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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 3 5 6 6 6 6 8 9 7 8 13 10 87
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/11/1, Topic: Integration Find$\displaystyle\int\left(x^{3}+\frac{1}{x^{3}}\right) \mathrm{d} x$.$[3]$Question 2 Code: 9709/11/M/J/14/4, Topic: Differentiation A curve has equation$\displaystyle y=\frac{4}{(3 x+1)^{2}}$. Find the equation of the tangent to curve at the point where the line$x=-1$intersects the curve.$[5]$Question 3 Code: 9709/13/M/J/14/4, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$.$[3]\text{(ii)}$Hence solve the equation$\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$for$0 \leqslant x \leqslant 2 \pi$.$[3]$Question 4 Code: 9709/11/M/J/19/4, Topic: Coordinate geometry The diagram shows a trapezium$A B C D$in which the coordinates of$A, B$and$C$are$(4,0),(0,2)$and$(h, 3 h)$respectively. The lines$B C$and$A D$are parallel, angle$A B C=90^{\circ}$and$C D$is parallel to the$x$-axis.$\text{(i)}$Find, by calculation, the value of$\mathrm{h}$.$[3]\text{(ii)}$Hence find the coordinates of$D$.$[3]$Question 5 Code: 9709/13/M/J/13/5, Topic: Trigonometry$\text{(i)}$Sketch, on the same diagram, the curves$y=\sin 2 x$and$y=\cos x-1$for$0 \leqslant x \leqslant 2 \pi$.$[4]\text{(ii)}$Hence state the number of solutions, in the interval$0 \leqslant x \leqslant 2 \pi$, of the equations$\text{(a)}2 \sin 2 x+1=0$,$[1]\text{(b)}\sin 2 x-\cos x+1=0$.$[1]$Question 6 Code: 9709/11/M/J/20/7, Topic: Trigonometry$\text{(a)}$Prove the identity$\displaystyle \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta} \equiv \frac{2}{\cos \theta}$.$[3]\text{(b)}$Hence solve the equation$\displaystyle \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=\frac{3}{\sin \theta}$, for$0 \leqslant \theta \leqslant 2 \pi$.$[3]$Question 7 Code: 9709/13/M/J/14/8, Topic: Quadratics$\text{(i)}$Express$2 x^{2}-10 x+8$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants, and use your answer to state the minimum value of$2 x^{2}-10 x+8$.$[4]\text{(ii)}$Find the set of values of$k$for which the equation$2 x^{2}-10 x+8=k x$has no real roots.$[4]$Question 8 Code: 9709/12/M/J/15/8, Topic: Series$\text{(a)}$The first, second and last terms in an arithmetic progression are 56,53 and$-22$respectively. Find the sum of all the terms in the progression.$[4]\text{(b)}$The first, second and third terms of a geometric progression are$2 k+6,2 k$and$k+2$respectively, where$k$is a positive constant.$\text{(i)}$Find the value of$k$.$[3]\text{(ii)}$Find the sum to infinity of the progression.$[2]$Question 9 Code: 9709/13/M/J/21/8, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined as follows: \begin{aligned}&\mathrm{f}: x \mapsto x^{2}-1 \text { for } x<0, \\&\mathrm{~g}: x \mapsto \frac{1}{2 x+1} \text { for } x<-\frac{1}{2}.\end{aligned}\text{(a)}$Solve the equation$\mathrm{fg}(x)=3$.$[4]\text{(b)}$Find an expression for$(\mathrm{fg})^{-1}(x)$.$[3]$Question 10 Code: 9709/13/M/J/13/9, Topic: Series$\text{(a)}$In an arithmetic progression, the sum,$S_{n}$, of the first$n$terms is given by$S_{n}=2 n^{2}+8 n$. Find the first term and the common difference of the progression.$[3]\text{(b)}$The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the$1 \mathrm{st}$term, the 9 th term and the$n$th term respectively of an arithmetic progression. Find the value of$n$.$[5]$Question 11 Code: 9709/13/M/J/19/10, Topic: Coordinate geometry, Integration The diagram shows part of the curve with equation$y=(3 x+4)^{\frac{1}{2}}$and the tangent to the curve at the point$A$. The$x$-coordinate of$A$is 4.$\text{(i)}$Find the equation of the tangent to the curve at$A$.$[5]\text{(ii)}$Find, showing all necessary working, the area of the shaded region.$[5]\text{(iii)}$A point is moving along the curve. At the point$P$the$y$-coordinate is increasing at half the rate at which the$x$-coordinate is increasing. Find the$x$-coordinate of$P$.$[3]$Question 12 Code: 9709/13/M/J/12/11, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=8-(x-2)^{2}$, for$x \in \mathbb{R}$.$\text{(i)}$Find the coordinates and the nature of the stationary point on the curve$y=\mathrm{f}(x)$.$[3]$The function$\mathrm{g}$is such that$\mathrm{g}(x)=8-(x-2)^{2}$, for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(ii)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$[1]$For this value of$k$,$\text{(iii)}$find an expression for$\mathrm{g}^{-1}(x)$,$[3]\text{(iv)}$sketch, on the same diagram, the graphs of$y=\mathrm{g}(x)$and$y=\mathrm{g}^{-1}(x)$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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