Group (2015-2019)
$\def\frac{\dfrac}$1. (2015/Myanmar /q2 )
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)
2. (2015/Myanmar /q7a )
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 $\left(\dot{g}^{-1} \circ f^{-1}\right)(2) . \quad(5$ marks $)$
3. (2015/FC /q2 )
A function $f$ is defined by $f(2 x+1)=x^{2}-3$. Find $a \in R$ such that $f(5)=a^{2}-8 . \quad \quad \quad \quad \quad \quad: \quad(3$ marks $)$
4. (2015/FC /q7a )
-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 $(g \circ f)^{-1} . \quad(5$ marks)
5. (2016/Myanmar /q2 )
The function $f$ is defined, for $x \in R$, by $f(x)=2 x-3$. Find the value of $x$ for which $f(x)=f^{-1}(x)$
6. (2016/Myanmar /q7a )
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 $b$.
7. (2016/FC /q2 )
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 $x^{2}-a x-b=0$
8. (2016/FC /q7a )
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 $\left(f \circ g^{-1}\right)(x)=x-4$
9. (2017/Myanmar /q2 )
Let $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=k x-1$, where $k$ is a constant and $g(x)=x+12 .$ Find the value of $k$ for which $$(g \circ f)(2)=(f \circ g)(2)$$
Q(2) Solution
10. (2017/Myanmar /q7a )
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 $(f \circ g)^{-1}(x)=\frac{5}{3}$
Q 7(a) Solution
11. (2017/FC /q2 )
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.
12. (2017/FC /q7a )
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 $(\mathrm{f} \circ \mathrm{g})^{-1}(8) . \quad(5 \mathrm{marks})$
13. (2018/Myanmar /q2 )
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 $\left(f^{-1} \circ f\right)(x)=x$
Click for Solution
14. (2018/Myanmar /q7a )
The functions $f$ and $g$ are defined for real $x$ by $f(x)=2 x-1$ and $g(x)=2 x+3 .$ Evaluate $\left(g^{-1} \circ f^{-1}\right)(2)$
Click for Solution
15. (2018/FC /q2 )
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.
16. (2018/FC /q7a )
7.(a) 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 $\left(g^{-1}\circ f\right)(2).$
Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}+1$, and $\mathrm{g}(\mathrm{x})=2 \mathrm{x}^{2}-\mathrm{x}+3$. Find the values of $\mathrm{x}$ which satisfy the equation $(\mathrm{f} \circ \mathrm{g})(\mathrm{x})=4 \mathrm{x}+1$. (3 marks)
18. (2019/Myanmar /q6a )
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 $(g \circ f)^{-1}(x)=\left(f^{-1} \circ g^{-1}\right)(x)$. $(5 \mathrm{marks})$ Click for Solution
19. (2019/Myanmar /q7a )
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 $2 \odot \mathrm{k}=-3(5 \mathrm{marks}$ Click for Solution
20. (2019/FC /q1a )
Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-1$, and $\mathrm{g}(\mathrm{x})=3 \mathrm{x}+1$. Find the values of $\mathrm{x}$ which satisfy the equation $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})=7 \mathrm{x}-4, \quad$ (3 marks) Click for Solution 1(a)
21. (2019/FC /q6a )
Functions $\mathrm{f}: \mathrm{R} \mapsto \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \mapsto \mathrm{R}$ are defined by $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-1$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}+2$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)-4 . \quad(5$ marks $)$ Click for Solution 6(a)
22. (2019/FC /q7a )
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 $)$ Click for Solution 7(a)
23. (2015/Myanmar /q7b )
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)
24. (2015/FC /q7b )
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)
25. (2016/Myanmar /q7b )
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 $k$
26. (2016/FC /q7b )
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 $\odot$ :s commutative. Solve the equation $(2 \odot 3) \odot x=(x \odot x) \odot 5$.
