$\require{\cancel}$ $\require{\stix[upint]}$

### 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 6 6 4 8 7 7 11 10 9 9 87
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/21/2, Topic: Differentiation The function$\mathrm{f}$is defined by$\displaystyle \mathrm{f}(x)=\frac{1}{3}(2 x-1)^{\frac{3}{2}}-2 x$for$\frac{1}{2}< x< a$. It is given that$\mathrm{f}$is a decreasing function. Find the maximum possible value of the constant$a$.$$Question 2 Code: 9709/11/M/J/17/3, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta} \equiv \frac{2}{\sin \theta}$.$\text{(ii)}$Hence solve the equation$\displaystyle\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{3}{\cos \theta}$for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$$Question 3 Code: 9709/11/M/J/10/4, Topic: Integration The diagram shows the curve$y=6 x-x^{2}$and the line$y=5$. Find the area of the shaded region.$$Question 4 Code: 9709/12/M/J/12/5, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\tan x+\frac{1}{\tan x} \equiv \frac{1}{\sin x \cos x}$.$\text{(ii)}$Solve the equation$\displaystyle\frac{2}{\sin x \cos x}=1+3 \tan x$, for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$$Question 5 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry Points$A$and$B$have coordinates$(8,3)$and$(p, q)$respectively. The equation of the perpendicular bisector of$A B$is$y=-2 x+4$. Find the values of$p$and$q$.$$Question 6 Code: 9709/13/M/J/10/7, Topic: Circular measure The diagram shows a metal plate$A B C D E F$which has been made by removing the two shaded regions from a circle of radius 10 cm and centre$O.$The parallel edges$A B$and$E D$are both of length 12 cm.$\text{(i)}$Show that angle$D O E$is 1.287 radians, correct to 4 significant figures.$\text{(ii)}$Find the perimeter of the metal plate.$\text{(iii)}$Find the area of the metal plate.$$Question 7 Code: 9709/11/M/J/19/7, Topic: Vectors The diagram shows a three-dimensional shape in which the base$O A B C$and the upper surface$D E F G$are identical horizontal squares. The parallelograms$O A E D$and$C B F G$both lie in vertical planes. The point$M$is the mid-point of$A F$. Unit vectors$\mathbf{i}$and$\mathbf{j}$are parallel to$O A$and$O C$respectively and the unit vector$\mathbf{k}$is vertically upwards. The position vectors of$A$and$D$are given by$\overrightarrow{O A}=8 \mathbf{i}$and$\overrightarrow{O D}=3 \mathbf{i}+10 \mathbf{k}$.$\text{(i)}$Express each of the vectors$\overrightarrow{A M}$and$\overrightarrow{G M}$in terms of$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$.$\text{(ii)}$Use a scalar product to find angle$G M A$correct to the nearest degree.$$Question 8 Code: 9709/11/M/J/11/8, Topic: Series A television quiz show takes place every day. On day 1 the prize money is$\$1000$. If this is not won the prize money is increased for day 2. The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money.

Model 1: Increase the prize money by $\$ 1000$each day. Model 2: Increase the prize money by$10 \%$each day. On each day that the prize money is not won the television company makes a donation to charity. The amount donated is$5 \%$of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity$\text{(i)}$if Model 1 is used,$\text{(ii)}$if Model 2 is used.$$Question 9 Code: 9709/13/M/J/10/9, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=x+\frac{4}{x}$which has a minimum point at$M .$The line$y=5$intersects the curve at the points$A$and$B$.$\text{(i)}$Find the coordinates of$A, B$and$M$.$\text{(ii)}$Find the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$$Question 10 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 11 Code: 9709/12/M/J/10/10, Topic: Differentiation The equation of a curve is$y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.$\text{(i)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$.$\text{(ii)}$Find the equation of the tangent to the curve at the point where the curve intersects the$y$-axis.$\text{(iii)}$Find the set of values of$x$for which$\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$is an increasing function of$x$.$$Question 12 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration Points$A(2,9)$and$B(3,0)$lie on the curve$y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at$A$intersects the$x$-axis at$C$. Showing all necessary working,$\text{(i)}$find the equation of the tangent$A C$and hence find the$x$-coordinate of$C$,$\text{(ii)}$find the area of the shaded region$A B C$.$\$

Worked solutions: P1, P3 & P6 (S1)

If you need worked solutions for P1, P3 & P6 (S1), contact us @ [email protected] | +254 721 301 418.

1. Send us the link to these questions ( https://stemcie.com/view/16 ).
2. We will solve the questions and provide you with the step by step worked solutions.
3. We will then schedule a one to one online session to take you through the solutions (optional).