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### MATHEMATICS 970Y

#### 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 4 5 6 6 6 6 8 8 8 12 9 81
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/17/1, Topic: Series The coefficients of$x$and$x^{2}$in the expansion of$(2+a x)^{7}$are equal. Find the value of the non-zero constant$a$.$$Question 2 Code: 9709/11/M/J/18/2, Topic: Differentiation A point is moving along the curve$\displaystyle y=2 x+\frac{5}{x}$in such a way that the$x$-coordinate is increasing at a constant rate of$0.02$units per second. Find the rate of change of the$y$-coordinate when$x=1$.$$Question 3 Code: 9709/11/M/J/16/3, Topic: Integration The diagram shows part of the curve$\displaystyle x=\frac{12}{y^{2}}-2$. The shaded region is bounded by the curve, the$y$-axis and the lines$y=1$and$y=2$. Showing all necessary working, find the volume, in terms of$\pi$, when this shaded region is rotated through$360^{\circ}$about the$y$-axis.$$Question 4 Code: 9709/12/M/J/12/4, Topic: Coordinate geometry The point$A$has coordinates$(-1,-5)$and the point$B$has coordinates$(7,1)$. The perpendicular bisector of$A B$meets the$x$-axis at$C$and the$y$-axis at$D$. Calculate the length of$C D$.$$Question 5 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$B$.$$Question 6 Code: 9709/11/M/J/19/5, Topic: Quadratics The function$\mathrm{f}$is defined by$\mathrm{f}(x)=-2 x^{2}+12 x-3$for$x \in \mathbb{R}$.$\text{(i)}$Express$-2 x^{2}+12 x-3$in the form$-2(x+a)^{2}+b$, where$a$and$b$are constants.$\text{(ii)}$State the greatest value of$\mathrm{f}(x)$. The function$\mathrm{g}$is defined by$\mathrm{g}(x)=2 x+5$for$x \in \mathbb{R}$.$\text{(iii)}$Find the values of$x$for which$\operatorname{gf}(x)+1=0$.$$Question 7 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$x$.$\text{(b)}$Given that$\mathrm{ff}(x)=\mathrm{f}^{-1}(x)$, find$x$in terms of$a$.$$Question 8 Code: 9709/13/M/J/10/7, Topic: Circular measure The diagram shows a metal plate$A B C D E F$which has been made by removing the two shaded regions from a circle of radius 10 cm and centre$O.$The parallel edges$A B$and$E D$are both of length 12 cm.$\text{(i)}$Show that angle$D O E$is 1.287 radians, correct to 4 significant figures.$\text{(ii)}$Find the perimeter of the metal plate.$\text{(iii)}$Find the area of the metal plate.$$Question 9 Code: 9709/12/M/J/17/8, Topic: Vectors Relative to an origin$O$, the position vectors of three points$A, B$and$C$are given by$\overrightarrow{O A}=3 \mathbf{i}+p \mathbf{j}-2 p \mathbf{k}, \quad \overrightarrow{O B}=6 \mathbf{i}+(p+4) \mathbf{j}+3 \mathbf{k} \quad$and$\quad \overrightarrow{O C}=(p-1) \mathbf{i}+2 \mathbf{j}+q \mathbf{k}$where$p$and$q$are constants.$\text{(i)}$In the case where$p=2$, use a scalar product to find angle$A O B$.$\text{(ii)}$In the case where$\overrightarrow{A B}$is parallel to$\overrightarrow{O C}$, find the values of$p$and$q$.$$Question 10 Code: 9709/12/M/J/11/9, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=3-4 \cos ^{k} x$, for$0 \leqslant x \leqslant \pi$, where$k$is a constant.$\text{(i)}$In the case where$k=2$,$\text{(a)}$find the range of$\mathrm{f}$,$\text{(b)}$find the exact solutions of the equation$\mathrm{f}(x)=1$.$\text{(ii)}$In the case where$k=1$,$\text{(a)}$sketch the graph of$y=\mathrm{f}(x)$,$\text{(b)}$state, with a reason, whether$\mathrm{f}$has an inverse.$$Question 11 Code: 9709/13/M/J/11/10, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned}\text{(i)}$Evaluate$\mathrm{fg(2)}$.$\text{(ii)}$Sketch in a single diagram the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$, making clear the relationship between the graphs.$\text{(iii)}$Obtain an expression for$\mathrm{g}^{\prime}(x)$and use your answer to explain why$\mathrm{g}$has an inverse.$\text{(iv)}$Express each of$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$in terms of$x$.$$Question 12 Code: 9709/13/M/J/16/10, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=2 x+3$for$x \geqslant 0$. The function$\mathrm{g}$is such that$\mathrm{g}(x)=a x^{2}+b$for$x \leqslant q$, where$a, b$and$q$are constants. The function fg is such that fg$(x)=6 x^{2}-21$for$x \leqslant q\text{(i)}$Find the values of$a$and$b$.$\text{(ii)}$Find the greatest possible value of$q$.$$It is now given that$q=-3$.$\text{(iii)}$Find the range of$\mathrm{fg}$.$\text{(iv)}$Find an expression for$\mathrm{(f g)^{-1}}(x)$and state the domain of$(f g)^{-1}$.$\$

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