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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Pure Mathematics 1 (P1) Variant(s) P11, P12, P13 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 5 4 6 7 8 7 6 8 8 11 8 10 88
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry Find the coordinates of the point at which the perpendicular bisector of the line joining$(2,7)$to$(10,3)$meets the$x$-axis.$[5]$Question 2 Code: 9709/11/M/J/21/1, Topic: Integration The equation of a curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{x^{4}}+32 x^{3}$. It is given that the curve passes through the point$\left(\frac{1}{2}, 4\right)$. Find the equation of the curve.$[4]$Question 3 Code: 9709/11/M/J/13/4, Topic: Series The third term of a geometric progression is$-108$and the sixth term is 32. Find$\text{(i)}$the common ratio,$[3]\text{(ii)}$the first term,$[1]\text{(iii)}$the sum to infinity.$[2]$Question 4 Code: 9709/11/M/J/15/5, Topic: Quadratics A piece of wire of length$24 \mathrm{~cm}$is bent to form the perimeter of a sector of a circle of radius$r \mathrm{~cm}$.$\text{(i)}$Show that the area of the sector,$A \mathrm{~cm}^{2}$, is given by$A=12 r-r^{2}$.$[3]\text{(ii)}$Express$A$in the form$a-(r-b)^{2}$, where$a$and$b$are constants.$[2]\text{(iii)}$Given that$r$can vary, state the greatest value of$A$and find the corresponding angle of the sector.$[2]$Question 5 Code: 9709/11/M/J/17/6, Topic: Differentiation The horizontal base of a solid prism is an equilateral triangle of side$x \mathrm{~cm}$. The sides of the prism are vertical. The height of the prism is$h \mathrm{~cm}$and the volume of the prism is$2000 \mathrm{~cm}^{3}$.$\text{(i)}$Express$\mathrm{h}$in terms of$x$and show that the total surface area of the prism,$A \mathrm{~cm}^{2}$, is given by$[3]$$$A=\frac{\sqrt{3}}{2} x^{2}+\frac{24000}{\sqrt{3}} x^{-1}$$$\text{(ii)}$Given that$x$can vary, find the value of$x$for which$A$has a stationary value.$[3]\text{(iii)}$Determine, showing all necessary working, the nature of this stationary value.$[2]$Question 6 Code: 9709/12/M/J/11/7, Topic: Coordinate geometry The line$L_{1}$passes through the points$A(2,5)$and$B(10,9)$. The line$L_{2}$is parallel to$L_{1}$and passes through the origin. The point$C$lies on$L_{2}$such that$A C$is perpendicular to$L_{2}.$Find$\text{(i)}$the coordinates of$C$,$[5]\text{(ii)}$the distance$A C$.$[2]$Question 7 Code: 9709/11/M/J/14/7, Topic: Coordinate geometry The coordinates of points$A$and$B$are$(a, 2)$and$(3, b)$respectively, where$a$and$b$are constants. The distance$A B$is$\sqrt{(} 125)$units and the gradient of the line$A B$is 2. Find the possible values of$a$and of$b$.$[6]$Question 8 Code: 9709/12/M/J/11/8, Topic: Vectors Relative to the origin$O$, the position vectors of the points$A, B$and$C$are given by $$\overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 3 \\ 5 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 4 \\ 2 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 10 \\ 0 \\ 6 \end{array}\right)$$$\text{(i)}$Find angle$A B C$.$[6]$The point$D$is such that$A B C D$is a parallelogram.$\text{(ii)}$Find the position vector of$D$.$[2]$Question 9 Code: 9709/12/M/J/17/8, Topic: Vectors Relative to an origin$O$, the position vectors of three points$A, B$and$C$are given by$\overrightarrow{O A}=3 \mathbf{i}+p \mathbf{j}-2 p \mathbf{k}, \quad \overrightarrow{O B}=6 \mathbf{i}+(p+4) \mathbf{j}+3 \mathbf{k} \quad$and$\quad \overrightarrow{O C}=(p-1) \mathbf{i}+2 \mathbf{j}+q \mathbf{k}$where$p$and$q$are constants.$\text{(i)}$In the case where$p=2$, use a scalar product to find angle$A O B$.$[4]\text{(ii)}$In the case where$\overrightarrow{A B}$is parallel to$\overrightarrow{O C}$, find the values of$p$and$q$.$[4]$Question 10 Code: 9709/13/M/J/11/9, Topic: Differentiation A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{\sqrt{x}}-1$and$P(9,5)$is a point on the curve.$\text{(i)}$Find the equation of the curve.$[4]\text{(ii)}$Find the coordinates of the stationary point on the curve.$[3]\text{(iii)}$Find an expression for$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$and determine the nature of the stationary point.$[2]\text{(iv)}$The normal to the curve at$P$makes an angle of$\tan ^{-1} k$with the positive$x$-axis. Find the value of$k$.$[2]$Question 11 Code: 9709/13/M/J/13/9, Topic: Series$\text{(a)}$In an arithmetic progression, the sum,$S_{n}$, of the first$n$terms is given by$S_{n}=2 n^{2}+8 n$. Find the first term and the common difference of the progression.$[3]\text{(b)}$The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the$1 \mathrm{st}$term, the 9 th term and the$n$th term respectively of an arithmetic progression. Find the value of$n$.$[5]$Question 12 Code: 9709/13/M/J/12/11, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=8-(x-2)^{2}$, for$x \in \mathbb{R}$.$\text{(i)}$Find the coordinates and the nature of the stationary point on the curve$y=\mathrm{f}(x)$.$[3]$The function$\mathrm{g}$is such that$\mathrm{g}(x)=8-(x-2)^{2}$, for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(ii)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$[1]$For this value of$k$,$\text{(iii)}$find an expression for$\mathrm{g}^{-1}(x)$,$[3]\text{(iv)}$sketch, on the same diagram, the graphs of$y=\mathrm{g}(x)$and$y=\mathrm{g}^{-1}(x)$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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