27. (2017/Myanmar /q7b )
A binary operation $\odot$ on $R$ is defined by $x \odot y=y^{x}+2 x^{y} y^{x}-x^{y}$. Evaluate $(2 \odot 1) \odot 1$
28. (2017/FC /q7b )
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 $\begin{array}{ll}\text { numerical value of y } \mathrm{y} & \text { y. } & (5 \mathrm{marks})\end{array}$
29. (2018/Myanmar /q7b )
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 $a \odot 2=a+2$
30. (2018/FC /q7b )
A binary operation $\odot$ on $R$ is defined by $x \odot y=(x+2 y)^{2}-3 y^{2}$. Show that the binary operation is commutative. Find the possible values of $k$ such that $(k-3) \odot(k+2)=25$
Answer (2015-2019)
1. $g(x)=2x-3$
2. $-9$
3. $a=\pm 3$
4. $(g \circ f)^{-1}=\frac{1+2x}{2x-4},x\not=2$ domain of $(g \circ f)^{-1} =\{x|x\not=2,x\in R\}$
5. $x=3$
6. $a=3, b=-2 \quad$
7. Show
8. $x=0$ or 9
9. $k=1$
10. $x=\frac{10}{7}$
11. $g(x)=x^2+x-3$
12. $(f\circ g)^{-1}(x)=\dfrac{x+1}{3}, 3$
13. Verify
14. $-\frac{3}{4}$
15. $p=3,27 x^{4}+18 x^{2}+4 $
16. $\left(f^{-1} \circ g\right)(x)=3 x-8,1,\left(g^{-1} \circ f\right)(x)=\frac{x+8}{3}, \frac{10}{3}$
17. $x=\frac 32$ or $x=1$
18. $(g\circ f)(x)=8x-1,g^{-1}(x)=\frac{x-3}{4}$
19. $k=3$ or $k=7$
20. $\dfrac 13$ or 2
21. $x=3$
22. 274
23. No solution
24. $(1\odot 0)\odot 2\not=1\odot (0\odot 2)$
25. $k=2$ or 3
26. $x=1 \pm \sqrt{2}$
27. $4,(x \neq 0, y \neq 0)$
28. 279,135,2
29. $a=0 $ or $6$
30. $k=3$ or $-2$
Group (2014)
1. $f: x \mapsto \frac{12}{a x+b}, f(0)=-3, f(2)=-6$, given. Find $a$ and $b$. Find $x$ for which $f(x)=x$. $\qquad\mbox{ (5 marks)}$
2. 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 $h(q)=q-1$. $\qquad\mbox{ (5 marks)}$
3. 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. The function $f$ is defined by $f(x)=7^{x}$. Prove that $f(x+2)-10 f(x+1)+21 f(x)=0$ $\qquad\mbox{ (3 marks)}$
5. 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 $f\left(p^{2}\right).$ $\qquad\mbox{ (3 marks)}$
6. A function $f: R \rightarrow R$ is defined by $f(3 x+1)=x^{2}+1$. Find $a \in R$ such that $f(10)=a^{2}-6.$ $\qquad\mbox{ (3 marks)}$
7. 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 $g^{-1}(3)$. $\qquad\mbox{ (3 marks)}$
8. 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 $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$. $\qquad\mbox{ (5 marks)}$
9. 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 $\left(g^{-1} \circ f^{-1}\right)(x)=(f \circ g)^{-1}(x)$. $\qquad\mbox{ (5 marks)}$
10. 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. $\qquad\mbox{ (3 marks)}$
11. 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 $f.$ $\qquad\mbox{ (3 marks)}$
12. A function $f$ is defined by $f: x \mapsto \frac{3-x}{2 x}, x \neq 0$. Find the value of $x$ for which $f(x)=f^{-1}(x).$ $\qquad\mbox{ (3 marks)}$
13. The function $f$ is given by $f(x)=\frac{4 x-9}{x-2}, x \neq 2$. Find the value of $x$ for which $4 f^{-1}(x)=x.$ $\qquad\mbox{ (3 marks)}$
