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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Pure Mathematics 1 (P1) Variant(s) P11, P12, P13 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 Total
Marks 6 5 5 5 9 9 9 12 60
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 8 questions Question 1 Code: 9709/12/O/N/17/2, Topic: Functions A function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 4-5 x$for$x \in \mathbb{R}$.$\text{(i)}$Find an expression for$\mathrm{f}^{-1}(x)$and find the point of intersection of the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$.$[3]\text{(ii)}$Sketch, on the same diagram, the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$, making clear the relationship between the graphs.$[3]$Question 2 Code: 9709/12/M/J/18/2, Topic: Quadratics The equation of a curve is$y=x^{2}-6 x+k$, where$k$is a constant.$\text{(i)}$Find the set of values of$k$for which the whole of the curve lies above the$x$-axis.$[2]\text{(ii)}$Find the value of$k$for which the line$y+2 x=7$is a tangent to the curve.$[3]$Question 3 Code: 9709/11/O/N/10/3, Topic: Quadratics Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+3 \\ &\mathrm{~g}: x \mapsto x^{2}-2 x \end{aligned} Express\operatorname{gf}(x)$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$[5]$Question 4 Code: 9709/12/M/J/11/6, Topic: Functions The function$\mathrm{f}$is defined by$\displaystyle\mathrm{f}: x \mapsto \frac{x+3}{2 x-1}, x \in \mathbb{R}, x \neq \frac{1}{2}$.$\text{(i)}$Show that$\mathrm{ff}(x)=x$.$[3]\text{(ii)}$Hence, or otherwise, obtain an expression for$\mathrm{f}^{-1}(x)$.$[2]$Question 5 Code: 9709/13/O/N/13/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto x^{2}+4 x$for$x \geqslant c$, where$c$is a constant. It is given that$\mathrm{f}$is a one-one function.$\text{(i)}$State the range of$\mathrm{f}$in terms of$c$and find the smallest possible value of$c$.$[3]$The function$\mathrm{g}$is defined by$g: x \mapsto a x+b$for$x \geqslant 0$, where$a$and$b$are positive constants. It is given that, when$c=0, \operatorname{gf}(1)=11$and$f g(1)=21$.$\text{(ii)}$Write down two equations in$a$and$b$and solve them to find the values of$a$and$b$.$[6]$Question 6 Code: 9709/13/M/J/18/10, Topic: Functions The one-one function$\mathrm{f}$is defined by$\mathrm{f}(x)=(x-2)^{2}+2$for$x \geqslant c$, where$c$is a constant.$\text{(i)}$State the smallest possible value of$c$.$[1]$In parts$\text{(ii)}$and$\text{(iii)}$the value of$c$is 4.$\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$.$[3]\text{(iii)}$Solve the equation$\mathrm{ff}(x)=51$, giving your answer in the form$a+\sqrt{b}$.$[5]$Question 7 Code: 9709/12/O/N/18/10, Topic: Quadratics, Coordinate geometry The equation of a curve is$\displaystyle y=2 x+\frac{12}{x}$and the equation of a line is$y+x=k$, where$k$is a constant.$\text{(i)}$Find the set of values of$k$for which the line does not meet the curve.$[3]$In the case where$k=15$, the curve intersects the line at points$A$and$B$.$\text{(ii)}$Find the coordinates of$A$and$B$.$[3]\text{(iii)}$Find the equation of the perpendicular bisector of the line joining$A$and$B$.$[3]$Question 8 Code: 9709/12/M/J/15/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x^{2}-6 x+5$for$x \in \mathbb{R}$.$\text{(i)}$Find the set of values of$p$for which the equation$\mathrm{f}(x)=p$has no real roots.$[3]$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 2 x^{2}-6 x+5$for$0 \leqslant x \leqslant 4$.$\text{(ii)}$Express$\mathrm{g}(x)$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$[3]\text{(iii)}$Find the range of$\mathrm{g}$.$[2]$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto 2 x^{2}-6 x+5$for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(iv)}$State the smallest value of$k$for which$\mathrm{h}$has an inverse.$[1]\text{(v)}$For this value of$k$, find an expression for$\mathrm{h}^{-1}(x)$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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