Group (9)
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| 1 | ( 2011 ) Given that $x \odot y=x^{2}+x y+y^{2}, x, y \in R$, solve the equation $(6 \odot k)-(k \odot 2)=8-8 k$. Is $\odot$ commutative? Why? (5 marks)
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| 2 | ( 2011 ) A binary operation $\odot$ on $R$ is defined by $x \odot y=x^{2}+y^{2}$, for all real numbers $x$ and $y$. Show that binary operation is commutative and find the value of $2 \odot(3 \odot 1)$. Solve the equation $x \odot 2 \sqrt{6}=3 \odot 4$. (5 marks)
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| 3 | ( (2016/FC/q07a) ) Functions $f$ and $g$ are defined by $f: \mathrm{x} \mapsto 2 x+1$ and $g: x \mapsto \frac{2 x+5}{3-x}, x \neq 3$. Find the values of $x$ for which $\left(f \circ g^{-1}\right)(x)=x-4$. (5 marks)
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| 4 | ( 2010 ) Let $N$ be the set of natural numbers.Is the function $\odot$ defined by $a \odot b=(a+b) b$ where $a, b \in N$, a binary operation? If it is a binary operation.find $(6 \odot 3) \odot 4$ and $6 \odot(3 \odot 4)$.$\text{ (5 marks)}$
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| 5 | ( 2014 ) The binary operations $\odot_{1}$ and $\odot_{2}$ on $R$ are defined by $x \odot_{1} y=x^{2}-y^{2}$ and $x \odot_{2} y=7 x+4 y.$ Find $\left(2 \odot_{2}, 1\right) \odot_{1} 4$ Find also $x$ if $\left(-3 \odot_{1}\right.$ 2) $\odot_{2}\left(1 \odot_{1} x\right)=3$. (5 marks)
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| 6 | ( 2010 ) Let $J^{+}$be the set of all positive integers.An operation $\odot$ on $J^{+}$is given by $x \odot y=x(2 x+y)$, for all positive integers $x$ and $y$.Prove that $\odot$ is a binary operation on $J^{+}$and calculate $(2 \odot 3) \odot 4$ and $2 \odot(3 \odot 4) .$ Is the binary operation commutative? $\text{ (5 marks)}$
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| 7 | ( 2010 ) Let $J^{+}$be the set of positive integers and a binary oneration $\odot$ be defined by $a \odot b=a(3 a+b)$ for $a, b \in \mathrm{J}^{+} .$Find the values of $2 \odot 1$ and $(2 \odot 1) \odot 4$.Find also the value of $p$ if $p \odot(p+1)=39$.$\text{ (5 marks)}$
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| 8 | ( 2010 ) An operation $\odot$ on $R$ is defined by $a \odot b=a(a+2 b), a, b \in R$.Is $\odot$ commutative? Ca'culate $(2 \odot 3) \odot 4$.Find the values of $x$ such that $x \odot 2=2 \odot 7$.$\text{ (5 marks)}$
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| 9 | ( 2010 ) A binary operation $\odot$ on $R$ is defined by $x \odot y=x+y+10 x y$.Show that the binary operation is commutative.Find the values of $b$ such that $(1 \odot b) \odot \mathrm{b}=485$.$\text{ (5 marks)}$
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| 10 | ( 2010 ) Abinary operation $\odot$ on $R$ is defined by $x \odot y=x+y+4 x y$.Show that the binary operation is commutative.Find the values of $a$ such that $(a \odot 3) \odot a=263$.$\text{ (5 marks)}$
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| 11 | ( 2012 ) A binary operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-2 b$. If $4 \odot(2 \odot k)=20$. find $k \odot 5$. (5 marks)
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| 12 | ( 2010 ) A binary operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-2 b$, for all $a, b \in R$.If $4 \odot(2 \odot k)=20$, find the value of $(k \odot 5) \odot k$.$\text{ (5 marks)}$
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| 13 | ( (2015/FC/q07b) ) Show that the mapping $\odot$ defined by $x \odot y=x y+x^{2}+y^{2}$ is a binary operation on the set $R$ and verify that it is commutative and but not associative. . $\quad$ (5 marks)
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| 14 | ( 2011 ) Let $N$ be the set of natural numbers. Is the function $\odot$ defined by $a \odot b=2 a(a+b)$, where $a, b \in N$ a binary operation? If it is a binary operation calculate $1 \odot 4$ and $4 \odot 1 .$ Is $1 \odot 4=4 \odot 1 ?$ (5 marks)
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| 15 | ( 2011 ) Let $N$ be the set of natural numbers. Is the function $\odot$ defined by $a \odot b=(2 a+b) b$, where $a, b \in N$ a binary operation? If it is a binary operation calculate $5 \odot 3$ and $3 \odot$ 5. Is $5 \odot 3=3 \odot 5 ?$ (5 marks)
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| 16 | ( 2011 ) A binary operation $\odot$ on the set $R$ of real numbers is defined by $x \odot y=x y+x+y$. Show that $(x \odot y) \odot z=x \odot(y \odot z)$ and calculate $(2 \odot 1) \odot 3$. (5 marks)
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| 17 | ( 2012 ) The mapping defined by $x \odot y=x y-x-y$ is a binary operation on the set $R$ of real numbers. Is the binary operation commutative? Find $(2 \odot 3) \odot 4$ and $2 \odot(3 \odot 4)$. Are they equal? (5 marks)
