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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 5 6 6 7 7 7 7 7 7 11 11 15 96
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/11/3, Topic: Coordinate geometry The line$\displaystyle\frac{x}{a}+\frac{y}{b}=1$, where$a$and$b$are positive constants, meets the$x$-axis at$P$and the$y$-axis at$Q$. Given that$P Q=\sqrt{(} 45)$and that the gradient of the line$P Q$is$-\frac{1}{2}$, find the values of$a$and$b$.$[5]$Question 2 Code: 9709/13/M/J/12/4, Topic: Trigonometry$\text{(i)}$Solve the equation$\sin 2 x+3 \cos 2 x=0$for$0^{\circ} \leqslant x \leqslant 360^{\circ}$.$[5]\text{(ii)}$How many solutions has the equation$\sin 2 x+3 \cos 2 x=0$for$0^{\circ} \leqslant x \leqslant 1080^{\circ}$?$[1]$Question 3 Code: 9709/13/M/J/21/4, Topic: Trigonometry$\text{(a)}$Show that the equation$[2]$$$\frac{\tan x+\sin x}{\tan x-\sin x}=k$$ where$k$is a constant, may be expressed as $$\frac{1+\cos x}{1-\cos x}=k$$$\text{(b)}$Hence express$\cos x$in terms of$k$.$[2]\text{(c)}$Hence solve the equation$\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$for$-\pi < x < \pi$.$[2]$Question 4 Code: 9709/13/M/J/10/5, Topic: Differentiation, Integration The equation of a curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{\sqrt{(} 3 x-2)}.$Given that the curve passes through the point$P(2,11)$, find$\text{(i)}$the equation of the normal to the curve at$P$,$[3]\text{(ii)}$the equation of the curve.$[4]$Question 5 Code: 9709/12/M/J/13/5, Topic: Trigonometry It is given that$a=\sin \theta-3 \cos \theta$and$b=3 \sin \theta+\cos \theta$, where$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$\text{(i)}$Show that$a^{2}+b^{2}$has a constant value for all values of$\theta$.$[3]\text{(ii)}$Find the values of$\theta$for which$2 a=b$.$[4]$Question 6 Code: 9709/12/M/J/17/6, Topic: Integration The diagram shows the straight line$x+y=5$intersecting the curve$\displaystyle y=\frac{4}{x}$at the points$A(1,4)$and$B(4,1)$. Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$[7]$Question 7 Code: 9709/12/M/J/15/7, Topic: Coordinate geometry The point$C$lies on the perpendicular bisector of the line joining the points$A(4,6)$and$B(10,2)$.$C$also lies on the line parallel to$A B$through$(3,11)$.$\text{(i)}$Find the equation of the perpendicular bisector of$A B$.$[4]\text{(ii)}$Calculate the coordinates of$C$.$[3]$Question 8 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 9 Code: 9709/12/M/J/20/7, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$2 r$, and angle$A O B=\frac{1}{6} \pi$radians. The point$C$is the midpoint of$O A$.$\text{(a)}$Show that the exact length of$B C$is$r \sqrt{5-2 \sqrt{3}}$.$[2]\text{(b)}$Find the exact perimeter of the shaded region.$[2]\text{(c)}$Find the exact area of the shaded region.$[3]$Question 10 Code: 9709/13/M/J/10/9, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=x+\frac{4}{x}$which has a minimum point at$M .$The line$y=5$intersects the curve at the points$A$and$B$.$\text{(i)}$Find the coordinates of$A, B$and$M$.$[5]\text{(ii)}$Find the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$[6]$Question 11 Code: 9709/11/M/J/13/9, Topic: Differentiation, Integration A curve has equation$y=\mathrm{f}(x)$and is such that$\mathrm{f}^{\prime}(x)=3 x^{\frac{1}{2}}+3 x^{-\frac{1}{2}}-10$.$\text{(i)}$By using the substitution$u=x^{\frac{1}{2}}$, or otherwise, find the values of$x$for which the curve$y=\mathrm{f}(x)$has stationary points.$[4]\text{(ii)}$Find$\mathrm{f}^{\prime \prime}(x)$and hence, or otherwise, determine the nature of each stationary point.$[3]\text{(iii)}$It is given that the curve$y=\mathrm{f}(x)$passes through the point$(4,-7)$. Find$\mathrm{f}(x)$.$[4]$Question 12 Code: 9709/12/M/J/14/10, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 2 x-3, \quad x \in \mathbb{R}, \\ &\mathrm{g}: x \mapsto x^{2}+4 x, \quad x \in \mathbb{R}. \end{aligned}\text{(i)}$Solve the equation$\mathrm{ff}(x)=11$.$[2]\text{(ii)}$Find the range of$\mathrm{g}$.$[2]\text{(iii)}$Find the set of values of$x$for which$\mathrm{g}(x)>12$.$[3]\text{(iv)}$Find the value of the constant$p$for which the equation$\operatorname{gf}(x)=p$has two equal roots.$[3]$Function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto x^{2}+4 x$for$x \geqslant k$, and it is given that$\mathrm{h}$has an inverse.$\text{(v)}$State the smallest possible value of$k$.$[1]\text{(vi)}$Find an expression for$\mathrm{h}^{-1}(x)$.$[4]\$

Worked solutions: P1, P3 & P6 (S1)

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