# Myanmar Matriculation (Function 4, 8)

### Group (4)

 $\quad$ 1 ( 2010 ) $f: R \rightarrow R, g: R \rightarrow R$ and $h: R \rightarrow R$ are functions defined by $f(x)=x^{2}+2$ $-g(x)=x-1$ and $h(x)=3 x-2 .$ Find the formulae of $f \circ g$ and $f \circ(h \circ g)$.$\text{ (5 marks)}$ 2 ( 2014 ) A function $f$ is defined by $f(x)=\frac{k x+5}{x-1}$ for all $x \neq 1$, where $k$ is a constant. If $f^{-1}(7)=4$, find the value of $k$. If $g(x)=2 x+3$, find the formula of $f^{-1} \circ(g \circ f)$ in simplified form. $\qquad\mbox{ (5 marks)}$ 3 ( 2013 ) Let $f: R \rightarrow R, g: R \rightarrow R$ be defined by $f(x)=x-2$ and $g(x)=x^{2}$ and $h(x)=x+8$. If $(h \circ g)(a)=(g \circ f)(a)$ then find the value of ' $a$ '. (5 marks) 4 ( 2013 ) Two functions are defined by $f(x)=\frac{1}{x+1}, x \neq-1$ and $g(x)=\frac{x}{x-2}, x \neq 2$. Find the values of $x$ for which $(f \circ g)(x)+(g \circ f)(x)=0$. (5 marks) 5 ( 2011 ) The functions $f$ and $g$ are defined by $f(x)=-x-2$ and $g(x)=m x+3$. Find the value of $m$ for which $(f \circ g)(x)=(g \circ f)(x)$. Hence find $g^{-1}(5)$. (5 marks) 6 ( 2010 ) Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=\frac{2 x}{x-3}, x \neq 3$ and $g(x)=2 x-3$.Find formulae for the inverse functions $f^{-1}$ and $g^{-1}$, Evaluate $\left(f^{-1} \circ g^{-1}\right)(5)$.$\text{ (5 marks)}$ 7 ( 2010 ) Find formula for $f^{-1}$, the inverse function of $f$ defined by $f(x)=\frac{2}{3-4 x}$; ( $x \neq \frac{3}{4}$.State the suitable domain of $f^{-1}$.Find also $\left(f^{-1} \circ f^{-1}\right)(2)$.$\text{ (5 marks)}$ 8 ( 2010 ) Find formula for $f^{-1}$, the inverse function of $f$ defined by $f(x)=\frac{5}{3-x}, x \neq 3$.State the suitable domain of $f^{-1}$.Find also $\left(f^{-1} \circ f^{-1}\right)(2)$.$\text{ (5 marks)}$ 9 ( 2010 ) Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=2 x+5$ and $g(x)=\frac{2 x}{x-3}, x \neq 3$.Find formulae for the inverse functions $f^{-1}$ and $g^{-1}$.Evaluate $\left(g^{-1} \circ f^{-1}\right)(7)$.$\text{ (5 marks)}$ 10 ( 2010 ) If $f$ and $g$ are functions such that $f(x)=x+1$ and $f(g(x))=3 x-1$.Find the formula of $(g \circ f)^{-1}$ and hence find $(g \circ f)^{-1}(4)$.$\text{ (5 marks)}$ 11 ( 2012 ) Find the formula for the inverse function $f_{1}^{-1}$ where $f: R \rightarrow R$ is defined by $f(x)=1+9 x .$ Find the image of 2 under $\left(f \circ f^{-1}\right)$. (5 marks) 12 ( 2012 ) Functions $f$ and $g$ are defined by $f(x)=3 x+2, g(x)=\frac{2 x}{3 x-2}, x \neq \frac{2}{3}$. Evaluate $(g \circ f)$ (3) and $\left(g^{-1} \circ f^{-1}\right)(1)$. (5 marks) 13 ( 2013 ) The functions $f$ and $g$ are defined by $f(x)=3 x-4$ and $g(x)=\frac{4x+1}{2-x},x\not= 2$ Evaluate $(g \circ f)(-1)$ and $\left(f^{-1} \circ g^{-1}\right)(-5)$. (5 marks) 14 ( (2015/Myanmar/q07a) ) The functions $f$ and $g$ are defined for real $x$ by $f(x)=2 x-1$ and $g(x)=\frac{2 x+3}{x-1}, x \neq 1 .$ Evaluate $\left(\dot{g}^{-1} \circ f^{-1}\right)(2).$ (5 marks) 15 ( (2018/Myanmar/q07a) ) The functions $f$ and $g$ are defined for real $x$ by $f(x)=2 x-1$ and $g(x)=2 x+3 .$ Evaluate $\left(g^{-1} \circ f^{-1}\right)(2)$. (5 marks) 16 ( 2010 ) Functions $f$ and $g$ are defined by $f(x)=2 x-1$ and $g(x)=\frac{2 x+3}{x-1}, x \neq 1$.Evaluate $\left(g^{-1} \circ f^{-1}\right)(2)$ and $(g \circ f)(2)$.$\text{ (5 marks)}$

 $\quad$ 1 $(f \circ g)(x)=x^{2}-2 x+3 \quad(f \circ(h \circ g))(x)=9 x^{2}-30 x+27$ 2 $k=4, \frac{16 x+2}{7 x+11}, x \neq-\frac{11}{7}$ 3 $a=-1 \quad$ 4 $x=0$ (or) $\frac{5}{2} \quad$ 5 $m=4, g^{-1}(5)=\frac{1}{2}$ 6 $f^{-1}(x)=\frac{3 x}{x-2}, x \neq 2 \quad g^{-1}(x)=\frac{x+3}{2} ; 6$ 7 $f^{-1}(x)=\frac{3 x-2}{4 x}, \mathrm{x} \neq 0\{x \mid x \in R \backslash\{0\}\} ;-\frac{1}{4}$ 8 $f^{-1}(x)=\frac{3 x-5}{x}, x \neq 0$ $\{x \mid x \in R \backslash\{0\}\} ;-7$ 9 $f^{-1}(x)=\frac{x-5}{2} ; g ^{-1}(x)=\frac{3 x}{x-2}, x \neq 2 ;-3 \quad$ 10 $(\mathrm{g} \circ \mathrm{f})^{-1}(x)=\frac{x-1}{3} ; 1$ 11 $f^{-1}(x)=\frac{x-1}{9} ;\left(f \circ f^{-1}\right)(2)=2$ 12 $(g \circ f)(3)=\frac{22}{31} ;\left(g^{-1} \circ f^{-1}\right)(1)=\frac{2}{9}$ 13 $(g \circ f)(-1)=-3 ;\left(f^{-1} \circ g^{-1}\right)(-5)=5 \quad$ 14 $-9$ 15 $-\frac{3}{4}$ 16 $-9 ; \frac{9}{2}$

### Group (8)

 $\quad$ 1 ( (2017/Myanmar/q07b) ) A binary operation $\odot$ on $R$ is defined by $x \odot y=y^{x}+2 x^{y} y^{x}-x^{y}$. Evaluate $(2 \odot 1) \odot 1$. (5 marks) 2 ( 2011 ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ be defined by $a \odot b=\frac{a^{2}+b^{2}}{2}-a b$ for $a, b \in R$. Find the values of $3 \odot 1$ and $(3 \odot 1) \odot 4$. Find the values of $x$ such that $x \odot 2=x+2$. (5 marks) 3 ( 2012 ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ be defined by$$x \odot y=\frac{4 x^{2}+y^{2}}{2}-2 x y \quad \text { for } x, y \in R$$ Find the values of $3 \odot 2$ and $(3 \odot 2) \odot 16$. If $a$ and $b$ are two real numbers such that $a \odot b=8$, find the relation between $a$ and $b$. (5 marks) 4 ( 2013 ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ be defined by $a \odot b=2 a b-a+4 b$ for $a, b \in R .$ Find the values of $3 \odot(2 \odot 4)$ and $(3 \odot 2) \odot 4 .