14. $f: x \mapsto 3 x+5, g: x \mapsto \frac{1}{3}(x-5)$, given. Show that $(g \circ f)^{-1}(x)=x.$ $\qquad\mbox{ (3 marks)}$
15. 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 $a$. $\qquad\mbox{ (5 marks)}$
16. Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=3 x-1$ and $g(x)=x+7$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)+8$. $\qquad\mbox{ (5 marks)}$
17. A function $f: R \rightarrow R$ is defined by $f(x)=p x+2$. If $f^{-1}(11)=3$, find the value of $p$ and hence show that $(f \circ f)^{-1}(x)=\left(f^{-1} \circ f^{-1}\right)(x)$. $\qquad\mbox{ (5 marks)}$
18. 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. $\qquad\mbox{ (5 marks)}$
19. Let $f(x)=\frac{3 x}{x-4}, x \neq 4$. Find the formula of $f^{-1}$. $\qquad\mbox{ (3 marks)}$
20. 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 $(f \circ g)^{-1}=\left(g^{-1} \circ f^{-1}\right)(x)$. $\qquad\mbox{ (5 marks)}$
21. 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 $\left(g^{-1} \circ f\right)(2) ?$ $\qquad\mbox{ (5 marks)}$
22. 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? $\qquad\mbox{ (3 marks)}$
23. 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 $x \odot 2=28$. $\qquad\mbox{ (5 marks)}$
24. Given that $a \odot b=a^{2}+\frac{6 a}{b}+4$, find the value of $(3 \odot 9) \odot 1$. Solve the equation $3 \odot \mathrm{y}=22$. $\qquad\mbox{ (5 marks)}$
25. 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 $2\odot(3\odot b)=512.$ $\qquad\mbox{ (5 marks)}$
26. 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 $4 \odot(x \odot 2)=185$ $\qquad\mbox{ (5 marks)}$
27. 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? $\qquad\mbox{ (5 marks)}$
28. 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 $(k+1) \odot(k-2)=109$. $\qquad\mbox{ (5 marks)}$
29. 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 $p \odot 3=p+10$. $\qquad\mbox{ (5 marks)}$
30. 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 $a * 2=a+2$. $\qquad\mbox{ (5 marks)}$
31. 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 $p \odot p$. $\qquad\mbox{ (5 marks)}$
32. 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) $\odot_{2}\left(1 \odot_{1} x\right)=3$. $\qquad\mbox{ (5 marks)}$
Answer (2014)
1. $a=1, b=-4, x=6$ (or) $-2$.
2. $q=5$
3. $p=7,q=-2$
4. Prove
5. $31 \quad$
6. $a=\pm 4$
7. 3
8. Show
9. $12 x-5$
10. $(f \circ g)(x)=4 x^{2}+8 x+1$
11. $f^{-1}(x)=\frac{3 x-2}{4 x}, x \neq 0,\left\{x \mid x \in R, x \neq \frac{3}{4}\right\} \quad$
12. $x=-\frac{3}{2}$ (or) $x=1 \quad$
13. $x=6$
14. Show
15. $f^{-1}(x)=\frac{a}{x-1},\{x \mid x \neq 1, x \in R\}, a=1$
16. $x=1$
17. $p=3$
18. $k=4, \frac{16 x+2}{7 x+11}, x \neq-\frac{11}{7}$
19. $f^{-1}(x)=\frac{4 x}{x-3}, x \neq 3$
20. $f^{-1}(x)=\frac{x+3}{2}, g^{-1}(x)=\frac{x-2}{3}$
21. $\left(f^{-1} \circ g\right)(x)=\frac{x+8}{3} /\left(g^{-1} \circ f\right)(x)=3 x-8; \frac{11}{3},-2$
