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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 |

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 $\mathrm{p}\left(\mathrm{e}^{y}+\mathrm{e}^{-y}\right)=0$. $[4]$