KC Sinha Mathematics Solution Class 11 Chapter 16 Binomial Theorem Exercise 16.1

 Exercise 16.1

Page no 16.11

Type 1

Question 1

Expand the following by binomial theorem :

(i) $(x+y)^{5}$

(ii) $(1-x)^{6}$

(iii) $\left(x+\frac{1}{x}\right)^{7}$

(iv) $\left(x^{2}+\frac{2}{x}\right)^{4}, x \neq 0$

(v) $\left(\frac{2 x}{3}-\frac{3}{2 x}\right)^{6}$

(vi) $(3 x-2 y)^{5}$

(vii) $\left(\frac{2}{x}-\frac{x}{2}\right)^{5}$

(viii) $\left(1+2 x+x^{2}\right)^{3}$

(ix) $\left(x^{2}+\frac{3}{x}\right)^{4}, x \neq 0$

(x) $(2 x-3)^{6}$

(xi) $(1-2 x)^{5}$

(xii) $\left(\frac{x}{3}+\frac{1}{x}\right)^{5}$

(xiii) $\left(1+\frac{x}{2}-\frac{2}{x}\right)^{4}, x \neq 0$

Question 2

Expand the following :

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

Question 3

(i) Express $\left(x+\sqrt{x^{2}+1}\right)^{6}+\left(x-\sqrt{x^{2}+1}\right)^{6}$ as a polynomial in $x$.

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

(iii) Using binomial theorem expand $\left\{(x+y)^{5}+(x-y)^{5}\right\}$ and hence find the value of $\left\{(\sqrt{2}+1)^{5}+(\sqrt{2}-1)^{5}\right\}$

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

Question 4

Evaluate

(i) $(\sqrt{3}+1)^{5}-(\sqrt{3}-1)^{5}$

(ii) $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}$

(iii) $(\sqrt{3}+\sqrt{2})^{3}+(\sqrt{3}-\sqrt{2})^{3}$

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

Page no 16.12

Question 5

Find the number of terms in the following expansions

(i) $\left(1-2 x+x^{2}\right)^{30}$

(ii) $(\sqrt{a}+\sqrt{b})^{10}+(\sqrt{a}-\sqrt{b})^{10}$

Question 6

If $A$ be the sum of odd terms and $B$ the sum of even terms in the expansion of $(x+a)^{n}$; show that $4 A B=(x+a)^{2 n}-(x-a)^{2 n}$.

Question 7

Find the value of $(0.99)^{10}$ correct to 4 places of decimal.

Question 8

Using Binomial theorem evaluate $(0.99)^{15}$ correct to four places of decimal.

Question 9

Evaluate :

(i) $(101)^{4}$

(ii) $(102)^{5}$

(iii) $(999)^{5}$

(iv) $(102)^{6}$

Question 10

Evaluate :

(i) $(51)^{6}$

(ii) $(98)^{4}$

(iii) $(96)^{3}$

(iv) $(98)^{5}$

Question 11

Find the value of $(1.01)^{10}+(0.99)^{10}$ correct to 7 places of decimal.

Question 12

Find the greater of the two numbers

(i) $(1.01)^{1000000}$ and 10000 (ii) $(1.1)^{10000}$ and 1000 .

Question 13

Prove that (i) ${ }^{n} C_{0}+2 \cdot{ }^{n} C_{1}+\ldots+2^{n} \cdot{ }^{n} C_{n}=3^{n}$ for every natural number $n$.

(ii) ${ }^{2 n} C_{0}-3 \cdot{ }^{2 n} C_{1}+3^{2} \cdot{ }^{2 n} C_{2}-\ldots+(-1)^{2 n} \cdot 3^{2 n} \quad{ }^{2 n} C_{2 n}=4^{n}$, for all $n \in N$

Question 14

Find the approximate value of $(0.99)^{5}$ using the first three terms of its expansion.