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### HENRYTAIGO

#### Cambridge International AS and A Level

 Name of student HENRYTAIGO Date Adm. number Year/grade HenryTaigo Stream HenryTaigo Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 5 5 5 5 5 5 30
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/14/3, Topic: Differentiation The parametric equations of a curve are $$x=\ln (2 t+3), \quad y=\frac{3 t+2}{2 t+3} .$$ Find the gradient of the curve at the point where it crosses the$y$-axis.$[6]$Question 2 Code: 9709/32/M/J/14/3, Topic: Trigonometry Solve the equation $$\cos \left(x+30^{\circ}\right)=2 \cos x$$ giving all solutions in the interval$-180^{\circ} < x < 180^{\circ}$.$[5]$Question 3 Code: 9709/33/M/J/14/3, Topic: Trigonometry$\text{(i)}$Show that the equation $$\tan \left(x-60^{\circ}\right)+\cot x=\sqrt{3}$$ can be written in the form$[3]$$$2 \tan ^{2} x+(\sqrt{3}) \tan x-1=0$$$\text{(ii)}$Hence solve the equation $$\tan \left(x-60^{\circ}\right)+\cot x=\sqrt{3},$$ for$0^{\circ} < x < 180^{\circ}$.$[3]$Question 4 Code: 9709/31/O/N/14/3, Topic: Algebra The polynomial$a x^{3}+b x^{2}+x+3$, where$a$and$b$are constants, is denoted by$\mathrm{p}(x)$. It is given that$(3 x+1)$is a factor of$\mathrm{p}(x)$, and that when$\mathrm{p}(x)$is divided by$(x-2)$the remainder is 21. Find the values of$a$and$b$.$[5]$Question 5 Code: 9709/32/O/N/14/3, Topic: Algebra Question 6 Code: 9709/33/O/N/14/3, Topic: Algebra The polynomial$4 x^{3}+a x^{2}+b x-2$, where$a$and$b$are constants, is denoted by$\mathrm{p}(x)$. It is given that$(x+1)$and$(x+2)$are factors of$\mathrm{p}(x)$.$\text{(i)}$Find the values of$a$and$b$.$[4]\text{(ii)}$When$a$and$b$have these values, find the remainder when$\mathrm{p}(x)$is divided by$\left(x^{2}+1\right)$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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