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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 5 4 6 7 6 8 7 7 8 8 9 79
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/12/1, Topic: Integration The diagram shows the region enclosed by the curve$\displaystyle y=\frac{6}{2 x-3}$, the$x$-axis and the lines$x=2$and$x=3$. Find, in terms of$\pi$, the volume obtained when this region is rotated through$360^{\circ}$about the$x$-axis.$[4]$Question 2 Code: 9709/11/M/J/18/1, Topic: Series$\text{(i)}$Find the first three terms in the expansion, in ascending powers of$x$, of$(1-2 x)^{5}$.$[2]\text{(ii)}$Given that the coefficient of$x^{2}$in the expansion of$\left(1+a x+2 x^{2}\right)(1-2 x)^{5}$is 12 , find the value of the constant$a$.$[3]$Question 3 Code: 9709/12/M/J/21/1, Topic: Quadratics$\text{(a)}$Express$16 x^{2}-24 x+10$in the form$(4 x+a)^{2}+b$.$[2]\text{(b)}$It is given that the equation$16 x^{2}-24 x+10=k$, where$k$is a constant, has exactly one root. Find the value of this root.$[2]$Question 4 Code: 9709/11/M/J/18/3, Topic: Integration, Coordinate geometry A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{12}{(2 x+1)^{2}}$. The point$(1,1)$lies on the curve. Find the coordinates of the point at which the curve intersects the$x$-axis.$[6]$Question 5 Code: 9709/11/M/J/15/4, Topic: Vectors Relative to the origin$O$, the position vectors of points$A$and$B$are given by $$\overrightarrow{O A}=\left(\begin{array}{r} 3 \\ 0 \\ -4 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -3 \\ 2 \end{array}\right)$$$\text{(i)}$Find the cosine of angle$A O B$.$[3]$The position vector of$C$is given by$\overrightarrow{O C}=\left(\begin{array}{c}k \\ -2 k \\ 2 k-3\end{array}\right)\text{(ii)}$Given that$A B$and$O C$have the same length, find the possible values of$k$.$[4]$Question 6 Code: 9709/13/M/J/17/4, Topic: Vectors Relative to an origin$O$, the position vectors of points$A$and$B$are given by $$\overrightarrow{O A}=\left(\begin{array}{l} 5 \\ 1 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 5 \\ 4 \\ -3 \end{array}\right)$$ The point$P$lies on$A B$and is such that$\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}$.$\text{(i)}$Find the position vector of$P$.$[3]\text{(ii)}$Find the distance$O P$.$[1]\text{(iii)}$Determine whether$O P$is perpendicular to$A B$. Justify your answer.$[2]$Question 7 Code: 9709/11/M/J/10/7, Topic: Coordinate geometry The diagram shows part of the curve$\displaystyle y=2-\frac{18}{2 x+3}$, which crosses the$x$-axis at$A$and the$y$-axis at$B$. The normal to the curve at$A$crosses the$y$-axis at$C$.$\text{(i)}$Show that the equation of the line$A C$is$9 x+4 y=27$.$[6]\text{(ii)}$Find the length of$B C$.$[2]$Question 8 Code: 9709/11/M/J/19/7, Topic: Vectors The diagram shows a three-dimensional shape in which the base$O A B C$and the upper surface$D E F G$are identical horizontal squares. The parallelograms$O A E D$and$C B F G$both lie in vertical planes. The point$M$is the mid-point of$A F$. Unit vectors$\mathbf{i}$and$\mathbf{j}$are parallel to$O A$and$O C$respectively and the unit vector$\mathbf{k}$is vertically upwards. The position vectors of$A$and$D$are given by$\overrightarrow{O A}=8 \mathbf{i}$and$\overrightarrow{O D}=3 \mathbf{i}+10 \mathbf{k}$.$\text{(i)}$Express each of the vectors$\overrightarrow{A M}$and$\overrightarrow{G M}$in terms of$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$.$[3]\text{(ii)}$Use a scalar product to find angle$G M A$correct to the nearest degree.$[4]$Question 9 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 10 Code: 9709/12/M/J/14/8, Topic: Integration, Differentiation The equation of a curve is such that$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 x-1$. Given that the curve has a minimum point at$(3,-10)$, find the coordinates of the maximum point.$[8]$Question 11 Code: 9709/13/M/J/15/8, Topic: Differentiation The function$\mathrm{f}$is defined by$\displaystyle\mathrm{f}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$for$x>-1$.$\text{(i)}$Find$\mathrm{f}^{\prime}(x)$.$[3]\text{(ii)}$State, with a reason, whether$\mathrm{f}$is an increasing function, a decreasing function or neither.$[1]$The function$\mathrm{g}$is defined by$\displaystyle\mathrm{g}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$for$x<-1\text{(iii)}$Find the coordinates of the stationary point on the curve$y=\mathrm{g}(x)$.$[4]$Question 12 Code: 9709/13/M/J/20/8, Topic: Series The first term of a progression is$\sin ^{2} \theta$, where$0 < \theta < \frac{1}{2} \pi$. The second term of the progression is$\sin ^{2} \theta \cos ^{2} \theta$.$\text{(a)}$Given that the progression is geometric, find the sum to infinity. It is now given instead that the progression is arithmetic.$[3]\text{(b)} \quad \text{(i)}$Find the common difference of the progression in terms of$\sin \theta$.$[3] \quad \quad \text{(ii)}$Find the sum of the first 16 terms when$\theta=\frac{1}{3} \pi$.$[3]\$

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