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

### 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 5 3 5 4 6 5 7 7 7 5 11 11 76
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/11/1, Topic: Trigonometry The coefficient of$x^{3}$in the expansion of$(a+x)^{5}+(1-2 x)^{6}$, where$a$is positive, is 90. Find the value of$a$.$$Question 2 Code: 9709/11/M/J/13/1, Topic: Differentiation It is given that$\mathrm{f}(x)=(2 x-5)^{3}+x$, for$x \in \mathbb{R}$. Show that$\mathrm{f}$is an increasing function.$$Question 3 Code: 9709/13/M/J/10/2, Topic: Series$\text{(i)}$Find the first three terms, in descending powers of$x$, in the expansion of$\displaystyle \left(x-\frac{2}{x}\right)^{6}$.$\text{(ii)}$Find the coefficient of$x^{4}$in the expansion of$\displaystyle \left(1+x^{2}\right)\left(x-\frac{2}{x}\right)^{6}$.$$Question 4 Code: 9709/12/M/J/16/3, Topic: Vectors Relative to an origin$O$, the position vectors of points$A$and$B$are given by $$\overrightarrow{O A}=2 \mathbf{i}-5 \mathbf{j}-2 \mathbf{k} \quad \text { and } \quad \overrightarrow{O B}=4 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k}$$ The point$C$is such that$\overrightarrow{A B}=\overrightarrow{B C}$. Find the unit vector in the direction of$\overrightarrow{O C}$.$$Question 5 Code: 9709/12/M/J/17/3, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2} \equiv \frac{1-\sin \theta}{1+\sin \theta}$.$\text{(ii)}$Hence solve the equation$\displaystyle\left(\frac{1}{\cos \theta}-\tan \theta\right)^{2}=\frac{1}{2}$, for$0^{\circ} \leqslant \theta \leqslant 360^{\circ}$.$$Question 6 Code: 9709/13/M/J/18/3, Topic: Series The common ratio of a geometric progression is$0.99$. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.$$Question 7 Code: 9709/11/M/J/11/7, Topic: Differentiation A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{(1+2 x)^{2}}$and the point$\left(1, \frac{1}{2}\right)$lies on the curve.$\text{(i)}$Find the equation of the curve.$\text{(ii)}$Find the set of values of$x$for which the gradient of the curve is less than$\frac{1}{3}$.$$Question 8 Code: 9709/12/M/J/11/7, Topic: Coordinate geometry The line$L_{1}$passes through the points$A(2,5)$and$B(10,9)$. The line$L_{2}$is parallel to$L_{1}$and passes through the origin. The point$C$lies on$L_{2}$such that$A C$is perpendicular to$L_{2}.$Find$\text{(i)}$the coordinates of$C$,$\text{(ii)}$the distance$A C$.$$Question 9 Code: 9709/11/M/J/16/7, Topic: Circular measure In the diagram,$A O B$is a quarter circle with centre$O$and radius$r$. The point$C$lies on the arc$A B$and the point$D$lies on$O B.$The line$C D$is parallel to$A O$and angle$A O C=\theta$radians.$\text{(i)}$Express the perimeter of the shaded region in terms of$r, \theta$and$\pi$.$\text{(ii)}$For the case where$r=5 \mathrm{~cm}$and$\theta=0.6$, find the area of the shaded region.$$Question 10 Code: 9709/11/M/J/21/7, Topic: Trigonometry$\text{(a)}$Prove the identity$\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta} \equiv 1-\tan ^{2} \theta$.$\text{(b)}$Hence solve the equation$\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta}=2 \tan ^{4} \theta$for$0^{\circ} \leqslant \theta \leqslant 180^{\circ}$.$$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$B$.$\text{(ii)}$Show, with all necessary working, that the areas of the regions marked$P$and$Q$are equal.$$Question 12 Code: 9709/12/M/J/16/11, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 6 x-x^{2}-5$for$x \in \mathbb{R}$.$\text{(i)}$Find the set of values of$x$for which$\mathrm{f}(x) \leqslant 3$.$\text{(ii)}$Given that the line$y=m x+c$is a tangent to the curve$y=\mathrm{f}(x)$, show that$4 c=m^{2}-12 m+16$.$$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$\mathrm{g}^{-1}(x)$.$\$

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/145 ).
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).