Mathematics · Textbook solutions

Binomial Theorem

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 24 questions

7.2 Binomial Theorem for Positive Integral Indices

18 q

Solved Examples

Worked · 4
  1. 7.1 Eg.1
    Expand (x2+3x)4\left(x^{2}+\dfrac{3}{x}\right)^{4}, x≠0x \neq 0.
  2. 7.1 Eg.2
    Compute (98)5(98)^{5}.
  3. 7.1 Eg.3
    Which is larger (1.01)1000000(1.01)^{1000000} or 10,00010{,}000?
  4. 7.1 Eg.4
    Using binomial theorem, prove that 6n−5n6^{n} - 5n always leaves remainder 1 when divided by 25.

Exercise 7.1

Practice · 14
  1. Ex 7.1 Q1
    Expand the expression (1−2x)5(1 - 2x)^{5}.
  2. Ex 7.1 Q2
    Expand the expression (2x−x2)5\left(\dfrac{2}{x} - \dfrac{x}{2}\right)^{5}.
  3. Ex 7.1 Q3
    Expand the expression (2x−3)6(2x - 3)^{6}.
  4. Ex 7.1 Q4
    Expand the expression (x3+1x)5\left(\dfrac{x}{3} + \dfrac{1}{x}\right)^{5}.
  5. Ex 7.1 Q5
    Expand the expression (x+1x)6\left(x + \dfrac{1}{x}\right)^{6}.
  6. Ex 7.1 Q6
    Using binomial theorem, evaluate (96)3(96)^{3}.
  7. Ex 7.1 Q7
    Using binomial theorem, evaluate (102)5(102)^{5}.
  8. Ex 7.1 Q8
    Using binomial theorem, evaluate (101)4(101)^{4}.
  9. Ex 7.1 Q9
    Using binomial theorem, evaluate (99)5(99)^{5}.
  10. Ex 7.1 Q10
    Using Binomial Theorem, indicate which number is larger (1.1)10000(1.1)^{10000} or 1000.
  11. Ex 7.1 Q11
    Find (a+b)4−(a−b)4(a + b)^{4} - (a - b)^{4}. Hence, evaluate (3+2)4−(3−2)4\left(\sqrt{3} + \sqrt{2}\right)^{4} - \left(\sqrt{3} - \sqrt{2}\right)^{4}.
  12. Ex 7.1 Q12
    Find (x+1)6+(x−1)6(x + 1)^{6} + (x - 1)^{6}. Hence or otherwise evaluate (2+1)6+(2−1)6\left(\sqrt{2} + 1\right)^{6} + \left(\sqrt{2} - 1\right)^{6}.
  13. Ex 7.1 Q13
    Show that 9n+1−8n−99^{n+1} - 8n - 9 is divisible by 64, whenever nn is a positive integer.
  14. Ex 7.1 Q14
    Prove that ∑r=0n3r nCr=4n\displaystyle\sum_{r=0}^{n} 3^{r}\,{}^{n}C_{r} = 4^{n}.

Miscellaneous Exercise on Chapter 7

6 q

Miscellaneous Exercise

Practice · 6
  1. Misc Q1
    If aa and bb are distinct integers, prove that a−ba - b is a factor of an−bna^{n} - b^{n}, whenever nn is a positive integer. [Hint: write an=(a−b+b)na^{n} = (a - b + b)^{n} and expand]
  2. Misc Q2
    Evaluate (3+2)6−(3−2)6\left(\sqrt{3} + \sqrt{2}\right)^{6} - \left(\sqrt{3} - \sqrt{2}\right)^{6}.
  3. Misc Q3
    Find the value of (a2+a2−1)4+(a2−a2−1)4\left(a^{2} + \sqrt{a^{2} - 1}\right)^{4} + \left(a^{2} - \sqrt{a^{2} - 1}\right)^{4}.
  4. Misc Q4
    Find an approximation of (0.99)5(0.99)^{5} using the first three terms of its expansion.
  5. Misc Q5
    Expand using Binomial Theorem (1+x2−2x)4\left(1 + \dfrac{x}{2} - \dfrac{2}{x}\right)^{4}, x≠0x \neq 0.
  6. Misc Q6
    Find the expansion of (3x2−2ax+3a2)3(3x^{2} - 2ax + 3a^{2})^{3} using binomial theorem.