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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 4 5 5 7 4 7 7 8 8 11 10 79
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/13/1, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{x^{2}}$and$(2,9)$is a point on the curve. Find the equation of the curve.$[3]$Question 2 Code: 9709/11/M/J/12/2, Topic: Series Find the coefficient of$x^{6}$in the expansion of$\displaystyle\left(2 x^{3}-\frac{1}{x^{2}}\right)^{7}$.$[4]$Question 3 Code: 9709/11/M/J/13/3, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$8 \mathrm{~cm}$. Angle$B O A$is$\alpha$radians.$O A C$is a semicircle with diameter$O A$. The area of the semicircle$O A C$is twice the area of the sector$O A B$.$\text{(i)}$Find$\alpha$in terms of$\pi$.$[3]\text{(ii)}$Find the perimeter of the complete figure in terms of$\pi$.$[2]$Question 4 Code: 9709/12/M/J/19/4, Topic: Trigonometry Angle$x$is such that$\sin x=a+b$and$\cos x=a-b$, where$a$and$b$are constants.$\text{(i)}$Show that$a^{2}+b^{2}$has a constant value for all values of$x$.$[3]\text{(ii)}$In the case where$\tan x=2$, express$a$in terms of$b$.$[2]$Question 5 Code: 9709/11/M/J/17/5, Topic: Coordinate geometry The equation of a curve is$y=2 \cos x$.$\text{(i)}$Sketch the graph of$y=2 \cos x$for$-\pi \leqslant x \leqslant \pi$, stating the coordinates of the point of intersection with the$y$-axis.$[2]$Points$P$and$Q$lie on the curve and have$x$-coordinates of$\frac{1}{3} \pi$and$\pi$respectively.$\text{(ii)}$Find the length of$P Q$correct to 1 decimal place.$[2]$The line through$P$and$Q$meets the$x$-axis at$H(h, 0)$and the$y$-axis at$K(0, k)$.$\text{(iii)}$Show that$h=\frac{5}{9} \pi$and find the value of$k$.$[3]$Question 6 Code: 9709/11/M/J/21/5, Topic: Series The fifth, sixth and seventh terms of a geometric progression are$8 k,-12$and$2 k$respectively. Given that$k$is negative, find the sum to infinity of the progression.$[4]$Question 7 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 8 Code: 9709/11/M/J/15/6, Topic: Coordinate geometry The line with gradient$-2$passing through the point$P(3 t, 2 t)$intersects the$x$-axis at$A$and the$y$-axis at$B$.$\text{(i)}$Find the area of triangle$A O B$in terms of$t$.$[3]$The line through$P$perpendicular to$A B$intersects the$x$-axis at$C$.$\text{(ii)}$Show that the mid-point of$P C$lies on the line$y=x$.$[4]$Question 9 Code: 9709/13/M/J/14/8, Topic: Quadratics$\text{(i)}$Express$2 x^{2}-10 x+8$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants, and use your answer to state the minimum value of$2 x^{2}-10 x+8$.$[4]\text{(ii)}$Find the set of values of$k$for which the equation$2 x^{2}-10 x+8=k x$has no real roots.$[4]$Question 10 Code: 9709/12/M/J/20/8, Topic: Integration, Coordinate geometry The diagram shows part of the curve$\displaystyle y=\frac{6}{x}$. The points$(1,6)$and$(3,2)$lie on the curve. The shaded region is bounded by the curve and the lines$y=2$and$x=1$.$\text{(a)}$Find the volume generated when the shaded region is rotated through$360^{\circ}$about the$y$-axis.$[5]\text{(b)}$The tangent to the curve at a point$X$is parallel to the line$y+2 x=0$. Show that$X$lies on the line$y=2 x$.$[3]$Question 11 Code: 9709/11/M/J/15/10, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{\sqrt{(} 3 x+4)}.$The curve intersects the$y$-axis at$A(0,4).$The normal to the curve at$A$intersects the line$x=4$at the point$B$.$\text{(i)}$Find the coordinates of$B$.$[5]\text{(ii)}$Show, with all necessary working, that the areas of the regions marked$P$and$Q$are equal.$[6]$Question 12 Code: 9709/13/M/J/15/11, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$r$. The point$C$on$O B$is such that angle$A C O$is a right angle. Angle$A O B$is$\alpha$radians and is such that$A C$divides the sector into two regions of equal area.$\text{(i)}$Show that$\sin \alpha \cos \alpha=\frac{1}{2} \alpha$.$[4]$It is given that the solution of the equation in part$\text{(i)}$is$\alpha=0.9477$, correct to 4 decimal places.$\text{(ii)}$Find the ratio perimeter of region$O A C$: perimeter of region$A C B$, giving your answer in the form$k: 1$, where$k$is given correct to 1 decimal place.$[5]\text{(iii)}$Find angle$A O B$in degrees.$[1]\$

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