$\require{\cancel}$ $\require{\stix[upint]}$

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 | 4 | 4 | 6 | 4 | 6 | 6 | 5 | 9 | 9 | 8 | 9 | 12 | 82 |

Score |

Question 1 Code: 9709/11/M/J/15/1, Topic: Trigonometry

Given that $\theta$ is an obtuse angle measured in radians and that $\sin \theta=k$, find, in terms of $k$, an expression for

$\text{(i)}$ $\cos \theta$, $[1]$

$\text{(ii)}$ $\tan \theta$, $[2]$

$\text{(iii)}$ $\sin (\theta+\pi)$. $[1]$

Question 2 Code: 9709/11/M/J/14/2, Topic: Quadratics

$\text{(i)}$ Express $4 x^{2}-12 x$ in the form $(2 x+a)^{2}+b$. $[2]$

$\text{(ii)}$ Hence, or otherwise, find the set of values of $x$ satisfying $4 x^{2}-12 x>7$. $[2]$

Question 3 Code: 9709/11/M/J/18/4, Topic: Trigonometry

$\text{(i)}$ Prove the identity $(\sin \theta+\cos \theta)(1-\sin \theta \cos \theta) \equiv \sin ^{3} \theta+\cos ^{3} \theta$. $[3]$

$\text{(ii)}$ Hence solve the equation $(\sin \theta+\cos \theta)(1-\sin \theta \cos \theta)=3 \cos ^{3} \theta$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

Question 4 Code: 9709/13/M/J/20/4, Topic: Series

$\text{(a)}$ Expand $(1+a)^{5}$ in ascending powers of $a$ up to and including the term in $a^{3}$. $[1]$

$\text{(b)}$ Hence expand $\left[1+\left(x+x^{2}\right)\right]^{5}$ in ascending powers of $x$ up to and including the term in $x^{3}$, simplifying your answer. $[3]$

Question 5 Code: 9709/11/M/J/11/5, Topic: Trigonometry

$\text{(i)}$ Show that the equation $2 \tan ^{2} \theta \sin ^{2} \theta=1$ can be written in the form $[2]$

$$ 2 \sin ^{4} \theta+\sin ^{2} \theta-1=0 $$$\text{(ii)}$ Hence solve the equation $2 \tan ^{2} \theta \sin ^{2} \theta=1$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[4]$

Question 6 Code: 9709/12/M/J/21/5, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2 x^{2}+3$ for $x \geqslant 0$.

$\text{(a)}$ Find and simplify an expression for $\mathrm{ff}(x)$. $[2]$

$\text{(b)}$ Solve the equation $\mathrm{ff}(x)=34 x^{2}+19$. $[4]$

Question 7 Code: 9709/12/M/J/21/7, Topic: Coordinate geometry

The point $A$ has coordinates $(1,5)$ and the line $l$ has gradient $-\frac{2}{3}$ and passes through $A$. A circle has centre $(5,11)$ and radius $\sqrt{52}$.

$\text{(a)}$ Show that $l$ is the tangent to the circle at $A$. $[2]$

$\text{(b)}$ Find the equation of the other circle of radius $\sqrt{52}$ for which l is also the tangent at $A$. $[3]$

Question 8 Code: 9709/11/M/J/18/8, Topic: Series

$\text{(a)}$ A geometric progression has a second term of 12 and a sum to infinity of 54. Find the possible values of the first term of the progression. $[4]$

$\text{(b)}$ The $n$th term of a progression is $p+q n$, where $p$ and $q$ are constants, and $S_{n}$ is the sum of the first $n$ terms.

$\text{(i)}$ Find an expression, in terms of $p, q$ and $n$, for $S_{n}$. $[3]$

$\text{(ii)}$ Given that $S_{4}=40$ and $S_{6}=72$, find the values of $p$ and $q$. $[2]$

Question 9 Code: 9709/11/M/J/12/10, Topic: Differentiation

It is given that a curve has equation $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=x^{3}-2 x^{2}+x$.

$\text{(i)}$ Find the set of values of $x$ for which the gradient of the curve is less than 5. $[4]$

$\text{(ii)}$ Find the values of $\mathrm{f}(x)$ at the two stationary points on the curve and determine the nature of each stationary point. $[5]$

Question 10 Code: 9709/12/M/J/13/10, Topic: Series

$\text{(a)}$ The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression. $[4]$

$\text{(b)}$ The third term of a geometric progression is four times the first term. The sum of the first six terms is $k$ times the first term. Find the possible values of $k$. $[4]$

Question 11 Code: 9709/13/M/J/20/10, Topic: Coordinate geometry

$\text{(a)}$ The coordinates of two points $A$ and $B$ are $(-7,3)$ and $(5,11)$ respectively.

Show that the equation of the perpendicular bisector of $A B$ is $3 x+2 y=11$. $[4]$

$\text{(b)}$ A circle passes through $A$ and $B$ and its centre lies on the line $12 x-5 y=70$.

Find an equation of the circle. $[5]$

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 $\mathrm{h}^{-1}(x)$. $[3]$