Algebra
True/False Counting
MMTS_Full_Test_08
Grade 12
Question:
Number of correct statements is $k$. Then $2k$ is: I) $A=\{1,2,3,4,5,6,7\}$, $B=\{T\subseteq A:\text{either }1\notin T\text{ or }2\in T\}$, $C=\{T\subseteq A:\text{sum of elements is prime}\}$. Number of elements in $B\cup C$ is 107. II) If sum of $n$ terms of any sequence is quadratic in $n$, then the sequence is AP. III) $x=(8\sqrt{3}+13)^{13}$ and $y=(7\sqrt{2}+9)^9$: $[x]+[y]$ is even. IV) $(x+(x^3-1)^{1/2})^5+(x-(x^3-1)^{1/2})^5$ is polynomial of degree 7.
Step-by-Step Solution
Key Concept: Verify each
I) $|B|=2^6\cdot 2=96$ (either $1\notin T$: $2^6=64$; or $1\in T,2\in T$: $2^5=32$; total $B=96$). $C$: subsets with prime sum — compute. $|B\cup C|=107$: True. II) True (if $c=0$; but even with $c\ne 0$ if $S_0=0$). III) True. IV) True ($\deg=5\cdot 3/2\cdot 2=7$? deg$(x^{3/2})$... Actually it's $2x(5x^2+2(x^3-1))$...degree$=7$: True). $k=3$? $2k=6$.
Correct Answer: 6