$\require{\cancel}$ $\require{\stix[upint]}$

### 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 9 10 11 12 Total
Marks 5 6 6 6 5 7 12 12 9 9 10 10 97
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/19/2, Topic: Quadratics The line$4 y=x+c$, where$c$is a constant, is a tangent to the curve$y^{2}=x+3$at the point$P$on the curve.$\text{(i)}$Find the value of$c$.$[3]\text{(ii)}$Find the coordinates of$P$.$[2]$Question 2 Code: 9709/13/M/J/21/4, Topic: Trigonometry$\text{(a)}$Show that the equation$[2]$$$\frac{\tan x+\sin x}{\tan x-\sin x}=k$$ where$k$is a constant, may be expressed as $$\frac{1+\cos x}{1-\cos x}=k$$$\text{(b)}$Hence express$\cos x$in terms of$k$.$[2]\text{(c)}$Hence solve the equation$\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$for$-\pi < x < \pi$.$[2]$Question 3 Code: 9709/13/M/J/13/5, Topic: Trigonometry$\text{(i)}$Sketch, on the same diagram, the curves$y=\sin 2 x$and$y=\cos x-1$for$0 \leqslant x \leqslant 2 \pi$.$[4]\text{(ii)}$Hence state the number of solutions, in the interval$0 \leqslant x \leqslant 2 \pi$, of the equations$\text{(a)}2 \sin 2 x+1=0$,$[1]\text{(b)}\sin 2 x-\cos x+1=0$.$[1]$Question 4 Code: 9709/11/M/J/19/5, Topic: Quadratics The function$\mathrm{f}$is defined by$\mathrm{f}(x)=-2 x^{2}+12 x-3$for$x \in \mathbb{R}$.$\text{(i)}$Express$-2 x^{2}+12 x-3$in the form$-2(x+a)^{2}+b$, where$a$and$b$are constants.$[2]\text{(ii)}$State the greatest value of$\mathrm{f}(x)$. The function$\mathrm{g}$is defined by$\mathrm{g}(x)=2 x+5$for$x \in \mathbb{R}$.$[1]\text{(iii)}$Find the values of$x$for which$\operatorname{gf}(x)+1=0$.$[3]$Question 5 Code: 9709/11/M/J/21/6, Topic: Quadratics The equation of a curve is$y=(2 k-3) x^{2}-k x-(k-2)$, where$k$is a constant. The line$y=3 x-4$is a tangent to the curve. Find the value of$k[5]$Question 6 Code: 9709/12/M/J/19/7, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} \mathrm{f}: x & \mapsto 3 x-2, \quad x \in \mathbb{R} \\ \mathrm{g}: x & \mapsto \frac{2 x+3}{x-1}, \quad x \in \mathbb{R}, x \neq 1 \end{aligned}\text{(i)}$Obtain expressions for$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$, stating the value of$x$for which$\mathrm{g}^{-1}(x)$is not defined.$[4]\text{(ii)}$Solve the equation$\mathrm{fg}(x)=\frac{7}{3}$.$[3]$Question 7 Code: 9709/13/M/J/11/10, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned}\text{(i)}$Evaluate$\mathrm{fg(2)}$.$[2]\text{(ii)}$Sketch in a single diagram the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$, making clear the relationship between the graphs.$[3]\text{(iii)}$Obtain an expression for$\mathrm{g}^{\prime}(x)$and use your answer to explain why$\mathrm{g}$has an inverse.$[3]\text{(iv)}$Express each of$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$in terms of$x$.$[4]$Question 8 Code: 9709/12/M/J/12/10, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned}\text{(i)}$Obtain expressions, in terms of$x$, for$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$, stating the value of$x$for which$\mathrm{g}^{-1}(x)$is not defined.$[4]\text{(ii)}$Sketch the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$on the same diagram, making clear the relationship between the two graphs.$[3]\text{(iii)}$Given that the equation$f g(x)=5-k x$, where$k$is a constant, has no solutions, find the set of possible values of$k$.$[5]$Question 9 Code: 9709/13/M/J/13/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where$k$is a constant.$\text{(i)}$In the case where$k=3$, solve the equation$\mathrm{ff}(x)=25$.$[2]$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.$\text{(ii)}$Find the set of values of$k$for which the equation$\mathrm{f}(x)=\mathrm{g}(x)$has no real solutions.$[3]$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.$\text{(iii)}$Find an expression for$\mathrm{h}^{-1}(x)$.$[4]$Question 10 Code: 9709/13/M/J/16/10, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=2 x+3$for$x \geqslant 0$. The function$\mathrm{g}$is such that$\mathrm{g}(x)=a x^{2}+b$for$x \leqslant q$, where$a, b$and$q$are constants. The function fg is such that fg$(x)=6 x^{2}-21$for$x \leqslant q\text{(i)}$Find the values of$a$and$b$.$[3]\text{(ii)}$Find the greatest possible value of$q$.$[2]$It is now given that$q=-3$.$\text{(iii)}$Find the range of$\mathrm{fg}$.$[1]\text{(iv)}$Find an expression for$\mathrm{(f g)^{-1}}(x)$and state the domain of$(f g)^{-1}$.$[3]$Question 11 Code: 9709/12/M/J/10/11, Topic: Functions The function$\mathrm{f}: x \mapsto 4-3 \sin x$is defined for the domain$0 \leqslant x \leqslant 2 \pi$.$\text{(i)}$Solve the equation$\mathrm{f}(x)=2$.$[3]\text{(ii)}$Sketch the graph of$y=\mathrm{f}(x)$.$[2]\text{(iii)}$Find the set of values of$k$for which the equation$\mathrm{f}(x)=k$has no solution.$[2]$The function$\mathrm{g}: x \mapsto 4-3 \sin x$is defined for the domain$\frac{1}{2} \pi \leqslant x \leqslant A$.$\text{(iv)}$State the largest value of$A$for which$\mathrm{g}$has an inverse.$[1]\text{(v)}$For this value of$A$, find the value of$g^{-1}(3)$.$[3]$Question 12 Code: 9709/13/M/J/12/11, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=8-(x-2)^{2}$, for$x \in \mathbb{R}$.$\text{(i)}$Find the coordinates and the nature of the stationary point on the curve$y=\mathrm{f}(x)$.$[3]$The function$\mathrm{g}$is such that$\mathrm{g}(x)=8-(x-2)^{2}$, for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(ii)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$[1]$For this value of$k$,$\text{(iii)}$find an expression for$\mathrm{g}^{-1}(x)$,$[3]\text{(iv)}$sketch, on the same diagram, the graphs of$y=\mathrm{g}(x)$and$y=\mathrm{g}^{-1}(x)$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

If you need worked solutions for P1, P3 & P6 (S1), contact us @ [email protected] | +254 721 301 418.

1. Send us the link to these questions ( https://stemcie.com/view/42 ).
2. We will solve the questions and provide you with the step by step worked solutions.
3. We will then schedule a one to one online session to take you through the solutions (optional).