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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 6 5 7 7 8 8 11 6 10 10 11 95
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/12/4, Topic: Coordinate geometry The point$A$has coordinates$(-1,-5)$and the point$B$has coordinates$(7,1)$. The perpendicular bisector of$A B$meets the$x$-axis at$C$and the$y$-axis at$D$. Calculate the length of$C D$.$$Question 2 Code: 9709/13/M/J/17/4, Topic: Vectors Relative to an origin$O$, the position vectors of points$A$and$B$are given by $$\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)$$ The point$P$lies on$A B$and is such that$\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}$.$\text{(i)}$Find the position vector of$P$.$\text{(ii)}$Find the distance$O P$.$\text{(iii)}$Determine whether$O P$is perpendicular to$A B$. Justify your answer.$$Question 3 Code: 9709/12/M/J/15/5, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta} \equiv \frac{\tan \theta-1}{\tan \theta+1}$.$\text{(ii)}$Hence solve the equation$\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{\tan \theta}{6}$, for$0^{\circ} \leqslant \theta \leqslant 180^{\circ}$.$$Question 4 Code: 9709/13/M/J/12/6, Topic: Series The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.$\text{(i)}$Find the common difference of the progression.$$The first term, the ninth term and the$n$th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.$\text{(ii)}$Find the common ratio of the geometric progression and the value of$n$.$$Question 5 Code: 9709/13/M/J/12/7, Topic: Coordinate geometry The curve$\displaystyle y=\frac{10}{2 x+1}-2$intersects the$x$-axis at$A$. The tangent to the curve at$A$intersects the$y$-axis at$C$.$\text{(i)}$Show that the equation of$A C$is$5 y+4 x=8$.$\text{(ii)}$Find the distance$A C$.$$Question 6 Code: 9709/13/M/J/13/7, Topic: Coordinate geometry The diagram shows three points$A(2,14), B(14,6)$and$C(7,2).$The point$X$lies on$A B$, and$C X$is perpendicular to$A B$. Find, by calculation,$\text{(i)}$the coordinates of$X$,$\text{(ii)}$the ratio$A X: X B$.$$Question 7 Code: 9709/12/M/J/16/8, Topic: Coordinate geometry Three points have coordinates$A(0,7), B(8,3)$and$C(3 k, k).$Find the value of the constant$k$for which$\text{(i)}C$lies on the line that passes through$A$and$B$,$\text{(ii)}C$lies on the perpendicular bisector of$A B$.$$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$\displaystyle\int y \mathrm{~d} x$.$\text{(ii)}$Find the equation of the tangent to the curve at$P$in the form$y=m x+c$.$\text{(iii)}$Find, showing all necessary working, the area of the shaded region.$$Question 9 Code: 9709/12/M/J/21/9, Topic: Integration The diagram shows part of the curve with equation$y^{2}=x-2$and the lines$x=5$and$y=1$. The shaded region enclosed by the curve and the lines is rotated through$360^{\circ}$about the$x$-axis. Find the volume obtained.$$Question 10 Code: 9709/13/M/J/21/10, Topic: Coordinate geometry Points$A(-2,3), B(3,0)$and$C(6,5)$lie on the circumference of a circle with centre$D$.$\text{(a)}$Show that angle$A B C=90^{\circ}$.$\text{(b)}$Hence state the coordinates of$D$.$\text{(c)}$Find an equation of the circle.$$The point$E$lies on the circumference of the circle such that$BE$is a diameter.$\text{(d)}$Find an equation of the tangent to the circle at$E$.$$Question 11 Code: 9709/13/M/J/12/11, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=8-(x-2)^{2}$, for$x \in \mathbb{R}$.$\text{(i)}$Find the coordinates and the nature of the stationary point on the curve$y=\mathrm{f}(x)$.$$The function$\mathrm{g}$is such that$\mathrm{g}(x)=8-(x-2)^{2}$, for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(ii)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$$For this value of$k$,$\text{(iii)}$find an expression for$\mathrm{g}^{-1}(x)$,$\text{(iv)}$sketch, on the same diagram, the graphs of$y=\mathrm{g}(x)$and$y=\mathrm{g}^{-1}(x)$.$$Question 12 Code: 9709/12/M/J/16/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 6 x-x^{2}-5$for$x \in \mathbb{R}$.$\text{(i)}$Find the set of values of$x$for which$\mathrm{f}(x) \leqslant 3$.$\text{(ii)}$Given that the line$y=m x+c$is a tangent to the curve$y=\mathrm{f}(x)$, show that$4 c=m^{2}-12 m+16$.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 6 x-x^{2}-5$for$x \geqslant k$, where$k$is a constant.$\text{(iii)}$Express$6 x-x^{2}-5$in the form$a-(x-b)^{2}$, where$a$and$b$are constants.$\text{(iv)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$\text{(v)}$For this value of$k$, find an expression for$\mathrm{g}^{-1}(x)$.$\$

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