22. The closure property is not satified, $\odot$ is not a binary operation.
23. $-135, x=-4$ (or) $12 $
24. $319, y=2 \quad$
25. $a=12, b=2.$
26. $x=\pm 3$
27. $2 \odot 4=22,4 \odot 2=22, \odot$ is not commulative
28. $k=4$ (or) $-3$
29. $p=-11$
30. $a=0$ (or) 6
31. $\odot$ is not associative, $p=1, p\odot p=1 \quad$
32. $308, x=\pm 3$
Group (2013)
1. 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)
2. Let $f: R \rightarrow R$ is defined by $f(x)=a x-4$. Given that $f(3)=5$, find $a$. Hence solve the equation $(f \circ f)(x)=f(x)$. (3 marks)
3. Let the function $f: R \rightarrow R$ and $g: R \rightarrow R$ be given by $f(x)=2 x+1$ and $g(x)=x^{2}+5$. Find the value of $a \in R$ for which $(f \circ g)(a)=f(a)+22$. (3 marks)
4. A function $f$ is defined, for $x \neq 0$, by $f(x)=\frac{a}{x}+1$, where $a$ is constant. Given that $6(f \circ f)(-1)+a=0$, find the possible values of $a$. (3 marks)
5. 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)
6. 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)
7. 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)
8. Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f: x \mapsto 3 x-1$ and $g: x \mapsto x+7$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)+8$. (5 marks)
9. The functions $f$ and $g$ are defined by $f(x)=3 x-4$ and $g(x)=\begin{gathered}{-\frac{1}{-} x}, x \neq 2 \text {. }\end{gathered}$ Evaluate $(g \circ f)(-1)$ and $\left(f^{-1} \circ g^{-1}\right)(-5)$. (5 marks)
10. 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)
11. 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)
12. 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)
13. A function $f$ is defined by $f(3 x-2)=5+6 x$. Find the value of $f^{-1}(29)$. (3 marks)
14. 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)
15. A function is defined by $f(x)=\frac{1}{3-2 x}$ for all values of $x$ except $x=\frac{3}{2}$. Find the values of $x$ which map on to themselves under the function $f$. Find also an expression for $f^{-1}$ and the value of $(f \circ f)(2)$. (5 marks)
16. A function $f$ is defined by $f(x)=\frac{a x-3}{x-1}$ for all $x \rightarrow h^{f}, f(3)=6$, find the value of a and the formula of $f^{-1}$ in simplified form. Verify 히 so that $\left(f^{-1} \circ f\right)(x)=x$. (5 marks)
17. 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)
18. 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)
19. 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)
20. If $f: x \mapsto 2 x+b$ and $g: x \mapsto 3 a-2 x$, such that $f \circ g=g \circ f$, find the relationship between $a$ and $b$. (3 marks)
21. 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)
22. 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)
23. 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)
24. 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)
25. 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)
26. 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)
27. 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)
28. 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)
29. 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)
30. 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)
31. 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)
32. 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)
33. 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 (2013)
1. $k=6$
2. $a=3, x=2$
3. $a=-2$ (or) 3
4. $a=-2$ (or) 3
5. $x=-2 \quad$
6. $a=-1 \quad$
7. $x=0$ (or) $\frac{5}{2} \quad$
8. $x=1 \quad$
9. $(g \circ f)(-1)=-3 ;\left(f^{-1} \circ g^{-1}\right)(-5)=5 \quad$
10. $g(x)=x^{2}+x-3$
