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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 5 5 7 7 6 10 9 8 6 9 9 84
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]$Question 2 Code: 9709/13/M/J/11/2, Topic: Coordinate geometry Find the set of values of$m$for which the line$y=m x+4$intersects the curve$y=3 x^{2}-4 x+7$at two distinct points.$[5]$Question 3 Code: 9709/12/M/J/11/3, Topic: Quadratics The equation$x^{2}+p x+q=0$, where$p$and$q$are constants, has roots$-3$and 5.$\text{(i)}$Find the values of$p$and$q$.$[2]\text{(ii)}$Using these values of$p$and$q$, find the value of the constant$r$for which the equation$x^{2}+p x+q+r=0$has equal roots.$[3]$Question 4 Code: 9709/11/M/J/15/4, Topic: Vectors Relative to the origin$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]$Question 5 Code: 9709/12/M/J/17/4, Topic: Circular measure The diagram shows a circle with radius$r \mathrm{~cm}$and centre$O$. Points$A$and$B$lie on the circle and$A B C D$is a rectangle. Angle$A O B=2 \theta$radians and$A D=r \mathrm{~cm}$.$\text{(i)}$Express the perimeter of the shaded region in terms of$r$and$\theta$.$[3]\text{(ii)}$In the case where$r=5$and$\theta=\frac{1}{6} \pi$, find the area of the shaded region.$[4]$Question 6 Code: 9709/13/M/J/21/6, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}$are both defined for$x \in \mathbb{R}and are given by \begin{aligned} &\mathrm{f}(x)=x^{2}-2 x+5 \\ &\mathrm{~g}(x)=x^{2}+4 x+13 \end{aligned}\text{(a)}$By first expressing each of$\mathrm{f}(x)$and$\mathrm{g}(x)$in completed square form, express$\mathrm{g}(x)$in the form$\mathrm{f}(x+p)+q$, where$p$and$q$are constants.$[4]\text{(b)}$Describe fully the transformation which transforms the graph of$y = f(x)$to the graph of$y = g(x)$.$[2]$Question 7 Code: 9709/11/M/J/13/8, Topic: Quadratics, Functions$\text{(i)}$Express$2 x^{2}-12 x+13$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$[3]\text{(ii)}$The function$\mathrm{f}$is defined by$\mathrm{f}(x)=2 x^{2}-12 x+13$for$x \geqslant k$, where$k$is a constant. It is given that$\mathrm{f}$is a one-one function. State the smallest possible value of$k$.$[1]$The value of$k$is now given to be 7.$\text{(iii)}$Find the range of$\mathrm{f}$.$[1]\text{(iv)}$Find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$.$[5]$Question 8 Code: 9709/11/M/J/12/9, Topic: Coordinate geometry The coordinates of$A$are$(-3,2)$and the coordinates of$C$are$(5,6).$The mid-point of$A C$is$M$and the perpendicular bisector of$A C$cuts the$x$-axis at$B$.$\text{(i)}$Find the equation of$M B$and the coordinates of$B$.$[5]\text{(ii)}$Show that$A B$is perpendicular to$B C$.$[2]\text{(iii)}$Given that$A B C D$is a square, find the coordinates of$D$and the length of$A D$.$[2]$Question 9 Code: 9709/13/M/J/16/9, Topic: Vectors The position vectors of$A, B$and$C$relative to an origin$O$are given by $$\overrightarrow{O A}=\left(\begin{array}{r} 2 \\ 3 \\ -4 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 1 \\ 5 \\ p \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 5 \\ 0 \\ 2 \end{array}\right)$$ where$p$is a constant.$\text{(i)}$Find the value of$p$for which the lengths of$A B$and$C B$are equal.$[4]\text{(ii)}$For the case where$p=1$, use a scalar product to find angle$A B C$.$[4]$Question 10 Code: 9709/12/M/J/21/9, Topic: Integration The diagram shows part of the curve with equation$y^{2}=x-2$and the lines$x=5$and$y=1$. The shaded region enclosed by the curve and the lines is rotated through$360^{\circ}$about the$x$-axis. Find the volume obtained.$[6]$Question 11 Code: 9709/13/M/J/21/9, Topic: Series$\text{(a)}$A geometric progression is such that the second term is equal to 24% of the sum to infinity. Find the possible values of the common ratio.$[3]\text{(b)}$An arithmetic progression$P$has first term$a$and common difference$d$. An arithmetic progression$Q$has first term$2(a+1)$and common difference$(d+1)$. It is given that $$\frac{\text { 5th term of } P}{12\text{th term of } Q}=\frac{1}{3} \quad \text { and } \quad \frac{\text { Sum of first } 5 \text { terms of } P}{\text { Sum of first } 5 \text { terms of } Q}=\frac{2}{3}$$ Find the value of$a$and the value of$d$.$[6]$Question 12 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]\$

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