Group (2)
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| 1 | (2010)A function $f$ is defined by $f(x)=3 x+5$ and $g(x)=3(x-5)$.Find the value of $a$ such that $(g \circ f)^{-1}(a)=10$.$\text{ (3 marks)}$
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| 2 | (2011)Functions $f$ and $g$ are defined by $f: x \mapsto \frac{x}{x-3}, x \neq 3, g: x \mapsto 3 x+5 .$ Find the value of $x$ for which $(f \circ g)^{-1}(x)=0$. (3 marks)
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| 3 | (2013) Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=\frac{2-4 x}{x+1}, x \neq-1$ and $g(x)=2 x-1$. If $\left(g \circ f^{-1}\right)(x)=3$, find the value of $x$. (3 marks)
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| 4 | (2011)Function $f$ is defined by $f(x)=\frac{3 x-2}{3-2 x}, x \neq \frac{3}{2}$. Find the formula for the inverse function and calculate $\left(f \circ f^{-1}\right)(2)$. (3 marks)
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| 5 | (2013) A function $f$ is defined by $f(3 x-2)=5+6 x$.Find the value of $f^{-1}(29)$. (3 marks)
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| 6 | (2014)If $f: R \rightarrow R$ and $g \circ f: R \rightarrow R$ are defined by $f(x)=x^{2}+3$ and $(g \circ f)(x)=2 x^{2}+3$ respectively, find $g^{-1}(3)$. $\qquad\mbox{ (3 marks)}$
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| 7 | (2013) Let $f: R \rightarrow R$ be defined by $f(x)=3 x-2$. Find the formula of $g$ such that $(g \circ f)^{-1}(x)=x+3$. (3 marks)
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| 8 | (2013) Let $f: R \rightarrow R$ be given by $f(x)=2 x-6$ and a function $g$ by $g(x)=\frac{1}{2}(x+6)$. Show that $(g \circ f)^{-1}(x)=x$. (3 marks)
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| 9 | (2018/Myanmar/q02)If the function $f: R \rightarrow R$ is a one-to-one correspondence, then verify that $\left(f \circ f^{-1}\right)(y)=y$ and $\left(f^{-1} \circ f\right)(x)=x$. (3 marks)
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| 10 | (2014)$f: x \mapsto 3 x+5, g: x \mapsto \frac{1}{3}(x-5)$, given. Show that $(g \circ f)^{-1}(x)=x.$ $\qquad\mbox{ (3 marks)}$
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| 11 | (2012)Functions $f$ and $g$ are defined by $f(x)=2x+5$ and $g(x)=\frac 13(x-4).$ Find the formulae of $g^{-1}$ and $g^{-1} \circ f.$ (3 marks)
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| 12 | (2012)A function $f: x \mapsto \frac{b}{x-a}, x \neq a$ and $a>0$ is such that $(f \circ f)(x)=x$. Show that $x^{2}- a x -b=0$. (3 marks)
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| 13 | (2010)Functions $f$ and $g$ are given by $f(x)=x^{2}+2$ and $g(x)=3 x+1$.Find the formulae of $f \circ g$ and $g \circ f$ in simplified forms.$\text{ (3 marks)}$
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| 14 | (2010)Functions $f$ and $g$ are given by $f(x)=2-x$ and $g(x)=5-x^{2}$, then find the formulae of $g \circ f$ and $f\circ g$.$\text{ (3 marks)}$
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| 15 | (2011)Functions $f$ and $g$ are given by $f(x)=2 x^{2}+3$ and $g(x)=2 x+1 .$ Find the formulae of $g \circ f$ and $f \circ f$ in simplified forms. (3 marks)
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| 16 | (2011)A function $f$ is defined by $f(x)=\frac{4 x+2}{x-5}$ where $x \neq 5$. Find the formula of $f \circ f$ in simplified form. (3 marks)
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| 17 | (2012)Find the formulae for the functions $f \circ g$ and $g \circ f$ where $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=x+2$ and $g(x)=\frac{x}{2}$. (3 marks)
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| 18 | (2012)A function $f$ is defined by $f: x \mapsto \frac{8}{x+4}, x \neq-4$. Express $(f \circ f)(x)$ in the form $\frac{a x+b}{c-x}$, stating the values of $a, b$ and $c$. (3 marks)
