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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Pure Mathematics 3 (P3) Variant(s) P31, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 4 3 7 7 8 8 8 9 10 10 10 10 94
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/31/O/N/12/1, Topic: Algebra Find the set of values of$x$satisfying the inequality$3|x-1|<|2 x+1|$.$[4]$Question 2 Code: 9709/31/O/N/13/1, Topic: Differentiation The equation of a curve is$y=\displaystyle\frac{1+x}{1+2 x}$for$x>-\frac{1}{2}$. Show that the gradient of the curve is always negative.$[3]$Question 3 Code: 9709/33/O/N/18/3, Topic: Numerical solutions of equations$\text{(i)}$By sketching a suitable pair of graphs, show that the equation$x^{3}=3-x$has exactly one real root.$[2]\text{(ii)}$Show that if a sequence of real values given by the iterative formula $$\displaystyle x_{n+1}=\frac{2 x_{n}^{3}+3}{3 x_{n}^{2}+1}$$ converges, then it converges to the root of the equation in$\text{part $\text{(i)}$}$.$[2]\text{(iii)}$Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.$[3]$Question 4 Code: 9709/33/O/N/18/5, Topic: Differential equations The coordinates$(x, y)$of a general point on a curve satisfy the differential equation $$x \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\left(2-x^{2}\right) y$$ The curve passes through the point$(1,1)$. Find the equation of the curve, obtaining an expression for$y$in terms of$x$.$[7]$Question 5 Code: 9709/31/O/N/11/6, Topic: Trigonometry$\text{(i)}$Express$\cos x+3 \sin x$in the form$R \cos (x-\alpha)$, where$R>0$and$0^{\circ}< \alpha <90^{\circ}$, giving the exact value of$R$and the value of$\alpha$correct to 2 decimal places.$[3]\text{(ii)}$Hence solve the equation$\cos 2 \theta+3 \sin 2 \theta=2$, for$0^{\circ}< \theta <90^{\circ}$.$[5]$Question 6 Code: 9709/31/O/N/17/6, Topic: Differential equations The variables$x$and$y$satisfy the differential equation $$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=4 \cos ^{2} y \tan x$$ for$0 \leqslant x < \frac{1}{2} \pi$, and$x=0$when$y=\frac{1}{4} \pi$. Solve this differential equation and find the value of$x$when$y=\frac{1}{3} \pi$.$[8]$Question 7 Code: 9709/33/O/N/12/7, Topic: Integration The diagram shows part of the curve$y=\sin ^{3} 2 x \cos ^{3} 2 x$. The shaded region shown is bounded by the curve and the$x$-axis and its exact area is denoted by$A$.$\text{(i)}$Use the substitution$u=\sin 2 x$in a suitable integral to find the value of$A$.$[6]\text{(ii)}$Given that$\displaystyle\int_{0}^{k \pi}\left|\sin ^{3} 2 x \cos ^{3} 2 x\right| \mathrm{d} x=40 A$, find the value of the constant$k$.$[2]$Question 8 Code: 9709/31/O/N/15/7, Topic: Vectors The points$A, B$and$C$have position vectors, relative to the origin$O$, given by $$\overrightarrow{O A}=\left(\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 3 \\ 0 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 1 \\ 1 \\ 4 \end{array}\right)$$ The plane$m$is perpendicular to$A B$and contains the point$C$.$\text{(i)}$Find a vector equation for the line passing through$A$and$B$.$[2]\text{(ii)}$Obtain the equation of the plane$m$, giving your answer in the form$a x+b y+c z=d$.$[2]\text{(iii)}$The line through$A$and$B$intersects the plane$m$at the point$N.$Find the position vector of$N$and show that$C N=\sqrt{(} 13)$.$[5]$Question 9 Code: 9709/31/O/N/11/9, Topic: Differentiation, Integration The diagram shows the curve$y=x^{2} \ln x$and its minimum point$M$.$\text{(i)}$Find the exact values of the coordinates of$M$.$[5]\text{(ii)}$Find the exact value of the area of the shaded region bounded by the curve, the$x$-axis and the line$x=\mathrm{e}$.$[5]$Question 10 Code: 9709/33/O/N/14/9, Topic: Numerical solutions of equations$\text{(i)}$Sketch the curve$y=\ln (x+1)$and hence, by sketching a second curve, show that the equation $$x^{3}+\ln (x+1)=40$$ has exactly one real root. State the equation of the second curve.$[3]\text{(ii)}$Verify by calculation that the root lies between 3 and 4.$[2]\text{(iii)}$Use the iterative formula $$\displaystyle x_{n+1}=\sqrt[3]{\left(40-\ln \left(x_{n}+1\right)\right)} \text {, }$$ with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.$[3]\text{(iv)}$Deduce the root of the equation $$\left(e^{y}-1\right)^{3}+y=40.$$ giving the answer correct to 2 decimal places.$[2]$Question 11 Code: 9709/33/O/N/13/10, Topic: Differential equations A particular solution of the differential equation $$3 y^{2} \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=4\left(y^{3}+1\right) \cos ^{2} x$$ is such that$y=2$when$x=0$. The diagram shows a sketch of the graph of this solution for$0 \leqslant x \leqslant 2 \pi$; the graph has stationary points at$A$and$B$. Find the$y$-coordinates of$A$and$B$, giving each coordinate correct to 1 decimal place.$[10]$Question 12 Code: 9709/33/O/N/16/10, Topic: Vectors The line$l$has vector equation$\mathbf{r}=\mathbf{i}+2 \mathbf{j}+\mathbf{k}+\lambda(2 \mathbf{i}-\mathbf{j}+\mathbf{k})$.$\text{(i)}$Find the position vectors of the two points on the line whose distance from the origin is$\sqrt{(} 10)$.$[5]\text{(ii)}$The plane$p$has equation$a x+y+z=5$, where$a$is a constant. The acute angle between the line$l$and the plane$p$is equal to$\sin ^{-1}\left(\frac{2}{3}\right)$. Find the possible values of$a$.$[5]\$

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