question 10.

Find the coefficient of a^{5}b^{7} in the expansion of (a – 2b)^{12}.

Solution:

Question 11.

If the 17^{th} and 18^{th} terms in the expansion of (2 + a)^{50} are equal, then find the value of a. (NCERT)

Solution:

In the expansion of (x + a)^{n}

T_{r+1} = ^{n}C_{r }x^{n – r }a^{r}

Here n = 50, x = 2, a = a

T_{17} = T_{16 + 1} = ^{50}C_{16 }(2)^{50 – 16 }(a)^{16}

⇒ T_{17} = ^{50}C_{16 }(2)^{34 }(a)^{16}

and T_{18} = T_{17 + 1} = ^{50}C_{17 }(2)^{50 – 17 }(a)^{17}

= ^{50}C_{17 }(2)^{33 }(a)^{17}

Question 12.

Prove that the value of the middle term in the expansion of (1+x)^{2n} is \(\frac { { 1.3.5 …….. (2n – 1)} }{ n! }\).2^{n} x^{n}

Solution:

Question 13.

In the expansion of (x + 1)^{n}, the coefficient of the (r – 1)^{th}, r^{th} and (r + 1)^{th} terms are in the ratio 1 : 3 : 5, then find the value of n and r.

Solution:

In the expansion of (x + 1)^{n},

T_{r + 1} = ^{n}C_{r}x^{n – r}(1)^{r}

T_{r – 1} = T_{r – 2 + 1} = ^{n}C_{r – 2}(x)^{n – (r – 2)}(1)^{r – 2}

Coefficient of T_{r – 1}^{th} term = ^{n}C_{r – 2}

T_{r} = T_{r – 1 + 1} = ^{n}C_{r – 1}(x)^{n – (r – 1)}(1)^{r – 1}

Coefficient of T_{r}^{th}term = ^{n}C_{r – 1}

T_{r + 1} = ^{n}C_{r }x^{n – r }(1)^{r}

Coefficient of T_{r+1}^{th} term = ^{n}C_{r}

Put n = 4r – 5 from equation (1) in equation (2),

3(4r – 5) – 8r = – 3

⇒ 12r – 15 – 8r = – 3

⇒ 4r = 12

∴ r = 3

Put r = 3 in equation (2),

n – 4 x 3 = – 5

⇒ n = 12 – 5

⇒ n = l

n = 7, r = 3

Question 14.

Prove that the coefficient of x^{n} in the expansion of (1 + x)^{2n} of in the expansion of (1 + x)^{2n – 1}.

Solution:

In the expansion of (x + a)^{n}

T_{r+1} = ^{n}C_{r }x^{n – r }a^{r}

Here x = 1, a = x, n = 2n

T_{r+1} = ^{2n}C_{r}(1)^{2n – r}(x)^{r}

For the coefficient of x^{n}, put r = n,

T_{n+1} = ^{2n}C_{n}(a)^{2n – n}(x)^{n}

and T_{18} = T_{17 + 1} = ^{50}C_{17 }(2)^{50 – 17 }(a)^{17}

= (^{2n}C_{n}) x^{n}

∴ In the expansion of (1 + x)^{2n}, the coefficient of x^{n} = ^{2n}C_{n} …. (1)

and in the expansion of (1 + x)^{2n – 1}, x = 1, a = x, n = 2n – 1

∴ T_{r+1} = ^{2n}C_{r }(1)^{2n – 1 -r }(x)^{n}

For the coefficient of x^{n}, put r = n, ‘

We get T_{n+1} = ^{2n – 1}C_{n }x^{n}

The coefficient of x^{n} in the expansion of (1 + x)^{2n – 1} = ^{n – 1}C_{n}

∴ The coefficient of x^{n} in the expansion of (1 + x)^{2n}

= 2 x The coefficient of x^{n} in the expansion of (1 + x)^{2n}, [from equation (1) and (2)]

Question 15.

Find the constant term in the expansion of (\(\frac { 3 }{ 2 }\)x^{2} – \(\frac { 1 }{ 3x }\))^{6}

Solution: