Myanmar Matriculation (Function 3, 7)

Group (3)

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1	( 2014 ) Let $N$ be the set of natural numbers. A function $f$ from $N$ to $N$ is given by, $f(x)=$ the sum of all factors of $x$. If $f(16)=8 p-9$, then find $f\left(p^{2}\right).$ $\qquad\mbox{ (3 marks)}$
2	( 2012 ) A function $f$ is defined by $f: x \mapsto \frac{x+4}{2 x-1}, x \neq \frac{1}{2}$. Find the value of $p$ if $f\left(\frac{1}{p}\right)=p$. (3 marks)
3	( 2013 ) A function $f$ from $A$ to $A$, where $A$ is the set of positive integers, is given by $f(x)=$ the sum of all positive divisors of $x$. Find the value of $k$, if $f(15)=3 k+6$. (3 marks)
4	( 2011 ) Let the function $f: R \rightarrow R$ be given by $f(x)=c x+d$, where $c$ and $d$ are fixed real numbers. If $f(0)=-3$ and $f(2)=1$, find $c$ and $d$, and then find $f(9)$. (3 marks)
5	( 2012 ) Let $f: R \rightarrow R$ be given by $f(x) =\frac{x+a}{x-2}, x \neq 2, f(8)=3$. Find the value of $a$ and $f^{-1}(7)$. (3 marks)
6	( 2012 ) A function $f$ is defined by $f(x)=3 x-5$. Find the formula of $f^{-1}$. Find also the value of $k$, such that $f(k)=f^{-1}(k)$. (3 marks)
7	( 2014 ) A function $f$ is defined by $f: x \mapsto \frac{3-x}{2 x}, x \neq 0$. Find the value of $x$ for which $f(x)=f^{-1}(x).$ $\qquad\mbox{ (3 marks)}$
8	( 2014 ) The function $f$ is given by $f(x)=\frac{4 x-9}{x-2}, x \neq 2$. Find the value of $x$ for which $4 f^{-1}(x)=x.$ $\qquad\mbox{ (3 marks)}$
9	( (2016/Myanmar/q02) ) The function $f$ is defined, for $x \in R$, by $f(x)=2 x-3$. Find the value of $x$ for which $f(x)=f^{-1}(x)$. (3 marks)
10	( 2010 ) Given that $f(x)=\frac{x+a}{x-3}, x \neq 3$, and $f(8)=3$, find the value of $a$ and $f^{-1}(11)$.$\text{ (3 marks)}$
11	( 2011 ) Let $f: R \rightarrow R$ be given by $f(x)=\frac{4 x+5}{a x-1}, x \neq \frac{1}{a}, f^{-1}(3)=1$, find $a$. (3 marks)
12	( 2012 ) A function $f$ is defined by $f(x)=\frac{x}{a}+a$. If $f^{-1}(3)=2$, find the values of $a$. (3 marks)
13	( 2012 ) A function $f$ is such that $f(x)=\frac{2}{k x+3}$ for all $x \neq-\frac{3}{k}$ where $k \neq 0$. If $f(-1)=2$, find the value of $k$ and the formula of $f^{-1}$. (3 marks)
14	( 2010 ) A function $f: R \rightarrow R$ is defined by $f(x)=\frac{a x-9}{x-1}, x \neq 1$.If $f^{-1}(-1)=6$, find the value of $a$ and evaluate the image of 3 under $f$.$\text{ (3 marks)}$
15	( 2010 ) A function $f$ is defined by $f(x)=\frac{5 x+3}{x-4}$ where $x \neq 4$.Find the formula of $f^{-1}$.$\text{ (3 marks)}$
16	( 2014 ) Let $f(x)=\frac{3 x}{x-4}, x \neq 4$. Find the formula of $f^{-1}$. $\qquad\mbox{ (3 marks)}$
17	( 2014 ) Find the formula for $f^{-1}$, the inverse function of $f$ defined by $f(x)=\frac{2}{3-4 x}$. State the suitable domain of $f.$ $\qquad\mbox{ (3 marks)}$
18	( 2014 ) Let the mapping $\odot$ be defined by $(x, y) \rightarrow x \odot y=x+2 y$, where $x$ and $y$ are in $A=\{0,1,2\}.$ Is this mapping a binary operation? $\qquad\mbox{ (3 marks)}$

