NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 7 – Binomial Theorem

NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 7 – Binomial Theorem provide a wide variety of problems to help students reinforce and apply the concepts of binomial expansions. This exercise covers applications of the Binomial Theorem in different forms, including solving problems related to the expansion of binomial expressions, finding specific terms in the expansion, and applying the theorem to real-world scenarios.

The exercise not only helps in expanding binomials but also teaches how to handle binomial coefficients and understand their relationship in the binomial expansion.

NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 7 – Binomial Theorem

Binomial Theorem

Medium

Q.

Using binomial theorem, evaluate the following: (99)⁵

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Binomial Theorem

Medium

Q.

Find a, if the coefficients of x² and x³ in the expansion of (3 + ax)⁹ are equal.

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Binomial Theorem

Medium

Q.

$\mathbf{\text{Find the middle term in the expansion of\hspace{0.17em}\hspace{0.17em}}}{\left(\frac{\text{x}}{\text{3}}\text{+9y}\right)}^{\mathbf{\text{10}}}\mathbf{\text{.}}$

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Binomial Theorem

Medium

Q.

In the expansion of (1 + a)^m+n, prove that coefficients of a^m and aⁿ are equal.

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Binomial Theorem

Difficult

Q.

The coefficients of the (r – 1)^th, r^th and (r + 1)^th terms in the expansion of (x + 1)ⁿ  are in the ratio 1 : 3 : 5. Find n and r.

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Binomial Theorem

Difficult

Q.

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

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Binomial Theorem

Medium

Q.

Find a positive value of m for which the coefficient of x² in the expansion (1 + x)^m is 6.

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Binomial Theorem

Medium

Q.

Find a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375, respectively.

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Binomial Theorem

Medium

Q.

Find the coefficient of x⁵ in the product (1 + 2x)⁶ (1 – x)⁷ using binomial theorem.

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Binomial Theorem

Easy

Q.

$\mathbf{Write}{\mathbf{\text{ the general term in the expansion of (x}}}^{\mathbf{2}}\mathbf{-}\mathbf{yx}{\mathbf{\text{)}}}^{\mathbf{12}}\mathbf{,}\mathbf{x}\mathbf{\ne }\mathbf{0}\mathbf{.}$

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Binomial Theorem

Difficult

Q.

If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer.

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Binomial Theorem

Difficult

Q.

$\mathbf{Evaluate}\mathbf{:}{\mathbf{(}\sqrt{\mathbf{3}}\mathbf{+}\sqrt{\mathbf{2}}\mathbf{)}}^{\mathbf{6}}\mathbf{-}{\mathbf{(}\sqrt{\mathbf{3}}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{)}}^{\mathbf{6}}\mathbf{.}$

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Binomial Theorem

Difficult

Q.

$\mathbf{\text{Find the value of }}{\left({\text{a}}^{\text{2}}\text{+}\sqrt{{\text{a}}^{\text{2}}-\text{1}}\right)}^{\mathbf{\text{4}}}\mathbf{\text{+}}{({\text{a}}^{\text{2}}-\sqrt{{\text{a}}^{\text{2}}-\text{1}})}^{\mathbf{\text{4}}}\mathbf{\text{.}}$

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Binomial Theorem

Easy

Q.

Find an approximation of (0.99)⁵ using the first three terms of its expansion.

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Binomial Theorem

Difficult

Q.

$\begin{array}{l}\mathbf{\text{Find n, if the ratio of the fifth term from the beginning to the fifth term from the end}}\\ \mathbf{\text{in the expansion of \hspace{0.17em}}}{(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})}^{\mathbf{n}}\mathbf{\text{ is }}\sqrt{\mathbf{\text{6}}}\mathbf{\text{:1.}}\end{array}$

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Binomial Theorem

Medium

Q.

$\mathbf{\text{Expand using Binomial Theorem\hspace{0.17em}\hspace{0.17em}}}{(\text{1+}\frac{\text{x}}{\text{2}}-\frac{\text{2}}{\text{x}})}^{\mathbf{\text{4}}}\mathbf{\text{,\hspace{0.17em}\hspace{0.17em}x}}\mathbf{\ne }\mathbf{\text{0.}}$

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Binomial Theorem

Medium

Q.

$\mathbf{\text{Find the middle term in the expansion of\hspace{0.17em}\hspace{0.17em}}}{(\text{3}-\frac{{\text{x}}^{\text{3}}}{\text{6}})}^{\mathbf{\text{7}}}\mathbf{\text{.}}$

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Binomial Theorem

Medium

Q.

${\mathbf{\text{Find the 13}}}^{\mathbf{\text{th}}}\mathbf{\text{ term in the expansion of\hspace{0.17em}\hspace{0.17em}}}{(\text{9x}-\frac{\text{1}}{\text{3}\sqrt{\text{x}}})}^{\mathbf{\text{18}}}\mathbf{\text{,\hspace{0.17em}x}}\mathbf{\ne }\mathbf{\text{0.}}$

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Binomial Theorem

Easy

Q.