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| 18 | ( 2013 ) An operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-2 a b+b^{2}$, for all real numbers ' $a$ ' and ' $b$ '. Show that $\odot$ is a binary operation and evaluate $3 \odot(2 \odot 1)$. (5 marks)
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| 19 | ( 2013 ) A binary operation $\odot$ on $N$ is defined by $x \odot y=$ the remainder when $x^{y}$ is divided by 5 . Is the binary operation commutative? Find $[(2 \odot 3) \odot 4]+[2 \odot$ $(3 \odot 4)]$. Is the binary operation associative? (5 marks) (5 marks)
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| 20 | ( 2014 ) Let $J$ be the set of positive integers. Show that the operation $\odot$ defined by $a \odot b=a^{\mathrm{b}}+a+b$ for $a, b \in J$ is a binary operation on $\mathrm{J}$. Find the values of $2 \odot 4$ and $4 \odot 2$. Is this binary operation commutative? Why? 5 marks) |
Answer Group (9)
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| 1 | $k=-2 ;$ Yes
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| 2 | $104 ; x=\pm 1$
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| 3 | $x=0$ or 9
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| 4 | Yes ; $124 ; 952$
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| 5 | $308, x=\pm 3$
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| 6 | $448 ; 68$; No
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| 7 | $14 ; 644 ; 3$
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| 8 | No ; $384 ;-8,4$
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| 9 | $-\frac{11}{5}, 2$
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| 10 | $-\frac{5}{2}, 2$
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| 11 | $-1 \quad $
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| 12 | $-5$
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| 13 | $(1\odot 0)\odot 2\not=1\odot (0\odot 2)$
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| 14 | Yes ; 10 ; 40 ;No
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| 15 | Yes; 39; 55; No
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| 16 | $23$
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| 17 | Yes $;(2 \odot 3) \odot 4=-1,2 \odot(3 \odot 4)=3,(2 \odot 3) \odot 4 \neq 2 \odot(3 \odot 4)$
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| 18 | $3 \odot(2 \odot 1)=4 \quad$
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| 19 | $ \mathrm{No} ; 3 ; \mathrm{No}$
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| 20 | $2 \odot 4=22,4 \odot 2=22, \odot$ is not commulative |
Group (10)
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| 1 | ( 2012 ) A function $\odot$ on the set $R$ of real numbers is defined by $x \odot y=y(3 x+2 y)$, $x, y \in R$. Prove that $\odot$ is binary operation and s il ve the equation $(3 x \odot x)=44$. (5 marks)
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| 2 | ( 2012 ) An operation $\odot$ is defined on $R$ by $x \odot y=x y-x+y$. Prove that $\odot$ is a binary operation on $R$. Is $\odot$ commutative? Why? Find the value of a such that $(a \odot 2)+(2 \odot a)=16$. (5 marks)
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| 3 | ( 2012 ) A binary operation $\odot$ on the set $R$ of real numbers is defined by $x \odot y=x^{2}-x y+y^{2}$. Prove that the binary operation is commutative. Find the values of $p$ such that $2 \odot p=12$. (5 marks)
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| 4 | ( 2012 ) A binary operation $\odot$ on the set $R$ of real numbers is defined by $x \odot y=x^{2}-x y+y^{2}$. Prove that the binary operation is commutative. Find the values of $a$ such that $2 \odot a=12$. (5 marks)
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| 5 | ( 2013 ) A binary operation $\odot$ on $R$ is defined by $x \odot y=(2 x-3 y)^{2}-5 y^{2}$. Show that the binary operation is commutative. Find the values of $k$ for which $(-2) \odot k=80$. (5 marks)
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| 6 | ( 2013 ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ is defined by $x \odot y=x+x y-y$ for all $x, y \in R$. Show that the operation $\odot$ is not associative. Solve the equation $(2 \odot 3) \odot x=(x \odot x)-7$. (5 marks)
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| 7 | ( 2014 ) Let $\odot$ be the binary operation on $R$ defined by $a \odot b=a^{2}+b^{2}$ for all $a, b \in R$. Show that $(a \odot b) \odot a=a \odot(b \odot a)$. Solve also the equation $4 \odot(x \odot 2)=185$ $\qquad\mbox{ (5 marks)}$
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| 8 | ( 2014 ) A binary operation $\odot$ on $R$ is defined by $x \odot y=(4 x+y)^{2}-15 x^{2}$, show that the binary operation is commutative. Find the possible values of $k$ such that $(k+1) \odot(k-2)=109$. $\qquad\mbox{ (5 marks)}$
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| 9 | ( 2014 ) A binary operation $\odot$ on $R$ is defined by $x \odot y=\frac{x^{2}+y^{2}}{2}+2 x y$. Show that $\odot$ is commutative. Find the values of $p$ such that $p \odot 3=p+10$. $\qquad\mbox{ (5 marks)}$
