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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 6 5 7 7 7 7 9 8 11 9 10 90
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/O/N/20/1, Topic: Quadratics$\text{(a)}$Express$x^{2}+6 x+5$in the form$(x+a)^{2}+b$, where$a$and$b$are constants.$[2]\text{(b)}$The curve with equation$y=x^{2}$is transformed to the curve with equation$y=x^{2}+6 x+5$. Describe fully the transformation(s) involved.$[2]$Question 2 Code: 9709/12/O/N/17/2, Topic: Functions A function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 4-5 x$for$x \in \mathbb{R}$.$\text{(i)}$Find an expression for$\mathrm{f}^{-1}(x)$and find the point of intersection of the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$.$[3]\text{(ii)}$Sketch, on the same diagram, the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$, making clear the relationship between the graphs.$[3]$Question 3 Code: 9709/11/O/N/19/4, Topic: Series A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run$x \mathrm{~km}$on day 1 , and on each subsequent day she will increase the distance by$10 \%$of the previous day's distance. On day 21 she will run$20 \mathrm{~km}$.$\text{(i)}$Find the distance she must run on day 1 in order to achieve this. Give your answer in$\mathrm{km}$correct to 1 decimal place.$[3]\text{(ii)}$Find the total distance she runs over the 21 days.$[2]$Question 4 Code: 9709/12/O/N/17/5, Topic: Trigonometry$\text{(i)}$Show that the equation$\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$can be expressed as$[3]$$$2 \cos ^{2} 2 x+3 \cos 2 x+1=0$$$\text{(ii)}$Hence solve the equation$\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$[4]$Question 5 Code: 9709/11/M/J/18/5, Topic: Coordinate geometry The diagram shows a kite$O A B C$in which$A C$is the line of symmetry. The coordinates of$A$and$C$are$(0,4)$and$(8,0)$respectively and$O$is the origin.$\text{(i)}$Find the equations of$A C$and$O B$.$[4]\text{(ii)}$Find, by calculation, the coordinates of$B$.$[3]$Question 6 Code: 9709/13/M/J/17/7, Topic: Circular measure The diagram shows two circles with centres$A$and$B$having radii$8 \mathrm{~cm}$and$10 \mathrm{~cm}$respectively. The two circles intersect at$C$and$D$where$C A D$is a straight line and$A B$is perpendicular to$C D$.$\text{(i)}$Find angle$A B C$in radians.$[1]\text{(ii)}$Find the area of the shaded region.$[6]$Question 7 Code: 9709/12/M/J/18/7, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 7-2 x^{2}-12 x$for$x \in \mathbb{R}$.$\text{(i)}$Express$7-2 x^{2}-12 x$in the form$a-2(x+b)^{2}$, where$a$and$b$are constants.$[2]\text{(ii)}$State the coordinates of the stationary point on the curve$\mathrm{y = f}(x)$.$[1]$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 7-2 x^{2}-12 x$for$x \geqslant k$.$\text{(iii)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$[1]\text{(iv)}$For this value of$k$, find$\mathrm{g}^{-1}(x)$.$[3]$Question 8 Code: 9709/11/M/J/18/8, Topic: Series$\text{(a)}$A geometric progression has a second term of 12 and a sum to infinity of 54. Find the possible values of the first term of the progression.$[4]\text{(b)}$The$n$th term of a progression is$p+q n$, where$p$and$q$are constants, and$S_{n}$is the sum of the first$n$terms.$\text{(i)}$Find an expression, in terms of$p, q$and$n$, for$S_{n}$.$[3]\text{(ii)}$Given that$S_{4}=40$and$S_{6}=72$, find the values of$p$and$q$.$[2]$Question 9 Code: 9709/13/O/N/20/8, Topic: Differentiation The equation of a curve is$\displaystyle y=2 x+1+\frac{1}{2 x+1}$for$x>-\frac{1}{2}$.$\text{(a)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$and$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$.$[3]\text{(b)}$Find the coordinates of the stationary point and determine the nature of the stationary point.$[5]$Question 10 Code: 9709/12/M/J/17/10, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}(x)=3 \tan \left(\frac{1}{2} x\right)-2$, for$-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.$\text{(i)}$Solve the equation$\mathrm{f}(x)+4=0$, giving your answer correct to 1 decimal place.$[3]\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$and find the domain of$\mathrm{f}^{-1}$.$[5]\text{(iii)}$Sketch, on the same diagram, the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{f}^{-1}(x)$.$[3]$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: 9709/13/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$A B C=90^{\circ}$.$[2]\text{(b)}$Hence state the coordinates of$D$.$[1]\text{(c)}$Find an equation of the circle.$[2]$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$E$.$[5]\$

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