Binomial Expansion for Class 12 Mathematics first part

Add. Mathematics

Grade 12

Fully Unlocked

Lesson Binomial Expansion By: Ghanashyam Adhikari

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Interactive Visualization & Simulation

Pascal's Triangle & Binomial Coefficients

Interactive Pascal's Triangle (Coefficients Generator)

Expand \((x + y)^n\):

Adjust index \(n\) to generate binomial coefficients dynamically.

Index \(n = 4\)

\((x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\)

Real-Life Applications

Binomial Probabilities & Compound Growth

Binomial expansions model situations with two independent outcomes (success/failure, heads/tails, coin tosses, genetics, and financial compounding).

Coin Tossing / Genetics

Probability Model

The coefficients in \((p + q)^n\) give exact probabilities of getting \(r\) successes in \(n\) trials, crucial in statistics and genetics.

Compound Interest & Approximation

Financial Growth

Using binomial theorem for negative or fractional indices \((1 + x)^n \approx 1 + nx\), economists quickly approximate compound interest over large periods.

Chapter 2 • Binomial Theorem

Formula Derivation & Proof

Solution by Ghanashyam Adhikari

Binomial Theorem Statement & Mathematical Induction Proof

Let \(a, x \in \mathbb{R}\) and \(n \in \mathbb{N}\). Then:

\[(a + x)^n = C(n, 0)a^n + C(n, 1)a^{n-1}x + C(n, 2)a^{n-2}x^2 + \dots + C(n, r)a^{n-r}x^r + \dots + C(n, n)x^n\]

Proof by Mathematical Induction:

• Step 1 (Base Cases): True for \(n = 1, 2, 3\) by direct expansion:
  \((a + x)^1 = 1.a + 1.x = C(1, 0)a + C(1, 1)x\)
  \((a + x)^2 = a^2 + 2ax + x^2 = C(2, 0)a^2 + C(2, 1)ax + C(2, 2)x^2\)
  \((a + x)^3 = a^3 + 3a^2x + 3ax^2 + x^3 = C(3, 0)a^3 + C(3, 1)a^2x + C(3, 2)ax^2 + C(3, 3)x^3\)
• Step 2 (Assumption): Assume the theorem is true for some \(n = k \in \mathbb{N}\):
  \((a + x)^k = C(k, 0)a^k + C(k, 1)a^{k-1}x + C(k, 2)a^{k-2}x^2 + \dots + C(k, r)a^{k-r}x^r + \dots + C(k, k)x^k\)
• Step 3 (Inductive Step): Prove for \(n = k + 1\).
  \((a + x)^{k+1} = (a + x)(a + x)^k\)
  \(= (a + x) [C(k, 0)a^k + C(k, 1)a^{k-1}x + \dots + C(k, k)x^k]\)
  Multiplying out and using Pascal's identity \(C(k, r-1) + C(k, r) = C(k+1, r)\) yields:
  \((a + x)^{k+1} = C(k+1, 0)a^{k+1} + C(k+1, 1)a^kx + \dots + C(k+1, k+1)x^{k+1}\).
• Conclusion: By mathematical induction, the theorem holds for all \(n \in \mathbb{N}\) (Newton's formula).

Binomial Theorem for Any Index (Negative or Fractional)

If \(n\) is any negative integer or fraction, the expansion has infinite terms and is valid for \(|x| < 1\):

\[(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots + \frac{n(n-1)\dots(n-r+1)}{r!}x^r + \dots\] \[(1 - x)^{-n} = 1 + nx + \frac{n(n+1)}{2!}x^2 + \frac{n(n+1)(n+2)}{3!}x^3 + \dots + \frac{n(n+1)\dots(n+r-1)}{r!}x^r + \dots\]

Summary Table

Formulas, General Terms & Middle Terms

Expansion	General Term (\(t_{r+1}\))
\((a + x)^n\)	\(t_{r+1} = C(n, r) a^{n-r} x^r\)
\((a - x)^n\)	\(t_{r+1} = (-1)^r C(n, r) a^{n-r} x^r\)
\((1 + x)^n\)	\(t_{r+1} = C(n, r) x^r\)
\((1 - x)^n\)	\(t_{r+1} = (-1)^r C(n, r) x^r\)

Middle Term Rules:

• When \(n\) is even: Total terms \(= n + 1\) (odd). There is only 1 middle term given by \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) term.
• When \(n\) is odd: Total terms \(= n + 1\) (even). There are 2 middle terms given by \(\left(\frac{n+1}{2}\right)^{\text{th}}\) and \(\left(\frac{n+1}{2} + 1\right)^{\text{th}}\) terms.

