CBSE 11th Mathematics Binomial Theorem Sample Question Papers Study Material - QB365 Set A | QB365

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CBSE 11th Mathematics Binomial Theorem Sample Question Papers Study Material - QB365 Set A

Published on: 18/08/2026

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Std: 11th Standard CBSE - Mathematics

chapter: Binomial Theorem - Sample Question Papers

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Questions:

Generic 2

5 x 2 = 10

Find the number of terms in the expansions of the following expressions.
$\left( 4+{ x }^{ 2 }+4x \right) ^{ 3 }$

Evaluate the following terms in the given expansion.
7th term in the binomial expansion (x+3y)⁸

Expand each of the expressions :
(1–2x)⁵

Expand each of the expressions :
(2x-3)⁶

Find the number of terms in the expansions of following expressions.
(x+3y)²

Generic 3

2 x 3 = 6

Using binomial theorem, evaluate each of the following
$\left( 53 \right) ^{ 7 }-\left( 47 \right) ^{ 7 }$

Using binomial theorem, evaluate each of the following
$\left( 12 \right) ^{ 5 }+\left( 8 \right) ^{ 5 }$

Generic 5

1 x 5 = 5

The coefficients of 5th, 6th and 7th terms in the expansion of(1 + x)ⁿ are in AP. Find the value of n.

Multiple Choice Question

16 x 1 = 16

The coefficient of x in the expansion of $\left(1-3 x+7 x^{2}\right)(1-x)^{16} \text { is }$ _____.

(a)

(b)

(c)

-17

(d)

-19

10.

If the coefficient of x⁷ and x⁸ in $\left(2+\frac{x}{3}\right)^{n}$ are equal, then n is equal to _____.

(a)

(b)

(c)

(d)

11.

The coefficient of 11th term from the end in the expansion of $\left(2 x-\frac{1}{x^{2}}\right)^{25} \text { is }$_____.

(a)

${ }^{25} C_{15} \times 2^{10}$

(b)

$-{ }^{25} C_{15} \times 2^{10}$

(c)

${ }^{25} C_{20} \times 2^{10}$

(d)

$-{ }^{25} C_{20} \times 2^{10}$

12.

The coefficient of x¹¹ in the expansion $\left(x^{3}-\frac{2}{x^{2}}\right)^{12} \text { is }$ _____.

(a)

${ }^{12} C_{11}(-1)^{11} 2^{11}$

(b)

${ }^{12} C_{6}(-1)^{6} 2^{6}$

(c)

${ }^{12} C_{5}(-1)^{5} 2^{5}$

(d)

None of these

13.

The number of terms in the expansion of (2x + 3y - 4z)ⁿ is _____.

(a)

n + 1

(b)

n + 2

(c)

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

(d)

(n + 1)(n + 2)

14.

Which among the following is /are true?
I. (1.1)10000 > 100 II. (1.1)10000 < 100
III. (1.1)10000 > 1000 IV. (1.1)10000 < 1000

(a)

I and IV are true

(b)

II and IV are true

(c)

I and III are true

(d)

Only IV is true

15.

If $x \neq 0, \text { then }\left(x^{2}+\frac{3}{x}\right)$ is equal to _____.

(a)

$x^{8}+12 x^{5}+\frac{108}{x}+\frac{81}{x^{4}} $

(b)

$x^{8}+12 x^{5}+54 x^{3}+\frac{180}{x}+\frac{81}{x^{3}} $

(c)

$x^{8}+12 x^{5}+54 x^{2}+\frac{108}{x}+\frac{81}{x^{4}}$

(d)

None of the above

16.

The number of terms in the expansion (1- 3x + 3x² - x³)⁹ is equal to _____.

(a)

(b)

(c)

(d)

17.

The total number of terms in the expansion of (a + x)⁵¹ - (a - x)⁵¹ after simplification is _____.

(a)

(b)

(c)

(d)

102

18.

Constant term in the expansion of $\left( x-\frac { 1 }{ x } \right) ^{ 14 }$ _____.

(a)

3032

(b)

3432

(c)

5042

(d)

-3442

19.

The coefficient of x⁵y⁸ in the expansion of (x + y)¹³ is _____.

(a)

13C₅

(b)

13C₈

(c)

8C₅

(d)

None of the these

20.

The total number of terms in the expansion of (x + a)⁸⁰ + (x - a)⁸⁰ is _____.

