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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 5 6 7 6 4 7 9 9 10 7 9 84
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/18/1, Topic: Series The coefficient of$x^{2}$in the expansion of$\displaystyle \left(2+\frac{x}{2}\right)^{6}+(a+x)^{5}$is 330 . Find the value of the constant$a$.$[5]$Question 2 Code: 9709/13/M/J/10/2, Topic: Series$\text{(i)}$Find the first three terms, in descending powers of$x$, in the expansion of$\displaystyle \left(x-\frac{2}{x}\right)^{6}$.$[3]\text{(ii)}$Find the coefficient of$x^{4}$in the expansion of$\displaystyle \left(1+x^{2}\right)\left(x-\frac{2}{x}\right)^{6}$.$[2]$Question 3 Code: 9709/13/M/J/18/4, Topic: Coordinate geometry A curve with equation$y=\mathrm{f}(x)$passes through the point$A(3,1)$and crosses the$y$-axis at$B$. It is given that$\mathrm{f}^{\prime}(x)=(3 x-1)^{-\frac{1}{3}}$. Find the$y$-coordinate of$B$.$[6]$Question 4 Code: 9709/11/M/J/18/5, Topic: Coordinate geometry The diagram shows a kite$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]$Question 5 Code: 9709/12/M/J/20/5, Topic: Functions The function$\mathrm{f}$is defined for$x \in \mathbb{R}$by $$\text { f: } x \mapsto a-2 x$$ where$a$is a constant.$\text{(a)}$Express$\mathrm{ff}(x)$and$\mathrm{f}^{-1}(x)$in terms of$a$and$x$.$[4]\text{(b)}$Given that$\mathrm{ff}(x)=\mathrm{f}^{-1}(x)$, find$x$in terms of$a$.$[2]$Question 6 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry Points$A$and$B$have coordinates$(8,3)$and$(p, q)$respectively. The equation of the perpendicular bisector of$A B$is$y=-2 x+4$. Find the values of$p$and$q$.$[4]$Question 7 Code: 9709/11/M/J/12/7, Topic: Series$\text{(a)}$The first two terms of an arithmetic progression are 1 and$\cos ^{2} x$respectively. Show that the sum of the first ten terms can be expressed in the form$a-b \sin ^{2} x$, where$a$and$b$are constants to be found.$[3]\text{(b)}$The first two terms of a geometric progression are 1 and$\frac{1}{3} \tan ^{2} \theta$respectively, where$0< \theta <\frac{1}{2} \pi$.$\text{(i)}$Find the set of values of$\theta$for which the progression is convergent.$[2]\text{(ii)}$Find the exact value of the sum to infinity when$\theta=\frac{1}{6} \pi$.$[2]$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: 9709/12/M/J/16/9, Topic: Series A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.$\text{(i)}$On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.$\text{(a)}$How many litres will be lost on the 30 th day after filling?$[2]\text{(b)}$The tank becomes empty during the$n$th day after filling. Find the value of$n$.$[3]\text{(ii)}$Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by$10 \%$on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30 th day after filling.$[4]$Question 10 Code: 9709/11/M/J/17/9, Topic: Functions The function$\mathrm{f}$is defined by$\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$for$x \in \mathbb{R}, x \neq \frac{3}{2}$.$\text{(i)}$Find an expression for$\mathrm{f}^{-1}(x)$.$[3]$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 4 x+a$for$x \in \mathbb{R}$, where$a$is a constant.$\text{(ii)}$Find the value of$a$for which$\operatorname{gf}(-1)=3$.$[3]\text{(iii)}$Find the possible values of$a$given that the equation$\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$has two equal roots.$[4]$Question 11 Code: 9709/11/M/J/19/9, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=2-3 \cos x$for$0 \leqslant x \leqslant 2 \pi$.$\text{(i)}$State the range of$\mathrm{f}$.$[2]\text{(ii)}$Sketch the graph of$y=\mathrm{f}(x)$. The function$\mathrm{g}$is defined by$\mathrm{g}(x)=2-3 \cos x$for$0 \leqslant x \leqslant p$, where$p$is a constant.$[2]\text{(iii)}$State the largest value of$p$for which$\mathrm{g}$has an inverse.$[1]\text{(iv)}$For this value of$p$, find an expression for$\mathrm{g}^{-1}(x)$.$[3]$Question 12 Code: 9709/13/M/J/21/9, Topic: Series$\text{(a)}$A geometric progression is such that the second term is equal to 24% of the sum to infinity. Find the possible values of the common ratio.$[3]\text{(b)}$An arithmetic progression$P$has first term$a$and common difference$d$. An arithmetic progression$Q$has first term$2(a+1)$and common difference$(d+1)$. It is given that $$\frac{\text { 5th term of } P}{12\text{th term of } Q}=\frac{1}{3} \quad \text { and } \quad \frac{\text { Sum of first } 5 \text { terms of } P}{\text { Sum of first } 5 \text { terms of } Q}=\frac{2}{3}$$ Find the value of$a$and the value of$d$.$[6]\$

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