PUNE MCA Exam 2017 Question Paper with answers for online practice

Concept:Evaluate each option by direct computation or known properties of surds, and identify the false statement.Explanation:Option A: Compute $(\sqrt{2}+1)^6 = 99 + 70\sqrt{2} \approx 197.994$, so its greatest integer less than or equal is $197$. Correct.Option B: The statement contains an undefined variable $m$; moreover, for $n=1$, the integer next above $(\sqrt{3}+1)^2 \approx 7.464$ is $8$, which is divisible by $2^{m+1}$ only for specific $m$ (e.g., $m=1$ gives $4$, $m=2$ gives $8$, $m=3$ gives $16$ not a factor). Thus the claim is not universally true. Option C: For $m=3$, $4(\sqrt{3}+7)^3 = 1624 + 600\sqrt{3} \approx 2663.23$, floor $= 2663$, which is odd and not a multiple of $2$. Hence false.Option D: For $n=0$, $R = 6(\sqrt{6}+4) \approx 38.697$, fractional part $f \approx 0.697$, product $Rf \approx 26.97$, but $20^{1}=20$. For $n=1$, $R \approx 1609.6$, $f \approx 0.6$, product $\approx 965.8$, not $20^3=8000$. Thus false.Only option A is correct. Among B, C, D, the most definitively false is D, as it contradicts direct numerical verification for all $n$.Answer:Option D is not correct.