Properties of Binomial Coefficients (\(C_0, C_1, C_2, \dots, C_n\)):

• Sum of all binomial coefficients: \(C_0 + C_1 + C_2 + \dots + C_n = 2^n\).
• Sum of odd coefficients = Sum of even coefficients \(= 2^{n-1}\).
• Symmetry property: \(C(n, r) = C(n, n-r)\).

Worked Examples (Q1 to Q10)

Complete Textbook Worked Examples

Solution by Ghanashyam Adhikari

Example 1

Expand \((2x + 3)^4\) using Binomial Theorem.

Solution: Here, \(n = 4, a = 2x, x = 3\).

\((2x + 3)^4 = C(4,0)(2x)^4 + C(4,1)(2x)^3(3) + C(4,2)(2x)^2(3)^2 + C(4,3)(2x)(3)^3 + C(4,4)(3)^4\)

\(= 1 \cdot 16x^4 + 4 \cdot 8x^3 \cdot 3 + 6 \cdot 4x^2 \cdot 9 + 4 \cdot 2x \cdot 27 + 1 \cdot 81\)

\(= 16x^4 + 96x^3 + 216x^2 + 216x + 81\).

Example 2

Find the 5th term in the expansion of \((2x - \frac{1}{x})^{10}\).

Solution: Here, \(n = 10, a = 2x, x = -\frac{1}{x}\). We need \(t_5 = t_{4+1}\), so \(r = 4\).

\(t_5 = C(10, 4) (2x)^{10-4} \left(-\frac{1}{x}\right)^4 = 210 \cdot 64 x^6 \cdot \frac{1}{x^4} = 13440 x^2\).

Final Answer: \(13440 x^2\).

Example 3

Find the coefficient of \(x^5\) in the expansion of \((2x^2 - \frac{1}{x})^7\).

Solution: Here, \(n = 7, a = 2x^2, x = -\frac{1}{x}\).

\(t_{r+1} = C(7, r)(2x^2)^{7-r}\left(-\frac{1}{x}\right)^r = C(7,r)2^{7-r}(-1)^r x^{14-3r}\)

Equating power of \(x\) to \(5\): \(14 - 3r = 5 \implies 3r = 9 \implies r = 3\).

Coefficient \(= C(7,3) 2^4 (-1)^3 = 35 \times 16 \times (-1) = -560\).

Example 4

Find the term independent of \(x\) in the expansion of \((x^2 + \frac{1}{x})^9\).

Solution: Here, \(n = 9, a = x^2, x = \frac{1}{x}\).

\(t_{r+1} = C(9, r)(x^2)^{9-r}\left(\frac{1}{x}\right)^r = C(9, r) x^{18 - 3r}\)

For term independent of \(x\), power of \(x\) is \(0\): \(18 - 3r = 0 \implies r = 6\).

Term \(= C(9, 6) = 84\).

Example 5

Find the middle term in the expansion of \((x - \frac{1}{x^2})^{12}\).

Solution: Here, \(n = 12\) (even). Total terms = \(13\). There is only 1 middle term given by \(\left(\frac{12}{2} + 1\right) = 7^{\text{th}}\) term.

\(t_7 = C(12, 6) (x)^{12-6} \left(-\frac{1}{x^2}\right)^6 = 924 x^6 \cdot \frac{1}{x^{12}} = \frac{924}{x^6}\).

Example 6

Find the middle terms in the expansion of \((2x + 3y)^7\).

Solution: Here, \(n = 7\) (odd). Total terms = \(8\). There are 2 middle terms: \(\left(\frac{7+1}{2}\right) = 4^{\text{th}}\) and \(5^{\text{th}}\) terms.

\(t_4 = C(7, 3)(2x)^4(3y)^3 = 35 \cdot 16x^4 \cdot 27y^3 = 15120 x^4 y^3\)

\(t_5 = C(7, 4)(2x)^3(3y)^4 = 35 \cdot 8x^3 \cdot 81y^4 = 22680 x^3 y^4\).

Example 7

Find the value of \(a\) if the middle term in the expansion of \((\frac{a}{2} + 2)^8\) is \(1120\).

Solution: Here, \(n = 8\) (even). Middle term is \(t_5\).

