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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 3 6 7 6 6 7 7 7 10 8 11 11 89
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/16/1, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 10-3 x, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{10}{3-2 x}, \quad x \in \mathbb{R}, x \neq \frac{3}{2} \end{aligned} Solve the equation\mathrm{ff}(x)=\operatorname{gf}(2)$.$[3]$Question 2 Code: 9709/12/M/J/10/5, Topic: Vectors Relative to an origin$O$, the position vectors of the points$A$and$B$are given by $$\overrightarrow{O A}=\left(\begin{array}{r} -2 \\ 3 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{c} 4 \\ 1 \\ p \end{array}\right).$$$\text{(i)}$Find the value of$p$for which$\overrightarrow{O A}$is perpendicular to$\overrightarrow{O B}$.$[2]\text{(ii)}$Find the values of$p$for which the magnitude of$\overrightarrow{A B}$is 7.$[4]$Question 3 Code: 9709/13/M/J/11/5, Topic: Vectors In the diagram,$O A B C D E F G$is a rectangular block in which$O A=O D=6 \mathrm{~cm}$and$A B=12 \mathrm{~cm}$. The unit vectors i,$\mathbf{j}$and$\mathbf{k}$are parallel to$\overrightarrow{O A}, \overrightarrow{O C}$and$\overrightarrow{O D}$respectively. The point$P$is the mid-point of$D G, Q$is the centre of the square face$C B F G$and$R$lies on$A B$such that$A R=4 \mathrm{~cm}$.$\text{(i)}$Express each of the vectors$\overrightarrow{P Q}$and$\overrightarrow{R Q}$in terms of$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$.$[3]\text{(ii)}$Use a scalar product to find angle$R Q P$.$[4]$Question 4 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry The diagram shows the curve$y=7 \sqrt{x}$and the line$y=6 x+k$, where$k$is a constant. The curve and the line intersect at the points$A$and$B$.$\text{(i)}$For the case where$k=2$, find the$x$-coordinates of$A$and$B$.$[4]\text{(ii)}$Find the value of$k$for which$y=6 x+k$is a tangent to the curve$y=7 \sqrt{x}$.$[2]$Question 5 Code: 9709/12/M/J/15/6, Topic: Trigonometry A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height,$h \mathrm{~m}$, of a passenger above the ground is given by the formula$h=60(1-\cos k t)$. In this formula,$k$is a constant,$t$is the time in minutes that has elapsed since the passenger started the ride at ground level and$k t$is measured in radians.$\text{(i)}$Find the greatest height of the passenger above the ground.$[1]$One complete revolution of the wheel takes 30 minutes.$\text{(ii)}$Show that$k=\frac{1}{15} \pi$.$[2]\text{(iii)}$Find the time for which the passenger is above a height of$90 \mathrm{~m}$.$[3]$Question 6 Code: 9709/12/M/J/19/6, Topic: Trigonometry The equation of a curve is$y=3 \cos 2 x$and the equation of a line is$\displaystyle 2 y+\frac{3 x}{\pi}=5$.$\text{(i)}$State the smallest and largest values of$y$for both the curve and the line for$0 \leqslant x \leqslant 2 \pi$.$[3]\text{(ii)}$Sketch, on the same diagram, the graphs of$y=3 \cos 2 x$and$\displaystyle 2 y+\frac{3 x}{\pi}=5$for$0 \leqslant x \leqslant 2 \pi$.$[3]\text{(iii)}$State the number of solutions of the equation$\displaystyle 6 \cos 2 x=5-\frac{3 x}{\pi}$for$0 \leqslant x \leqslant 2 \pi$.$[1]$Question 7 Code: 9709/11/M/J/16/7, Topic: Circular measure In the diagram,$A O B$is a quarter circle with centre$O$and radius$r$. The point$C$lies on the arc$A B$and the point$D$lies on$O B.$The line$C D$is parallel to$A O$and angle$A O C=\theta$radians.$\text{(i)}$Express the perimeter of the shaded region in terms of$r, \theta$and$\pi$.$[4]\text{(ii)}$For the case where$r=5 \mathrm{~cm}$and$\theta=0.6$, find the area of the shaded region.$[3]$Question 8 Code: 9709/13/M/J/12/8, Topic: Circular measure In the diagram,$A B$is an arc of a circle with centre$O$and radius$r$. The line$X B$is a tangent to the circle at$B$and$A$is the mid-point of$O X$.$\text{(i)}$Show that angle$A O B=\frac{1}{3} \pi$radians.$[2]$Express each of the following in terms of$r, \pi$and$\sqrt{3}$:$\text{(ii)}$the perimeter of the shaded region,$[3]\text{(iii)}$the area of the shaded region.$[2]$Question 9 Code: 9709/13/M/J/19/9, Topic: Trigonometry The function$\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$is defined for$0 \leqslant x \leqslant \pi$, where$p$and$q$are positive constants. The diagram shows the graph of$y=\mathrm{f}(x)$.$\text{(i)}$In terms of$p$and$q$, state the range of$\mathrm{f}$.$[2]\text{(ii)}$State the number of solutions of the following equations.$\quad\text{(a)}\mathrm{f}(x)=p+q[1]\quad\text{(b)}\mathrm{f}(x)=q[1]\quad\text{(c)}\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q[1]\text{(iii)}$For the case where$p=3$and$q=2$, solve the equation$\mathrm{f}(x)=4$, showing all necessary working.$[5]$Question 10 Code: 9709/11/M/J/21/10, Topic: Coordinate geometry The equation of a circle is$x^{2}+y^{2}-4 x+6 y-77=0$.$\text{(a)}$Find the$x$-coordinates of the points$A$and$B$where the circle intersects the$x$-axis.$[2]\text{(b)}$Find the point of intersection of the tangents to the circle at A and B.$[6]$Question 11 Code: 9709/13/M/J/18/11, Topic: Differentiation, Integration The diagram shows part of the curve$y=(x+1)^{2}+(x+1)^{-1}$and the line$x=1$. The point$A$is the minimum point on the curve.$\text{(i)}$Show that the$x$-coordinate of$A$satisfies the equation$2(x+1)^{3}=1$and find the exact value of$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$at$A$.$[5]\text{(ii)}$Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$[6]$Question 12 Code: 9709/12/M/J/21/12, Topic: Circular measure The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are$A, B, C, D$,$E$and$F$. Points$P$and$Q$are situated where straight sections of the rope meet the pipe with centre$A.\text{(a)}$Show that angle$P A Q=\frac{1}{3} \pi$radians.$[2]\text{(b)}$Find the length of the rope.$[4]\text{(c)}$Find the area of the hexagon$A B C D E F$, giving your answer in terms of$\sqrt{3}$.$[2]\text{(d)}$Find the area of the complete region enclosed by the rope.$[3]\$

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