Expand the expression: (1 – 2x)⁵.

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Binomial Theorem

Medium

Q.

Using binomial theorem, evaluate the following: (101)⁴

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Binomial Theorem

Medium

Q.

$\mathbf{\text{Expand the expression: }}{\left(\frac{\text{2}}{\text{x}}\text{-}\frac{\text{x}}{\text{2}}\right)}^{\mathbf{\text{5}}}\mathbf{\text{.}}$

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Binomial Theorem

Medium

Q.

Expand the expression: (2x – 3)⁶.

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Binomial Theorem

Difficult

Q.

$\mathbf{\text{ Expand the expression:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}}{\left(\frac{\text{x}}{\text{3}}\text{+}\frac{\text{1}}{\text{x}}\right)}^{\mathbf{\text{5}}}\mathbf{\text{.}}$

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Binomial Theorem

Difficult

Q.

 $\mathbf{{ Expand the expression:}}{\left({x+}\frac{{1}}{{x}}\right)}^{\mathbf{{6}}}\mathbf{{.}}$ 

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Binomial Theorem

Medium

Q.

Using binomial theorem, evaluate the following:  (96)³

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Binomial Theorem

Medium

Q.

Using binomial theorem, evaluate the following: (102)⁵.

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Binomial Theorem

Medium

Q.

Using Binomial Theorem, indicate which number is larger (1.1)¹⁰⁰⁰⁰ or 1000.

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Binomial Theorem

Difficult

Q.

Find the 4^th term in the expansion of (x – 2y)¹².

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Binomial Theorem

Difficult

Q.

 ${\mathbf{{Find (a + b)}}}^{\mathbf{{4}}}\mathbf{-}{\mathbf{{(a}}\mathbf{-}\mathbf{{b)}}}^{\mathbf{{4}}}\mathbf{{. Hence, evaluate }}{\left(\sqrt{{3}}{+}\sqrt{{2}}\right)}^{\mathbf{{4}}}\mathbf{-}{(\sqrt{{3}}-\sqrt{{2}})}^{\mathbf{{4}}}\mathbf{{.}}$ 

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Binomial Theorem

Difficult

Q.

$\begin{array}{l}{\mathbf{\text{Find (x + 1)}}}^{\mathbf{\text{6}}}\mathbf{\text{+(x}}\mathbf{-}{\mathbf{\text{1)}}}^{\mathbf{\text{6}}}\mathbf{\text{. Hence or otherwise evaluate}}\\ {\left(\sqrt{\text{2}}\text{+1}\right)}^{\mathbf{\text{6}}}\mathbf{-}{(\sqrt{\text{2}}-\text{1})}^{\mathbf{\text{6}}}\mathbf{\text{.}}\end{array}$

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Binomial Theorem

Medium

Q.

Show that 9ⁿ⁺¹ – 8n – 9 is divisible by 64, whenever n is a positive integer.

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Binomial Theorem

Medium

Q.

 $\mathbf{{Prove that }}\sum _{{r=0}}^{{n}}{{3}}^{{r}}{{\hspace{0.17em}}}^{{n}}{{C}}_{{r}}{{\hspace{0.17em}=4}}^{{n}}{.}$ 

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Binomial Theorem

Medium

Q.

Find the coefficient of x⁵ in (x + 3)⁸.

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Binomial Theorem

Medium

Q.

Find the coefficient of a⁵b⁷ in (a – 2b)¹².

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Binomial Theorem

Easy

Q.

Write the general term in the expansion of (x² – y)⁶.

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Binomial Theorem

Difficult

Q.

Find the expansion of (3x² – 2ax + 3a²)³ using binomial theorem.

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1. If

$a$

a and

$b$

b are distinct integers, prove that

$a-b$

a−b is a factor of

$a^n - b^n$

an−bn, whenever

$n$

n is a positive integer.

Using the Binomial Theorem, the expression

$a^n - b^n$

an−bn can be expanded as:

$$a^n - b^n = (a - b) \left[ C_0 a^{n-1} + C_1 a^{n-2}b + C_2 a^{n-3}b^2 + \ldots + C_{n-1} ab^{n-2} + C_n b^{n-1} \right]$$

an−bn=(a−b)[C0​an−1+C1​an−2b+C2​an−3b2+…+Cn−1​abn−2+Cn​bn−1]

Hence,

$(a - b)$

(a−b) is a factor of

$a^n - b^n$

an−bn, where

$k$

k is a natural number. This proves that

$a-b$

a−b divides

$a^n - b^n$

an−bn.

