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First Year Math Mathematical Induction and Binomial Theorem 1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.xi) \left(1-2 x+3 x^{2}\right)^{\frac{1}{2}}


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1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.xi) \left(1-2 x+3 x^{2}\right)^{\frac{1}{2}}

Example 4: Evaluate \sqrt[3]{30} correct to three places of decimal.
Example 4: Evaluate  \sqrt[3]{30}  correct to three places of decimal.

Example 4: Evaluate \sqrt[3]{30} correct to three places of decimal.

Example 2: Evaluate (9.9)^{5}
Example 2: Evaluate  (9.9)^{5}

Example 2: Evaluate (9.9)^{5}

5. Expand the following in descending powers of x :i) \left(x^{2}+x-1\right)^{3}
5. Expand the following in descending powers of  x  :i)  \left(x^{2}+x-1\right)^{3}

5. Expand the following in descending powers of x :i) \left(x^{2}+x-1\right)^{3}

Use mathematical induction to prove the following formulae for every positive integer n . 25. x+1 is a factor of x^{2 n}-1 ;(x \neq-1)
Use mathematical induction to prove the following formulae for every positive integer n . 25.  x+1  is a factor of  x^{2 n}-1 ;(x \neq-1)

Use mathematical induction to prove the following formulae for every positive integer n . 25. x+1 is a factor of x^{2 n}-1 ;(x \neq-1)

3. Find the coefficient of x^{n} in the expansion ofv) \left(1-x+x^{2}-x^{3}+\ldots\right)^{2}
3. Find the coefficient of  x^{n}  in the expansion ofv)  \left(1-x+x^{2}-x^{3}+\ldots\right)^{2}

3. Find the coefficient of x^{n} in the expansion ofv) \left(1-x+x^{2}-x^{3}+\ldots\right)^{2}

Example 5: Show that: \left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\ 2\end{array}\right)+3\left(\begin{array}{l}n \\ 3\end{array}\right)+\ldots .+n\left(\begin{array}{l}n \\ n\end{array}\right)=n .2^{n-1}
Example 5: Show that:  \left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\ 2\end{array}\right)+3\left(\begin{array}{l}n \\ 3\end{array}\right)+\ldots .+n\left(\begin{array}{l}n \\ n\end{array}\right)=n .2^{n-1}

Example 5: Show that: \left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\ 2\end{array}\right)+3\left(\begin{array}{l}n \\ 3\end{array}\right)+\ldots .+n\left(\begin{array}{l}n \\ n\end{array}\right)=n .2^{n-1}

3. Find the coefficient of x^{n} in the expansion ofiii) \frac{(1+x)^{3}}{(1-x)^{2}}
3. Find the coefficient of  x^{n}  in the expansion ofiii)  \frac{(1+x)^{3}}{(1-x)^{2}}

3. Find the coefficient of x^{n} in the expansion ofiii) \frac{(1+x)^{3}}{(1-x)^{2}}

13. \frac{1}{2 \times 5}+\frac{1}{5 \times 8}+\frac{1}{8 \times 11}+\ldots .+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{2(3 n+2)}
13.  \frac{1}{2 \times 5}+\frac{1}{5 \times 8}+\frac{1}{8 \times 11}+\ldots .+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{2(3 n+2)}

13. \frac{1}{2 \times 5}+\frac{1}{5 \times 8}+\frac{1}{8 \times 11}+\ldots .+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{2(3 n+2)}

2. Calculate the following by means of binomial theorem:i) (0.97)^{3}
2. Calculate the following by means of binomial theorem:i)  (0.97)^{3}

2. Calculate the following by means of binomial theorem:i) (0.97)^{3}

1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.x) \left(1+x-2 x^{2}\right)^{\frac{1}{2}}
1. Expand the following upto 4 terms taking the values of  x  such that the expansion in each case is valid.x)  \left(1+x-2 x^{2}\right)^{\frac{1}{2}}

1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.x) \left(1+x-2 x^{2}\right)^{\frac{1}{2}}

