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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 6 4 6 8 9 11 9 9 11 10 12 11 106
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/10/4, Topic: Coordinate geometry In the diagram,$A$is the point$(-1,3)$and$B$is the point$(3,1)$. The line$L_{1}$passes through$A$and is parallel to$O B$. The line$L_{2}$passes through$B$and is perpendicular to$A B$. The lines$L_{1}$and$L_{2}$meet at$C$. Find the coordinates of$C$.$[6]$Question 2 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry Points$A$and$B$have coordinates$(8,3)$and$(p, q)$respectively. The equation of the perpendicular bisector of$A B$is$y=-2 x+4$. Find the values of$p$and$q$.$[4]$Question 3 Code: 9709/11/M/J/14/7, Topic: Coordinate geometry The coordinates of points$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]$Question 4 Code: 9709/11/M/J/18/7, Topic: Vectors Relative to an origin$O$, the position vectors of the points$A, B$and$C$are given by $$\overrightarrow{O A}=\left(\begin{array}{r} 1 \\ -3 \\ 2 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} -1 \\ 3 \\ 5 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 3 \\ 1 \\ -2 \end{array}\right)$$$\text{(i)}$Find$\overrightarrow{A C}$.$[1]\text{(ii)}$The point$M$is the mid-point of$A C$. Find the unit vector in the direction of$\overrightarrow{O M}$.$[3]\text{(iii)}$Evaluate$\overrightarrow{A B} \cdot \overrightarrow{A C}$and hence find angle$B A C$.$[4]$Question 5 Code: 9709/11/M/J/17/8, Topic: Circular measure In the diagram,$O A X B$is a sector of a circle with centre$O$and radius$10 \mathrm{~cm}.$The length of the chord$A B$is$12 \mathrm{~cm}$. The line$O X$passes through$M$, the mid-point of$A B$, and$O X$is perpendicular to$A B$. The shaded region is bounded by the chord$A B$and by the arc of a circle with centre$X$and radius$X A$.$\text{(i)}$Show that angle$A X B$is$2.498$radians, correct to 3 decimal places.$[3]\text{(ii)}$Find the perimeter of the shaded region.$[3]\text{(iii)}$Find the area of the shaded region.$[3]$Question 6 Code: 9709/11/M/J/10/10, Topic: Vectors The diagram shows the parallelogram$O A B C$. Given that$\overrightarrow{O A}=\mathbf{i}+3 \mathbf{j}+3 \mathbf{k}$and$\overrightarrow{O C}=3 \mathbf{i}-\mathbf{j}+\mathbf{k}$, find$\text{(i)}$the unit vector in the direction of$\overrightarrow{O B}$,$[3]\text{(ii)}$the acute angle between the diagonals of the parallelogram,$[5]\text{(iii)}$the perimeter of the parallelogram, correct to 1 decimal place.$[3]$Question 7 Code: 9709/13/M/J/13/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where$k$is a constant.$\text{(i)}$In the case where$k=3$, solve the equation$\mathrm{ff}(x)=25$.$[2]$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.$\text{(ii)}$Find the set of values of$k$for which the equation$\mathrm{f}(x)=\mathrm{g}(x)$has no real solutions.$[3]$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.$\text{(iii)}$Find an expression for$\mathrm{h}^{-1}(x)$.$[4]$Question 8 Code: 9709/12/M/J/20/10, Topic: Differentiation The equation of a curve is$y=54 x-(2 x-7)^{3}$.$\text{(a)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.$[4]\text{(b)}$Find the coordinates of each of the stationary points on the curve.$[3]\text{(c)}$Determine the nature of each of the stationary points.$[2]$Question 9 Code: 9709/11/M/J/11/11, Topic: Functions, Coordinate geometry Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned}\text{(i)}$Find and simplify expressions for$\mathrm{fg}(x)$and$\operatorname{gf}(x)$.$[2]\text{(ii)}$Hence find the value of$a$for which$\mathrm{fg}(a)=\operatorname{gf}(a)$.$[3]\text{(iii)}$Find the value of$b(b \neq a)$for which$\mathrm{g}(b)=b$.$[2]\text{(iv)}$Find and simplify an expression for$\mathrm{f}^{-1} \mathrm{~g}(x)$.$[2]$The function$\mathrm{h}$is defined by $$\mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0$$$\text{(v)}$Find an expression for$\mathrm{h}^{-1}(x)$.$[2]$Question 10 Code: 9709/13/M/J/15/11, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$r$. The point$C$on$O B$is such that angle$A C O$is a right angle. Angle$A O B$is$\alpha$radians and is such that$A C$divides the sector into two regions of equal area.$\text{(i)}$Show that$\sin \alpha \cos \alpha=\frac{1}{2} \alpha$.$[4]$It is given that the solution of the equation in part$\text{(i)}$is$\alpha=0.9477$, correct to 4 decimal places.$\text{(ii)}$Find the ratio perimeter of region$O A C$: perimeter of region$A C B$, giving your answer in the form$k: 1$, where$k$is given correct to 1 decimal place.$[5]\text{(iii)}$Find angle$A O B$in degrees.$[1]$Question 11 Code: 9709/11/M/J/20/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{x+2}$and the line$2 y+x=8$, intersecting at points$A$and$B$. The point$C$lies on the curve and the tangent to the curve at$C$is parallel to$A B$.$\text{(a)}$Find, by calculation, the coordinates of$A, B$and$C$.$[6]\text{(b)}$Find the volume generated when the shaded region, bounded by the curve and the line, is rotated through$360^{\circ}$about the$x$-axis.$[6]$Question 12 Code: 9709/11/M/J/14/12, Topic: Integration, Differentiation A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{\frac{1}{2}}-x^{-\frac{1}{2}}$. The curve passes through the point$\left(4, \frac{2}{3}\right)$.$\text{(i)}$Find the equation of the curve.$[4]\text{(ii)}$Find$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.$[2]\text{(iii)}$Find the coordinates of the stationary point and determine its nature.$[5]\$

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