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### ABC INTERNATIONAL SCHOOL

#### Cambridge International AS and A Level

 Name of student JOHN DOE Date Adm. number A22/0051/2006 Year/grade 13 Stream Red Subject Pure Mathematics 3 (P3) Variant(s) P31, P32 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 Total
Marks 5 4 6 5 7 8 8 8 9 9 69
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 10 questions Question 1 Code: 9709/31/M/J/12/2, Topic: Algebra$\text{(i)}$Expand$\displaystyle\frac{1}{\sqrt{(} 1-4 x)}$in ascending powers of$x$, up to and including the term in$x^{2}$, simplifying the coefficients.$[3]\text{(ii)}$Hence find the coefficient of$x^{2}$in the expansion of$\displaystyle\frac{1+2 x}{\sqrt{(} 4-16 x)}$.$[2]$Question 2 Code: 9709/32/M/J/15/2, Topic: Logarithmic and exponential functions Using the substitution$u=4^{x}$, solve the equation$4^{x}+4^{2}=4^{x+2}$, giving your answer correct to 3 significant figures.$[4]$Question 3 Code: 9709/31/M/J/18/2, Topic: Trigonometry$\text{(i)}$Given that$\sin \left(x-60^{\circ}\right)=3 \cos \left(x-45^{\circ}\right)$, find the exact value of$\tan x$.$[4]\text{(ii)}$Hence solve the equation$\sin \left(x-60^{\circ}\right)=3 \cos \left(x-45^{\circ}\right)$, for$0^{\circ} < x < 360^{\circ}$.$[2]$Question 4 Code: 9709/32/M/J/16/3, Topic: Integration Find the exact value of$\displaystyle\int_{0}^{\frac{1}{2} \pi} x^{2} \sin 2 x \mathrm{~d} x$.$[5]$Question 5 Code: 9709/31/O/N/14/5, Topic: Complex numbers Throughout this question the use of a calculator is not permitted. The complex numbers$w$and$z$satisfy the relation $$w=\frac{z+\mathrm{i}}{\mathrm{i} z+2}.$$$\text{(i)}$Given that$z=1+\mathrm{i}$, find$w$, giving your answer in the form$x+\mathrm{i} y$, where$x$and$y$are real.$[4]\text{(ii)}$Given instead that$w=z$and the real part of$z$is negative, find$z$, giving your answer in the form$x+\mathrm{i} y$, where$x$and$y$are real.$[4]$Question 6 Code: 9709/31/M/J/17/5, Topic: Numerical solutions of equations The diagram shows a semicircle with centre$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)$.$[3]\text{(ii)}$Verify by calculation that$x$lies between 1 and$1.5.[2]\text{(iii)}$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.$[3]$Question 7 Code: 9709/31/M/J/12/7, Topic: Differential equations The variables$x$and$y$are related by the differential equation $$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6 x \mathrm{e}^{3 x}}{y^{2}}$$ It is given that$y=2$when$x=0$. Solve the differential equation and hence find the value of$y$when$x=0.5$, giving your answer correct to 2 decimal places.$[7]$Question 8 Code: 9709/31/O/N/12/7, Topic: Differentiation The equation of a curve is$\ln (x y)-y^{3}=1$.$\text{(i)}$Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y}{x\left(3 y^{3}-1\right)}$.$[4]\text{(ii)}$Find the coordinates of the point where the tangent to the curve is parallel to the$y$-axis, giving each coordinate correct to 3 significant figures.$[4]$Question 9 Code: 9709/31/O/N/16/7, Topic: Differentiation, Integration The diagram shows part of the curve$y=\left(2 x-x^{2}\right) \mathrm{e}^{\frac{1}{2} x}$and its maximum point$M$.$\text{(i)}$Find the exact$x$-coordinate of$M$.$[4]\text{(ii)}$Find the exact value of the area of the shaded region bounded by the curve and the positive$x$-axis.$[5]$Question 10 Code: 9709/32/M/J/18/9, Topic: Algebra Let$\displaystyle\mathrm{f}(x)=\frac{x-4 x^{2}}{(3-x)\left(2+x^{2}\right)}$.$\text{(i)}$Express$\displaystyle\mathrm{f}(x)$in the form$\displaystyle\frac{A}{3-x}+\frac{B x+C}{2+x^{2}}$.$[4]\text{(ii)}$Hence obtain the expansion of$\mathrm{f}(x)$in ascending powers of$x$, up to and including the term in$x^{3}$.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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