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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 4 6 8 8 8 9 8 9 11 10 88
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/10/1, Topic: Algebra Solve the inequality$2|x-3|>|3 x+1|$.$[4]$Question 2 Code: 9709/31/O/N/16/1, Topic: Logarithmic and exponential functions Solve the equation$\displaystyle\frac{3^{x}+2}{3^{x}-2}=8$, giving your answer correct to 3 decimal places.$[3]$Question 3 Code: 9709/31/O/N/14/2, Topic: Numerical solutions of equations$\text{(i)}$Use the trapezium rule with 3 intervals to estimate the value of $$\displaystyle\int_{\frac{1}{6} \pi}^{\frac{2}{3} \pi} \operatorname{cosec} x \mathrm{~d} x \text {, }$$ giving your answer correct to 2 decimal places.$[3]\text{(ii)}$Using a sketch of the graph of$y=\operatorname{cosec} x$, explain whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part$\text{(i)}$.$[2]$Question 4 Code: 9709/31/O/N/17/3, Topic: Algebra, Numerical solutions of equations The equation$x^{3}=3 x+7$has one real root, denoted by$\alpha$.$\text{(i)}$Show by calculation that$\alpha$lies between 2 and 3.$[2]$Two iterative formulae,$A$and$B, derived from this equation are as follows: \begin{aligned} &\displaystyle x_{n+1}=\left(3 x_{n}+7\right)^{\frac{1}{3}} \\ &\displaystyle x_{n+1}=\frac{x_{n}^{3}-7}{3} \end{aligned} Each formula is used with initial valuex_{1}=2.5$.$\text{(ii)}$Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate$\alpha$correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[4]$Question 5 Code: 9709/33/O/N/11/5, Topic: Integration, Numerical solutions of equations It is given that$\displaystyle\int_{1}^{a} x \ln x \mathrm{~d} x=22$, where$a$is a constant greater than$1.\text{(i)}$Show that$\displaystyle a=\sqrt{\left(\frac{87}{2 \ln a-1}\right)}$.$[5]\text{(ii)}$Use an iterative formula based on the equation in part$\text{(i)}$to find the value of$a$correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.$[3]$Question 6 Code: 9709/33/O/N/15/5, Topic: Integration Use the substitution$u=4-3 \cos x$to find the exact value of$\displaystyle\int_{0}^{\frac{1}{2} \pi} \frac{9 \sin 2 x}{\sqrt{(} 4-3 \cos x)} \mathrm{d} x$.$[8]$Question 7 Code: 9709/31/O/N/17/5, Topic: Differentiation The equation of a curve is$2 x^{4}+x y^{3}+y^{4}=10$.$\text{(i)}$Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{8 x^{3}+y^{3}}{3 x y^{2}+4 y^{3}}$.$[4]\text{(ii)}$Hence show that there are two points on the curve at which the tangent is parallel to the$x$-axis and find the coordinates of these points.$[4]$Question 8 Code: 9709/31/O/N/16/6, Topic: Numerical solutions of equations$\text{(i)}$By sketching a suitable pair of graphs, show that the equation $$\operatorname{cosec} \frac{1}{2} x=\frac{1}{3} x+1$$ has one root in the interval$0$[2]$

$\text{(ii)}$ Show by calculation that this root lies between $1.4$ and $1.6$. $[2]$

$\text{(iii)}$ Show that, if a sequence of values in the interval $0 < x \leqslant \pi$ given by the iterative formula $[2]$

$$\displaystyle x_{n+1}=2 \sin ^{-1}\left(\frac{3}{x_{n}+3}\right)$$

converges, then it converges to the root of the equation in part $\text{(i)}$. $[2]$

$\text{(iv)}$ Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$

Question 9 Code: 9709/31/O/N/18/6, Topic: Trigonometry

$\text{(i)}$ Show that the equation $(\sqrt{2}) \operatorname{cosec} x+\cot x=\sqrt{3}$ can be expressed in the form $R \sin (x-\alpha)=\sqrt{2}$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$. $[4]$

$\text{(ii)}$ Hence solve the equation $(\sqrt{2}) \operatorname{cosec} x+\cot x=\sqrt{3}$, for $0^{\circ} < x < 180^{\circ}$. $[4]$

Question 10 Code: 9709/33/O/N/18/8, Topic: Complex numbers

$\text{(a)}$ Showing all necessary working, express the complex number $\displaystyle\frac{2+3 \mathrm{i}}{1-2 \mathrm{i}}$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$, where $r>0$ and $-\pi<\theta \leqslant \pi$. Give the values of $r$ and $\theta$ correct to 3 significant figures. $[5]$

$\text{(b)}$ On an Argand diagram sketch the locus of points representing complex numbers $z$ satisfying the equation $|z-3+2 i|=1$. Find the least value of $|z|$ for points on this locus, giving your answer in an exact form. $[4]$

Question 11 Code: 9709/31/O/N/14/9, Topic: Algebra

Let $\displaystyle \mathrm{f}(x)=\frac{x^{2}-8 x+9}{(1-x)(2-x)^{2}}$.

$\text{(i)}$ Express $\mathrm{f}(x)$ in partial fractions. $[5]$

$\text{(ii)}$ Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^{2}$. $[5]$

Question 12 Code: 9709/31/O/N/10/10, Topic: Differentiation

A certain substance is formed in a chemical reaction. The mass of substance formed $t$ seconds after the start of the reaction is $x$ grams. At any time the rate of formation of the substance is proportional to $(20-x).$ When $t=0, x=0$ and $\displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=1$

$\text{(i)}$ Show that $x$ and $t$ satisfy the differential equation $[2]$

$$\displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=0.05(20-x)$$

$\text{(ii)}$ Find, in any form, the solution of this differential equation. $[5]$

$\text{(iii)}$ Find $x$ when $t=10$, giving your answer correct to 1 decimal place. $[2]$

$\text{(iv)}$ State what happens to the value of $x$ as $t$ becomes very large. $[1]$

Worked solutions: P1, P3 & P6 (S1)

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