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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 3 4 4 7 5 5 6 6 6 8 11 69
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/O/N/12/1, Topic: Series The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first$n$terms is$n$. Find the value of the positive integer$n$.$[4]$Question 2 Code: 9709/12/O/N/13/1, Topic: Trigonometry Given that$\cos x=p$, where$x$is an acute angle in degrees, find, in terms of$p$,$\text{(i)}\sin x$,$[1]\text{(ii)}\tan x$,$[1]\text{(iii)}\tan \left(90^{\circ}-x\right)$.$[1]$Question 3 Code: 9709/13/O/N/11/2, Topic: Series The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is$\text{(i)}$an arithmetic progression,$[2]\text{(ii)}$a geometric progression.$[2]$Question 4 Code: 9709/11/O/N/16/2, Topic: Series Find the term independent of$x$in the expansion of$\displaystyle\left(2 x+\frac{1}{2 x^{3}}\right)^{8}$.$[4]$Question 5 Code: 9709/11/M/J/15/3, Topic: Series$\text{(i)}$Find the first three terms, in ascending powers of$x$, in the expansion of$\text{(a)}(1-x)^{6}[2]\text{(b)}(1+2 x)^{6}$.$[2]\text{(ii)}$Hence find the coefficient of$x^{2}$in the expansion of$[(1-x)(1+2 x)]^{6}$.$[3]$Question 6 Code: 9709/12/M/J/15/3, Topic: Series$\text{(i)}$Find the coefficients of$x^{2}$and$x^{3}$in the expansion of$(2-x)^{6}$.$[3]\text{(ii)}$Find the coefficient of$x^{3}$in the expansion of$(3 x+1)(2-x)^{6}$.$[2]$Question 7 Code: 9709/12/M/J/18/3, Topic: Series A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by$2 \%$of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained$8000 \mathrm{~kg}$of salt.$\text{(i)}$Find the amount of salt obtained in the 12 th week after the change.$[3]\text{(ii)}$Find the total amount of salt obtained in the first 12 weeks after the change.$[2]$Question 8 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$x$.$[3]\text{(b)}$Hence find the coefficient of$x^{2}$in the expansion of$(4+x)^{2}(3-2 x)^{5}$.$[3]$Question 9 Code: 9709/12/O/N/10/5, Topic: Series$\text{(a)}$The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum of the first$m$terms is zero. Find the value of$m$.$[3]\text{(b)}$A geometric progression, in which all the terms are positive, has common ratio$r$. The sum of the first$n$terms is less than$90 \%$of the sum to infinity. Show that$r^{n}>0.1[3]$Question 10 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 11 Code: 9709/12/M/J/11/9, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=3-4 \cos ^{k} x$, for$0 \leqslant x \leqslant \pi$, where$k$is a constant.$\text{(i)}$In the case where$k=2$,$\text{(a)}$find the range of$\mathrm{f}$,$[2]\text{(b)}$find the exact solutions of the equation$\mathrm{f}(x)=1$.$[3]\text{(ii)}$In the case where$k=1$,$\text{(a)}$sketch the graph of$y=\mathrm{f}(x)$,$[2]\text{(b)}$state, with a reason, whether$\mathrm{f}$has an inverse.$[1]$Question 12 Code: 9709/11/O/N/16/11, Topic: Coordinate geometry, Differentiation The point$P(3,5)$lies on the curve$\displaystyle y=\frac{1}{x-1}-\frac{9}{x-5}$.$\text{(i)}$Find the$x$-coordinate of the point where the normal to the curve at$P$intersects the$x$-axis.$[5]\text{(ii)}$Find the$x$-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers.$[6]\$

Worked solutions: P1, P3 & P6 (S1)

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