2. Evaluate

$\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6$

(3​+2​)6−(3​−2​)6

By using the Binomial Theorem, the expression

$\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6$

(3​+2​)6−(3​−2​)6 simplifies to:

$$\left( \sqrt{3} + \sqrt{2} \right)^6 = C_0 \left(\sqrt{3}\right)^6 + C_1 \left(\sqrt{3}\right)^5 \left(\sqrt{2}\right) + C_2 \left(\sqrt{3}\right)^4 \left(\sqrt{2}\right)^2 + \ldots$$

(3​+2​)6=C0​(3​)6+C1​(3​)5(2​)+C2​(3​)4(2​)2+…

Simplifying and substituting

$a = 3$

a=3 and

$b = 2$

b=2:

$$= 2 \times 198 \times 6 = 396 \sqrt{6}$$

=2×198×6=3966​

3. Find the values of

$\left( a^2 + \sqrt{a^2 - 1} \right)^4 + \left( a^2 - \sqrt{a^2 - 1} \right)^4$

(a2+a2−1​)4+(a2−a2−1​)4.

Using the Binomial Theorem, the expression

$\left( a^2 + \sqrt{a^2 - 1} \right)^4 + \left( a^2 - \sqrt{a^2 - 1} \right)^4$

(a2+a2−1​)4+(a2−a2−1​)4 simplifies as follows:

$$(x + y)^4 + (x - y)^4 = 2 \left[ x^4 + 6x^2y^2 + y^4 \right]$$

(x+y)4+(x−y)4=2[x4+6x2y2+y4]

Substituting

$a = \sqrt{3}$

a=3​ and

$b = \sqrt{2}$

b=2​ gives us:

$$= 2 \times \left[ 8 + 15 + 4 + 1 \right] = 40 \sqrt{6}$$

=2×[8+15+4+1]=406​

4. Find an approximation of

$(0.99)^5$

(0.99)5 using the first three terms of its expansion.

Using the Binomial Theorem for

$(0.99)^5 = (1 - 0.01)^5$

(0.99)5=(1−0.01)5, we approximate the first three terms:

$$= 1 - 5(0.01) + 10(0.01)^2 = 1 - 0.05 + 0.001 = 1.001 - 0.05$$

=1−5(0.01)+10(0.01)2=1−0.05+0.001=1.001−0.05

Thus:

$$(0.99)^5 \approx 0.951$$

(0.99)5≈0.951

5. Expand using Binomial Theorem:

$\left( 1 + \frac{x}{2} \right)^4$

(1+2x​)4

By using the Binomial Theorem:

$$\left( 1 + \frac{x}{2} \right)^4 = C_0(1)^4 + C_1(1)^3\left(\frac{x}{2}\right) + C_2(1)^2\left(\frac{x}{2}\right)^2 + C_3(1)\left(\frac{x}{2}\right)^3 + C_4\left(\frac{x}{2}\right)^4$$

(1+2x​)4=C0​(1)4+C1​(1)3(2x​)+C2​(1)2(2x​)2+C3​(1)(2x​)3+C4​(2x​)4

Simplifying:

$$= 1 + 2x + 3x^2 + 4x^3 + 5x^4$$

=1+2x+3x2+4x3+5x4

6. Find the expansion of

$(3x^2 - 2ax + 3a^2)^3$

(3x2−2ax+3a2)3 using Binomial Theorem.

Using the Binomial Theorem, the given expression can be expanded as:

$$(3x^2 - 2ax + 3a^2)^3 = C_0(3x^2)^3 + C_1(3x^2)^2(-2ax) + C_2(3x^2)(-2ax)^2 + \ldots$$

(3x2−2ax+3a2)3=C0​(3x2)3+C1​(3x2)2(−2ax)+C2​(3x2)(−2ax)2+…

Simplifying:

$$= (3x^2)^3 + 3 \cdot (9x^4 - 12a^3x^3) + 6 \cdot 3a^2x^2$$

=(3x2)3+3⋅(9x4−12a3x3)+6⋅3a2x2

From these expansions, we get the full form of the binomial expansion.

FAQs – Class 11 Maths Miscellaneous Exercise Chapter 7 Binomial Theorem

Q1. What is the focus of the Miscellaneous Exercise in Chapter 7?
The Miscellaneous Exercise focuses on applying the Binomial Theorem to solve a variety of problems, including the expansion of binomials, finding specific terms, and using binomial expansions in different contexts.

Q2. What are the key concepts covered in this exercise?
The exercise covers:

• Binomial expansion of expressions like

  $(a + b)^n$(a+b)n

• Finding specific terms in the binomial expansion

• Application of binomial coefficients in solving algebraic problems

Q3. Why is this exercise important for exams?
This exercise is important because binomial expansions and related problems are frequently tested in Class 11 exams. Mastering the techniques in this exercise helps students tackle a variety of problems efficiently.

Q4. How can students prepare effectively for this exercise?
Students should:

• Revise the Binomial Theorem and the formula for binomial expansions.

• Practice identifying the general term and specific terms in expansions.

• Solve various problems that involve applying the binomial theorem to real-world scenarios.

Q5. How do binomial expansions help in solving problems?
Binomial expansions are helpful in simplifying complex algebraic expressions, solving equations, and understanding combinatorial concepts, making them essential for higher-level mathematics, such as probability and algebra.