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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 4 6 6 6 5 6 8 7 11 11 11 86
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/11/2, Topic: Series$\text{(i)}$Find the terms in$x^{2}$and$x^{3}$in the expansion of$\displaystyle\left(1-\frac{3}{2} x\right)^{6}$.$[3]\text{(ii)}$Given that there is no term in$x^{3}$in the expansion of$\displaystyle (k+2 x)\left(1-\frac{3}{2} x\right)^{6}$, find the value of the constant$k$.$[2]$Question 2 Code: 9709/13/M/J/20/2, Topic: Integration The equation of a curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-3 x^{-\frac{1}{2}}.$It is given that the point$(4,7)$lies on the curve. Find the equation of the curve.$[4]$Question 3 Code: 9709/13/M/J/14/4, Topic: Trigonometry$\text{(i)}$Prove the identity$\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$.$[3]\text{(ii)}$Hence solve the equation$\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$for$0 \leqslant x \leqslant 2 \pi$.$[3]$Question 4 Code: 9709/12/M/J/18/4, Topic: Functions The function$\mathrm{f}$is such that$\mathrm{f}(x)=a+b \cos x$for$0 \leqslant x \leqslant 2 \pi$. It is given that$\mathrm{f}\left(\frac{1}{3} \pi\right)=5$and$\mathrm{f}(\pi)=11$.$\text{(i)}$Find the values of the constants$a$and$b$.$[3]\text{(ii)}$Find the set of values of$k$for which the equation$\mathrm{f}(x)=k$has no solution.$[3]$Question 5 Code: 9709/13/M/J/21/4, Topic: Trigonometry$\text{(a)}$Show that the equation$[2]$$$\frac{\tan x+\sin x}{\tan x-\sin x}=k$$ where$k$is a constant, may be expressed as $$\frac{1+\cos x}{1-\cos x}=k$$$\text{(b)}$Hence express$\cos x$in terms of$k$.$[2]\text{(c)}$Hence solve the equation$\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$for$-\pi < x < \pi$.$[2]$Question 6 Code: 9709/11/M/J/14/5, Topic: Series An arithmetic progression has first term$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]$Question 7 Code: 9709/12/M/J/15/6, Topic: Trigonometry A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height,$h \mathrm{~m}$, of a passenger above the ground is given by the formula$h=60(1-\cos k t)$. In this formula,$k$is a constant,$t$is the time in minutes that has elapsed since the passenger started the ride at ground level and$k t$is measured in radians.$\text{(i)}$Find the greatest height of the passenger above the ground.$[1]$One complete revolution of the wheel takes 30 minutes.$\text{(ii)}$Show that$k=\frac{1}{15} \pi$.$[2]\text{(iii)}$Find the time for which the passenger is above a height of$90 \mathrm{~m}$.$[3]$Question 8 Code: 9709/13/M/J/14/7, Topic: Vectors The position vectors of points$A, B$and$C$relative to an origin$O$are given by $$\overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -1 \\ 7 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 2 \\ 4 \\ 7 \end{array}\right)$$$\text{(i)}$Show that angle$B A C=\cos ^{-1}\left(\frac{1}{3}\right)$.$[5]\text{(ii)}$Use the result in part$\text{(i)}$to find the exact value of the area of triangle$A B C$.$[3]$Question 9 Code: 9709/12/M/J/20/7, Topic: Circular measure In the diagram,$O A B$is a sector of a circle with centre$O$and radius$2 r$, and angle$A O B=\frac{1}{6} \pi$radians. The point$C$is the midpoint of$O A$.$\text{(a)}$Show that the exact length of$B C$is$r \sqrt{5-2 \sqrt{3}}$.$[2]\text{(b)}$Find the exact perimeter of the shaded region.$[2]\text{(c)}$Find the exact area of the shaded region.$[3]$Question 10 Code: 9709/11/M/J/21/9, Topic: Functions Functions$\mathrm{f}$and$\mathrm{g}are defined as follows: \begin{aligned} &\mathrm{f}(x)=(x-2)^{2}-4 \text { for } x \geqslant 2, \\ &\mathrm{~g}(x)=a x+2 \text { for } x \in \mathbb{R}, \end{aligned} wherea$is a constant.$\text{(a)}$State the range of$\mathrm{f}$.$[1]\text{(b)}$Find$\mathrm{f}^{-1}(x)$.$[2]\text{(c)}$Given that$a=-\frac{5}{3}$, solve the equation$\mathrm{f}(x)=\mathrm{g}(x)$.$[3]\text{(d)}$Given instead that$\operatorname{ggf}^{-1}(12)=62$, find the possible values of$a$.$[5]$Question 11 Code: 9709/11/M/J/10/10, Topic: Vectors The diagram shows the parallelogram$O A B C$. Given that$\overrightarrow{O A}=\mathbf{i}+3 \mathbf{j}+3 \mathbf{k}$and$\overrightarrow{O C}=3 \mathbf{i}-\mathbf{j}+\mathbf{k}$, find$\text{(i)}$the unit vector in the direction of$\overrightarrow{O B}$,$[3]\text{(ii)}$the acute angle between the diagonals of the parallelogram,$[5]\text{(iii)}$the perimeter of the parallelogram, correct to 1 decimal place.$[3]$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$\mathrm{f}(x) \leqslant 3$.$[3]\text{(ii)}$Given that the line$y=m x+c$is a tangent to the curve$y=\mathrm{f}(x)$, show that$4 c=m^{2}-12 m+16$.$[3]$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.$[2]\text{(iv)}$State the smallest value of$k$for which$\mathrm{g}$has an inverse.$[1]\text{(v)}$For this value of$k$, find an expression for$\mathrm{g}^{-1}(x)$.$[2]\$

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