NCERT Solutions
Class 11 - MathematicsChapter 7: BINOMIALTHEOREM
Class 11 - Mathematics
EXERCISE 7.1
Question. 1
Expand \((1 - 2x)^5\).
Answer:
\(1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5\)
Question. 2
Expand \(\left( \dfrac{2}{x} - \dfrac{x}{2} \right)^5\).
Answer:
\(\dfrac{32}{x^5} - \dfrac{40}{x^3} + \dfrac{20}{x} - 5x + \dfrac{5}{8}x^3 - \dfrac{x^5}{32}\)
Question. 3
Expand \((2x - 3)^6\).
Answer:
\(64x^6 - 576x^5 + 2160x^4 - 4320x^3 + 4860x^2 - 2916x + 729\)
Question. 4
Expand \(\left( \dfrac{x}{3} + \dfrac{1}{x} \right)^5\).
Answer:
\(\dfrac{x^5}{243} + \dfrac{5x^3}{81} + \dfrac{10x}{27} + \dfrac{10}{9x} + \dfrac{5}{3x^3} + \dfrac{1}{x^5}\)
Question. 5
Expand \(\left( x + \dfrac{1}{x} \right)^6\).
Answer:
\(x^6 + 6x^4 + 15x^2 + 20 + \dfrac{15}{x^2} + \dfrac{6}{x^4} + \dfrac{1}{x^6}\)
Question. 6
Using Binomial Theorem, evaluate \(96^3\).
Answer:
884736
Question. 7
Using Binomial Theorem, evaluate \(102^5\).
Answer:
11040808032
Question. 8
Using Binomial Theorem, evaluate \(101^4\).
Answer:
104060401
Question. 9
Using Binomial Theorem, evaluate \(99^5\).
Answer:
9509900499
Question. 10
Using Binomial Theorem, indicate which number is larger: \((1.1)^{10000}\) or 1000.
Answer:
\((1.1)^{10000} > 1000\)
Question. 11
Find \((a + b)^4 - (a - b)^4\). Hence, evaluate \((\sqrt{3} + \sqrt{2})^4 - (\sqrt{3} - \sqrt{2})^4\).
Answer:
\(8(a^3b + ab^3);\ 40\sqrt{6}\)
Question. 12
Find \((x + 1)^6 + (x - 1)^6\). Hence evaluate \((\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6\).
Answer:
\(2(x^6 + 15x^4 + 15x^2 + 1),\ 198\)
Question. 13
Show that \(9^{n+1} - 8n - 9\) is divisible by 64 whenever \(n\) is a positive integer.
Answer:
40\sqrt{6}
Question. 14
Prove that \(\sum_{r=0}^{n} 3^r \binom{n}{r} = 4^n\).
Answer:
Answer not provided in the screenshot.
Miscellaneous Exercise on Chapter 7
Question. 1
If \(a\) and \(b\) are distinct integers, prove that \(a - b\) is a factor of \(a^n - b^n\), whenever \(n\) is a positive integer.
Answer:
Use expansion: \(a^n = (a - b + b)^n\). Hence \(a^n - b^n\) is divisible by \(a - b\).
Question. 2
Evaluate \((\sqrt{3} + \sqrt{2})^6 - (\sqrt{3} - \sqrt{2})^6\).
Answer:
396\(\sqrt{6}\)
Question. 3
Find the value of \((a^2 + \sqrt{a^2 - 1})^4 + (a^2 - \sqrt{a^2 - 1})^4\).
Answer:
\(2a^8 + 12a^6 - 10a^4 - 4a^2 + 2\)
Question. 4
Find an approximation of \((0.99)^5\) using the first three terms of its expansion.
Answer:
0.9510
Question. 5
Expand using Binomial Theorem: \(\left(1 + \dfrac{x}{2} - \dfrac{2}{x}\right)^4\), \(x \neq 0\).
Answer:
\(\dfrac{16}{x} + \dfrac{8}{x^2} - \dfrac{32}{x^3} + \dfrac{16}{x^4} - 4x + \dfrac{x^2}{2} + \dfrac{x^3}{2} + \dfrac{x^4}{16} - 5\)
Question. 6
Find the expansion of \((3x^2 - 2ax + 3a^2)^3\) using binomial theorem.
Answer:
27x^6 - 54ax^5 + 117a^2x^4 - 116a^3x^3 + 117a^4x^2 - 54a^5x + 27a^6