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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 | 3 | 3 | 5 | 6 | 5 | 7 | 7 | 8 | 9 | 9 | 11 | 11 | 84 |

Score |

Question 1 Code: 9709/11/M/J/13/1, Topic: Differentiation

It is given that $\mathrm{f}(x)=(2 x-5)^{3}+x$, for $x \in \mathbb{R}$. Show that $\mathrm{f}$ is an increasing function. $[3]$

Question 2 Code: 9709/12/M/J/13/1, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{x^{2}}$ and $(2,9)$ is a point on the curve. Find the equation of the curve. $[3]$

Question 3 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 4 Code: 9709/12/M/J/16/4, Topic: Series

Find the term that is independent of $x$ in the expansion of

$\text{(i)}$ $\displaystyle\left(x-\frac{2}{x}\right)^{6}$, $[2]$

$\text{(ii)}$ $\displaystyle\left(2+\frac{3}{x^{2}}\right)\left(x-\frac{2}{x}\right)^{6}$. $[4]$

Question 5 Code: 9709/11/M/J/14/5, Topic: Series

An arithmetic progression has first term $a$ and common difference $d$. It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.

$\text{(i)}$ Find $d$ in terms of $a$. $[3]$

$\text{(ii)}$ Find the 100 th term in terms of $a$. $[2]$

Question 6 Code: 9709/12/M/J/14/6, Topic: Series

The $1 \mathrm{st}, 2 \mathrm{nd}$ and $3 \mathrm{rd}$ terms of a geometric progression are the $1 \mathrm{st}, 9$ th and $21 \mathrm{st}$ terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is $r$, where $r \neq 1$. Find

$\text{(i)}$ the value of $r$, $[4]$

$\text{(ii)}$ the 4 th term of each progression. $[3]$

Question 7 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 8 Code: 9709/12/M/J/11/8, Topic: Vectors

Relative to the origin $O$, the position vectors of the points $A, B$ and $C$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 3 \\ 5 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 4 \\ 2 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 10 \\ 0 \\ 6 \end{array}\right) $$$\text{(i)}$ Find angle $A B C$. $[6]$

The point $D$ is such that $A B C D$ is a parallelogram.

$\text{(ii)}$ Find the position vector of $D$. $[2]$

Question 9 Code: 9709/11/M/J/11/9, Topic: Circular measure

In the diagram, $O A B$ is an isosceles triangle with $O A=O B$ and angle $A O B=2 \theta$ radians. Arc $P S T$ has centre $O$ and radius $r$, and the line $A S B$ is a tangent to the $\operatorname{arc} P S T$ at $S$.

$\text{(i)}$ Find the total area of the shaded regions in terms of $r$ and $\theta$. $[4]$

$\text{(ii)}$ In the case where $\theta=\frac{1}{3} \pi$ and $r=6$, find the total perimeter of the shaded regions, leaving your answer in terms of $\sqrt{3}$ and $\pi$. $[5]$

Question 10 Code: 9709/13/M/J/21/9, Topic: Series

$\text{(a)}$ A geometric progression is such that the second term is equal to 24% of the sum to infinity.

Find the possible values of the common ratio. $[3]$

$\text{(b)}$ An arithmetic progression $P$ has first term $a$ and common difference $d$. An arithmetic progression $Q$ has first term $2(a+1)$ and common difference $(d+1)$. It is given that

$$ \frac{\text { 5th term of } P}{12\text{th term of } Q}=\frac{1}{3} \quad \text { and } \quad \frac{\text { Sum of first } 5 \text { terms of } P}{\text { Sum of first } 5 \text { terms of } Q}=\frac{2}{3} $$Find the value of $a$ and the value of $d$. $[6]$

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

The coordinates of the points $A$ and $B$ are $(-1,-2)$ and $(7,4)$ respectively.

$\text{(a)}$ Find the equation of the circle, $C$, for which $A B$ is a diameter. $[4]$

$\text{(b)}$ Find the equation of the tangent, $T$, to circle $C$ at the point $B$. $[4]$

$\text{(c)}$ Find the equation of the circle which is the reflection of circle $C$ in the line $T$. $[3]$

Question 12 Code: 9709/13/M/J/18/11, Topic: Differentiation, Integration

The diagram shows part of the curve $y=(x+1)^{2}+(x+1)^{-1}$ and the line $x=1$. The point $A$ is the minimum point on the curve.

$\text{(i)}$ Show that the $x$-coordinate of $A$ satisfies the equation $2(x+1)^{3}=1$ and find the exact value of $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ at $A$. $[5]$

$\text{(ii)}$ Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[6]$