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| 19 | (2013) If $f$ and $g$ are functions such that $f(x)=2 x-1$ and $(g \circ f)(x)=4 x^{2}-2 x-3$, find the formula of $g$ in simplified form. (3 marks)
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| 20 | (2013) Functions $f$ and $g$ are defined by $f: x \mapsto \frac{3 x-1}{x-2}, x \neq 2$ and $g: x \mapsto \frac{2 x-1}{x-3}$, $x \neq 3$. Find the formula for $f \circ g$. (3 marks)
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| 21 | (2014)The function $f$ is defined by $f(x)=7^{x}$. Prove that $f(x+2)-10 f(x+1)+21 f(x)=0$ $\qquad\mbox{ (3 marks)}$
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| 22 | (2014)If $f$ and $g$ are functions such that $g(x)=2 x+1$ and $(g \circ f)(x)=2 x^{2}+4 x-3$, find the formula of $f \circ g$ in simplified form. $\qquad\mbox{ (3 marks)}$
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| 23 | (2015/Myanmar/q02)If $f: R \rightarrow R$ is defined by $f(x)=\cdot x^{2}+3$, find the function $g$ such that $(g \circ f)(x)=2 x^{2}+3$ (3 marks)
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| 24 | (2017/FC/q02)If $\mathrm{f}$ and $\mathrm{g}$ are functions such that $\mathrm{f}(\mathrm{x})=2 \mathrm{x}-1$ and $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})=4 \mathrm{x}^{2}-2 \mathrm{x}-3$, find the formula of g in simplified form. (3 marks)
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| 25 | (2018/FC/q02)If $f(x)=p x^{2}+1$ where $p$ is a constant and $f(3)=28$, find the value of $p$. Find also the formula of $f \circ f$ in simplified form. (3 marks)
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| 26 | (2012)The functions $f$ and $g$ are defined by $f(x)=3 x+1$ and $g(x)=\frac{2 x+3}{x+1}, x \neq-1$, find the composite function $f \circ g$ and hence find $(f \circ g)(2)$. (3 marks) |
Answer Group (2)
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| 1 | 90
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| 2 | $x=\frac{5}{2}$
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| 3 | $x=-2 \quad$
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| 4 | $f^{-1}(x)=\frac{3 x+2}{2 x+3}, x \neq-\frac{3}{2},\left(f \circ f^{-1}\right)(2)=2$
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| 5 | $f^{-1}(29)=10$
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| 6 | 3
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| 7 | $g(x)=\frac{x-7}{3} \quad$
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| 8 | Show
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| 9 | Verify
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| 10 | Show
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| 11 | $g^{-1}(x)=3 x+4 ;\left(g^{-1} \circ f\right)(x)=6 x+19$
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| 12 | Show
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| 13 | $9 x^{2}+6 x+3 ; 3 x^{2}+7$
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| 14 | $1+4 x-x^{2} ; x^{2}-3$
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| 15 | $(g \circ f)(x)=4 x^{2}+7,(f \circ g)(x)=8 x^{4}+24 x^{2}+21$
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| 16 | $(f \circ f)(x)=\frac{18 x-2}{27-x}$
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| 17 | $(f \circ g)(x)=\frac{x+4}{2};(g \circ f)(x)=\frac{x+2}{2}$
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| 18 | $(f \circ f)(x)=\frac{-2 x-8}{-6-x} ;$ $a=-2, b=-8, c=-6$
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| 19 | $g(x)=x^{2}+x-3$
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| 20 | $(f \circ g)(x)=x \quad$
