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ANTHONYFEP

Cambridge International AS and A Level

 Name of student ANTHONYFEP Date Adm. number Year/grade 1992 Stream AnthonyFep Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 Total
Marks 11 10 9 10 40
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 4 questions Question 1 Code: 9709/31/M/J/16/9, Topic: Vectors With respect to the origin$O$, the points$A, B, C, D$have position vectors given by $$\overrightarrow{O A}=\mathbf{i}+3 \mathbf{j}+2 \mathbf{k}, \quad \overrightarrow{O B}=2 \mathbf{i}+\mathbf{j}-\mathbf{k}, \quad \overrightarrow{O C}=2 \mathbf{i}+4 \mathbf{j}+\mathbf{k}, \quad \overrightarrow{O D}=-3 \mathbf{i}+\mathbf{j}+2 \mathbf{k}$$$\text{(i)}$Find the equation of the plane containing$A, B$and$C$, giving your answer in the form$a x+b y+c z=d$.$[6]\text{(ii)}$The line through$D$parallel to$O A$meets the plane with equation$x+2 y-z=7$at the point$P$. Find the position vector of$P$and show that the length of$D P$is$2 \sqrt{(} 14)$.$[5]$Question 2 Code: 9709/32/M/J/16/9, Topic: Vectors The points$A, B$and$C$have position vectors, relative to the origin$O$, given by$\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\overrightarrow{O B}=4 \mathbf{j}+\mathbf{k}$and$\overrightarrow{O C}=2 \mathbf{i}+5 \mathbf{j}-\mathbf{k}.$A fourth point$D$is such that the quadrilateral$A B C D$is a parallelogram.$\text{(i)}$Find the position vector of$D$and verify that the parallelogram is a rhombus.$[5]\text{(ii)}$The plane$p$is parallel to$O A$and the line$B C$lies in$p$. Find the equation of$p$, giving your answer in the form$a x+b y+c z=d$.$[5]$Question 3 Code: 9709/33/M/J/16/9, Topic: Complex numbers Throughout this question the use of a calculator is not permitted. The complex numbers$-1+3 \mathrm{i}$and$2-\mathrm{i}$are denoted by$u$and$v$respectively. In an Argand diagram with origin$O$, the points$A, B$and$C$represent the numbers$u, v$and$u+v$respectively.$\text{(i)}$Sketch this diagram and state fully the geometrical relationship between$O B$and$A C$.$[4]\text{(ii)}$Find, in the form$x+\mathrm{i} y$, where$x$and$y$are real, the complex number$\displaystyle\frac{u}{v}$.$[3]\text{(iii)}$Prove that angle$A O B=\frac{3}{4} \pi$.$[2]$Question 4 Code: 9709/31/O/N/16/9, Topic: Complex numbers Throughout this question the use of a calculator is not permitted.$\text{(a)}$Solve the equation$(1+2 \mathrm{i}) w^{2}+4 w-(1-2 \mathrm{i})=0$, giving your answers in the form$x+\mathrm{i} y$, where$x$and$y$are real.$[5]\text{(b)}$On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities$|z-1-\mathrm{i}| \leqslant 2$and$-\frac{1}{4} \pi \leqslant \arg z \leqslant \frac{1}{4} \pi$.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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