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

### 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 Total
Marks 7 6 8 9 8 9 9 11 67
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 8 questions Question 1 Code: 9709/12/O/N/11/4, Topic: Quadratics The equation of a curve is$y^{2}+2 x=13$and the equation of a line is$2 y+x=k$, where$k$is a constant.$\text{(i)}$In the case where$k=8$, find the coordinates of the points of intersection of the line and the curve.$[4]\text{(ii)}$Find the value of$k$for which the line is a tangent to the curve.$[3]$Question 2 Code: 9709/13/M/J/17/6, Topic: Quadratics The line$3 y+x=25$is a normal to the curve$y=x^{2}-5 x+k$. Find the value of the constant$k$.$[6]$Question 3 Code: 9709/12/O/N/17/6, Topic: Functions$\text{(a)}$The function$\mathrm{f}$, defined by$\mathrm{f}: x \mapsto a+b \sin x$for$x \in \mathbb{R}$, is such that$\mathrm{f}\left(\frac{1}{6} \pi\right)=4$and$\mathrm{f}\left(\frac{1}{2} \pi\right)=3$.$\text{(i)}$Find the values of the constants$a$and$b$.$[3]\text{(ii)}$Evaluate$\mathrm{ff}(0)$.$[2]\text{(b)}$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto c+d \sin x$for$x \in \mathbb{R}$. The range of$\mathrm{g}$is given by$-4 \leqslant \mathrm{~g}(x) \leqslant 10$. Find the values of the constants$c$and$d$.$[3]$Question 4 Code: 9709/11/M/J/13/7, Topic: Quadratics, Differentiation, Coordinate geometry A curve has equation$y=x^{2}-4 x+4$and a line has equation$y=m x$, where$m$is a constant.$\text{(i)}$For the case where$m=1$, the curve and the line intersect at the points$A$and$B$. Find the coordinates of the mid-point of$A B$.$[4]\text{(ii)}$Find the non-zero value of$m$for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.$[5]$Question 5 Code: 9709/11/M/J/17/7, Topic: Integration, Quadratics, Differentiation A curve for which$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=7-x^{2}-6 x$passes through the point$(3,-10)$.$\text{(i)}$Find the equation of the curve.$[3]\text{(ii)}$Express$7-x^{2}-6 x$in the form$a-(x+b)^{2}$, where$a$and$b$are constants.$[2]\text{(iii)}$Find the set of values of$x$for which the gradient of the curve is positive.$[3]$Question 6 Code: 9709/13/M/J/17/9, Topic: Functions$\text{(i)}$Express$9 x^{2}-6 x+6$in the form$(a x+b)^{2}+c$, where$a, b$and$c$are constants.$[3]$The function$\mathrm{f}$is defined by$\mathrm{f}(x)=9 x^{2}-6 x+6$for$x \geqslant p$, where$p$is a constant.$\text{(ii)}$State the smallest value of$p$for which$\mathrm{f}$is a one-one function.$[1]\text{(iii)}$For this value of$p$, obtain an expression for$\mathrm{f}^{-1}(x)$, and state the domain of$\mathrm{f}^{-1}$.$[4]\text{(iv)}$State the set of values of$q$for which the equation$\mathrm{f}(x)=q$has no solution.$[1]$Question 7 Code: 9709/11/M/J/14/10, Topic: Functions The diagram shows the function$\mathrm{f}$defined for$-1 \leqslant x \leqslant 4$, where $$\mathrm{f}(x)= \begin{cases}3 x-2 & \text { for }-1 \leqslant x \leqslant 1 \\ \frac{4}{5-x} & \text { for } 1 < x \leqslant 4\end{cases}$$$\text{(i)}$State the range of$\mathrm{f}$.$[1]\text{(ii)}$Copy the diagram and on your copy sketch the graph of$y=\mathrm{f}^{-1}(x)$.$[2]\text{(iii)}$Obtain expressions to define the function$\mathrm{f}^{-1}$, giving also the set of values for which each expression is valid.$[6]$Question 8 Code: 9709/11/M/J/16/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 4 \sin x-1$for$-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.$\text{(i)}$State the range of$\mathrm{f}$.$[2]\text{(ii)}$Find the coordinates of the points at which the curve$y=\mathrm{f}(x)$intersects the coordinate axes.$[3]\text{(iii)}$Sketch the graph of$y=\mathrm{f}(x)$.$[2]\text{(iv)}$Obtain an expression for$\mathrm{f}^{-1}(x)$, stating both the domain and range of$\mathrm{f}^{-1}$.$[4]\$

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/43 ).
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).