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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 5 5 6 5 7 7 7 10 10 10 10 12 94
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/13/3, Topic: Quadratics The straight line$y=m x+14$is a tangent to the curve$\displaystyle y=\frac{12}{x}+2$at the point$P$. Find the value of the constant$m$and the coordinates of$P$.$$Question 2 Code: 9709/13/M/J/16/3, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=6 x^{2}+\frac{k}{x^{3}}$and passes through the point$P(1,9)$. The gradient of the curve at$P$is$2.\text{(i)}$Find the value of the constant$k$.$\text{(ii)}$Find the equation of the curve.$$Question 3 Code: 9709/12/M/J/11/4, Topic: Differentiation, Coordinate geometry A curve has equation$\displaystyle y=\frac{4}{3 x-4}$and$P(2,2)$is a point on the curve.$\text{(i)}$Find the equation of the tangent to the curve at$P$.$\text{(ii)}$Find the angle that this tangent makes with the$x$-axis.$$Question 4 Code: 9709/12/M/J/15/5, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta} \equiv \frac{\tan \theta-1}{\tan \theta+1}$.$\text{(ii)}$Hence solve the equation$\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{\tan \theta}{6}$, for$0^{\circ} \leqslant \theta \leqslant 180^{\circ}$.$$Question 5 Code: 9709/12/M/J/17/6, Topic: Integration The diagram shows the straight line$x+y=5$intersecting the curve$\displaystyle y=\frac{4}{x}$at the points$A(1,4)$and$B(4,1)$. Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$$Question 6 Code: 9709/12/M/J/15/7, Topic: Coordinate geometry The point$C$lies on the perpendicular bisector of the line joining the points$A(4,6)$and$B(10,2)$.$C$also lies on the line parallel to$A B$through$(3,11)$.$\text{(i)}$Find the equation of the perpendicular bisector of$A B$.$\text{(ii)}$Calculate the coordinates of$C$.$$Question 7 Code: 9709/11/M/J/14/9, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\sin \theta}{1-\cos \theta}-\frac{1}{\sin \theta} \equiv \frac{1}{\tan \theta}$.$\text{(ii)}$Hence solve the equation$\displaystyle\frac{\sin \theta}{1-\cos \theta}-\frac{1}{\sin \theta}=4 \tan \theta$for$0^{\circ} < \theta < 180^{\circ}$.$$Question 8 Code: 9709/11/M/J/15/9, Topic: Differentiation The equation of a curve is$y=x^{3}+p x^{2}$, where$p$is a positive constant.$\text{(i)}$Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of$p$.$\text{(ii)}$Find the nature of each of the stationary points.$$Another curve has equation$y=x^{3}+p x^{2}+p x$.$\text{(iii)}$Find the set of values of$p$for which this curve has no stationary points.$$Question 9 Code: 9709/11/M/J/17/9, Topic: Functions The function$\mathrm{f}$is defined by$\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$for$x \in \mathbb{R}, x \neq \frac{3}{2}$.$\text{(i)}$Find an expression for$\mathrm{f}^{-1}(x)$.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 4 x+a$for$x \in \mathbb{R}$, where$a$is a constant.$\text{(ii)}$Find the value of$a$for which$\operatorname{gf}(-1)=3$.$\text{(iii)}$Find the possible values of$a$given that the equation$\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$has two equal roots.$$Question 10 Code: 9709/13/M/J/19/9, Topic: Trigonometry The function$\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$is defined for$0 \leqslant x \leqslant \pi$, where$p$and$q$are positive constants. The diagram shows the graph of$y=\mathrm{f}(x)$.$\text{(i)}$In terms of$p$and$q$, state the range of$\mathrm{f}$.$\text{(ii)}$State the number of solutions of the following equations.$\quad\text{(a)}\mathrm{f}(x)=p+q\quad\text{(b)}\mathrm{f}(x)=q\quad\text{(c)}\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q\text{(iii)}$For the case where$p=3$and$q=2$, solve the equation$\mathrm{f}(x)=4$, showing all necessary working.$$Question 11 Code: 9709/12/M/J/13/11, Topic: Coordinate geometry, Integration The diagram shows the curve$y=\sqrt{(} 1+4 x)$, which intersects the$x$-axis at$A$and the$y$-axis at$B$. The normal to the curve at$B$meets the$x$-axis at$C$. Find$\text{(i)}$the equation of$B C$,$\text{(ii)}$the area of the shaded region.$$Question 12 Code: 9709/12/M/J/15/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x^{2}-6 x+5$for$x \in \mathbb{R}$.$\text{(i)}$Find the set of values of$p$for which the equation$\mathrm{f}(x)=p$has no real roots.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 2 x^{2}-6 x+5$for$0 \leqslant x \leqslant 4$.$\text{(ii)}$Express$\mathrm{g}(x)$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$\text{(iii)}$Find the range of$\mathrm{g}$.$$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto 2 x^{2}-6 x+5$for$k \leqslant x \leqslant 4$, where$k$is a constant.$\text{(iv)}$State the smallest value of$k$for which$\mathrm{h}$has an inverse.$\text{(v)}$For this value of$k$, find an expression for$\mathrm{h}^{-1}(x)$.$\$

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