11. $(f \circ g)(x)=x \quad$
12. $g(x)=\frac{x-7}{3} \quad$
13. $f^{-1}(29)=10$
14. $f^{-1}(x)=\frac{x-5}{4} ;\left(f^{-1} \circ f^{-1}\right)(x)=\frac{x-25}{16}$
15. $ x=\frac{1}{2}$ (or) 1 ; $ f^{-1}(x)=\frac{3 x-1}{2 x}, x \neq 0 ;$ $\frac{1}{5} $
16. $a=5$ ; $f^{-1}(x)=\frac{x-3}{x-5}, x \neq 5$
17. $f^{-1}(x)=\frac{x+3}{4}$ ; $g^{-1}(x)=\frac{2-x}{x+5}, x \neq-5 ;(g \circ f)^{-1}(x)=\frac{2 x+17}{4 x+20}$ $x \neq-5 \quad$
18. $f^{-1}(x)=\frac{3 x}{x-2}, x \neq 2 ; g^{-1}(x)=\frac{x+3}{2} ;\left(f^{-1} \circ g^{-1}\right)(5)=6$
19. $(g \circ f)^{-1}(x)=\frac{x-1}{3} ;$ $(g \circ f)^{-1}(4)=1 $
20. $a+b=0$
21. $2 \odot 1=8 ; x=-5($ or $)-1$
22. $x=\frac{1}{3}$ (or) $1 ;(2 \odot 1) \odot 3=-9 \quad$
23. $(4 \odot 8) \odot 1=671 ; x=-4$ (or) 2
24. $(-2 \odot 1) \odot 4=32 ; p=5$ (or) $-2 \quad$
25. $(3 \odot 2) \odot 4=17 ; k=3$ (or) 2
26. $3 \odot(2 \odot 4)=297 ;(3 \odot 2) \odot 4=135 ; y \odot y=2$
27. $3 \odot(2 \odot 1)=4 \quad$
28. $k=-8$ (or) 2
29. $x=6$ (or) $-2 \quad$
30. $ \mathrm{No} ; 3 ; \mathrm{No}$
31. $a=-5, b=16, c=-7 $
32. Show
33. Show
| $\,$ | ||
|---|---|---|
| 1. | 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) | |
| 2. | 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) | |
| 3. | 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) | |
| 4. | 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) | |
| 5. | Given that $f(x)=3 x-4, g(x)=x^{2}-1$. Find the values of $x$ which satisfy the equation $(g \circ f)(x)=9-3 x$. (3 marks) | |
| 6. | Find the forwinulae 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) | |
| 7. | 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) | |
| 8. | 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) | |
| 9. | 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) | |
| 10. | A function $f$ is defined by $f(x)=\frac{x}{a}+a$. If $f^{-1}(3)=2$, find the values of $a$. (3 marks) | |
| 11. | 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) | |
| 12. | 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) | |
| 13. | 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) | |
| 14. | 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) | |
| 15. | 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) | |
| 16. | 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) | |
| 17. | Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=x+7$ and $g(x)=3 x-1 .$ Find the value of $x$ for which $\left(g^{-1} \circ f\right)(x)=\left(f^{-1} \circ g\right)(x)+8$. (5 marks) | |
| 18. | 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) | |
| 19. | 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 what is the value of $b \in R$ for which $\left(f^{-1} \circ g\right)(b)=4$. (5 marks) | |
| 20. | Functions $f$ and $g$ are defined by $f: x \mapsto 2 x+1$ 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) | |
| 21. | 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) | |
| 22. | Let $f(x)=3 x+2$ and $g(x)=\frac{2 x-3}{x-2}, x \neq 2$. Find the formulae of $f \circ g$ and $g^{-1}$ Solve the equation $g^{-1}(x)=x$. (5 marks) | |
| 23. | 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) | |
| 24. | 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) | |
| 25. | 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) | |
| 26. | 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) | |
| 27. | 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) | |
| 28. | 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) | |
| 29. | Given $(3 a-b) \odot(a+3 b)=a^{2}-3 a b+4 b^{2}$, evaluate $4 \odot 8$. (5 marks) | |
| 30. | 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) | |