(a)

(b)

(c)

(d)

161

21.

If in the expansion of (1 +x)¹⁵, the coefficient of (2x + 3)^th and (r - 1)^th terms are equal then r is equal to _____.

(a)

(b)

(c)

(d)

none of the these

22.

The middle term in the expansion of $\left( \frac { { 2x }^{ 2 } }{ 3 } +\frac { 3 }{ { 2x }^{ 2 } } \right) ^{ 10 }$is _____.

(a)

240

(b)

280

(c)

262

(d)

252

23.

If in the expansion of (1 +x)ⁿ, the coefficients of fifth, sixth and seventh terms are inA.P. then n is equal to _____.

(a)

5,7

(b)

7,16

(c)

7,14

(d)

8,15

24.

If in the expansion of (a + b)ⁿ and (a + b)^{n + 3}, the ratio of the coefficients of second and third terms and third and fourth terms respectively are equal then n is _____.

(a)

(b)

(c)

(d)

none of these

Case Study Questions

1 x 4 = 4

25.

An algebraic expression containing two terms is called a binomial expression $(a+b)^n$ is a binomal expression. The number of terms in a binomial expansion is always one greater than index.
On the basis of above information answer the following questions.
(i) The number of terms in the expansion of $(2 y-3 z)^4$ is

(a) 3

(b) 4

(c) 5

(d) 6

(ii) The number of terms in the expansion of $\left(x^2-\frac{1}{3 x}\right)^9$ is

(a) 11

(b) 10

(c) 9

(d) 8

(iii) The number of terms in the expansion of $\left(x^3-3 x^2+3 x+1\right)^4$ is

(a) 4

(b) 5

(c) 12

(d) 13

(iv) The number of terms in the expansion of $\left[\left(4 x^2+12 x y+9 y^2\right)^5\right]^{\frac{1}{2}}$ is

(a) 5

(b) 6

(c) 7

(d) 8

(v) The number of terms in the expansion of $\left[\left[\left(x^2-\frac{1}{2 x}\right)^{10}\right]^{1 / 3}\right]^9$ is

(a) 9

(b) 10

(c) 30

(d) 31

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Answers Key:

Generic 2

5 x 2 = 10

$\left(4+x^2+4 x\right)^3=\left\{(x+2)^2\right\}^3(x+2)^6 = 7 $

Here, a=x, b=3y and n=8
General term, T_r+1 =⁸C_r (x)^8-r (3r)^r
Put r+1=7 ⇒ r =6
Then,T₇=T₆₊₁ =⁸C_r (x)^8-6 (3y)⁶ =20412x² y⁶

By using Binomial Theorem, the expression (1– 2x)⁵can be expanded as,
$(1-2x)^{ 5 }=\left[ 1+\left( -2x \right) \right] ^{ 5 }$
$=^{ 5 }{ C }_{ o }+^{ 5 }{ C }_{ 1 }(-2x)+^{ 5 }{ C }_{ 2 }(-2x)^{ 2 }+^{ 5 }{ C }_{ 3 }(-2x)^{ 3 }+^{ 5 }{ C }_{ 4 }(-2x)^{ 4 }+^{ 5 }{ C }_{ 5 }(-2x)^{ 5 }$
$=1+5(-2x)+10({ 4x }^{ 2 })+10({ -8x }^{ 3 })+5({ 16x }^{ 4 })+1(-{ 32x }^{ 5 })$
$=1-10x+{ 40x }^{ 2 }-{ 80x }^{ 3 }+{ 80x }^{ 4 }-{ 32x }^{ 5 }$
which is the required expansion.

By using Binomial Theorem, the expression (2x – 3)⁶can be expanded as
(2x-3)⁶ = ⁶C_o (2x)₆ - ⁶C₁(2x)^{5 $\times $}3 + ⁶C₄(2x)⁴(3)² - ⁶C₃(2x)³(3)³+⁶C₄ (2x)² (3)⁴ - ⁶C₅ (2x) 3⁵ + ⁶C₆3⁶
=⁶C_o 64x⁶- ⁶C₁32x⁵^{$\times $}3 + ⁶C₂16x⁴^{$\times $}9 -⁶C₃8x³^{$\times $}27 + ⁶C₂4x²^{$\times $}81 - ⁶C₁ 2x ^{$\times $} 243 + ⁶C_o729 [$\because $ ⁿC_r= ⁿC_n-r]
$=64x^{ 6 }-6\times 32\times 3\times { x }^{ 5 }+\frac { 6\times 5 }{ 2 } \times 16x^{ 4 }\times 9$$-\frac { 6\times 5\times 4\times 8{ x }^{ 3 }\times 27 }{ 6 } +\frac { 6\times 5 }{ 2 } \times { 4x }^{ 2 }\times 81-6\times 2x+243+729$
$={ 64x }^{ 2 }-{ 576x }^{ 5 }+{ 2160x }^{ 4 }-{ 4320x }^{ 3 }-{ 2916x }+729$

