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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 | 4 | 6 | 7 | 6 | 7 | 8 | 9 | 9 | 11 | 10 | 87 |

Score |

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

The coefficient of $x^{3}$ in the expansion of $(a+x)^{5}+(2-x)^{6}$ is 90. Find the value of the positive constant $a$. $[5]$

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

$\text{(i)}$ Express the equation $2 \cos ^{2} \theta=\tan ^{2} \theta$ as a quadratic equation in $\cos ^{2} \theta$. $[2]$

$\text{(ii)}$ Solve the equation $2 \cos ^{2} \theta=\tan ^{2} \theta$ for $0 \leqslant \theta \leqslant \pi$, giving solutions in terms of $\pi$. $[3]$

Question 3 Code: 9709/13/M/J/20/3, Topic: Coordinate geometry

In each of parts $\text{(a)}, \text{(b)}$ and $\text{(c)}$, the graph shown with solid lines has equation $y=\mathrm{f}(x)$. The graph shown with broken lines is a transformation of $y=\mathrm{f}(x)$.

$\text{(a)}$

State, in terms of $\mathrm{f}$, the equation of the graph shown with broken lines. $[1]$

$\text{(b)}$

State, in terms of f, the equation of the graph shown with broken lines. $[1]$

$\text{(c)}$

State, in terms of $\mathrm{f}$, the equation of the graph shown with broken lines. $[2]$

Question 4 Code: 9709/13/M/J/15/4, Topic: Trigonometry

$\text{(i)}$ Express the equation $3 \sin \theta=\cos \theta$ in the form $\tan \theta=k$ and solve the equation for $0^{\circ}< \theta <180^{\circ}$. $[2]$

$\text{(ii)}$ Solve the equation $3 \sin ^{2} 2 x=\cos ^{2} 2 x$ for $0^{\circ} < x < 180^{\circ}$. $[4]$

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

A curve has equation $\displaystyle y=3+\frac{12}{2-x}$.

$\text{(i)}$ Find the equation of the tangent to the curve at the point where the curve crosses the $x$-axis. $[5]$

$\text{(ii)}$ A point moves along the curve in such a way that the $x$-coordinate is increasing at a constant rate of $0.04$ units per second. Find the rate of change of the $y$-coordinate when $x=4$. $[2]$

Question 6 Code: 9709/12/M/J/20/6, Topic: Quadratics

The equation of a curve is $y=2 x^{2}+k x+k-1$, where $k$ is a constant.

$\text{(a)}$ Given that the line $y=2 x+3$ is a tangent to the curve, find the value of $k$. $[3]$

It is now given that $k=2$.

$\text{(b)}$ Express the equation of the curve in the form $y=2(x+a)^{2}+b$, where $a$ and $b$ are constants, and hence state the coordinates of the vertex of the curve. $[3]$

Question 7 Code: 9709/13/M/J/21/7, Topic: Series

$\text{(a)}$ Write down the first four terms of the expansion, in ascending powers of $x$, of $(a-x)^{6}$. $[2]$

$\text{(b)}$ Given that the coefficient of $x^{2}$ in the expansion of $\displaystyle \left(1+\frac{2}{a x}\right)(a-x)^{6}$ is $-20$, find in exact form the possible values of the constant $a$. $[5]$

Question 8 Code: 9709/12/M/J/10/9, Topic: Integration

The diagram shows the curve $y=(x-2)^{2}$ and the line $y+2 x=7$, which intersect at points $A$ and $B$. Find the area of the shaded region. $[8]$

Question 9 Code: 9709/11/M/J/12/9, Topic: Coordinate geometry

The coordinates of $A$ are $(-3,2)$ and the coordinates of $C$ are $(5,6).$ The mid-point of $A C$ is $M$ and the perpendicular bisector of $A C$ cuts the $x$-axis at $B$.

$\text{(i)}$ Find the equation of $M B$ and the coordinates of $B$. $[5]$

$\text{(ii)}$ Show that $A B$ is perpendicular to $B C$. $[2]$

$\text{(iii)}$ Given that $A B C D$ is a square, find the coordinates of $D$ and the length of $A D$. $[2]$

Question 10 Code: 9709/13/M/J/20/9, Topic: Functions

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}(x)=x^{2}-4 x+3 \text { for } x>c \text {, where } c \text { is a constant, } \\ &\mathrm{g}(x)=\frac{1}{x+1} \text { for } x>-1 \end{aligned} $$$\text{(a)}$ Express $\mathrm{f}(x)$ in the form $(x-a)^{2}+b$. $[2]$

It is given that $\mathrm{f}$ is a one-one function.

$\text{(b)}$ State the smallest possible value of $c$. $[1]$

It is now given that $c=5$.

$\text{(c)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$. $[3]$

$\text{(d)}$ Find an expression for $\mathrm{gf}(x)$ and state the domain of $\mathrm{gf}$. $[3]$

Question 11 Code: 9709/11/M/J/11/11, Topic: Functions, Coordinate geometry

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned} $$$\text{(i)}$ Find and simplify expressions for $\mathrm{fg}(x)$ and $\operatorname{gf}(x)$. $[2]$

$\text{(ii)}$ Hence find the value of $a$ for which $\mathrm{fg}(a)=\operatorname{gf}(a)$. $[3]$

$\text{(iii)}$ Find the value of $b(b \neq a)$ for which $\mathrm{g}(b)=b$. $[2]$

$\text{(iv)}$ Find and simplify an expression for $\mathrm{f}^{-1} \mathrm{~g}(x)$. $[2]$

The function $\mathrm{h}$ is defined by

$$ \mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0 $$$\text{(v)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[2]$

Question 12 Code: 9709/11/M/J/14/11, Topic: Quadratics, Differentiation, Integration

A line has equation $y=2 x+c$ and a curve has equation $y=8-2 x-x^{2}$.

$\text{(i)}$ For the case where the line is a tangent to the curve, find the value of the constant $c$. $[3]$

$\text{(ii)}$ For the case where $c=11$, find the $x$-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve. $[7]$