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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 2 3 5 5 6 5 7 7 8 7 10 10 75
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/14/1, Topic: Trigonometry The diagram shows part of the graph of$y=a+b \sin x.$State the values of the constants$a$and$b$.$[2]$Question 2 Code: 9709/11/M/J/16/1, Topic: Series Find the term independent of$x$in the expansion of$\displaystyle\left(x-\frac{3}{2 x}\right)^{6}$.$[3]$Question 3 Code: 9709/13/M/J/19/1, Topic: Quadratics The function$\mathrm{f}$is defined by$\mathrm{f}(x)=x^{2}-4 x+8$for$x \in \mathbb{R}$.$\text{(i)}$Express$x^{2}-4 x+8$in the form$(x-a)^{2}+b$.$[2]\text{(ii)}$Hence find the set of values of$x$for which$\mathrm{f}(x)<9$, giving your answer in exact form.$[3]$Question 4 Code: 9709/13/M/J/13/2, Topic: Circular measure The diagram shows a circle$C$with centre$O$and radius$3 \mathrm{~cm}$. The radii$O P$and$O Q$are extended to$S$and$R$respectively so that$O R S$is a sector of a circle with centre$O$. Given that$P S=6 \mathrm{~cm}$and that the area of the shaded region is equal to the area of circle$C$,$\text{(i)}$show that angle$P O Q=\frac{1}{4} \pi$radians,$[3]\text{(ii)}$find the perimeter of the shaded region.$[2]$Question 5 Code: 9709/11/M/J/10/3, Topic: Series The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is$49$.$\text{(i)}$Find the first term of the progression and the common difference.$[4]$The$n$th term of the progression is 46.$\text{(ii)}$Find the value of$n$.$[2]$Question 6 Code: 9709/13/M/J/10/3, Topic: Functions The function$\mathrm{f}: x \mapsto a+b \cos x$is defined for$0 \leqslant x \leqslant 2 \pi$. Given that$\mathrm{f}(0)=10$and that$\mathrm{f}\left(\frac{2}{3} \pi\right)=1$, find$\text{(i)}$the values of$a$and$b$,$[2]\text{(ii)}$the range of$\mathrm{f}$,$[1]\text{(iii)}$the exact value of$\mathrm{f}\left(\frac{5}{6} \pi\right)$.$[2]$Question 7 Code: 9709/11/M/J/15/3, Topic: Series$\text{(i)}$Find the first three terms, in ascending powers of$x$, in the expansion of$\text{(a)}(1-x)^{6}[2]\text{(b)}(1+2 x)^{6}$.$[2]\text{(ii)}$Hence find the coefficient of$x^{2}$in the expansion of$[(1-x)(1+2 x)]^{6}$.$[3]$Question 8 Code: 9709/12/M/J/13/6, Topic: Vectors Relative to an origin$O$, the position vectors of points$A$and$B$are given by $$\overrightarrow{O A}=\mathbf{i}-2 \mathbf{j}+2 \mathbf{k} \quad \text { and } \quad \overrightarrow{O B}=3 \mathbf{i}+p \mathbf{j}+q \mathbf{k}$$ where$p$and$q$are constants.$\text{(i)}$State the values of$p$and$q$for which$\overrightarrow{O A}$is parallel to$\overrightarrow{O B}$.$[2]\text{(ii)}$In the case where$q=2 p$, find the value of$p$for which angle$B O A$is$90^{\circ}$.$[2]\text{(iii)}$In the case where$p=1$and$q=8$, find the unit vector in the direction of$\overrightarrow{A B}$.$[3]$Question 9 Code: 9709/13/M/J/10/7, Topic: Circular measure The diagram shows a metal plate$A B C D E F$which has been made by removing the two shaded regions from a circle of radius 10 cm and centre$O.$The parallel edges$A B$and$E D$are both of length 12 cm.$\text{(i)}$Show that angle$D O E$is 1.287 radians, correct to 4 significant figures.$[2]\text{(ii)}$Find the perimeter of the metal plate.$[3]\text{(iii)}$Find the area of the metal plate.$[3]$Question 10 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry The point$A$has coordinates$(p, 1)$and the point$B$has coordinates$(9,3 p+1)$, where$p$is a constant.$\text{(i)}$For the case where the distance$A B$is 13 units, find the possible values of$p$.$[3]\text{(ii)}$For the case in which the line with equation$2 x+3 y=9$is perpendicular to$A B$, find the value of$p$.$[4]$Question 11 Code: 9709/11/M/J/15/9, Topic: Differentiation The equation of a curve is$y=x^{3}+p x^{2}$, where$p$is a positive constant.$\text{(i)}$Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of$p$.$[4]\text{(ii)}$Find the nature of each of the stationary points.$[3]$Another curve has equation$y=x^{3}+p x^{2}+p x$.$\text{(iii)}$Find the set of values of$p$for which this curve has no stationary points.$[3]$Question 12 Code: 9709/11/M/J/17/10, Topic: Differentiation, Integration The diagram shows part of the curve$\displaystyle y=\frac{4}{5-3 x}$.$\text{(i)}$Find the equation of the normal to the curve at the point where$x=1$in the form$y=m x+c$, where$m$and$c$are constants.$[5]$The shaded region is bounded by the curve, the coordinate axes and the line$x=1$.$\text{(ii)}$Find, showing all necessary working, the volume obtained when this shaded region is rotated through$360^{\circ}$about the$x$-axis.$[5]\$

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