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### JARODNOM

#### Cambridge International AS and A Level

 Name of student JARODNOM Date Adm. number Year/grade 1981 Stream Jarodnom Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 8 8 8 9 9 8 50
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/10/6, Topic: Numerical solutions of equations The diagram shows a semicircle$A C B$with centre$O$and radius$r$. The angle$B O C$is$x$radians. The area of the shaded segment is a quarter of the area of the semicircle.$\text{(i)}$Show that$x$satisfies the equation$[3]$$$x=\frac{3}{4} \pi-\sin x$$$\text{(ii)}$This equation has one root. Verify by calculation that the root lies between$1.3$and$1.5$.$[2]\text{(iii)}$Use the iterative formula $$\displaystyle x_{n+1}=\frac{3}{4} \pi-\sin x_{n}$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[3]$Question 2 Code: 9709/32/M/J/10/6, Topic: Differentiation The equation of a curve is $$x \ln y=2 x+1$$$\text{(i)}$Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{y}{x^{2}}$.$[4]\text{(ii)}$Find the equation of the tangent to the curve at the point where$y=1$, giving your answer in the form$a x+b y+c=0$.$[4]$Question 3 Code: 9709/33/M/J/10/6, Topic: Numerical solutions of equations The curve$\displaystyle y=\displaystyle\frac{\ln x}{x+1}$has one stationary point.$\text{(i)}$Show that the$x$-coordinate of this point satisfies the equation $$x=\frac{x+1}{\ln x}$$ and that this$x$-coordinate lies between 3 and 4.$[5]\text{(ii)}$Use the iterative formula $$\displaystyle x_{n+1}=\frac{x_{n}+1}{\ln x_{n}}$$ to determine the$x$-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[3]$Question 4 Code: 9709/31/O/N/10/6, Topic: Complex numbers The complex number$z$is given by $$z=(\sqrt{3})+\mathrm{i}$$$\text{(i)}$Find the modulus and argument of$z$.$[2]\text{(ii)}$The complex conjugate of$z$is denoted by$z^{*}$. Showing your working, express in the form$x+\mathrm{i} y$, where$x$and$y$are real,$\text{(a)}2 z+z^{*}$,$\text{(b)}\displaystyle\frac{\mathrm{i} z^{*}}{z}4\text{(iii)}$On a sketch of an Argand diagram with origin$O$, show the points$A$and$B$representing the complex numbers$z$and$\mathrm{i} z^{*}$respectively. Prove that angle$A O B=\frac{1}{6} \pi$.$[3]$Question 5 Code: 9709/32/O/N/10/6, Topic: Complex numbers Question 6 Code: 9709/33/O/N/10/6, Topic: Vectors The straight line$l$passes through the points with coordinates$(-5,3,6)$and$(5,8,1)$. The plane$p$has equation$2 x-y+4 z=9$.$\text{(i)}$Find the coordinates of the point of intersection of$l$and$p$.$[4]\text{(ii)}$Find the acute angle between$l$and$p$.$[4]\$

Worked solutions: P1, P3 & P6 (S1)

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