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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 4 5 3 3 5 6 8 7 8 9 11 11 80
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/10/1, Topic: Trigonometry The acute angle$x$radians is such that$\tan x=k$, where$k$is a positive constant. Express, in terms of$k$,$\text{(i)}\tan (\pi-x)$,$[1]\text{(ii)}\tan \left(\frac{1}{2} \pi-x\right)$,$[1]\text{(iii)}\sin x$.$[2]$Question 2 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry Find the coordinates of the point at which the perpendicular bisector of the line joining$(2,7)$to$(10,3)$meets the$x$-axis.$[5]$Question 3 Code: 9709/13/M/J/14/1, Topic: Series Find the coefficient of$x$in the expansion of$\displaystyle\left(x^{2}-\frac{2}{x}\right)^{5}$.$[3]$Question 4 Code: 9709/11/M/J/16/1, Topic: Series Find the term independent of$x$in the expansion of$\displaystyle\left(x-\frac{3}{2 x}\right)^{6}$.$[3]$Question 5 Code: 9709/13/M/J/11/2, Topic: Coordinate geometry Find the set of values of$m$for which the line$y=m x+4$intersects the curve$y=3 x^{2}-4 x+7$at two distinct points.$[5]$Question 6 Code: 9709/11/M/J/20/6, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-a \\ &\mathrm{~g}: x \mapsto 3 x+b \end{aligned} wherea$and$b$are constants.$\text{(a)}$Given that$\operatorname{gg}(2)=10$and$\mathrm{f}^{-1}(2)=14$, find the values of$a$and$b$.$[4]\text{(b)}$Using these values of$a$and$b$, find an expression for$\operatorname{gf}(x)$in the form$c x+d$, where$c$and$d$are constants.$[2]$Question 7 Code: 9709/11/M/J/15/7, Topic: Series$\text{(a)}$The third and fourth terms of a geometric progression are$\frac{1}{3}$and$\frac{2}{9}$respectively. Find the sum to infinity of the progression.$[4]\text{(b)}$A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector.$[4]$Question 8 Code: 9709/13/M/J/12/8, Topic: Circular measure In the diagram,$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]$Question 9 Code: 9709/13/M/J/13/8, Topic: Vectors The diagram shows a parallelogram$O A B C$in which $$\overrightarrow{O A}=\left(\begin{array}{r} 3 \\ 3 \\ -4 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{l} 5 \\ 0 \\ 2 \end{array}\right)$$$\text{(i)}$Use a scalar product to find angle$B O C$.$[6]\text{(ii)}$Find a vector which has magnitude 35 and is parallel to the vector$\overrightarrow{O C}$.$[2]$Question 10 Code: 9709/13/M/J/17/9, Topic: Functions$\text{(i)}$Express$9 x^{2}-6 x+6$in the form$(a x+b)^{2}+c$, where$a, b$and$c$are constants.$[3]$The function$\mathrm{f}$is defined by$\mathrm{f}(x)=9 x^{2}-6 x+6$for$x \geqslant p$, where$p$is a constant.$\text{(ii)}$State the smallest value of$p$for which$\mathrm{f}$is a one-one function.$[1]\text{(iii)}$For this value of$p$, obtain an expression for$\mathrm{f}^{-1}(x)$, and state the domain of$\mathrm{f}^{-1}$.$[4]\text{(iv)}$State the set of values of$q$for which the equation$\mathrm{f}(x)=q$has no solution.$[1]$Question 11 Code: 9709/12/M/J/11/10, Topic: Series$\text{(a)}$A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is$5 \mathrm{~cm}$, find the perimeter of the smallest sector.$[6]\text{(b)}$The first, second and third terms of a geometric progression are$2 k+3, k+6$and$k$, respectively. Given that all the terms of the geometric progression are positive, calculate$\text{(i)}$the value of the constant$k$,$[3]\text{(ii)}$the sum to infinity of the progression.$[2]$Question 12 Code: 9709/12/M/J/17/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=3 \tan \left(\frac{1}{2} x\right)-2$, for$-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.$\text{(i)}$Solve the equation$\mathrm{f}(x)+4=0$, giving your answer correct to 1 decimal place.$[3]\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$and find the domain of$\mathrm{f}^{-1}$.$[5]\text{(iii)}$Sketch, on the same diagram, the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$.$[3]\$

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