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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 4 7 7 7 5 10 6 8 10 12 11 91
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}$.$[4]$Question 2 Code: 9709/12/M/J/16/2, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{8}{(5-2 x)^{2}}.$Given that the curve passes through$(2,7)$, find the equation of the curve.$[4]$Question 3 Code: 9709/11/M/J/18/5, Topic: Coordinate geometry The diagram shows a kite$O A B C$in which$A C$is the line of symmetry. The coordinates of$A$and$C$are$(0,4)$and$(8,0)$respectively and$O$is the origin.$\text{(i)}$Find the equations of$A C$and$O B$.$[4]\text{(ii)}$Find, by calculation, the coordinates of$B$.$[3]$Question 4 Code: 9709/12/M/J/14/6, Topic: Series The$1 \mathrm{st}, 2 \mathrm{nd}$and$3 \mathrm{rd}$terms of a geometric progression are the$1 \mathrm{st}, 9$th and$21 \mathrm{st}$terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is$r$, where$r \neq 1$. Find$\text{(i)}$the value of$r$,$[4]\text{(ii)}$the 4 th term of each progression.$[3]$Question 5 Code: 9709/13/M/J/17/7, Topic: Circular measure The diagram shows two circles with centres$A$and$B$having radii$8 \mathrm{~cm}$and$10 \mathrm{~cm}$respectively. The two circles intersect at$C$and$D$where$C A D$is a straight line and$A B$is perpendicular to$C D$.$\text{(i)}$Find angle$A B C$in radians.$[1]\text{(ii)}$Find the area of the shaded region.$[6]$Question 6 Code: 9709/11/M/J/21/7, Topic: Trigonometry$\text{(a)}$Prove the identity$\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta} \equiv 1-\tan ^{2} \theta$.$[2]\text{(b)}$Hence solve the equation$\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta}=2 \tan ^{4} \theta$for$0^{\circ} \leqslant \theta \leqslant 180^{\circ}$.$[3]$Question 7 Code: 9709/11/M/J/10/8, Topic: Coordinate geometry The diagram shows a triangle$A B C$in which$A$is$(3,-2)$and$B$is$(15,22)$. The gradients of$A B, A C$and$B C$are$2 m,-2 m$and$m$respectively, where$m$is a positive constant.$\text{(i)}$Find the gradient of$A B$and deduce the value of$m$.$[2]\text{(ii)}$Find the coordinates of$C$.$[4]$The perpendicular bisector of$A B$meets$B C$at$D$.$\text{(iii)}$Find the coordinates of$D$.$[4]$Question 8 Code: 9709/11/M/J/14/8, Topic: Vectors 8 Relative to an origin$O$, the position vectors of points$A$and$B$are given by $$\overrightarrow{O A}=\left(\begin{array}{c} 3 p \\ 4 \\ p^{2} \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{c} -p \\ -1 \\ p^{2} \end{array}\right)$$$\text{(i)}$Find the values of$p$for which angle$A O B$is$90^{\circ}$.$[3]\text{(ii)}$For the case where$p=3$, find the unit vector in the direction of$\overrightarrow{B A}$.$[3]$Question 9 Code: 9709/13/M/J/15/9, Topic: Series$\text{(a)}$The first term of an arithmetic progression is$-2222$and the common difference is 17. Find the value of the first positive term.$[3]\text{(b)}$The first term of a geometric progression is$\sqrt{3}$and the second term is$2 \cos \theta$, where$0 < \theta < \pi$. Find the set of values of$\theta$for which the progression is convergent.$[5]$Question 10 Code: 9709/13/M/J/21/10, Topic: Coordinate geometry Points$A(-2,3), B(3,0)$and$C(6,5)$lie on the circumference of a circle with centre$D$.$\text{(a)}$Show that angle$A B C=90^{\circ}$.$[2]\text{(b)}$Hence state the coordinates of$D$.$[1]\text{(c)}$Find an equation of the circle.$[2]$The point$E$lies on the circumference of the circle such that$BE$is a diameter.$\text{(d)}$Find an equation of the tangent to the circle at$E$.$[5]$Question 11 Code: 9709/12/M/J/19/11, Topic: Differentiation, Integration The diagram shows part of the curve$\displaystyle y=\sqrt{(} 4 x+1)+\frac{9}{\sqrt{(4 x+1)}}$and the minimum point$M$.$\text{(i)}$Find expressions for$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and$\displaystyle\int y \mathrm{~d} x$.$[6]\text{(ii)}$Find the coordinates of$M$.$[3]$The shaded region is bounded by the curve, the$y$-axis and the line through$M$parallel to the$x$-axis.$\text{(iii)}$Find, showing all necessary working, the area of the shaded region.$[3]$Question 12 Code: 9709/12/M/J/20/11, Topic: Coordinate geometry The equation of a circle with centre$C$is$x^{2}+y^{2}-8 x+4 y-5=0$.$\text{(a)}$Find the radius of the circle and the coordinates of$C$.$[3]$The point$P(1,2)$lies on the circle.$\text{(b)}$Show that the equation of the tangent to the circle at$P$is$4 y=3 x+5$. The point$Q$also lies on the circle and$P Q$is parallel to the$x$-axis.$[3]\text{(c)}$Write down the coordinates of$Q$.$[2]$The tangents to the circle at$P$and$Q$meet at$T$.$\text{(d)}$Find the coordinates of$T$.$[3]\$

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