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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 5 7 6 9 7 11 9 12 9 12 96
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/13/1, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$and$(2,5)$is a point on the curve. Find the equation of the curve.$[4]$Question 2 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry Find the coordinates of the point at which the perpendicular bisector of the line joining$(2,7)$to$(10,3)$meets the$x$-axis.$[5]$Question 3 Code: 9709/12/M/J/15/3, Topic: Series$\text{(i)}$Find the coefficients of$x^{2}$and$x^{3}$in the expansion of$(2-x)^{6}$.$[3]\text{(ii)}$Find the coefficient of$x^{3}$in the expansion of$(3 x+1)(2-x)^{6}$.$[2]$Question 4 Code: 9709/13/M/J/19/5, Topic: Series Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of$13 \mathrm{~kg}$. At the end of week 1 they have each recorded a weight loss of$1 \mathrm{~kg}$and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer$A$'s weight loss in week 2 is$0.98 \mathrm{~kg}$. It is given that his weekly weight loss follows an arithmetic progression.$\text{(i)}$Write down an expression for his total weight loss after$x$weeks.$[1]\text{(ii)}$He reaches his$13 \mathrm{~kg}$target during week$n$. Use your answer to part$\text{(i)}$to find the value of$n$. Boxer$B$'s weight loss in week 2 is$0.92 \mathrm{~kg}$and it is given that his weekly weight loss follows a geometric progression.$[2]\text{(iii)}$Calculate his total weight loss after 20 weeks and show that he can never reach his target.$[4]$Question 5 Code: 9709/11/M/J/20/6, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-a \\ &\mathrm{~g}: x \mapsto 3 x+b \end{aligned} wherea$and$b$are constants.$\text{(a)}$Given that$\operatorname{gg}(2)=10$and$\mathrm{f}^{-1}(2)=14$, find the values of$a$and$b$.$[4]\text{(b)}$Using these values of$a$and$b$, find an expression for$\operatorname{gf}(x)$in the form$c x+d$, where$c$and$d$are constants.$[2]$Question 6 Code: 9709/11/M/J/13/7, Topic: Quadratics, Differentiation, Coordinate geometry A curve has equation$y=x^{2}-4 x+4$and a line has equation$y=m x$, where$m$is a constant.$\text{(i)}$For the case where$m=1$, the curve and the line intersect at the points$A$and$B$. Find the coordinates of the mid-point of$A B$.$[4]\text{(ii)}$Find the non-zero value of$m$for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.$[5]$Question 7 Code: 9709/11/M/J/16/8, Topic: Coordinate geometry A curve has equation$\displaystyle y=3 x-\frac{4}{x}$and passes through the points$A(1,-1)$and$B(4,11)$. At each of the points$C$and$D$on the curve, the tangent is parallel to$A B$. Find the equation of the perpendicular bisector of$C D$.$[7]$Question 8 Code: 9709/12/M/J/14/9, Topic: Differentiation, Integration The diagram shows part of the curve$y=8-\sqrt{(} 4-x)$and the tangent to the curve at$P(3,7)$.$\text{(i)}$Find expressions for$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and$\displaystyle\int y \mathrm{~d} x$.$[5]\text{(ii)}$Find the equation of the tangent to the curve at$P$in the form$y=m x+c$.$[2]\text{(iii)}$Find, showing all necessary working, the area of the shaded region.$[4]$Question 9 Code: 9709/13/M/J/18/9, Topic: Vectors The diagram shows a pyramid$O A B C D$with a horizontal rectangular base$O A B C$. The sides$O A$and$A B$have lengths of 8 units and 6 units respectively. The point$E$on$O B$is such that$O E=2$units. The point$D$of the pyramid is 7 units vertically above$E$. Unit vectors$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$are parallel to$O A$,$O C$and$E D$respectively.$\text{(i)}$Show that$\overrightarrow{O E}=1.6 \mathbf{i}+1.2 \mathbf{j}$.$[2]\text{(ii)}$Use a scalar product to find angle$B D O$.$[7]$Question 10 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$\mathrm{fg(2)}$.$[2]\text{(ii)}$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.$[3]\text{(iii)}$Obtain an expression for$\mathrm{g}^{\prime}(x)$and use your answer to explain why$\mathrm{g}$has an inverse.$[3]\text{(iv)}$Express each of$\mathrm{f}^{-1}(x)$and$\mathrm{g}^{-1}(x)$in terms of$x$.$[4]$Question 11 Code: 9709/12/M/J/20/10, Topic: Differentiation The equation of a curve is$y=54 x-(2 x-7)^{3}$.$\text{(a)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.$[4]\text{(b)}$Find the coordinates of each of the stationary points on the curve.$[3]\text{(c)}$Determine the nature of each of the stationary points.$[2]$Question 12 Code: 9709/11/M/J/20/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{x+2}$and the line$2 y+x=8$, intersecting at points$A$and$B$. The point$C$lies on the curve and the tangent to the curve at$C$is parallel to$A B$.$\text{(a)}$Find, by calculation, the coordinates of$A, B$and$C$.$[6]\text{(b)}$Find the volume generated when the shaded region, bounded by the curve and the line, is rotated through$360^{\circ}$about the$x$-axis.$[6]\$

Worked solutions: P1, P3 & P6 (S1)

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