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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 | 5 | 6 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 12 | 94 |

Score |

Question 1 Code: 9709/12/M/J/13/3, Topic: Quadratics

The straight line $y=m x+14$ is a tangent to the curve $\displaystyle y=\frac{12}{x}+2$ at the point $P$. Find the value of the constant $m$ and the coordinates of $P$. $[5]$

Question 2 Code: 9709/13/M/J/16/3, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=6 x^{2}+\frac{k}{x^{3}}$ and passes through the point $P(1,9)$. The gradient of the curve at $P$ is $2.$

$\text{(i)}$ Find the value of the constant $k$. $[1]$

$\text{(ii)}$ Find the equation of the curve. $[4]$

Question 3 Code: 9709/12/M/J/11/4, Topic: Differentiation, Coordinate geometry

A curve has equation $\displaystyle y=\frac{4}{3 x-4}$ and $P(2,2)$ is a point on the curve.

$\text{(i)}$ Find the equation of the tangent to the curve at $P$. $[4]$

$\text{(ii)}$ Find the angle that this tangent makes with the $x$-axis. $[2]$

Question 4 Code: 9709/12/M/J/15/5, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta} \equiv \frac{\tan \theta-1}{\tan \theta+1}$. $[1]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{\tan \theta}{6}$, for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[4]$

Question 5 Code: 9709/12/M/J/17/6, Topic: Integration

The diagram shows the straight line $x+y=5$ intersecting the curve $\displaystyle y=\frac{4}{x}$ at the points $A(1,4)$ and $B(4,1)$. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[7]$

Question 6 Code: 9709/12/M/J/15/7, Topic: Coordinate geometry

The point $C$ lies on the perpendicular bisector of the line joining the points $A(4,6)$ and $B(10,2)$. $C$ also lies on the line parallel to $A B$ through $(3,11)$.

$\text{(i)}$ Find the equation of the perpendicular bisector of $A B$. $[4]$

$\text{(ii)}$ Calculate the coordinates of $C$. $[3]$

Question 7 Code: 9709/11/M/J/14/9, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\sin \theta}{1-\cos \theta}-\frac{1}{\sin \theta} \equiv \frac{1}{\tan \theta}$. $[4]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\sin \theta}{1-\cos \theta}-\frac{1}{\sin \theta}=4 \tan \theta$ for $0^{\circ} < \theta < 180^{\circ}$. $[3]$

Question 8 Code: 9709/11/M/J/15/9, Topic: Differentiation

The equation of a curve is $y=x^{3}+p x^{2}$, where $p$ is a positive constant.

$\text{(i)}$ Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of $p$. $[4]$

$\text{(ii)}$ Find the nature of each of the stationary points. $[3]$

Another curve has equation $y=x^{3}+p x^{2}+p x$.

$\text{(iii)}$ Find the set of values of $p$ for which this curve has no stationary points. $[3]$

Question 9 Code: 9709/11/M/J/17/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$ for $x \in \mathbb{R}, x \neq \frac{3}{2}$.

$\text{(i)}$ Find an expression for $\mathrm{f}^{-1}(x)$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 4 x+a$ for $x \in \mathbb{R}$, where $a$ is a constant.

$\text{(ii)}$ Find the value of $a$ for which $\operatorname{gf}(-1)=3$. $[3]$

$\text{(iii)}$ Find the possible values of $a$ given that the equation $\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$ has two equal roots. $[4]$

Question 10 Code: 9709/13/M/J/19/9, Topic: Trigonometry

The function $\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$ is defined for $0 \leqslant x \leqslant \pi$, where $p$ and $q$ are positive constants. The diagram shows the graph of $y=\mathrm{f}(x)$.

$\text{(i)}$ In terms of $p$ and $q$, state the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ State the number of solutions of the following equations.

$\quad\text{(a)}$ $\mathrm{f}(x)=p+q$ $[1]$

$\quad\text{(b)}$ $\mathrm{f}(x)=q$ $[1]$

$\quad\text{(c)}$ $\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q$ $[1]$

$\text{(iii)}$ For the case where $p=3$ and $q=2$, solve the equation $\mathrm{f}(x)=4$, showing all necessary working. $[5]$

Question 11 Code: 9709/12/M/J/13/11, Topic: Coordinate geometry, Integration

The diagram shows the curve $y=\sqrt{(} 1+4 x)$, which intersects the $x$-axis at $A$ and the $y$-axis at $B$. The normal to the curve at $B$ meets the $x$-axis at $C$. Find

$\text{(i)}$ the equation of $B C$, $[5]$

$\text{(ii)}$ the area of the shaded region. $[5]$

Question 12 Code: 9709/12/M/J/15/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x^{2}-6 x+5$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find the set of values of $p$ for which the equation $\mathrm{f}(x)=p$ has no real roots. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 2 x^{2}-6 x+5$ for $0 \leqslant x \leqslant 4$.

$\text{(ii)}$ Express $\mathrm{g}(x)$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

$\text{(iii)}$ Find the range of $\mathrm{g}$. $[2]$

The function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto 2 x^{2}-6 x+5$ for $k \leqslant x \leqslant 4$, where $k$ is a constant.

$\text{(iv)}$ State the smallest value of $k$ for which $\mathrm{h}$ has an inverse. $[1]$

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{h}^{-1}(x)$. $[3]$