Class 8-12 Mathematics Practice Questions, Solutions | ICSE ISC CBSE Math Mathematics Made Easy for School Students

$\displaystyle \textbf{Question 1. }\text{State the first principle of mathematical induction.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{If a statement }P(n)\text{ is true for }n=1,\text{ and }P(k)\Rightarrow P(k+1),$
$\displaystyle \text{then }P(n)\text{ is true for all }n\in N.$
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$\displaystyle \textbf{Question 2. }\text{Write the set of value of }n\text{ for which the statement }\\ P(n):2n<n!\text{ is true.}$
$\displaystyle \text{Answer:}$
$\displaystyle 2n<n!$
$\displaystyle \text{For }n=1,2,3,\text{ the statement is false.}$
$\displaystyle \text{For }n=4,\ 2n=8\text{ and }n!=24,\text{ so the statement is true.}$
$\displaystyle \therefore \text{The statement is true for all }n\geq4.$
$\displaystyle \therefore \{n\in N:n\geq4\}$
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$\displaystyle \textbf{Question 3. }\text{State the second principle of mathematical induction.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{If }P(1),P(2),\ldots,P(k)\text{ being true implies }P(k+1)\text{ is true,}$
$\displaystyle \text{then }P(n)\text{ is true for all }n\in N.$
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$\displaystyle \textbf{Question 4. }\text{If }P(n):2\times4^{2n+1}+3^{3n+1}\text{ is divisible by }\lambda\text{ for all }n\in N\text{ is true, then find }\lambda.$
$\displaystyle \text{Answer:}$
$\displaystyle 2\times4^{2n+1}+3^{3n+1}$
$\displaystyle =8\cdot16^n+3\cdot27^n$
$\displaystyle \text{Now, }16\equiv5\pmod{11}\text{ and }27\equiv5\pmod{11}$
$\displaystyle \therefore 8\cdot16^n+3\cdot27^n\equiv8\cdot5^n+3\cdot5^n\pmod{11}$
$\displaystyle \equiv11\cdot5^n\pmod{11}$
$\displaystyle \equiv0\pmod{11}$
$\displaystyle \therefore \lambda=11$
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