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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 6 4 6 5 6 7 8 8 9 12 11 85
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/21/1, Topic: Integration A curve with equation$y=\mathrm{f}(x)$is such that$\displaystyle \mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}.$It is given that the curve passes through the point$(2,7)$. Find$\mathrm{f}(x)$.$[3]$Question 2 Code: 9709/11/M/J/17/2, Topic: Vectors Relative to an origin$O$, the position vectors of points$A$and$B$are given by $$\overrightarrow{O A}=\left(\begin{array}{r} 3 \\ -6 \\ p \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 2 \\ -6 \\ -7 \end{array}\right)$$ and angle$A O B=90^{\circ}$.$\text{(i)}$Find the value of$p$.$[2]$The point$C$is such that$\overrightarrow{O C}=\frac{2}{3} \overrightarrow{O A}\text{(ii)}$Find the unit vector in the direction of$\overrightarrow{B C}$.$[4]$Question 3 Code: 9709/12/M/J/21/3, Topic: Coordinate geometry The equation of a curve is$y=(x-3) \sqrt{x+1}+3$. The following points lie on the curve. Non-exact values are rounded to 4 decimal places. $$\begin{array}{lllll} A(2, k) & B(2.9,2.8025) & C(2.99,2.9800) & D(2.999,2.9980) & E(3,3) \end{array}$$$\text{(a)}$Find$k$, giving your answer correct to 4 decimal places.$[1]\text{(b)}$Find the gradient of$A E$, giving your answer correct to 4 decimal places.$[1]$The gradients of$B E, C E$and$D E$, rounded to 4 decimal places, are 1.9748,$1.9975$and$1.9997$respectively.$\text{(c)}$State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point$E$.$[2]$Question 4 Code: 9709/11/M/J/11/4, Topic: Vectors The diagram shows a prism$A B C D P Q R S$with a horizontal square base$A P S D$with sides of length$6 \mathrm{~cm}$. The cross-section$A B C D$is a trapezium and is such that the vertical edges$A B$and$D C$are of lengths$5 \mathrm{~cm}$and$2 \mathrm{~cm}$respectively. Unit vectors$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$are parallel to$A D, A P$and$A B$respectively.$\text{(i)}$Express each of the vectors$\overrightarrow{C P}$and$\overrightarrow{C Q}$in terms of$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$.$[2]\text{(ii)}$Use a scalar product to calculate angle$P C Q$.$[4]$Question 5 Code: 9709/12/M/J/20/4, Topic: Series The$n$th term of an arithmetic progression is$\frac{1}{2}(3 n-15)$. Find the value of$n$for which the sum of the first$n$terms is 84.$[5]$Question 6 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 7 Code: 9709/12/M/J/19/7, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} \mathrm{f}: x & \mapsto 3 x-2, \quad x \in \mathbb{R} \\ \mathrm{g}: x & \mapsto \frac{2 x+3}{x-1}, \quad x \in \mathbb{R}, x \neq 1 \end{aligned}\text{(i)}$Obtain expressions for$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$, stating the value of$x$for which$\mathrm{g}^{-1}(x)$is not defined.$[4]\text{(ii)}$Solve the equation$\mathrm{fg}(x)=\frac{7}{3}$.$[3]$Question 8 Code: 9709/13/M/J/17/8, Topic: Coordinate geometry$A(-1,1)$and$P(a, b)$are two points, where$a$and$b$are constants. The gradient of$A P$is 2.$\text{(i)}$Find an expression for$b$in terms of$a$.$[2]\text{(ii)}B(10,-1)$is a third point such that$A P=A B$. Calculate the coordinates of the possible positions of$P$.$[6]$Question 9 Code: 9709/12/M/J/10/9, Topic: Integration The diagram shows the curve$y=(x-2)^{2}$and the line$y+2 x=7$, which intersect at points$A$and$B$. Find the area of the shaded region.$[8]$Question 10 Code: 9709/12/M/J/10/10, Topic: Differentiation The equation of a curve is$y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.$\text{(i)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$.$[3]\text{(ii)}$Find the equation of the tangent to the curve at the point where the curve intersects the$y$-axis.$[3]\text{(iii)}$Find the set of values of$x$for which$\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$is an increasing function of$x$.$[3]$Question 11 Code: 9709/11/M/J/13/10, Topic: Coordinate geometry, Integration The diagram shows part of the curve$y=(x-2)^{4}$and the point$A(1,1)$on the curve. The tangent at$A$cuts the$x$-axis at$B$and the normal at$A$cuts the$y$-axis at$C$.$\text{(i)}$Find the coordinates of$B$and$C$.$[6]\text{(ii)}$Find the distance$A C$, giving your answer in the form$\frac{\sqrt{a}}{b}$, where$a$and$b$are integers.$[2]\text{(iii)}$Find the area of the shaded region.$[4]$Question 12 Code: 9709/11/M/J/11/11, Topic: Functions, Coordinate geometry Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned}\text{(i)}$Find and simplify expressions for$\mathrm{fg}(x)$and$\operatorname{gf}(x)$.$[2]\text{(ii)}$Hence find the value of$a$for which$\mathrm{fg}(a)=\operatorname{gf}(a)$.$[3]\text{(iii)}$Find the value of$b(b \neq a)$for which$\mathrm{g}(b)=b$.$[2]\text{(iv)}$Find and simplify an expression for$\mathrm{f}^{-1} \mathrm{~g}(x)$.$[2]$The function$\mathrm{h}$is defined by $$\mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0$$$\text{(v)}$Find an expression for$\mathrm{h}^{-1}(x)$.$[2]\$

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