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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 6 7 6 7 7 8 7 7 5 9 9 14 92
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/17/2, Topic: Vectors Relative to an origin$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 2 Code: 9709/13/M/J/10/5, Topic: Differentiation, Integration The equation of a curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{\sqrt{(} 3 x-2)}.$Given that the curve passes through the point$P(2,11)$, find$\text{(i)}$the equation of the normal to the curve at$P$,$[3]\text{(ii)}$the equation of the curve.$[4]$Question 3 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry The diagram shows the curve$y=7 \sqrt{x}$and the line$y=6 x+k$, where$k$is a constant. The curve and the line intersect at the points$A$and$B$.$\text{(i)}$For the case where$k=2$, find the$x$-coordinates of$A$and$B$.$[4]\text{(ii)}$Find the value of$k$for which$y=6 x+k$is a tangent to the curve$y=7 \sqrt{x}$.$[2]$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/12/6, Topic: Vectors Two vectors$\mathbf{u}$and$\mathbf{v}$are such that$\mathbf{u}=\left(\begin{array}{c}p^{2} \\ -2 \\ 6\end{array}\right)$and$\mathbf{v}=\left(\begin{array}{c}2 \\ p-1 \\ 2 p+1\end{array}\right)$, where$p$is a constant.$\text{(i)}$Find the values of$p$for which$\mathbf{u}$is perpendicular to$\mathbf{v}$.$[3]\text{(ii)}$For the case where$p=1$, find the angle between the directions of$\mathbf{u}$and$\mathbf{v}$.$[4]$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/12/7, Topic: Series$\text{(a)}$The first two terms of an arithmetic progression are 1 and$\cos ^{2} x$respectively. Show that the sum of the first ten terms can be expressed in the form$a-b \sin ^{2} x$, where$a$and$b$are constants to be found.$[3]\text{(b)}$The first two terms of a geometric progression are 1 and$\frac{1}{3} \tan ^{2} \theta$respectively, where$0< \theta <\frac{1}{2} \pi$.$\text{(i)}$Find the set of values of$\theta$for which the progression is convergent.$[2]\text{(ii)}$Find the exact value of the sum to infinity when$\theta=\frac{1}{6} \pi$.$[2]$Question 8 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 9 Code: 9709/12/M/J/21/7, Topic: Coordinate geometry The point$A$has coordinates$(1,5)$and the line$l$has gradient$-\frac{2}{3}$and passes through$A$. A circle has centre$(5,11)$and radius$\sqrt{52}$.$\text{(a)}$Show that$l$is the tangent to the circle at$A$.$[2]\text{(b)}$Find the equation of the other circle of radius$\sqrt{52}$for which l is also the tangent at$A$.$[3]$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: 9709/13/M/J/13/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{\sqrt{x}}-x$and points$A(1,7)$and$B(4,0)$which lie on the curve. The tangent to the curve at$B$intersects the line$x=1$at the point$C$.$\text{(i)}$Find the coordinates of$C$.$[4]\text{(ii)}$Find the area of the shaded region.$[5]$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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