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| 21 | Prove
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| 22 | $(f \circ g)(x)=4 x^{2}+8 x+1$
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| 23 | $g(x)=2x-3$
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| 24 | $g(x)=x^2+x-3$
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| 25 | $p=3,27 x^{4}+18 x^{2}+4 $
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| 26 | $(f \circ g)(x)=\frac{7 x+10}{x+1}, x \neq-1 ;(f \circ g)(2)=8$ |
Group (6)
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| 1 | ( 2011 ) Functions $f$ and $g$ are defined by $f(x)=4 x-3$ and $g(x)=2 x+1$. Find $(f \circ g)(x)$ and $f^{-1}(x)$ in simplified forms. Show also that $(f \circ g)^{-1}(x)=g^{-1}\left(f^{-1}(x)\right)$. (5 marks)
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| 2 | ( 2011 ) A function $f$ is defined by $f(x)=4 x-3 .$ Find $(f \circ f)(x)$ and $f^{-1}(x)$ in simplified forms. Show also that $(f \circ f)^{-1}(x)=f^{-1}\left(f^{-1}(x)\right)$. (5 marks)
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| 3 | ( 2011 ) The functions $f$ and $g$ are defined by $f(x)=3 x-5$ and $g(x)=4 x-5$. Verify that $\left(g^{-1} \circ f^{-1}\right)(x)=(f \circ g)^{-1}(x)$. (5 marks)
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| 4 | ( 2014 ) Let $f: R \rightarrow R$ be defined by $f(x)=2 x$ and $g: R \rightarrow R$ be defined by $g(x)=x-1$. Show that $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$. $\qquad\mbox{ (5 marks)}$
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| 5 | ( 2014 ) The functions $f$ and $g$ are defined by $f(x)=3 x+10$ and $g(x)=4 x-5$. Find $(f \circ g)(x)$ and verify that $\left(g^{-1} \circ f^{-1}\right)(x)=(f \circ g)^{-1}(x)$. (5 marks)}
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| 6 | ( 2014 ) The functions $f$ and $g$ are defined by $f(x)=2 x-3$ and $g(x)=3 x+2$. Find the inverse functions $f^{-1}$ and $g^{-1}$. Show that $(f \circ g)^{-1}=\left(g^{-1} \circ f^{-1}\right)(x)$. (5 marks)
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| 7 | ( (2019/Myanmar/q06a) ) The functions $f$ and $g$ are defined by $f(x)=2 x-1$ and $g(x)=4 x+3$. Find $(g \circ f)(x)$ and $g^{-1}(x)$ in simplified form. Show also that $(g \circ f)^{-1}(x)=\left(f^{-1} \circ g^{-1}\right)(x)$. $(5 \mathrm{marks})$
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| 8 | ( 2010 ) Let $f$ and $g$ be functions such that $f(x)=2 x+1$ and $(g \circ f)(x)=4 x^{2}-1$.Find the formulae of $g$ and $f^{-1} \circ g$.$\text{ (5 marks)}$
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| 9 | ( 2011 ) Functions $f$ and $g$ are defined on the set of real numbers by $f(x)=\frac{3}{x-2}, x \neq k$, and $g(x)=4 x+5$. State the value of $k$. Find the formulae for $g \circ f$ and $f^{-1}$. (5 marks)
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| 10 | ( 2012 ) The functions $f$ and $g$ are defined for real $x$ as follows: $$f(x)=2 x-1 \text { and } g(x)=\frac{2 x+3}{x-1}, x \neq 1$$ Find the formulae of $g \circ f$ and $f \circ g^{-1}$ in simplified forms. State also a suitable domain of $f \circ g^{-1}$. (5 marks)
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| 11 | ( 2013 ) A function $f$ is defined by $f(x)=4 x+5$, find the formulae of $f^{-1}$ and $f^{-1} \circ f^{-1}$, giving your answer in simplified form. (5 marks)
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| 12 | ( 2013 ) The functions $f$ and $g$ are defined by $f(x)=4 x-3$ and $g(x)=\frac{2-5 x}{x+1}, x \neq-1$. Find the inverse functions $f^{-1}$ and $g^{-1}$. Find also the formula for $(g \circ f)^{-1}$. (5 marks)
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| 13 | ( (2015/FC/q07a) ) Let $f(x)=2 x-1, g(x)=\frac{2 x+3}{x-1}, x \neq 1$. Find the formula for $(g \circ f)^{-1}$ and state the domain of $(g \circ f)^{-1}.$ (5 marks)