| 31. | 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) | |
| 32. | 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) | |
| 33. | 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) |
Answer (2012)
1. $p=-1$
2. $a=1, b=-18$
3. $a=1, b=-1$
4. Show
5. $x=\frac{1}{3}$ (or) $x=2$
6. $(f \circ g)(x)=\frac{x+4}{2};(g \circ f)(x)=\frac{x+2}{2}$
7. $(f \circ g)(x)=\frac{7 x+10}{x+1}, x \neq-1 ;(f \circ g)(2)=8$
8. $(f \circ f)(x)=\frac{-2 x-8}{-6-x} ;$ $a=-2, b=-8, c=-6$
9. $a=-1, b=1, f(x)=x-1$
10. $a=1$ (or) $a=2$
11. $a=10 ; f^{-1}(7)=4$
12. $k=2 ; f^{-1}(x)=\frac{2-3 x}{2 x}, x \neq 0$
13. $f^{-1}(x)=\frac{x+5}{3};k=\frac{5}{2}$
14. $g^{-1}(x)=3 x+4 ;\left(g^{-1} \circ f\right)(x)=6 x+19$
15. $f^{-1}(x)=\frac{x-1}{9} ;\left(f \circ f^{-1}\right)(2)=2$
16. $a=-8, b=12$
17. $x=1$
18. $(g \circ f)(3)=\frac{22}{31} ;\left(g^{-1} \circ f^{-1}\right)(1)=\frac{2}{9}$
19. $\left(f^{-1} \circ g\right)(x)=3 x-8 ; b=4$
20. $x=0$ (or) $x=9$
21. $(g \circ f)(x)=\frac{4 x+1}{2 x-2}, x \neq 1\left(f \circ g^{-1}\right)(x)=\frac{x+8}{x-2}, x \neq 2$ $\{x \mid x \neq 2, x \in \mathbb{R}\}$
22. $(f \circ g)(x)=\frac{8 x-13}{x-2}, x \neq 2, g^{-1}(x)=\frac{2 x-3}{x-2}, x \neq 2, x=1$ (or) $x=3$
23. $-1 \quad $
24. $x=\pm 2$
25. No ; $a=4 \quad$
26. Yes $;(2 \odot 3) \odot 4=-1,2 \odot(3 \odot 4)=3,(2 \odot 3) \odot 4 \neq 2 \odot(3 \odot 4)$
27. $p=4$ (or) $p=-2 \quad$
28. $a=4$ (or) $a=-2 \quad$
29. $4 \odot 8=8$
30. $3 \odot 2=8 ;(3 \odot 2) \odot 16=0,2 a-b=\pm 4$
31. $a=-5, b=16, c=-7$,
32.Yes; Yes
33.Yes;Yes;No
Group (2011)
| $\quad\;$ | $\,$ | |
|---|---|---|
| 1. | Let the function $f: R \rightarrow R$ be defined by $f(x)=2^{x}$. What are the images of $-2$ and 2? Find $a \in R$ such that $f(a)=256$. (3 marks) | |
| 2. | 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) | |
| 3. | A function $f$ is defined by $f(x+1)=4 x+5$. Find $a \in R$ such that $f(14)=a+14$. (3 marks) | |
| 4. | A function $f$ is defined by $f(2 x+1)=x^{2}-3 .$ Find $a \in R$ such that $f(5)=a^{2}-8$. (3 marks) | |
| 5. | 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) | |
| 6. | 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) | |
| 7. | 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) | |
| 8. | 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) | |
| 9. | 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) | |
| 10. | Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be $f(x)=p x+5$ and $g(x)=q x-3$, where $p \neq 0$, $q \neq 0$. If $g \circ f: R \rightarrow R$ is the identity function on $R$, find the value of $p$. (3 marks) | |
| 11. | 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) | |
| 12. | Functions $f$ and $g$ are defined on the set of real numbers by $f(x)=\frac{3}{x-2}, x \neq k$, and $g(x)=4 x+5$. | |
| 13. | The functions $f$ and $g$ are defined by $f(x)=-x-2$ and $g(x)=m x+3$. | |
| 14. | 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) | |
| 15. | 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) | |
| 16. | The functions $f$ and $g$ are defined by $f(x)=3 x-5$ and $g(x)=4 x-5$. | |
| 17. | A function $f$ is defined by $f: x \mapsto \frac{a}{x}+1, x \neq 0$, where $a$ is constant. Given that $6(f \circ f)(-1)+f^{-1}(2)=0$, find the possible values of $a$. (5 marks) | |
| 18. | 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) | |
| 19. | 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) | |
| 20. | 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) | |
| 21. | 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) | |
| 22. | 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) | |
