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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade Stream Subject Pure Mathematics 2 (P2) Variant(s) P21 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 4 5 6 6 8 7 7 6 8 10 10 9 86
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/21/M/J/10/2, Topic: Integration Show that$\displaystyle\int_{0}^{6} \frac{1}{x+2} \mathrm{~d} x=2 \ln 2$.$[4]$Question 2 Code: 9709/21/M/J/15/2, Topic: Logarithmic and exponential functions The variables$x$and$y$satisfy the equation $$y=A \mathrm{e}^{p(x-1)},$$ where$A$and$p$are constants. The graph of$\ln y$against$x$is a straight line passing through the points$(2,1.60)$and$(5,2.92)$, as shown in the diagram. Find the values of$A$and$p$correct to 2 significant figures.$[5]$Question 3 Code: 9709/21/M/J/10/3, Topic: Trigonometry$\text{(i)}$Show that the equation$\tan \left(x+45^{\circ}\right)=6 \tan x$can be written in the form$[3]$$$6 \tan ^{2} x-5 \tan x+1=0.$$$\text{(ii)}$Hence solve the equation$\tan \left(x+45^{\circ}\right)=6 \tan x$, for$0^{\circ} < x < 180^{\circ}$.$[3]$Question 4 Code: 9709/21/M/J/14/3, Topic: Algebra$\text{(i)}$Find the quotient when$6 x^{4}-x^{3}-26 x^{2}+4 x+15$is divided by$\left(x^{2}-4\right)$, and confirm that the remainder is 7.$[3]\text{(ii)}$Hence solve the equation$6 x^{4}-x^{3}-26 x^{2}+4 x+8=0$.$[3]$Question 5 Code: 9709/21/M/J/13/4, Topic: Algebra The polynomial$a x^{3}-5 x^{2}+b x+9$, where$a$and$b$are constants, is denoted by$\mathrm{p}(x)$. It is given that$(2 x+3)$is a factor of$\mathrm{p}(x)$, and that when$\mathrm{p}(x)$is divided by$(x+1)$the remainder is 8.$\text{(i)}$Find the values of$a$and$b$.$[5]\text{(ii)}$When$a$and$b$have these values, factorise$\mathrm{p}(x)$completely.$[3]$Question 6 Code: 9709/21/M/J/15/4, Topic: Algebra The polynomials$\mathrm{f}(x)$and$\mathrm{g}(x)$are defined by $$\mathrm{f}(x)=x^{3}+a x^{2}+b \quad \text { and } \quad \mathrm{g}(x)=x^{3}+b x^{2}-a$$ where$a$and$b$are constants. It is given that$(x+2)$is a factor of$\mathrm{f}(x)$. It is also given that, when$\mathrm{g}(x)$is divided by$(x+1)$, the remainder is$-18$.$\text{(i)}$Find the values of$a$and$b$.$[5]\text{(ii)}$When$a$and$b$have these values, find the greatest possible value of$\mathrm{g}(x)-\mathrm{f}(x)$as$x$varies.$[2]$Question 7 Code: 9709/21/M/J/16/4, Topic: Algebra The polynomial$\mathrm{p}(x)$is defined by $$\mathrm{p}(x)=8 x^{3}+30 x^{2}+13 x-25$$$\text{(i)}$Find the quotient when$\mathrm{p}(x)$is divided by$(x+2)$, and show that the remainder is 5.$[3]\text{(ii)}$Hence factorise$\mathrm{p}(x)-5$completely.$[3]\text{(iii)}$Write down the roots of the equation$\mathrm{p}(|x|)-5=0$.$[1]$Question 8 Code: 9709/21/M/J/17/4, Topic: Numerical solutions of equations The sequence of values given by the iterative formula $$\displaystyle x_{n+1}=\frac{2 x_{n}^{2}+x_{n}+9}{\left(x_{n}+1\right)^{2}}$$ with$x_{1}=2$, converges to$\alpha$.$\text{(i)}$Find the value of$\alpha$correct to 2 decimal places, giving the result of each iteration to 4 decimal places.$[3]\text{(ii)}$Determine the exact value of$\alpha$.$[3]$Question 9 Code: 9709/21/M/J/13/6, Topic: Numerical solutions of equations$\text{(i)}$By sketching a suitable pair of graphs, show that the equation$[2]$$$\cot x=4 x-2,$$ where$x$is in radians, has only one root for$0 \leqslant x \leqslant \frac{1}{2} \pi$.$\text{(ii)}$Verify by calculation that this root lies between$x=0.7$and$x=0.9$.$[2]\text{(iii)}$Show that this root also satisfies the equation$[1]$$$x=\frac{1+2 \tan x}{4 \tan x}.$$$\text{(iv)}$Use the iterative formula$\displaystyle x_{n+1}=\frac{1+2 \tan x_{n}}{4 \tan x_{n}}$to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[3]$Question 10 Code: 9709/21/M/J/20/6, Topic: Trigonometry, Integration$\text{(a)}$Prove that$[5]$$$\sin 2 \theta(\operatorname{cosec} \theta-\sec \theta) \equiv \sqrt{8} \cos \left(\theta+\frac{1}{4} \pi\right)$$$\text{(b)}$Solve the equation $$\sin 2 \theta(\operatorname{cosec} \theta-\sec \theta)=1$$ for$0 < \theta < \frac{1}{2} \pi$. Give the answer correct to 3 significant figures.$[2]\text{(c)}$Find$\displaystyle\int \sin x\left(\operatorname{cosec} \frac{1}{2} x-\sec \frac{1}{2} x\right) \, \mathrm{d} x$.$[3]$Question 11 Code: 9709/21/M/J/14/7, Topic: Differentiation The equation of a curve is $$2 x^{2}+3 x y+y^{2}=3.$$$\text{(i)}$Find the equation of the tangent to the curve at the point$(2,-1)$, giving your answer in the form$a x+b y+c=0$, where$a, b$and$c$are integers.$[6]\text{(ii)}$Show that the curve has no stationary points.$[4]$Question 12 Code: 9709/21/M/J/21/7, Topic: Algebra The polynomial$\mathrm{p}(x)$is defined by $$\mathrm{p}(x)=a x^{3}-11 x^{2}-19 x-a$$ where$a$is a constant. It is given that$(x-3)$is a factor of$\mathrm{p}(x)$.$\text{(a)}$Find the value of$a$.$[2]\text{(b)}$When$a$has this value, factorise$\mathrm{p}(x)$completely.$[3]\text{(c)}$Hence find the exact values of$y$that satisfy the equation$\mathrm{p}\left(\mathrm{e}^{y}+\mathrm{e}^{-y}\right)=0$.$[4]\$

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