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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 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 3 4 4 5 3 6 6 6 7 7 8 9 68
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/O/N/14/1, Topic: Series In the expansion of$(2+a x)^{7}$, the coefficient of$x$is equal to the coefficient of$x^{2}.$Find the value of the non-zero constant$a$.$[3]$Question 2 Code: 9709/11/O/N/17/1, Topic: Differentiation A curve has equation$y=2 x^{\frac{3}{2}}-3 x-4 x^{\frac{1}{2}}+4$. Find the equation of the tangent to the curve at the point$(4,0)$.$[4]$Question 3 Code: 9709/12/O/N/10/3, Topic: Differentiation The length,$x$metres, of a Green Anaconda snake which is$t$years old is given approximately by the$0.4$formula $$x=0.7 \sqrt{(}2 t-1)$$ where$1 \leqslant t \leqslant 10$. Using this formula, find$\text{(i)}\displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}$,$[2]\text{(ii)}$the rate of growth of a Green Anaconda snake which is 5 years old.$[2]$Question 4 Code: 9709/12/O/N/12/3, Topic: Differentiation The diagram shows a plan for a rectangular park$A B C D$, in which$A B=40 \mathrm{~m}$and$A D=60 \mathrm{~m}$. Points$X$and$Y$lie on$B C$and$C D$respectively and$A X, X Y$and$Y A$are paths that surround a triangular playground. The length of$D Y$is$x \mathrm{~m}$and the length of$X C$is$2 x \mathrm{~m}$.$\text{(i)}$Show that the area,$A \mathrm{~m}^{2}$, of the playground is given by$[2]$$$A=x^{2}-30 x+1200.$$$\text{(ii)}$Given that$x$can vary, find the minimum area of the playground.$[3]$Question 5 Code: 9709/11/O/N/20/4, Topic: Trigonometry In the diagram, the lower curve has equation$y=\cos \theta$. The upper curve shows the result of applying a combination of transformations to$y=\cos \theta$. Find, in terms of a cosine function, the equation of the upper curve.$[3]$Question 6 Code: 9709/11/O/N/10/5, Topic: Vectors The diagram shows a pyramid$O A B C$with a horizontal base$O A B$where$O A=6 \mathrm{~cm}, O B=8 \mathrm{~cm}$and angle$A O B=90^{\circ}$. The point$C$is vertically above$O$and$O C=10 \mathrm{~cm}$. Unit vectors$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$are parallel to$O A, O B$and$O C$as shown. Use a scalar product to find angle$A C B$.$[6]$Question 7 Code: 9709/12/O/N/14/5, Topic: Trigonometry$\text{(i)}$Show that the equation$1+\sin x \tan x=5 \cos x$can be expressed as$[3]$$$6 \cos ^{2} x-\cos x-1=0 .$$$\text{(ii)}$Hence solve the equation$1+\sin x \tan x=5 \cos x$for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$[3]$Question 8 Code: 9709/12/O/N/20/5, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}$are defined by $$\begin{array}{ll} \mathrm{f}(x)=4 x-2, & \text { for } x \in \mathbb{R}, \\ \displaystyle \mathrm{g}(x)=\frac{4}{x+1}, & \text { for } x \in \mathbb{R}, x \neq-1. \end{array}$$$\text{(a)}$Find the value of$\mathrm{fg}(7)$.$[1]\text{(b)}$Find the values of$x$for which$\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$.$[5]$Question 9 Code: 9709/11/O/N/11/6, Topic: Series$\text{(a)}$The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find the seventh term.$[4]\text{(b)}$A geometric progression has first term 1 and common ratio$r$. A second geometric progression has first term 4 and common ratio$\frac{1}{4} r$. The two progressions have the same sum to infinity,$S$. Find the values of$r$and$S$.$[3]$Question 10 Code: 9709/12/O/N/20/7, Topic: Differentiation, Integration The point$(4,7)$lies on the curve$y=\mathrm{f}(x)$and it is given that$\mathrm{f}^{\prime}(x)=6 x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}$.$\text{(a)}$A point moves along the curve in such a way that the$x$-coordinate is increasing at a constant rate of$0.12$units per second.$[3]$Find the rate of increase of the$y$-coordinate when$x=4$.$\text{(b)}$Find the equation of the curve.$[4]$Question 11 Code: 9709/12/O/N/10/9, Topic: Vectors The diagram shows a pyramid$O A B C P$in which the horizontal base$O A B C$is a square of side$10 \mathrm{~cm}$and the vertex$P$is$10 \mathrm{~cm}$vertically above$O$. The points$D, E, F, G$lie on$O P, A P, B P, C P$respectively and$D E F G$is a horizontal square of side$6 \mathrm{~cm}$. The height of$D E F G$above the base is$a \mathrm{~cm}$. Unit vectors$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$are parallel to$O A, O C$and$O D$respectively.$\text{(i)}$Show that$a=4$.$[2]\text{(ii)}$Express the vector$\overrightarrow{B G}$in terms of$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$.$[2]\text{(iii)}$Use a scalar product to find angle$G B A$.$[4]$Question 12 Code: 9709/11/O/N/12/9, Topic: Vectors The position vectors of points$A$and$B$relative to an origin$O$are$\mathbf{a}$and$\mathbf{b}$respectively. The position vectors of points$C$and$D$relative to$O$are$3 \mathbf{a}$and$2 \mathbf{b}$respectively. It is given that $$\mathbf{a}=\left(\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right) \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{l} 4 \\ 0 \\ 6 \end{array}\right)$$$\text{(i)}$Find the unit vector in the direction of$\overrightarrow{C D}$.$[3]\text{(ii)}$The point$E$is the mid-point of$C D$. Find angle$E O D$.$[6]\$

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