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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 3 (P3) | Variant(s) | P31, P33 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 4 | 3 | 7 | 7 | 8 | 8 | 8 | 9 | 10 | 10 | 10 | 10 | 94 |

Score |

Question 1 Code: 9709/31/O/N/12/1, Topic: Algebra

Find the set of values of $x$ satisfying the inequality $3|x-1|<|2 x+1|$. $[4]$

Question 2 Code: 9709/31/O/N/13/1, Topic: Differentiation

The equation of a curve is $y=\displaystyle\frac{1+x}{1+2 x}$ for $x>-\frac{1}{2}$. Show that the gradient of the curve is always negative. $[3]$

Question 3 Code: 9709/33/O/N/18/3, Topic: Numerical solutions of equations

$\text{(i)}$ By sketching a suitable pair of graphs, show that the equation $x^{3}=3-x$ has exactly one real root. $[2]$

$\text{(ii)}$ Show that if a sequence of real values given by the iterative formula

$$ \displaystyle x_{n+1}=\frac{2 x_{n}^{3}+3}{3 x_{n}^{2}+1} $$converges, then it converges to the root of the equation in $\text{part $\text{(i)}$}$. $[2]$

$\text{(iii)}$ Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$

Question 4 Code: 9709/33/O/N/18/5, Topic: Differential equations

The coordinates $(x, y)$ of a general point on a curve satisfy the differential equation

$$ x \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\left(2-x^{2}\right) y $$The curve passes through the point $(1,1)$. Find the equation of the curve, obtaining an expression for $y$ in terms of $x$. $[7]$

Question 5 Code: 9709/31/O/N/11/6, Topic: Trigonometry

$\text{(i)}$ Express $\cos x+3 \sin x$ in the form $R \cos (x-\alpha)$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$, giving the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places. $[3]$

$\text{(ii)}$ Hence solve the equation $\cos 2 \theta+3 \sin 2 \theta=2$, for $0^{\circ}< \theta <90^{\circ}$. $[5]$

Question 6 Code: 9709/31/O/N/17/6, Topic: Differential equations

The variables $x$ and $y$ satisfy the differential equation

$$ \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=4 \cos ^{2} y \tan x $$for $0 \leqslant x < \frac{1}{2} \pi$, and $x=0$ when $y=\frac{1}{4} \pi$. Solve this differential equation and find the value of $x$ when $y=\frac{1}{3} \pi$. $[8]$

Question 7 Code: 9709/33/O/N/12/7, Topic: Integration

The diagram shows part of the curve $y=\sin ^{3} 2 x \cos ^{3} 2 x$. The shaded region shown is bounded by the curve and the $x$-axis and its exact area is denoted by $A$.

$\text{(i)}$ Use the substitution $u=\sin 2 x$ in a suitable integral to find the value of $A$. $[6]$

$\text{(ii)}$ Given that $\displaystyle\int_{0}^{k \pi}\left|\sin ^{3} 2 x \cos ^{3} 2 x\right| \mathrm{d} x=40 A$, find the value of the constant $k$. $[2]$

Question 8 Code: 9709/31/O/N/15/7, Topic: Vectors

The points $A, B$ and $C$ have position vectors, relative to the origin $O$, given by

$$ \overrightarrow{O A}=\left(\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 3 \\ 0 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 1 \\ 1 \\ 4 \end{array}\right) $$The plane $m$ is perpendicular to $A B$ and contains the point $C$.

$\text{(i)}$ Find a vector equation for the line passing through $A$ and $B$. $[2]$

$\text{(ii)}$ Obtain the equation of the plane $m$, giving your answer in the form $a x+b y+c z=d$. $[2]$

$\text{(iii)}$ The line through $A$ and $B$ intersects the plane $m$ at the point $N.$ Find the position vector of $N$ and show that $C N=\sqrt{(} 13)$. $[5]$

Question 9 Code: 9709/31/O/N/11/9, Topic: Differentiation, Integration

The diagram shows the curve $y=x^{2} \ln x$ and its minimum point $M$.

$\text{(i)}$ Find the exact values of the coordinates of $M$. $[5]$

$\text{(ii)}$ Find the exact value of the area of the shaded region bounded by the curve, the $x$-axis and the line $x=\mathrm{e}$. $[5]$

Question 10 Code: 9709/33/O/N/14/9, Topic: Numerical solutions of equations

$\text{(i)}$ Sketch the curve $y=\ln (x+1)$ and hence, by sketching a second curve, show that the equation

$$ x^{3}+\ln (x+1)=40 $$has exactly one real root. State the equation of the second curve. $[3]$

$\text{(ii)}$ Verify by calculation that the root lies between 3 and 4. $[2]$

$\text{(iii)}$ Use the iterative formula

$$ \displaystyle x_{n+1}=\sqrt[3]{\left(40-\ln \left(x_{n}+1\right)\right)} \text {, } $$with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. $[3]$

$\text{(iv)}$ Deduce the root of the equation

$$ \left(e^{y}-1\right)^{3}+y=40. $$giving the answer correct to 2 decimal places. $[2]$

Question 11 Code: 9709/33/O/N/13/10, Topic: Differential equations

A particular solution of the differential equation

$$ 3 y^{2} \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=4\left(y^{3}+1\right) \cos ^{2} x $$is such that $y=2$ when $x=0$. The diagram shows a sketch of the graph of this solution for $0 \leqslant x \leqslant 2 \pi$; the graph has stationary points at $A$ and $B$. Find the $y$-coordinates of $A$ and $B$, giving each coordinate correct to 1 decimal place. $[10]$

Question 12 Code: 9709/33/O/N/16/10, Topic: Vectors

The line $l$ has vector equation $\mathbf{r}=\mathbf{i}+2 \mathbf{j}+\mathbf{k}+\lambda(2 \mathbf{i}-\mathbf{j}+\mathbf{k})$.

$\text{(i)}$ Find the position vectors of the two points on the line whose distance from the origin is $\sqrt{(} 10)$. $[5]$

$\text{(ii)}$ The plane $p$ has equation $a x+y+z=5$, where $a$ is a constant. The acute angle between the line $l$ and the plane $p$ is equal to $\sin ^{-1}\left(\frac{2}{3}\right)$. Find the possible values of $a$. $[5]$