\(t_5 = C(8, 4)\left(\frac{a}{2}\right)^4(2)^4 = 70 \cdot \frac{a^4}{16} \cdot 16 = 70a^4\)

Given \(70a^4 = 1120 \implies a^4 = 16 \implies a = \pm 2\).

Example 8

Prove that \(C_0 + C_1 + C_2 + \dots + C_n = 2^n\).

Proof: We know the binomial expansion for \((1 + x)^n\):

\((1 + x)^n = C_0 + C_1 x + C_2 x^2 + \dots + C_n x^n\)

Putting \(x = 1\) on both sides:

\((1 + 1)^n = C_0 + C_1(1) + C_2(1)^2 + \dots + C_n(1)^n \implies 2^n = C_0 + C_1 + C_2 + \dots + C_n\).

Example 9

Approximate \((1.02)^5\) using Binomial Theorem correct to 4 decimal places.

Solution: \((1.02)^5 = (1 + 0.02)^5\)

\(= 1 + 5(0.02) + 10(0.02)^2 + 10(0.02)^3 + 5(0.02)^4 + (0.02)^5\)

\(= 1 + 0.1 + 10(0.0004) + 10(0.000008) + \dots\)

\(\approx 1 + 0.1 + 0.004 + 0.00008 = 1.1041\).

Example 10

Find the coefficient of \(x^r\) and \(x^{r+1}\) equal in \((1 + x)^{11}\).

Solution: In \((1 + x)^{11}\), coefficient of \(x^r\) is \(C(11, r)\) and coefficient of \(x^{r+1}\) is \(C(11, r+1)\).

Given \(C(11, r) = C(11, r+1) \implies r + (r+1) = 11 \implies 2r = 10 \implies r = 5\).

Chapter Assessment • 15 Uploaded Quiz Items

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Textbook Exercise Solutions

Exercise 2.1 (Questions 9 to 20)

Official CDC Nepal Model Grid

Examination Section (Full Model Paper)

Total Marks: 50 5 (2m) + 5 (3m) + 5 (5m)

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CDC Model Question Paper (Binomial Expansion)

Group A: Short Answer Questions (5 Questions × 2 Marks = 10 Marks)

1. Find the number of terms in the expansion of \(\left(x - \frac{1}{x^2}\right)^{12}\).
2. Write the general term of the binomial expansion \((a + x)^n\).
3. Find the middle term in the expansion of \((x + a)^{10}\).
4. State the sum of all binomial coefficients in the expansion of \((1 + x)^n\).
5. Write down the expansion of \((1 - x)^{-1}\) up to 3 terms.

Group B: Medium Answer Questions (5 Questions × 3 Marks = 15 Marks)

6. Find the coefficient of \(x^9\) in the expansion of \(\left(2x^2 - \frac{1}{x}\right)^{12}\).
7. Find the term independent of \(x\) in the expansion of \(\left(x^2 + \frac{1}{x}\right)^9\).
8. Find the value of \(a\) if the middle term in the expansion of \(\left(\frac{a}{2} + 2\right)^8\) is \(1120\).
9. For what value of \(r\) are the coefficients of \(x^r\) and \(x^{r+1}\) equal in the expansion of \((1 + x)^{11}\)?
10. Find the value of \(n\) if \(220\), \(495\) and \(792\) are three successive coefficients in the expansion of \((1 + x)^n\).

Group C: Long Answer / Higher-Order Questions (5 Questions × 5 Marks = 25 Marks)

11. Prove by mathematical induction that \((a + x)^n = C(n,0)a^n + C(n,1)a^{n-1}x + \dots + C(n,n)x^n\) for all \(n \in \mathbb{N}\).
12. If \((1 + x)^n = C_0 + C_1 x + C_2 x^2 + \dots + C_n x^n\), prove that \(C_0 + 4C_1 + 7C_2 + \dots + (3n+1)C_n = (3n+2)2^{n-1}\).
13. Prove that \(C_0^2 + C_1^2 + C_2^2 + \dots + C_n^2 = \frac{(2n)!}{(n!)^2}\).
14. Show that the middle term in the expansion of \(\left(a - \frac{1}{a}\right)^{2n}\) is \(\frac{1 \cdot 3 \cdot 5 \dots (2n-1)}{n!} (-2)^n a^n\).
15. State and prove Binomial Theorem for any index (negative or fractional exponent) with its convergence condition.