$ If $x \odot y=2$ and $x \neq-2$, find the numerical value of $y \odot y$. (5 marks) 5 ( 2014 ) The operation $\odot$ is defined by $x \odot y=x^{2}-4 x y-5 y^{2}.$ Calculate $5 \odot 4$. Find the possible values of $x$ such that $x \odot 2=28$. $\qquad\mbox{ (5 marks)}$ 6 ( 2014 ) Given that $a \odot b=a^{2}+\frac{6 a}{b}+4$, find the value of $(3 \odot 9) \odot 1$. Solve the equation $3 \odot \mathrm{y}=22$. $\qquad\mbox{ (5 marks)}$ 7 ( (2016/Myanmar/q07b) ) A binary operation $\odot$ on $R$ is defined by $x \odot y=x^{2}-2 x y+2 y^{2}$. Find $(3 \odot 2) \odot 4$. If $(3 \odot k)-(k \odot 1)=k+1$, find the values of $k$. (5 marks) 8 ( (2017/FC/q07b) ) Let $\mathrm{R}$ be the set of real numbers and a binary operation $\odot$ on $\mathrm{R}$ be defined by $a \odot b=2 a b-a+4 b$ for $a, b \in R$. Find the values of $3 \odot(2 \odot 4)$ and $(3 \odot 2) \odot 4$. If $x \odot y=2$ and $x \neq-2$, find the $\begin{array}{ll}\text { numerical value of y } \mathrm{y} & \text { y. } & (5 \mathrm{marks})\end{array}$ 9 ( 2011 ) A binary operation $\odot$ on the set of integers is defined by $a \odot b=$ the remainder when $(a+2 b)$ is divided by 4. Find $(1 \odot 3) \odot 2$ and $1 \odot(3 \odot 2)$. Is $\odot$ commutative? Why? (5 marks) 10 ( 2011 ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ be defined by $x \odot y=x y-x+y$ for $x, y \in R$. Find the values of $(2 \odot 1) \odot 3$ and $2 \odot(1 \odot 3)$. Is this binary operation associative? Prove your answer. (5 marks) 11 ( 2011 ) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ be defined by $a \odot b=a b+a+b$ for $a, b \in R$. Find the values of $2 \odot(3 \odot 4)$ and (2 \odot 3) $\odot 4$. Is this binary operation associative? Prove your answer. (5 marks) 12 ( (2019/FC/q07a) ) A binary operation $\odot$ on the.set $\mathrm{R}$ of real numbers is defined by $\mathrm{x} \odot \mathrm{y}=\mathrm{x}^{2}+\mathrm{y}^{2}$. Evaluate $[(1 \odot 3) \odot 2]+[1 \odot(3 \odot 2)]$. Show that $x \odot(y \odot x)=(x \odot y) \odot x.$ (5 marks) 13 ( 2012 ) Given $(3 a-b) \odot(a+3 b)=a^{2}-3 a b+4 b^{2}$, evaluate $4 \odot 8$. (5 marks) 14 ( 2010 ) The binary operation $\odot$ on $R$ is defined by $a \odot b=(2 a+3 b) b$ where $a, b \in R$.Calculate $6 \odot(3 \odot 4)$.Find the values of $y$ if $2 \odot y=95$.$\text{ (5 marks)}$ 15 ( 2010 ) A binary operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-2 a b+b^{2}$.Show that $\odot$ is commutative.If $(3 \odot k)-(2 k \odot 1)=k-28$, find the values of $k$.