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| 10 | ( 2014 ) The binary operation $*$ on $R$ is defined by $x * y=\frac{x^{2}+y^{2}}{2}-x y$, for all real numbers $x$ and $y$. Show that the operation is commutative, and find the possible values of a such that $a * 2=a+2$. (5 marks)
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| 11 | ( 2014 ) An operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-a b+b^{2}$, for all real numbers $a$ and $b$. Is $\odot$ associative? Why? Find the value of $p$ such that $p \odot 2=3$ and hence evaluate $p \odot p$. (5 marks)}
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| 12 | ( (2015/Myanmar/q07b) ) Let $J^{+}$be the set of all positive integers. Is the function $\odot$ defined by $x \odot y=x+3 y$ a binary operation on $J^{+} ?$ If it is a binary operation, solve the equation $(k \odot$ 5) $-(3 \odot k)=2 k+13$ (5 marks)
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| 13 | ( (2016/FC/q07b) ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ is defined by $x \odot y=x y-x-y$ for all $x, y$ in $R$. Show that the operation $\odot$ is commutative. Solve the equation $(2 \odot 3) \odot x=(x \odot x) \odot 5$. (5 marks)
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| 14 | ( (2018/Myanmar/q07b) ) The binary operation $\odot$ on $R$ is defined by $x \odot y=\frac{x^{2}+y^{2}}{2}-x y$, for all real numbers $x$ and $y$. Show that the operation is commutative, and find the possible values of $a$ such that $a \odot 2=a+2$. (5 marks)
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| 15 | ( (2019/Myanmar/q07a) ) A binary operation $\odot$ on R is defined by $x\odot y = (3y−x)^2 −8y^2.$ Show that the binary operation is commutative. Find the possible values of $k$ such that $2 \odot \mathrm{k}=-31.$ (5 marks)
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| 16 | ( 2012 ) Let $R$ be the set of real numbers. Is the function $\odot$ defined by $a \odot b=$ $a^{2}-2 a b+3 b^{2}$ for all $a, b \in R$, a binary operation? Is $\odot$ commutative? Why? (5 marks)
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| 17 | ( 2012 ) Let $R$ be the set of real numbers. Is the function $\odot$ defined by $a \odot b=a^{2}-4 a b+b^{2}$, for all $a, b \in R$, a binary operation? Is $\odot$ commutative? Is $\odot$ associative? (5 marks)
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| 18 | ( 2012 ) The binary operation $\odot$ on $R$ is defined by $x \odot y=a x^{2}+b x+c y$, for all real numbers $x$ and $y .$ If $1 \odot 1=4,2 \odot 1=5$ and $1 \odot 2=-3$ then find the value of $a$, $b$ and $c$. (5 marks)
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| 19 | ( 2013 ) The binary operation $\odot$ on $R$ is defined by $x \odot y=a x^{2}+b x+c y$, for all real numbers $x$ and $y .$ If $1 \odot 1=4,2 \odot 1=5$ and $1 \odot 2=-3$, then find the values of $a, b$ and $c$. (5 marks)
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| 20 | ( 2014 ) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=3 x-1$ and $g(x)=x+7$. Find $\left(f^{-1} \circ \mathrm{g}\right)(x)$ and $\left(g^{-1} \circ f\right)(x)$. What are the values of $\left(f^{-1} \circ g\right)(3)$ and $\left(g^{-1} \circ f\right)(2) ?$ $\qquad\mbox{ (5 marks)}$ |
Answer Group (10)
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| 1 | $x=\pm 2$
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| 2 | No ; $a=4 \quad$
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| 3 | $p=4$ (or) $p=-2 \quad$
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| 4 | $a=4$ (or) $a=-2 \quad$
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| 5 | $k=-8$ (or) 2
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| 6 | $x=6$ (or) $-2 \quad$
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| 7 | $x=\pm 3$
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| 8 | $k=4$ (or) $-3$
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| 9 | $p=-11$
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| 10 | $a=0$ (or) 6
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| 11 | $\odot$ is not associative, $p=1, p\odot p=1 \quad$
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| 12 | No solution
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| 13 | $x=1 \pm \sqrt{2}$
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| 14 | $a=0 $ or $6$
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| 15 | $k=3$ or $k=7$
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| 16 | Yes; Yes
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| 17 | Yes;Yes;No
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| 18 | $a=-5, b=16, c=-7$,
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| 19 | $a=-5, b=16, c=-7 $
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| 20 | $\left(f^{-1} \circ g\right)(x)=\frac{x+8}{3} /\left(g^{-1} \circ f\right)(x)=3 x-8; \frac{11}{3},-2$ |
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