Given expression is (x+3y)² . Here, n = 2
$\therefore $ The number of terms in expansions is (n+1)
i.e 2+1 = 3

Generic 3

2 x 3 = 6

$\left( 53 \right) ^{ 7 }-\left( 47 \right) ^{ 7 }=\left( 50+3 \right) ^{ 7 }+\left( 50-3 \right) ^{ 7 }\\ \left( 50+3 \right) ^{ 7 }=^{ 7 }{ C }_{ 0 }{ \left( 50 \right) }^{ 7 }+^{ 7 }{ C }_{ 1 }{ \left( 5 \right) }^{ 6 }{ \left( 3 \right) }^{ 1 }+^{ 7 }{ C }_{ 2 }{ \left( 50 \right) }^{ 5 }{ \left( 2 \right) }^{ 2 }.........(i)\\ \left( 50-3 \right) ^{ 7 }=^{ 7 }{ C }_{ 0 }{ \left( 50 \right) }^{ 7 }-^{ 7 }{ C }_{ 1 }{ \left( 50 \right) }^{ 6 }{ \left( 3 \right) }^{ 1 }+^{ 7 }{ C }_{ 2 }{ \left( 3 \right) }^{ 2 }.........(ii)\\ Sub\quad equ(i)\quad and\quad (ii)\\ \left( 50+3 \right) ^{ 7 }-\left( 50-3 \right) ^{ 7 }=2\left[ ^{ 7 }{ C }_{ 1 }{ \left( 50 \right) }^{ 6 }{ \left( 3 \right) }^{ 1 }+^{ 7 }{ C }_{ 3 }{ \left( 50 \right) }^{ 4 }{ { \left( 3 \right) }^{ 3 }+^{ 7 }{ C }_{ 5 }{ \left( 50 \right) }^{ 2 }{ { \left( 3 \right) }^{ 5 }+^{ 7 }{ C }_{ 7 }{ { \left( 3 \right) }^{ 7 } } } } \right] \\ =668088019374$

$\left( 12 \right) ^{ 5 }+\left( 8 \right) ^{ 5 }=\left( 10+2 \right) ^{ 5 }+\left( 10-2 \right) ^{ 5 }\\ \left( 10+2 \right) ^{ 5 }=^{ 5 }{ C }_{ 0 }\times { 10 }^{ 5 }+^{ 5 }{ C }_{ 1 }\times { 10 }^{ 4 }\times 2+^{ 5 }{ C }_{ 2 }\times { 10 }^{ 3 }\times { 2 }^{ 2 }+.........(i)\\ \left( 10-2 \right) ^{ 5 }=^{ 10 }{ C }_{ 5 }\times 5-^{ 5 }{ C }_{ 1 }\times { 10 }^{ 4 }\times 2+^{ 5 }{ C }_{ 2 }\times { 10 }^{ 3 }\times { 2 }^{ 2 }+.........(ii)\\ Now\quad adding\quad equ(i)\quad and\quad (ii)\\ \left( 12 \right) ^{ 5 }+\left( 8 \right) ^{ 5 }=2\left[ ^{ 5 }{ C }_{ 0 }10^{ 5 }+^{ 5 }{ C }_{ 2 }\left( 10 \right) ^{ 3 }\left( 2 \right) ^{ 2 }+^{ 5 }{ C }_{ 4 }\left( 10 \right) ^{ 1 }\left( 2 \right) ^{ 4 } \right] \\ =281600$

Generic 5

1 x 5 = 5

T₅ = ⁿC₄ x⁴, T₆ = ⁿC₅ x⁵ and T₇ = ⁿC₆ x⁶
$\because $ Coefficient of these terms are in AP.
$\therefore $ 2ⁿC₅ = ⁿC₄ + ⁿC₆
Solve it.