Answer Group (3)

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1	$31 \quad$
2	$p=-1$
3	$k=6$
4	$c=2, d=-3, f(9)=15$
5	$a=10 ; f^{-1}(7)=4$
6	$f^{-1}(x)=\frac{x+5}{3};k=\frac{5}{2}$
7	$x=-\frac{3}{2}$ (or) $x=1 \quad$
8	$x=6$
9	$x=3$
10	7;4
11	$a=4$
12	$a=1$ (or) $a=2$
13	$k=2 ; f^{-1}(x)=\frac{2-3 x}{2 x}, x \neq 0$
14	$\frac{2}{3} ;-\frac{7}{2}$
15	$\frac{4 x+3}{x-5}, x \neq 5$
16	$f^{-1}(x)=\frac{4 x}{x-3}, x \neq 3$
17	$f^{-1}(x)=\frac{3 x-2}{4 x}, x \neq 0,\left\{x \mid x \in R, x \neq \frac{3}{4}\right\} \quad$
18	The closure property is not satified, $\odot$ is not a binary operation.

Group (7)

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1	( 2012 ) Functions $f$ and $g$ are defined by $f(x)=3 x+a, g(x)=-3 x + b$. Given that $(f \circ f)(4)=4$ and $g(3)=g^{-1}(3)$, find the value of $a$ and of $b$. (5 marks)
2	( ) Functions $f$ and $g$ are defined by $f(x)=\frac{x}{2-x}, x \neq 2$ and $g(x)=a x+b$. Given that $g^{-1}(7)=3$ and $(g \circ f)(5)=-7$, calculate the value of $a$ and of $b$. (5 marks)
3	( 2014 ) Functions $f$ and $g$ are defined by $f: x \mapsto \frac{x}{x+2}, x \neq-2$ and $g: x \mapsto p x+q$, where $p$ and $q$ are constants. Given that $g(2)=12$ and $(g \circ f)(-3)=19$, find the values of $p$ and $q$. $\qquad\mbox{ (5 marks)}$
4	( 2011 ) Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=a x+b$, where $a$ and $b$ are constants, $g(x)=x+7,(g \circ f)(1)=5$ and $(f \circ g)(1)=19 .$ Find the values of $a$ and $b$ and hence find the formula for $g \circ f$. (5 marks)
5	( 2012 ) Let $f: x \mapsto a+b x, f(2 b)=b,(f \circ f)(b)=ab.$ If $f$ is not a constant function, find formula for $f$. (5 marks)
6	( 2012 ) The functions $f:x\mapsto ax^3+bx+30.$ Then the values $x=2$ and $x=3$ which are unchanged by the mapping. Find the value of $a$ and $b$. (5 marks)
7	( 2014 ) $f: x \mapsto \frac{12}{a x+b}, f(0)=-3, f(2)=-6$, given. Find $a$ and $b$. Find $x$ for which $f(x)=x$. $\qquad\mbox{ (5 marks)}$
8	( 2014 ) A function $h$ is defined by $h: x \rightarrow \frac{x+3}{x-3}, x \neq 3$. Show that $h(3+p)+h(3-p)=2$ where $p$ is positive and find the positive number $q$ such that $h(q)=q-1$. $\qquad\mbox{ (5 marks)}$
9	( 2014 ) Given that $f(x)=\frac{a}{x}+1, x \neq 0$. Find the formula for $f^{-1}$, state the suitable domain of $f^{-1}$. If $f^{-1}(2)=1$, find $a$. $\qquad\mbox{ (5 marks)}$
10	( 2013 ) Given that $f: x \mapsto \frac{x}{p}+q, f(8)=1, f^{-1}(-2)=2$, show that $\frac{p}{2}+q^{2}=10$. (5 marks)

Answer Group (7)

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1	$a=-8, b=12$
2	$a=3, b=-2 \quad$
3	$p=7,q=-2$
4	$a=3, b=-5,(g \circ f)(x)=3 x+2$
5	$a=-1, b=1, f(x)=x-1$
6	$a=1, b=-18$
7	$a=1, b=-4, x=6$ (or) $-2$.
8	$q=5$
9	$f^{-1}(x)=\frac{a}{x-1},\{x \mid x \neq 1, x \in R\}, a=1$
10	Show