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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 3 6 5 5 6 8 8 9 10 10 10 13 93
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/33/O/N/10/1, Topic: Algebra Expand$(1+2 x)^{-3}$in ascending powers of$x$, up to and including the term in$x^{2}$, simplifying the coefficients.$$Question 2 Code: 9709/31/O/N/11/3, Topic: Algebra The polynomial$x^{4}+3 x^{3}+a x+3$is denoted by$\mathrm{p}(x)$. It is given that$\mathrm{p}(x)$is divisible by$x^{2}-x+1$.$\text{(i)}$Find the value of$a$.$\text{(ii)}$When$a$has this value, find the real roots of the equation$\mathrm{p}(x)=0$.$$Question 3 Code: 9709/31/O/N/20/3, Topic: Differentiation The parametric equations of a curve are $$x=3-\cos 2 \theta, \quad y=2 \theta+\sin 2 \theta$$ for$0 < \theta < \frac{1}{2} \pi$. Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\cot \theta$.$$Question 4 Code: 9709/33/O/N/13/4, Topic: Differentiation A curve has equation$3 \mathrm{e}^{2 x} y+\mathrm{e}^{x} y^{3}=14$. Find the gradient of the curve at the point$(0,2)$.$$Question 5 Code: 9709/31/O/N/14/4, Topic: Differentiation The parametric equations of a curve are $$x=\frac{1}{\cos ^{3} t}, \quad y=\tan ^{3} t,$$ where$0 \leqslant t < \frac{1}{2} \pi$.$\text{(i)}$Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sin t$.$\text{(ii)}$Hence show that the equation of the tangent to the curve at the point with parameter$t$is$y=x \sin t-\tan t$.$$Question 6 Code: 9709/33/O/N/14/5, Topic: Complex numbers The complex numbers$w$and$z$are defined by$w=5+3 \mathrm{i}$and$z=4+\mathrm{i}$.$\text{(i)}$Express$\displaystyle\frac{\mathrm{i} w}{z}$in the form$x+\mathrm{i} y$, showing all your working and giving the exact values of$x$and$y$.$\text{(ii)}$Find$w z$and hence, by considering arguments, show that$$$$\tan ^{-1}\left(\frac{3}{5}\right)+\tan ^{-1}\left(\frac{1}{4}\right)=\frac{1}{4} \pi.$$ Question 7 Code: 9709/33/O/N/15/5, Topic: Integration Use the substitution$u=4-3 \cos x$to find the exact value of$\displaystyle\int_{0}^{\frac{1}{2} \pi} \frac{9 \sin 2 x}{\sqrt{(} 4-3 \cos x)} \mathrm{d} x$.$$Question 8 Code: 9709/33/O/N/19/7, Topic: Vectors The plane$m$has equation$x+4 y-8 z=2$. The plane$n$is parallel to$m$and passes through the point$P$with coordinates$(5,2,-2)$.$\text{(i)}$Find the equation of$n$, giving your answer in the form$a x+b y+c z=d$.$\text{(ii)}$Calculate the perpendicular distance between$m$and$n$.$\text{(iii)}$The line$l$lies in the plane$n$, passes through the point$P$and is perpendicular to$O P$, where$O$is the origin. Find a vector equation for$l$.$$Question 9 Code: 9709/33/O/N/12/8, Topic: Vectors Two lines have equations $$\mathbf{r}=\left(\begin{array}{r} 5 \\ 1 \\ -4 \end{array}\right)+s\left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right) \quad \text { and } \quad \mathbf{r}=\left(\begin{array}{r} p \\ 4 \\ -2 \end{array}\right)+t\left(\begin{array}{r} 2 \\ 5 \\ -4 \end{array}\right)$$ where$p$is a constant. It is given that the lines intersect.$\text{(i)}$Find the value of$p$and determine the coordinates of the point of intersection.$\text{(ii)}$Find the equation of the plane containing the two lines, giving your answer in the form$a x+b y+c z=d$, where$a, b, c$and$d$are integers.$$Question 10 Code: 9709/31/O/N/18/9, Topic: Algebra, Integration Let$\displaystyle\mathrm{f}(x)=\frac{6 x^{2}+8 x+9}{(2-x)(3+2 x)^{2}}\text{(i)}$Express$\mathrm{f}(x)$in partial fractions.$\text{(ii)}$Hence, showing all necessary working, show that$\displaystyle\int_{-1}^{0} \mathrm{f}(x) \, \mathrm{d} x=1+\frac{1}{2} \ln \left(\frac{3}{4}\right)$.$$Question 11 Code: 9709/33/O/N/20/9, Topic: Algebra Let$\displaystyle\mathrm{f}(x)=\frac{8+5 x+12 x^{2}}{(1-x)(2+3 x)^{2}}\text{(a)}$Express$\mathrm{f}(x)$in partial fractions.$\text{(b)}$Hence obtain the expansion of$\mathrm{f}(x)$in ascending powers of$x$, up to and including the term in$x^{2}$.$$Question 12 Code: 9709/33/O/N/10/10, Topic: Vectors The polynomial$\mathrm{p}(z)$is defined by $$\mathrm{p}(z)=z^{3}+m z^{2}+24 z+32$$ where$m$is a constant. It is given that$(z+2)$is a factor of$\mathrm{p}(z)$.$\text{(i)}$Find the value of$m$.$\text{(ii)}$Hence, showing all your working, find$\text{(a)}$the three roots of the equation$\mathrm{p}(z)=0$,$\text{(b)}$the six roots of the equation$\mathrm{p}\left(z^{2}\right)=0$.$\$

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