\displaystyle \textbf{Question 1. }\text{If }x^\lambda-1\text{ is divisible by }x-\lambda,\text{ then the least positive integral value of }\lambda\text{ is}
\displaystyle \text{(a) }1\qquad \text{(b) }2\qquad \text{(c) }3\qquad \text{(d) }4
\displaystyle \text{Answer:}
\displaystyle x^\lambda-1\text{ is divisible by }x-\lambda
\displaystyle \therefore \lambda^\lambda-1=0
\displaystyle \lambda^\lambda=1
\displaystyle \therefore \lambda=1
\displaystyle \therefore\text{Correct option is (a).}
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\displaystyle \textbf{Question 2. }\text{For all }n\in N,\ 3\times5^{2n+1}+2^{3n+1}\text{ is divisible by}
\displaystyle \text{(a) }19\qquad \text{(b) }17\qquad \text{(c) }23\qquad \text{(d) }25
\displaystyle \text{Answer:}
\displaystyle 3\times5^{2n+1}+2^{3n+1}
\displaystyle =15\times25^n+2\times8^n
\displaystyle \text{Now, }25\equiv8\pmod{17}
\displaystyle \therefore 15\times25^n+2\times8^n\equiv15\times8^n+2\times8^n\pmod{17}
\displaystyle \equiv17\times8^n\pmod{17}
\displaystyle \equiv0\pmod{17}
\displaystyle \therefore\text{Correct option is (b).}
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\displaystyle \textbf{Question 3. }\text{If }10^n+3\times4^{n+2}+\lambda\text{ is divisible by }9\text{ for all }n\in N, \\ \text{ then the least positive integral value of }\lambda\text{ is}
\displaystyle \text{(a) }5\qquad \text{(b) }3\qquad \text{(c) }7\qquad \text{(d) }1
\displaystyle \text{Answer:}
\displaystyle 10^n+3\times4^{n+2}+\lambda
\displaystyle 10\equiv1\pmod{9}
\displaystyle \therefore 10^n\equiv1\pmod{9}
\displaystyle \text{Also, }4^{n+2}\equiv1,4,\text{ or }7\pmod{9}
\displaystyle \therefore 3\times4^{n+2}\equiv3\pmod{9}
\displaystyle \therefore 10^n+3\times4^{n+2}+\lambda\equiv1+3+\lambda\pmod{9}
\displaystyle \equiv4+\lambda\pmod{9}
\displaystyle \therefore 4+\lambda\equiv0\pmod{9}
\displaystyle \lambda\equiv5\pmod{9}
\displaystyle \therefore\text{Least positive integral value of }\lambda=5
\displaystyle \therefore\text{Correct option is (a).}
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\displaystyle \textbf{Question 4. }\text{Let }P(n):2^n<(1\times2\times3\times\cdots\times n).\text{ Then the smallest positive integer for which } \\ P(n)\text{ is true is}
\displaystyle \text{(a) }1\qquad \text{(b) }2\qquad \text{(c) }3\qquad \text{(d) }4
\displaystyle \text{Answer:}
\displaystyle P(n):2^n<n!
\displaystyle \text{For }n=1,\ 2^1=2\text{ and }1!=1,\text{ so }P(1)\text{ is false.}
\displaystyle \text{For }n=2,\ 2^2=4\text{ and }2!=2,\text{ so }P(2)\text{ is false.}
\displaystyle \text{For }n=3,\ 2^3=8\text{ and }3!=6,\text{ so }P(3)\text{ is false.}
\displaystyle \text{For }n=4,\ 2^4=16\text{ and }4!=24,\text{ so }P(4)\text{ is true.}
\displaystyle \therefore\text{Smallest positive integer}=4
\displaystyle \therefore\text{Correct option is (d).}
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\displaystyle \textbf{Question 5. }\text{A student was asked to prove a statement }P(n)\text{ by induction. He proved } \\ P(k+1)\text{ is true whenever }P(k)\text{ is true for all }k>5,\ k\in N\text{ and also }P(5)\text{ is true.} \\ \text{On the basis of this he could conclude that }P(n)\text{ is true.}
\displaystyle \text{(a) for all }n\in N\qquad \text{(b) for all }n>5
\displaystyle \text{(c) for all }n\geq5\qquad \text{(d) for all }n<5
\displaystyle \text{Answer:}
\displaystyle \text{Since }P(5)\text{ is true and }P(k)\Rightarrow P(k+1)\text{ for }k\geq5,
\displaystyle \text{we can conclude that }P(n)\text{ is true for all }n\geq5.
\displaystyle \therefore\text{Correct option is (c).}
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\displaystyle \textbf{Question 6. }\text{If }P(n):49^n+16^n+\lambda\text{ is divisible by }64\text{ for } n\in N\text{ is true, then the} \\ \text{least negative integral value of }\lambda\text{ is}
\displaystyle \text{(a) }-3\qquad \text{(b) }-2\qquad \text{(c) }-1\qquad \text{(d) }-4
\displaystyle \text{Answer:}
\displaystyle 49^n+16^n+\lambda
\displaystyle \text{For }n=1,\ 49^1+16^1+\lambda=65+\lambda
\displaystyle \text{For divisibility by }64,\ 65+\lambda\equiv0\pmod{64}
\displaystyle 1+\lambda\equiv0\pmod{64}
\displaystyle \lambda\equiv-1\pmod{64}
\displaystyle \therefore\text{Least negative integral value of }\lambda=-1
\displaystyle \therefore\text{Correct option is (c).}
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