4. If x is so small that its square and higher powers can be neglected then show thativ) \frac{\sqrt{4+x}}{(1-x)^{3}} \approx 2+\frac{25}{4} x
4. If  x  is so small that its square and higher powers can be neglected then show thativ)  \frac{\sqrt{4+x}}{(1-x)^{3}} \approx 2+\frac{25}{4} x

4. If x is so small that its square and higher powers can be neglected then show thativ) \frac{\sqrt{4+x}}{(1-x)^{3}} \approx 2+\frac{25}{4} x

Use mathematicl induction to prove the following formula for every positive inceger5. 1+\frac{1}{2}+\frac{1}{4}+++\frac{1}{2 n}=2\left[-1-\frac{1}{2}\right]
Use mathematicl induction to prove the following formula for every positive inceger5.  1+\frac{1}{2}+\frac{1}{4}+++\frac{1}{2 n}=2\left[-1-\frac{1}{2}\right]

Use mathematicl induction to prove the following formula for every positive inceger5. 1+\frac{1}{2}+\frac{1}{4}+++\frac{1}{2 n}=2\left[-1-\frac{1}{2}\right]

1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.xi) \left(1-2 x+3 x^{2}\right)^{\frac{1}{2}}
1. Expand the following upto 4 terms taking the values of  x  such that the expansion in each case is valid.xi)  \left(1-2 x+3 x^{2}\right)^{\frac{1}{2}}
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1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.xi) \left(1-2 x+3 x^{2}\right)^{\frac{1}{2}}

Use mathematical induction to prove the following formulae for every positive integen n . 24. 1^{3}+3^{3}+5^{3}+\ldots .+(2 n-1)^{3}=n^{2}\left[2 n^{2}-1\right]
Use mathematical induction to prove the following formulae for every positive integen n . 24.   1^{3}+3^{3}+5^{3}+\ldots .+(2 n-1)^{3}=n^{2}\left[2 n^{2}-1\right]

Use mathematical induction to prove the following formulae for every positive integen n . 24. 1^{3}+3^{3}+5^{3}+\ldots .+(2 n-1)^{3}=n^{2}\left[2 n^{2}-1\right]

Use the principle of extended mathematical induction to prove that:38. 1+n x \leq(1+x)^{0} for n \geq 2 and x>-1
Use the principle of extended mathematical induction to prove that:38.  1+n x \leq(1+x)^{0}  for  n \geq 2  and  x>-1

Use the principle of extended mathematical induction to prove that:38. 1+n x \leq(1+x)^{0} for n \geq 2 and x>-1

Example 7: If m and n are nearly equal show that\[\left(\frac{5 m-2 n}{3 n}\right)^{1 / 3} \approx \frac{m}{m+2 n}+\frac{n+m}{3 n}\]
Example 7: If  m  and  n  are nearly equal show that\[\left(\frac{5 m-2 n}{3 n}\right)^{1 / 3} \approx \frac{m}{m+2 n}+\frac{n+m}{3 n}\]

Example 7: If m and n are nearly equal show that\[\left(\frac{5 m-2 n}{3 n}\right)^{1 / 3} \approx \frac{m}{m+2 n}+\frac{n+m}{3 n}\]

3. Find the coefficient of x^{n} in the expansion ofi) \frac{1+x^{2}}{(1+x)^{2}}
3. Find the coefficient of  x^{n}  in the expansion ofi)  \frac{1+x^{2}}{(1+x)^{2}}

3. Find the coefficient of x^{n} in the expansion ofi) \frac{1+x^{2}}{(1+x)^{2}}

Example 3: Find the specified term in the expansion of \left(\frac{3}{2} x-\frac{1}{3 x}\right)^{11} i) the term involving x^{5} ii) the fifth termiii) the sixth term from the end. iv) coefficient of term involving x^{-1}
Example 3: Find the specified term in the expansion of  \left(\frac{3}{2} x-\frac{1}{3 x}\right)^{11} i) the term involving  x^{5} ii) the fifth termiii) the sixth term from the end. iv) coefficient of term involving  x^{-1}

Example 3: Find the specified term in the expansion of \left(\frac{3}{2} x-\frac{1}{3 x}\right)^{11} i) the term involving x^{5} ii) the fifth termiii) the sixth term from the end. iv) coefficient of term involving x^{-1}

