Algebra

Functions

Symbolic Operators

         

For positive integer kk, let k=k2(k1)2\bullet k\bullet = k^{2}-(k-1)^{2}. What is the value of 7\bullet 7\bullet?

(A)   36\ \ -36
(B)   7\ \ 7
(C)   13\ \ 13
(D)   49\ \ 49
(E)   85\ \ 85

For positive integers mm and nn, let mnm \nabla n be defined as mn=(mn)1(nm)1m \nabla n = (\frac{m}{n})^{-1} - (\frac{n}{m})^{-1}. What is the value of 2152 \nabla 15?

(A)   152\ \ -\frac{15}{2}

(B)   22130\ \ -\frac{221}{30}

(C)   22130\ \ \frac{221}{30}

(D)   215\ \ \frac{2}{15}

(E)   22930\ \ \frac{229}{30}

For positive integers mm, kk, and nn, let mk˘nm\breve{k}n be defined as mk˘n=kmnm\breve{k}n = k\frac{m}{n}, where kmnk\frac{m}{n} is a mixed fraction. What is the value of 64˘1+14˘66\breve{4}1 + 1\breve{4}6?

(A)   0\ \ 0

(B)   5\ \ 5

(C)   856\ \ \frac{85}{6}

(D)   700\ \ 700

(E)   787\ \ 787

For positive integers, xx, yy, and zz, let x,y,z^\widehat{x,y,z} be defined by x,y,z^=zxyzxy\widehat{x,y,z} = zx - yz - xy. What is the value of mm if 10,m,10^=7,3,1^? \widehat{10,m,10} = \widehat{7,3,1}?

(A)   63120\ \ -\frac{631}{20}

(B)   11720\ \ -\frac{117}{20}

(C)   0\ \ 0

(D)   21100\ \ \frac{21}{100}

(E)   11720\ \ \frac{117}{20}

For a positive integer nn, let n=12+34n\fbox{n} = 1-2+3-4 \ldots n. For example, 5=12+34+5=3.\fbox{5}=1-2+3-4+5=3. Which of the following is equal to 648 \fbox{6}-\fbox{48}?

(A)   1155\ \ -1155
(B)   27\ \ -27
(C)   21\ \ -21
(D)   6\ \ -\fbox{6}
(E)   21\ \ 21

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