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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, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 5 5 6 6 8 9 9 9 9 10 12 11 99
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/32/M/J/20/2, Topic: Logarithmic and exponential functions The variables$x$and$y$satisfy the equation$y^{2}=A \mathrm{e}^{k x}$, where$A$and$k$are constants. The graph of$\ln y$against$x$is a straight line passing through the points$(1.5,1.2)$and$(5.24,2.7)$as shown in the diagram. Find the values of$A$and$k$correct to 2 decimal places.$$Question 2 Code: 9709/33/M/J/13/3, Topic: Trigonometry Solve the equation$\tan 2 x=5 \cot x$, for$0^{\circ}< x <180^{\circ}$.$$Question 3 Code: 9709/31/M/J/10/4, Topic: Trigonometry, Integration$\text{(i)}$Using the expansions of$\cos (3 x-x)$and$\cos (3 x+x)$, prove that$$$$\frac{1}{2}(\cos 2 x-\cos 4 x) \equiv \sin 3 x \sin x$$$\text{(ii)}$Hence show that$$$$\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \sin 3 x \sin x \mathrm{~d} x=\frac{1}{8} \sqrt{3}$$ Question 4 Code: 9709/32/M/J/16/4, Topic: Differentiation The curve with equation$\displaystyle y=\frac{(\ln x)^{2}}{x}$has two stationary points. Find the exact values of the coordinates of these points.$$Question 5 Code: 9709/31/M/J/17/5, Topic: Numerical solutions of equations The diagram shows a semicircle with centre$O$, radius$r$and diameter$A B$. The point$P$on its circumference is such that the area of the minor segment on$A P$is equal to half the area of the minor segment on$B P$. The angle$A O P$is$x$radians.$\text{(i)}$Show that$x$satisfies the equation$x=\frac{1}{3}(\pi+\sin x)$.$\text{(ii)}$Verify by calculation that$x$lies between 1 and$1.5.\text{(iii)}$Use an iterative formula based on the equation in part$\text{(i)}$to determine$x$correct to 3 decimal places. Give the result of each iteration to 5 decimal places.$$Question 6 Code: 9709/32/M/J/11/6, Topic: Differential equations A certain curve is such that its gradient at a point$(x, y)$is proportional to$x y$. At the point$(1,2)$the gradient is$4.\text{(i)}$By setting up and solving a differential equation, show that the equation of the curve is$y=2 \mathrm{e}^{x^{2}-1}$.$\text{(ii)}$State the gradient of the curve at the point$(-1,2)$and sketch the curve.$$Question 7 Code: 9709/31/M/J/13/6, Topic: Vectors The points$P$and$Q$have position vectors, relative to the origin$O$, given by $$\overrightarrow{O P}=7 \mathbf{i}+7 \mathbf{j}-5 \mathbf{k} \quad \text { and } \quad \overrightarrow{O Q}=-5 \mathbf{i}+\mathbf{j}+\mathbf{k}$$ The mid-point of$P Q$is the point$A$. The plane$\Pi$is perpendicular to the line$P Q$and passes through$A$.$\text{(i)}$Find the equation of$\Pi$, giving your answer in the form$a x+b y+c z=d$.$\text{(ii)}$The straight line through$P$parallel to the$x$-axis meets$\Pi$at the point$B$. Find the distance$A B$, correct to 3 significant figures.$$Question 8 Code: 9709/33/M/J/12/7, Topic: Integration, Numerical solutions of equations The diagram shows part of the curve$y=\cos (\sqrt{x})$for$x \geqslant 0$, where$x$is in radians. The shaded region between the curve, the axes and the line$x=p^{2}$, where$p>0$, is denoted by$R$. The area of$R$is equal to 1.$\text{(i)}$Use the substitution$x=u^{2}$to find$\displaystyle\int_{0}^{p^{2}} \cos (\sqrt{x}) \mathrm{d} x$. Hence show that$\displaystyle \sin p=\frac{3-2 \cos p}{2 p}$.$\text{(ii)}$Use the iterative formula$\displaystyle p_{n+1}=\sin ^{-1}\left(\frac{3-2 \cos p_{n}}{2 p_{n}}\right)$, with initial value$p_{1}=1$, to find the value of$p$correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$$Question 9 Code: 9709/33/M/J/21/7, Topic: Differential equations For the curve shown in the diagram, the normal to the curve at the point$P$with coordinates$(x, y)$meets the$x$-axis at$N.$The point$M$is the foot of the perpendicular from$P$to the$x$-axis. The curve is such that for all values of$x$in the interval$0 \leqslant x < \frac{1}{2} \pi$, the area of triangle$P M N$is equal to$\tan x$.$\text{(a)} \, \, \text{(i)}$Show that$\displaystyle\frac{M N}{y}=\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$.$\text{(ii)}$Hence show that$x$and$y$satisfy the differential equation$\displaystyle\frac{1}{2} y^{2} \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\tan x$.$\text{(b)}$Given that$y=1$when$x=0$, solve this differential equation to find the equation of the curve, expressing$y$in terms of$x$.$$Question 10 Code: 9709/31/M/J/13/9, Topic: Trigonometry, Integration$\text{(i)}$Express$4 \cos \theta+3 \sin \theta$in the form$R \cos (\theta-\alpha)$, where$R>0$and$0 < \alpha < \frac{1}{2} \pi$. Give the value of$\alpha$correct to 4 decimal places.$\text{(ii)}$Hence$\text{(a)}$solve the equation$4 \cos \theta+3 \sin \theta=2$for$0< \theta <2 \pi$,$\text{(b)}$find$\displaystyle\int \frac{50}{(4 \cos \theta+3 \sin \theta)^{2}} \mathrm{~d} \theta$.$$Question 11 Code: 9709/31/M/J/10/10, Topic: Vectors The lines$l$and$m$have vector equations $$\mathbf{r}=\mathbf{i}+\mathbf{j}+\mathbf{k}+s(\mathbf{i}-\mathbf{j}+2 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=4 \mathbf{i}+6 \mathbf{j}+\mathbf{k}+t(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k})$$ respectively.$\text{(i)}$Show that$l$and$m$intersect.$\text{(ii)}$Calculate the acute angle between the lines.$\text{(iii)}$Find the equation of the plane containing$l$and$m$, giving your answer in the form$a x+b y+c z=d$.$$Question 12 Code: 9709/32/M/J/17/10, Topic: Numerical solutions of equations, Integration The diagram shows the curve$y=x^{2} \cos 2 x$for$0 \leqslant x \leqslant \frac{1}{4} \pi .$The curve has a maximum point at$M$where$x=p$.$\text{(i)}$Show that$p$satisfies the equation$\displaystyle p=\frac{1}{2} \tan ^{-1}\left(\frac{1}{p}\right)$.$\text{(ii)}$Use the iterative formula$\displaystyle p_{n+1}=\frac{1}{2} \tan ^{-1}\left(\frac{1}{p_{n}}\right)$to determine the value of$p$correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$\text{(iii)}$Find, showing all necessary working, the exact area of the region bounded by the curve and the$x$-axis.$\$

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