2. Using Binomial theorem find the value of the following to three places of decimals.i) \sqrt{99}
2. Using Binomial theorem find the value of the following to three places of decimals.i)  \sqrt{99}
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2. Using Binomial theorem find the value of the following to three places of decimals.i) \sqrt{99}

2. Using Binomial theorem find the value of the following to three places of decimals.v) \sqrt[4]{17}
2. Using Binomial theorem find the value of the following to three places of decimals.v)  \sqrt[4]{17}
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2. Using Binomial theorem find the value of the following to three places of decimals.v) \sqrt[4]{17}

1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.ii) (1+2 x)^{-1}
1. Expand the following upto 4 terms taking the values of  x  such that the expansion in each case is valid.ii)  (1+2 x)^{-1}
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1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.ii) (1+2 x)^{-1}

Example 3: Show that \frac{n^{3}+2 n}{3} represents an integer \forall n \in N .
Example 3: Show that  \frac{n^{3}+2 n}{3}  represents an integer  \forall n \in N .
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Example 3: Show that \frac{n^{3}+2 n}{3} represents an integer \forall n \in N .

3. Expand and simplify the following:iii) (2+i)^{5}-(2-i)^{5}
3. Expand and simplify the following:iii)  (2+i)^{5}-(2-i)^{5}
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3. Expand and simplify the following:iii) (2+i)^{5}-(2-i)^{5}

1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.ix) \frac{(4+2 x)^{1 / 2}}{2-x}
1. Expand the following upto 4 terms taking the values of  x  such that the expansion in each case is valid.ix)  \frac{(4+2 x)^{1 / 2}}{2-x}
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1. Expand the following upto 4 terms taking the values of x such that the expansion in each case is valid.ix) \frac{(4+2 x)^{1 / 2}}{2-x}

2. Using Binomial theorem find the value of the following to three places of decimals.xi) \frac{1}{\sqrt[6]{486}}
2. Using Binomial theorem find the value of the following to three places of decimals.xi)  \frac{1}{\sqrt[6]{486}}
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2. Using Binomial theorem find the value of the following to three places of decimals.xi) \frac{1}{\sqrt[6]{486}}

4. If x is so small that its square and higher powers can be neglected then show thatiii) \frac{(9+7 x)^{1 / 2}-(16+3 x)^{1 / 4}}{4+5 x} \simeq \frac{1}{4}-\frac{17}{384} x
4. If  x  is so small that its square and higher powers can be neglected then show thatiii)  \frac{(9+7 x)^{1 / 2}-(16+3 x)^{1 / 4}}{4+5 x} \simeq \frac{1}{4}-\frac{17}{384} x
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4. If x is so small that its square and higher powers can be neglected then show thatiii) \frac{(9+7 x)^{1 / 2}-(16+3 x)^{1 / 4}}{4+5 x} \simeq \frac{1}{4}-\frac{17}{384} x

Example 4: Find the following in the expansion of \left(\frac{x}{2}+\frac{2}{x^{2}}\right)^{12} i) the term independent of x .ii) the middle term
Example 4: Find the following in the expansion of  \left(\frac{x}{2}+\frac{2}{x^{2}}\right)^{12} i) the term independent of  x .ii) the middle term
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Example 4: Find the following in the expansion of \left(\frac{x}{2}+\frac{2}{x^{2}}\right)^{12} i) the term independent of x .ii) the middle term

2. Using Binomial theorem find the value of the following to three places of decimals.iv) \sqrt[3]{65}
2. Using Binomial theorem find the value of the following to three places of decimals.iv)  \sqrt[3]{65}
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2. Using Binomial theorem find the value of the following to three places of decimals.iv) \sqrt[3]{65}

8. Find 6 th term in the expansion of \left(x^{2}-\frac{3}{2 x}\right)^{10}
8. Find 6 th term in the expansion of  \left(x^{2}-\frac{3}{2 x}\right)^{10}
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8. Find 6 th term in the expansion of \left(x^{2}-\frac{3}{2 x}\right)^{10}

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