$\text{ (5 marks)}$ 16 ( 2011 ) A binary operation $\odot$ is defined on $R$ by $a \odot b=a(2 a+3 b)$, for all real numbers $a$ and $b$. Find $(1 \odot 1) \odot 2$ and $1 \odot(1 \odot 2)$. Find the values of $b$ such that $b \odot 3=26$. (5 marks) 17 ( 2011 ) Let $J^{+}$be the set of all positive integers. A binary operation $\odot$ on the set $J^{+}$is defined by $a \odot b=a^{2}+a b+b^{2}$. Prove that the binary operation is commutative. Find the value of $x$ such that $2 \odot x=12$. (5 marks) 18 ( 2013 ) The binary operation $\odot$ on $R$ is defined by $x \odot y=x^{2}+3 x y-2 y^{2}$. Find $2 \odot 1$. If $x \odot 2=-13$, find the values of $x$. (5 marks) 19 ( 2013 ) Giving that $a \odot b=a^{2}+\frac{6 a}{b}+4, b \neq 0$. Find the value of $(4 \odot 8) \odot 1$ and solve the equation $x \odot 3=12$. (5 marks) 20 ( 2013 ) If $a \odot b=a^{2}-3 a b+2 b^{2}$, find $(-2 \odot 1) \odot 4$. Find $p$ if $(p \odot 3)-(5 \odot p)=3 p-17$. (5 marks) 21 ( 2014 ) The operation $\odot$ on the set $N$ of natural numbers is defined by $x \odot y=x^{y}$. Find the value of a such that $2 \odot a=(2 \odot$ 3) $\odot 4$. Find also $b$ such that $2\odot(3\odot b)=512.$ (5 marks) 22 ( 2010 ) The operation $\odot$ is defined by $x \odot y=x^{2}+x y-3 y^{2}, x, y \in R$.If $4 \odot x=17$, find the possible values of $x$.Find also $(2 \odot 1) \odot 3$.$\text{ (5 marks)}$ 23 ( 2010 ) The operation $\odot$ is defined by $x \odot y=x^{2}+3 x y-y^{2}$ for $x, y \in R$.Find the possible values of $x$ such that $x \odot 2=3 .$ Find also $(5 \odot 4) \odot 2$.$\text{ (5 marks)}$ 24 ( 2013 ) The operation $\odot$ is defined by $x \odot y=x^{2}+x y-3 y^{2}, x, y \in R$. If $4 \odot x=17$. find the possible values of $x$. Find also $(2 \odot 1) \odot 3$. (5 marks) 25 ( 2013 ) A binary operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-2 a b+2 b^{2}$ Find $(3 \odot 2) \odot 4 .$ If $(3 \odot k)-(k \odot 1)=k+1$, find the value of $k$. (5 marks)

 $\quad$ 1 $4,(x \neq 0, y \neq 0)$ 2 $2 ; 2 ; x=0$ (or) 6 3 $3 \odot 2=8 ;(3 \odot 2) \odot 16=0,2 a-b=\pm 4$ 4 $3 \odot(2 \odot 4)=297 ;(3 \odot 2) \odot 4=135 ; y \odot y=2$ 5 $-135, x=-4$ (or) $12$ 6 $319, y=2 \quad$ 7 $k=2$ or 3 8 279,135,2 9 $3;3;$ No 10 5;13;No $(2\odot 1)\odot 3\not= 2\odot (1\odot 3)$ 11 59; 59; Yes $(a\odot b)\odot c=a\odot (b\odot c)$ 12 274 13 $4 \odot 8=8$ 14 $16416 ;-\frac{19}{3}, 5$ 15 $-4,3$ 16 $80 ; 26 ; 2$ (or) $-\frac{13}{2}$ 17 $x=2$ 18 $2 \odot 1=8 ; x=-5($ or $)-1$ 19 $(4 \odot 8) \odot 1=671 ; x=-4$ (or) 2 20 $(-2 \odot 1) \odot 4=32 ; p=5$ (or) $-2 \quad$ 21 $a=12, b=2.$ 22 $\frac{1}{3}, 1 ;-9$ 23 $-7,1 ; 5171$ 24 $x=\frac{1}{3}$ (or) $1 ;(2 \odot 1) \odot 3=-9 \quad$ 25 $(3 \odot 2) \odot 4=17 ; k=3$ (or) 2