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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 |

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 value $x_{1}=2.5$.

$\text{(ii)}$ 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

$\text{(ii)}$ 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. $[1]$