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| 14 | ( 2013 ) 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 $\left(f^{-1} \circ g^{-1}\right)(5)$. (5 marks)
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| 15 | ( 2013 ) Let $f$ and $g$ be two functions defined by $f(x)=x+1$ and $f(g(x))=3 x-1$.Find the formula of $(g \circ f)^{-1}$ and hence find $(g \circ f)^{-1}(4)$. (5 marks)
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| 16 | ( (2017/FC/q07a) ) Let $\mathrm{f}$ and $\mathrm{g}$ be two functions defined by $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+1$ and $\mathrm{f}(\mathrm{g}(\mathrm{x}))=3 \mathrm{x}-1$. Find the formula of $(\mathrm{f} \circ \mathrm{g})^{-1}$ and hence find $(\mathrm{f} \circ \mathrm{g})^{-1}(8).$ (5 marks)
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| 17 | ( (2018/FC/q07a) ) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x+7$ and $g(x)=3 x-1$. Find $\left(f^{-1}\circ g\right)(x)$ and $\left(g^{-1}\circ f\right)(x).$ What are the values of $\left(f^{-1}\circ g\right)(3)$ and $\left(g^{-1}\circ f\right)(2).$(5 marks)
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| 18 | ( 2012 ) Given that $f:x \mapsto \frac{2}{a x+b}, x \neq-\frac{b}{a}$, such that $f(0)=-2$ and $f(2)=2$, find the values of $a$ and $b$. Show that $f(p)+f(-p)=2 f\left(p^{2}\right).$ (5 marks) |
Answer Group (6)
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| 1 | $(f \circ g)(x)=8 x+1, f^{-1}(x)=\frac{x+3}{4}$
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| 2 | $(f \circ f)(x)=16 x-15, f^{-1}(x)=\frac{x+3}{4}$
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| 3 | Verify
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| 4 | Show
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| 5 | $12 x-5$
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| 6 | $f^{-1}(x)=\frac{x+3}{2}, g^{-1}(x)=\frac{x-2}{3}$
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| 7 | $(g\circ f)(x)=8x-1,g^{-1}(x)=\frac{x-3}{4}$
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| 8 | $g(x)=x^{2}-2 x;\left(f^{-1} \circ g\right)(x)=\frac{x^{2}-2 x-1}{2}$
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| 9 | $k=2,(g \circ f)(x)=\frac{5 x+2}{x-2}, x \neq 2, \quad f^{-1}(x)=\frac{2 x+3}{x}, x \neq 0$
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| 10 | $(g \circ f)(x)=\frac{4 x+1}{2 x-2}, x \neq 1\left(f \circ g^{-1}\right)(x)=\frac{x+8}{x-2}, x \neq 2$ $\{x \mid x \neq 2, x \in \mathbb{R}\}$
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| 11 | $f^{-1}(x)=\frac{x-5}{4} ;\left(f^{-1} \circ f^{-1}\right)(x)=\frac{x-25}{16}$
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| 12 | $f^{-1}(x)=\frac{x+3}{4} $; $g^{-1}(x)=\frac{2-x}{x+5}, x \neq-5 ;$ $(g \circ f)^{-1}(x)=\frac{2 x+17}{4 x+20}$ $x \neq-5 \quad$
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| 13 | $(g \circ f)^{-1}=\frac{1+2x}{2x-4},x\not=2$ domain of $(g \circ f)^{-1} =\{x|x\not=2,x\in R\}$
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| 14 | $f^{-1}(x)=\frac{3 x}{x-2}, x \neq 2 ; g^{-1}(x)=\frac{x+3}{2} ;\left(f^{-1} \circ g^{-1}\right)(5)=6$
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| 15 | $(g \circ f)^{-1}(x)=\frac{x-1}{3} ;$ $(g \circ f)^{-1}(4)=1 $
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| 16 | $(f\circ g)^{-1}(x)=\dfrac{x+1}{3}, 3$
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| 17 | $\left(f^{-1} \circ g\right)(x)=3 x-8,1,\left(g^{-1} \circ f\right)(x)=\frac{x+8}{3}, \frac{10}{3}$
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| 18 | $a=1, b=-1$ |
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