Algebra

# Symbolic Operators

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

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

For positive integers $$m$$ and $$n$$, let $$m \nabla n$$ be defined as $$m \nabla n = (\frac{m}{n})^{-1} - (\frac{n}{m})^{-1}$$. What is the value of $$2 \nabla 15$$?

(A) $$\ \ -\frac{15}{2}$$

(B) $$\ \ -\frac{221}{30}$$

(C) $$\ \ \frac{221}{30}$$

(D) $$\ \ \frac{2}{15}$$

(E) $$\ \ \frac{229}{30}$$

For positive integers $$m$$, $$k$$, and $$n$$, let $$m\breve{k}n$$ be defined as $$m\breve{k}n = k\frac{m}{n}$$, where $$k\frac{m}{n}$$ is a mixed fraction. What is the value of $$6\breve{4}1 + 1\breve{4}6$$?

(A) $$\ \ 0$$

(B) $$\ \ 5$$

(C) $$\ \ \frac{85}{6}$$

(D) $$\ \ 700$$

(E) $$\ \ 787$$

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

(A) $$\ \ -\frac{631}{20}$$

(B) $$\ \ -\frac{117}{20}$$

(C) $$\ \ 0$$

(D) $$\ \ \frac{21}{100}$$

(E) $$\ \ \frac{117}{20}$$

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

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

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