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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 5 7 6 8 6 7 9 11 10 83
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/17/2, Topic: Series The common ratio of a geometric progression is$r$. The first term of the progression is$\left(r^{2}-3 r+2\right)$and the sum to infinity is$S$.$\text{(i)}$Show that$S=2-r$.$\text{(ii)}$Find the set of possible values that$S$can take.$$Question 2 Code: 9709/11/M/J/19/2, Topic: Quadratics The line$4 y=x+c$, where$c$is a constant, is a tangent to the curve$y^{2}=x+3$at the point$P$on the curve.$\text{(i)}$Find the value of$c$.$\text{(ii)}$Find the coordinates of$P$.$$Question 3 Code: 9709/13/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$b$.$$Question 4 Code: 9709/12/M/J/13/3, Topic: Quadratics The straight line$y=m x+14$is a tangent to the curve$\displaystyle y=\frac{12}{x}+2$at the point$P$. Find the value of the constant$m$and the coordinates of$P$.$$Question 5 Code: 9709/13/M/J/14/5, Topic: Differentiation, Quadratics A function$\mathrm{f}$is such that$\displaystyle\mathrm{f}(x)=\frac{15}{2 x+3}$for$0 \leqslant x \leqslant 6\text{(i)}$Find an expression for$\mathrm{f}^{\prime}(x)$and use your result to explain why$\mathrm{f}$has an inverse.$\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$, and state the domain and range of$\mathrm{f}^{-1}$.$$Question 6 Code: 9709/12/M/J/21/5, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=2 x^{2}+3$for$x \geqslant 0$.$\text{(a)}$Find and simplify an expression for$\mathrm{ff}(x)$.$\text{(b)}$Solve the equation$\mathrm{ff}(x)=34 x^{2}+19$.$$Question 7 Code: 9709/12/M/J/12/6, Topic: Circular measure The diagram shows a metal plate made by removing a segment from a circle with centre$O$and radius$8 \mathrm{~cm}.$The line$A B$is a chord of the circle and angle$A O B=2.4$radians. Find$\text{(i)}$the length of$A B$,$\text{(ii)}$the perimeter of the plate,$\text{(iii)}$the area of the plate.$$Question 8 Code: 9709/12/M/J/16/6, Topic: Circular measure The diagram shows a circle with radius$r \mathrm{~cm}$and centre$O$. The line$P T$is the tangent to the circle at$P$and angle$P O T=\alpha$radians. The line$O T$meets the circle at$Q.\text{(i)}$Express the perimeter of the shaded region$P Q T$in terms of$r$and$\alpha$.$\text{(ii)}$In the case where$\alpha=\frac{1}{3} \pi$and$r=10$, find the area of the shaded region correct to 2 significant figures.$$Question 9 Code: 9709/12/M/J/18/8, Topic: Coordinate geometry Points$A$and$B$have coordinates$(h, h)$and$(4 h+6,5 h)$respectively. The equation of the perpendicular bisector of$A B$is$3 x+2 y=k$. Find the values of the constants$\mathrm{h}$and$k$.$$Question 10 Code: 9709/13/M/J/20/8, Topic: Series The first term of a progression is$\sin ^{2} \theta$, where$0 < \theta < \frac{1}{2} \pi$. The second term of the progression is$\sin ^{2} \theta \cos ^{2} \theta$.$\text{(a)}$Given that the progression is geometric, find the sum to infinity. It is now given instead that the progression is arithmetic.$\text{(b)} \quad \text{(i)}$Find the common difference of the progression in terms of$\sin \theta$.$ \quad \quad \text{(ii)}$Find the sum of the first 16 terms when$\theta=\frac{1}{3} \pi$.$$Question 11 Code: 9709/12/M/J/16/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 6 x-x^{2}-5$for$x \in \mathbb{R}$.$\text{(i)}$Find the set of values of$x$for which$\mathrm{f}(x) \leqslant 3$.$\text{(ii)}$Given that the line$y=m x+c$is a tangent to the curve$y=\mathrm{f}(x)$, show that$4 c=m^{2}-12 m+16$.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 6 x-x^{2}-5$for$x \geqslant k$, where$k$is a constant.$\text{(iii)}$Express$6 x-x^{2}-5$in the form$a-(x-b)^{2}$, where$a$and$b$are constants.$\text{(iv)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$\text{(v)}$For this value of$k$, find an expression for$\mathrm{g}^{-1}(x)$.$$Question 12 Code: 9709/12/M/J/21/11, Topic: Differentiation, Integration The gradient of a curve is given by$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=6(3 x-5)^{3}-k x^{2}$, where$k$is a constant. The curve has a stationary point at$(2,-3.5)$.$\text{(a)}$Find the value of$k$.$\text{(b)}$Find the equation of the curve.$\text{(c)}$Find$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.$\text{(d)}$Determine the nature of the stationary point at$(2,-3.5)$.$\$

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