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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 5 6 7 8 6 8 8 7 11 11 86
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/12/2, Topic: Series Find the coefficient of$x^{6}$in the expansion of$\displaystyle\left(2 x^{3}-\frac{1}{x^{2}}\right)^{7}$.$$Question 2 Code: 9709/11/M/J/19/2, Topic: Quadratics The line$4 y=x+c$, where$c$is a constant, is a tangent to the curve$y^{2}=x+3$at the point$P$on the curve.$\text{(i)}$Find the value of$c$.$\text{(ii)}$Find the coordinates of$P$.$$Question 3 Code: 9709/12/M/J/11/3, Topic: Quadratics The equation$x^{2}+p x+q=0$, where$p$and$q$are constants, has roots$-3$and 5.$\text{(i)}$Find the values of$p$and$q$.$\text{(ii)}$Using these values of$p$and$q$, find the value of the constant$r$for which the equation$x^{2}+p x+q+r=0$has equal roots.$$Question 4 Code: 9709/11/M/J/21/3, Topic: Series$\text{(a)}$Find the first three terms in the expansion of$(3-2 x)^{5}$in ascending powers of$x$.$\text{(b)}$Hence find the coefficient of$x^{2}$in the expansion of$(4+x)^{2}(3-2 x)^{5}$.$$Question 5 Code: 9709/11/M/J/15/5, Topic: Quadratics A piece of wire of length$24 \mathrm{~cm}$is bent to form the perimeter of a sector of a circle of radius$r \mathrm{~cm}$.$\text{(i)}$Show that the area of the sector,$A \mathrm{~cm}^{2}$, is given by$A=12 r-r^{2}$.$\text{(ii)}$Express$A$in the form$a-(r-b)^{2}$, where$a$and$b$are constants.$\text{(iii)}$Given that$r$can vary, state the greatest value of$A$and find the corresponding angle of the sector.$$Question 6 Code: 9709/11/M/J/17/6, Topic: Differentiation The horizontal base of a solid prism is an equilateral triangle of side$x \mathrm{~cm}$. The sides of the prism are vertical. The height of the prism is$h \mathrm{~cm}$and the volume of the prism is$2000 \mathrm{~cm}^{3}$.$\text{(i)}$Express$\mathrm{h}$in terms of$x$and show that the total surface area of the prism,$A \mathrm{~cm}^{2}$, is given by$$$$A=\frac{\sqrt{3}}{2} x^{2}+\frac{24000}{\sqrt{3}} x^{-1}$$$\text{(ii)}$Given that$x$can vary, find the value of$x$for which$A$has a stationary value.$\text{(iii)}$Determine, showing all necessary working, the nature of this stationary value.$$Question 7 Code: 9709/13/M/J/17/6, Topic: Quadratics The line$3 y+x=25$is a normal to the curve$y=x^{2}-5 x+k$. Find the value of the constant$k$.$$Question 8 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.$\text{(b)}$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.$$Question 9 Code: 9709/12/M/J/10/8, Topic: Differentiation A solid rectangular block has a square base of side$x \mathrm{~cm}$. The height of the block is$h \mathrm{~cm}$and the total surface area of the block is 96 cm$^{2}$.$\text{(i)}$Express$\mathrm{h}$in terms of$x$and show that the volume,$V \mathrm{~cm}^{3}$, of the block is given by$$$$V=24 x-\frac{1}{2} x^{3}$$ Given that$x$can vary,$\text{(ii)}$find the stationary value of$V$,$\text{(iii)}$determine whether this stationary value is a maximum or a minimum.$$Question 10 Code: 9709/13/M/J/12/8, Topic: Circular measure In the diagram,$A B$is an arc of a circle with centre$O$and radius$r$. The line$X B$is a tangent to the circle at$B$and$A$is the mid-point of$O X$.$\text{(i)}$Show that angle$A O B=\frac{1}{3} \pi$radians.$$Express each of the following in terms of$r, \pi$and$\sqrt{3}$:$\text{(ii)}$the perimeter of the shaded region,$\text{(iii)}$the area of the shaded region.$$Question 11 Code: 9709/12/M/J/17/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=3 \tan \left(\frac{1}{2} x\right)-2$, for$-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.$\text{(i)}$Solve the equation$\mathrm{f}(x)+4=0$, giving your answer correct to 1 decimal place.$\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$and find the domain of$\mathrm{f}^{-1}$.$\text{(iii)}$Sketch, on the same diagram, the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$.$$Question 12 Code: 9709/11/M/J/19/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{3}{\sqrt{(1+4 x)}}$and a point$P(2,1)$lying on the curve. The normal to the curve at$P$intersects the$x$-axis at$Q$.$\text{(i)}$Show that the$x$-coordinate of$Q$is$\frac{16}{9}$.$\text{(ii)}$Find, showing all necessary working, the area of the shaded region.$\$

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