| 23. | 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) | |
| 24. | 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) | |
| 25. | 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) | |
| 26. | 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) | |
| 27. | 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) | |
| 28. | 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) |
Answer (2011)
| $\quad$ | $\,$ | |
|---|---|---|
| 1. | $f(-2)=\frac{1}{4}, f(2)=4, a=8$ | |
| 2. | $c=2, d=-3, f(9)=15$ | |
| 3. | $a=43$ | |
| 4. | $a=\pm 3$ | |
| 5. | $(g \circ f)(x)=4 x^{2}+7,(f \circ g)(x)=8 x^{4}+24 x^{2}+21$ | |
| 6. | $(f \circ f)(x)=\frac{18 x-2}{27-x}$ | |
| 7. | $a=4$ | |
| 8. | $x=\frac{5}{2}$ | |
| 9. | $f^{-1}(x)=\frac{3 x+2}{2 x+3}, x \neq-\frac{3}{2},\left(f \circ f^{-1}\right)(2)=2$ | |
| 10. | $p=\frac 53$ | |
| 11. | $a=3, b=-5,(g \circ f)(x)=3 x+2$ | |
| 12. | $k=2,(g \circ f)(x)=\frac{5 x+2}{x-2}, x \neq 2, \quad f^{-1}(x)=\frac{2 x+3}{x}, x \neq 0$ | |
| 13. | $m=4, g^{-1}(5)=\frac{1}{2}$ | |
| 14. | $(f \circ g)(x)=8 x+1, f^{-1}(x)=\frac{x+3}{4}$ | |
| 15. | $(f \circ f)(x)=16 x-15, f^{-1}(x)=\frac{x+3}{4}$ | |
| 16. | Verify | |
| 17. | $a=-2$ (or) 3 | |
| 18. | $80 ; 26 ; 2$ (or) $-\frac{13}{2}$ | |
| 19. | $2 ; 2 ; x=0$ (or) 6 | |
| 20. | Yes ; 10 ; 40 ;No | |
| 21. | Yes; 39; 55; No | |
| 22. | $x=2$ | |
| 23. | $104 ; x=\pm 1$ | |
| 24. | $k=-2 ;$ Yes | |
| 25. | $3;3;$ No | |
| 26. | $23$ | |
| 27. | 5;13;No $(2\odot 1)\odot 3\not= 2\odot (1\odot 3)$ | |
| 28. | 59; 59; Yes $(a\odot b)\odot c=a\odot (b\odot c)$ |
Group (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | A function $f$ is defined by $f(x)=1+2 x$.Find the value of $x$ such that $(f \circ f)(x)=4 f(x)$.$\text{ (3 marks)}$ | |
| 2. | Functions $f$ and $g$ are defined by $f(x)=2 x+p$, where $p$ is a constant, and $g(x)=4 x+6 .$ Find the value of $p$ for which $(f \circ g)(x)=(g \circ f)(x)$.$\text{ (3 marks)}$ | |
| 3. | Functions $g$ and $h$ are defined by $g(x)=a x+10$, where $a$ is constant, and $h(x)=3 x+5$.Find the value of $a$ for which $(h \circ g)(x)=(g \circ h)(x)$.$\text{ (3 marks)}$ | |
| 4. | 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 $(g \circ f)(2)$.$\text{ (5 marks)}$ | |
| 5. | Given that $f(x)=\frac{x+a}{x-3}, x \neq 3$, and $f(8)=3$, find the value of $a$ and $f^{-1}(11)$.$\text{ (3 marks)}$ | |
| 6. | 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 $f$.$\text{ (3 marks)}$ | |
| 7. | A function $f$ is defined by $f(x)=3 x+5$ and $g(x)=3(x-5)$.Find the value of $a$ such that $(g \circ f)^{-1}(a)=10$.$\text{ (3 marks)}$ | |
| 8. | A function $f$ is defined by $f: x \mapsto \frac{2 x}{x-4}, x \neq 4$.Find the non-zero value of $x$ for which $(f \circ f)(x)=f^{-1}(x)$.$\text{ (5 marks)}$ | |
| 9. | Functions $h$ and $g$ are defined by $g: x \mapsto \frac{x+1}{x-2}, x \neq 2, h: x \mapsto \frac{a x+3}{x}$, $x \neq 0$, find the value of $a$ for which $\left(h \circ g^{-1}\right)(4)=g^{-1}(2)$.$\text{ (5 marks)}$ | |
| 10. | A functions $f$ is defined by $f(x)=a x+1 .$ If $f^{-1}(3)=1$, find the value of $a$ and hence show that $(f \circ f)^{-1}(x)=\left(f^{-1} \circ f^{-1}\right)(x)$.$\text{ (5 marks)}$.$\text{ (5 marks)}$ | |
| 11. | 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 fo $g$.$\text{ (3 marks)}$ | |
| 12. | 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.$\text{ (3 marks)}$ | |
| 13. | Let $f: R \rightarrow R$ be defined by $f(x)=4 x+1$.Find the formula for a function $g: R \rightarrow R$ such that $(f \circ g)(x)=21-12 x$.$\text{ (3 marks)}$ | |
