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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 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 4 6 5 7 7 8 7 8 7 9 10 14 92
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/12/1, Topic: Trigonometry Solve the equation$\sin 2 x=2 \cos 2 x$, for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$[4]$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: 9709/11/M/J/12/3, Topic: Circular measure In the diagram,$A B C$is an equilateral triangle of side$2 \mathrm{~cm}$. The mid-point of$B C$is$Q.$An arc of a circle with centre$A$touches$B C$at$Q$, and meets$A B$at$P$and$A C$at$R.$Find the total area of the shaded regions, giving your answer in terms of$\pi$and$\sqrt{3}$.$[5]$Question 4 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 5 Code: 9709/11/M/J/15/6, Topic: Coordinate geometry The line with gradient$-2$passing through the point$P(3 t, 2 t)$intersects the$x$-axis at$A$and the$y$-axis at$B$.$\text{(i)}$Find the area of triangle$A O B$in terms of$t$.$[3]$The line through$P$perpendicular to$A B$intersects the$x$-axis at$C$.$\text{(ii)}$Show that the mid-point of$P C$lies on the line$y=x$.$[4]$Question 6 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 7 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 8 Code: 9709/11/M/J/12/8, Topic: Functions The function$\mathrm{f}: x \mapsto x^{2}-4 x+k$is defined for the domain$x \geqslant p$, where$k$and$p$are constants.$\text{(i)}$Express$\mathrm{f}(x)$in the form$(x+a)^{2}+b+k$, where$a$and$b$are constants.$[2]\text{(ii)}$State the range of$\mathrm{f}$in terms of$k$.$[1]\text{(iii)}$State the smallest value of$p$for which$\mathrm{f}$is one-one.$[1]\text{(iv)}$For the value of$p$found in part$\text{(iii)}$, find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$, giving your answers in terms of$k$.$[4]$Question 9 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 10 Code: 9709/11/M/J/14/10, Topic: Functions The diagram shows the function$\mathrm{f}$defined for$-1 \leqslant x \leqslant 4$, where $$\mathrm{f}(x)= \begin{cases}3 x-2 & \text { for }-1 \leqslant x \leqslant 1 \\ \frac{4}{5-x} & \text { for } 1 < x \leqslant 4\end{cases}$$$\text{(i)}$State the range of$\mathrm{f}$.$[1]\text{(ii)}$Copy the diagram and on your copy sketch the graph of$y=\mathrm{f}^{-1}(x)$.$[2]\text{(iii)}$Obtain expressions to define the function$\mathrm{f}^{-1}$, giving also the set of values for which each expression is valid.$[6]$Question 11 Code: 9709/11/M/J/14/11, Topic: Quadratics, Differentiation, Integration A line has equation$y=2 x+c$and a curve has equation$y=8-2 x-x^{2}$.$\text{(i)}$For the case where the line is a tangent to the curve, find the value of the constant$c$.$[3]\text{(ii)}$For the case where$c=11$, find the$x$-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.$[7]$Question 12 Code: 9709/11/M/J/21/11, Topic: Differentiation, Integration The equation of a curve is$y=2 \sqrt{3 x+4}-x\text{(a)}$Find the equation of the normal to the curve at the point$(4,4)$, giving your answer in the form$y=m x+c[5]\text{(b)}$Find the coordinates of the stationary point.$[3]\text{(c)}$Determine the nature of the stationary point.$[2]\text{(d)}$Find the exact area of the region bounded by the curve, the$x$-axis and the lines$x = 0$and$x = 4$.$[4]\$

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