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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 4 3 3 4 7 8 7 9 10 10 10 10 85
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/33/M/J/11/1, Topic: Logarithmic and exponential functions Use logarithms to solve the equation$5^{2 x-1}=2\left(3^{x}\right)$, giving your answer correct to 3 significant figures.$[4]$Question 2 Code: 9709/31/M/J/13/1, Topic: Algebra Find the quotient and remainder when$2 x^{2}$is divided by$x+2$.$[3]$Question 3 Code: 9709/33/M/J/17/1, Topic: Trigonometry Prove the identity$\displaystyle\frac{\cot x-\tan x}{\cot x+\tan x} \equiv \cos 2 x$.$[3]$Question 4 Code: 9709/33/M/J/17/3, Topic: Logarithmic and exponential functions Using the substitution$u=\mathrm{e}^{x}$, solve the equation$4 \mathrm{e}^{-x}=3 \mathrm{e}^{x}+4$. Give your answer correct to 3 significant figures.$[4]$Question 5 Code: 9709/31/M/J/14/5, Topic: Complex numbers The complex number$z$is defined by$\displaystyle z=\frac{9 \sqrt{3}+91}{\sqrt{3}-\mathrm{i}}$. Find, showing all your working,$\text{(i)}$an expression for$z$in the form$r \mathrm{e}^{\mathrm{i} \theta}$, where$r>0$and$-\pi < \theta \leqslant \pi$,$[5]\text{(ii)}$the two square roots of$z$, giving your answers in the form$r \mathrm{e}^{\mathrm{i} \theta}$, where$r>0$and$-\pi < \theta \leqslant \pi$.$[3]$Question 6 Code: 9709/32/M/J/15/5, Topic: Numerical solutions of equations The diagram shows a circle with centre$O$and radius$r$. The tangents to the circle at the points$A$and$B$meet at$T$, and the angle$A O B$is$2 x$radians. The shaded region is bounded by the tangents$A T$and$B T$, and by the minor$\operatorname{arc} A B$. The perimeter of the shaded region is equal to the circumference of the circle.$\text{(i)}$Show that$x$satisfies the equation$[3]$$$\tan x=\pi-x$$$\text{(ii)}$This equation has one root in the interval$0 < x < \frac{1}{2} \pi$. Verify by calculation that this root lies between 1 and$1.3$.$[2]\text{(iii)}$Use the iterative formula $$\displaystyle x_{n+1}=\tan ^{-1}\left(\pi-x_{n}\right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[3]$Question 7 Code: 9709/33/M/J/20/6, Topic: Numerical solutions of equations$\text{(a)}$By sketching a suitable pair of graphs, show that the equation$x^{5}=2+x$has exactly one real root.$[2]\text{(b)}$Show that if a sequence of values given by the iterative formula $$\displaystyle x_{n+1}=\frac{4 x_{n}^{5}+2}{5 x_{n}^{4}-1}$$ converges, then it converges to the root of the equation in part$\text{(a)}$.$[2]\text{(c)}$Use the iterative formula with initial value$x_{1}=1.5$to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.$[3]$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$\displaystyle \sin p=\frac{3-2 \cos p}{2 p}$.$[6]\text{(ii)}$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.$[3]$Question 9 Code: 9709/31/M/J/13/8, Topic: Integration$\text{(a)}$Show that$\displaystyle\int_{2}^{4} 4 x \ln x \mathrm{~d} x=56 \ln 2-12$.$[5]\text{(b)}$Use the substitution$u=\sin 4 x$to find the exact value of$\displaystyle\int_{0}^{\frac{1}{24} \pi} \cos ^{3} 4 x \mathrm{~d} x$.$[5]$Question 10 Code: 9709/33/M/J/10/9, Topic: Algebra$\text{(i)}$Express$\displaystyle\frac{4+5 x-x^{2}}{(1-2 x)(2+x)^{2}}$in partial fractions.$[5]\text{(ii)}$Hence obtain the expansion of$\displaystyle\frac{4+5 x-x^{2}}{(1-2 x)(2+x)^{2}}$in ascending powers of$x$, up to and including the term in$x^{2}$.$[5]$Question 11 Code: 9709/33/M/J/13/9, Topic: Differentiation, Integration The diagram shows the curve$y=\sin ^{2} 2 x \cos x$for$0 \leqslant x \leqslant \frac{1}{2} \pi$, and its maximum point$M$.$\text{(i)}$Find the$x$-coordinate of$M$.$[6]\text{(ii)}$Using the substitution$u=\sin x$, find by integration the area of the shaded region bounded by the curve and the$x$-axis.$[4]$Question 12 Code: 9709/32/M/J/18/10, Topic: Vectors Two lines$l$and$m$have equations$\mathbf{r}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}+s(2 \mathbf{i}+3 \mathbf{j}-\mathbf{k})$and$\mathbf{r}=\mathbf{i}+3 \mathbf{j}+4 \mathbf{k}+t(\mathbf{i}+2 \mathbf{j}+\mathbf{k})$respectively.$\text{(i)}$Show that the lines are skew.$[4]$A plane$p$is parallel to the lines$l$and$m$.$\text{(ii)}$Find a vector that is normal to$p$.$[3]\text{(iii)}$Given that$p$is equidistant from the lines$l$and$m$, find the equation of$p$. Give your answer in the form$a x+b y+c z=d$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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