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

### 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 3 3 5 6 5 7 7 8 9 9 11 11 84
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/11/M/J/13/1, Topic: Differentiation It is given that$\mathrm{f}(x)=(2 x-5)^{3}+x$, for$x \in \mathbb{R}$. Show that$\mathrm{f}$is an increasing function.$[3]$Question 2 Code: 9709/12/M/J/13/1, Topic: Integration A curve is such that$\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]$Question 3 Code: 9709/13/M/J/13/3, Topic: Trigonometry$\text{(i)}$Express the equation$2 \cos ^{2} \theta=\tan ^{2} \theta$as a quadratic equation in$\cos ^{2} \theta$.$[2]\text{(ii)}$Solve the equation$2 \cos ^{2} \theta=\tan ^{2} \theta$for$0 \leqslant \theta \leqslant \pi$, giving solutions in terms of$\pi$.$[3]$Question 4 Code: 9709/12/M/J/16/4, Topic: Series Find the term that is independent of$x$in the expansion of$\text{(i)}\displaystyle\left(x-\frac{2}{x}\right)^{6}$,$[2]\text{(ii)}\displaystyle\left(2+\frac{3}{x^{2}}\right)\left(x-\frac{2}{x}\right)^{6}$.$[4]$Question 5 Code: 9709/11/M/J/14/5, Topic: Series An arithmetic progression has first term$a$and common difference$d$. It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.$\text{(i)}$Find$d$in terms of$a$.$[3]\text{(ii)}$Find the 100 th term in terms of$a$.$[2]$Question 6 Code: 9709/12/M/J/14/6, Topic: Series The$1 \mathrm{st}, 2 \mathrm{nd}$and$3 \mathrm{rd}$terms of a geometric progression are the$1 \mathrm{st}, 9$th and$21 \mathrm{st}$terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is$r$, where$r \neq 1$. Find$\text{(i)}$the value of$r$,$[4]\text{(ii)}$the 4 th term of each progression.$[3]$Question 7 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.$[7]$Question 8 Code: 9709/12/M/J/11/8, Topic: Vectors Relative to the origin$O$, the position vectors of the points$A, B$and$C$are given by $$\overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 3 \\ 5 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 4 \\ 2 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 10 \\ 0 \\ 6 \end{array}\right)$$$\text{(i)}$Find angle$A B C$.$[6]$The point$D$is such that$A B C D$is a parallelogram.$\text{(ii)}$Find the position vector of$D$.$[2]$Question 9 Code: 9709/11/M/J/11/9, Topic: Circular measure In the diagram,$O A B$is an isosceles triangle with$O A=O B$and angle$A O B=2 \theta$radians. Arc$P S T$has centre$O$and radius$r$, and the line$A S B$is a tangent to the$\operatorname{arc} P S T$at$S$.$\text{(i)}$Find the total area of the shaded regions in terms of$r$and$\theta$.$[4]\text{(ii)}$In the case where$\theta=\frac{1}{3} \pi$and$r=6$, find the total perimeter of the shaded regions, leaving your answer in terms of$\sqrt{3}$and$\pi$.$[5]$Question 10 Code: 9709/13/M/J/21/9, Topic: Series$\text{(a)}$A geometric progression is such that the second term is equal to 24% of the sum to infinity. Find the possible values of the common ratio.$[3]\text{(b)}$An arithmetic progression$P$has first term$a$and common difference$d$. An arithmetic progression$Q$has first term$2(a+1)$and common difference$(d+1)$. It is given that $$\frac{\text { 5th term of } P}{12\text{th term of } Q}=\frac{1}{3} \quad \text { and } \quad \frac{\text { Sum of first } 5 \text { terms of } P}{\text { Sum of first } 5 \text { terms of } Q}=\frac{2}{3}$$ Find the value of$a$and the value of$d$.$[6]$Question 11 Code: 9709/11/M/J/20/10, Topic: Coordinate geometry, Differentiation The coordinates of the points$A$and$B$are$(-1,-2)$and$(7,4)$respectively.$\text{(a)}$Find the equation of the circle,$C$, for which$A B$is a diameter.$[4]\text{(b)}$Find the equation of the tangent,$T$, to circle$C$at the point$B$.$[4]\text{(c)}$Find the equation of the circle which is the reflection of circle$C$in the line$T$.$[3]$Question 12 Code: 9709/13/M/J/18/11, Topic: Differentiation, Integration The diagram shows part of the curve$y=(x+1)^{2}+(x+1)^{-1}$and the line$x=1$. The point$A$is the minimum point on the curve.$\text{(i)}$Show that the$x$-coordinate of$A$satisfies the equation$2(x+1)^{3}=1$and find the exact value of$\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$at$A$.$[5]\text{(ii)}$Find, showing all necessary working, the volume obtained when the shaded region is rotated through$360^{\circ}$about the$x$-axis.$[6]\$

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/12 ).
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).