| 14. | $f: R \rightarrow R, g: R \rightarrow R$ 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 $f \circ(h \circ g)$.$\text{ (5 marks)}$ | |
| 15. | 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 $\left(f^{-1} \circ g^{-1}\right)(5)$.$\text{ (5 marks)}$ | |
| 16. | A function $f$ is defined by $f(x)=\frac{5 x+3}{x-4}$ where $x \neq 4$.Find the formula of $f^{-1}$.$\text{ (3 marks)}$ | |
| 17. | 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 $\left(f^{-1} \circ f^{-1}\right)(2)$.$\text{ (5 marks)}$ | |
| 18. | 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 $\left(f^{-1} \circ f^{-1}\right)(2)$.$\text{ (5 marks)}$ | |
| 19. | 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 $\left(g^{-1} \circ f^{-1}\right)(7)$.$\text{ (5 marks)}$ | |
| 20. | 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 $f^{-1} \circ g$.$\text{ (5 marks)}$ | |
| 21. | 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 $(g \circ f)^{-1}(4)$.$\text{ (5 marks)}$ | |
| 22. | 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 $2 \odot y=95$.$\text{ (5 marks)}$ | |
| 23. | 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 $(2 \odot 1) \odot 3$.$\text{ (5 marks)}$ | |
| 24. | 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 $(5 \odot 4) \odot 2$.$\text{ (5 marks)}$ | |
| 25. | 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 $p \odot(p+1)=39$.$\text{ (5 marks)}$ | |
| 26. | 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 $x \odot 2=2 \odot 7$.$\text{ (5 marks)}$ | |
| 27. | 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 $(1 \odot b) \odot \mathrm{b}=485$.$\text{ (5 marks)}$ | |
| 28. | 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 $(k \odot 5) \odot k$.$\text{ (5 marks)}$ | |
| 29. | 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 $(a \odot 3) \odot a=263$.$\text{ (5 marks)}$ | |
| 30. | A binary operation $\odot$ on $R$ is defined by $a \odot b=a^{2}-2 a b+b^{2}$.Show that $\odot$ is commuatative.If $(3 \odot k)-(2 k \odot 1)=k-28$, find the values of $k$.$\text{ (5 marks)}$ | |
| 31. | 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 $6 \odot(3 \odot 4)$.$\text{ (5 marks)}$ | |
| 32. | 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? $\text{ (5 marks)}$ |
Answer (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | $-\frac{1}{4}$ | |
| 2. | 2 | |
| 3. | 5 | |
| 4. | $-9 ; \frac{9}{2}$ | |
| 5. | 7;4 | |
| 6. | $\frac{2}{3} ;-\frac{7}{2}$ | |
| 7. | 90 | |
| 8. | 6 | |
| 9. | 4 | |
| 10. | 2 | |
| 11. | $1+4 x-x^{2} ; x^{2}-3$ | |
| 12. | $9 x^{2}+6 x+3 ; 3 x^{2}+7$ | |
| 13. | $5-3 x$ | |
| 14. | $(f \circ g)(x)=x^{2}-2 x+3 \quad(f \circ(h \circ g))(x)=9 x^{2}-30 x+27$ | |
| 15. | $f^{-1}(x)=\frac{3 x}{x-2}, x \neq 2 \quad g^{-1}(x)=\frac{x+3}{2} ; 6$ | |
| 16. | $\frac{4 x+3}{x-5}, x \neq 5$ | |
| 17. | $f^{-1}(x)=\frac{3 x-2}{4 x}, \mathrm{x} \neq 0\{x \mid x \in R \backslash\{0\}\} ;-\frac{1}{4}$ | |
| 18. | $f^{-1}(x)=\frac{3 x-5}{x}, x \neq 0$ $\{x \mid x \in R \backslash\{0\}\} ;-7$ | |
| 19. | $f^{-1}(x)=\frac{x-5}{2} ; g ^{-1}(x)=\frac{3 x}{x-2}, x \neq 2 ;-3 \quad$ | |
| 20. | $g(x)=x^{2}-2 x;\left(f^{-1} \circ g\right)(x)=\frac{x^{2}-2 x-1}{2}$ | |
| 21. | $(\mathrm{g} \circ \mathrm{f})^{-1}(x)=\frac{x-1}{3} ; 1$ | |
| 22. | $16416 ;-\frac{19}{3}, 5$ | |
| 23. | $\frac{1}{3}, 1 ;-9$ | |
| 24. | $-7,1 ; 5171$ | |
| 25. | $14 ; 644 ; 3$ | |
| 26. | No ; $384 ;-8,4$ | |
| 27. | $-\frac{11}{5}, 2$ | |
| 28. | $-5$ | |
| 29. | $-\frac{5}{2}, 2$ | |
| 30. | $-4,3$ | |
| 31. | Yes ; $124 ; 952$ | |
| 32. | $448 ; 68$; No |
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