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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 | 4 | 5 | 6 | 7 | 7 | 6 | 6 | 8 | 9 | 11 | 9 | 11 | 89 |

Score |

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

The diagram shows the region enclosed by the curve $\displaystyle y=\frac{6}{2 x-3}$, the $x$-axis and the lines $x=2$ and $x=3$. Find, in terms of $\pi$, the volume obtained when this region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

Question 2 Code: 9709/11/M/J/21/2, Topic: Series

The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410.

Find the 60th term of the progression. $[5]$

Question 3 Code: 9709/11/M/J/10/4, Topic: Integration

The diagram shows the curve $y=6 x-x^{2}$ and the line $y=5$. Find the area of the shaded region. $[6]$

Question 4 Code: 9709/13/M/J/14/5, Topic: Differentiation, Quadratics

A function $\mathrm{f}$ is such that $\displaystyle\mathrm{f}(x)=\frac{15}{2 x+3}$ for $0 \leqslant x \leqslant 6$

$\text{(i)}$ Find an expression for $\mathrm{f}^{\prime}(x)$ and use your result to explain why $\mathrm{f}$ has an inverse. $[3]$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$, and state the domain and range of $\mathrm{f}^{-1}$. $[4]$

Question 5 Code: 9709/11/M/J/16/6, Topic: Coordinate geometry

$\text{(a)}$ Find the values of the constant $m$ for which the line $y=m x$ is a tangent to the curve $y=2 x^{2}-4 x+8$. $[3]$

$\text{(b)}$ The function $\mathrm{f}$ is defined for $x \in \mathbb{R}$ by $\mathrm{f}(x)=x^{2}+a x+b$, where $a$ and $b$ are constants. The solutions of the equation $\mathrm{f}(x)=0$ are $x=1$ and $x=9$. Find

$\text{(i)}$ the values of $a$ and $b$, $[2]$

$\text{(ii)}$ the coordinates of the vertex of the curve $y=\mathrm{f}(x)$. $[2]$

Question 6 Code: 9709/12/M/J/16/6, Topic: Circular measure

The diagram shows a circle with radius $r \mathrm{~cm}$ and centre $O$. The line $P T$ is the tangent to the circle at $P$ and angle $P O T=\alpha$ radians. The line $O T$ meets the circle at $Q.$

$\text{(i)}$ Express the perimeter of the shaded region $P Q T$ in terms of $r$ and $\alpha$. $[3]$

$\text{(ii)}$ In the case where $\alpha=\frac{1}{3} \pi$ and $r=10$, find the area of the shaded region correct to 2 significant figures. $[3]$

Question 7 Code: 9709/13/M/J/21/6, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}(x)=x^{2}-2 x+5 \\ &\mathrm{~g}(x)=x^{2}+4 x+13 \end{aligned} $$$\text{(a)}$ By first expressing each of $\mathrm{f}(x)$ and $\mathrm{g}(x)$ in completed square form, express $\mathrm{g}(x)$ in the form $\mathrm{f}(x+p)+q$, where $p$ and $q$ are constants. $[4]$

$\text{(b)}$ Describe fully the transformation which transforms the graph of $y = f(x)$ to the graph of $y = g(x)$. $[2]$

Question 8 Code: 9709/12/M/J/17/7, Topic: Series

$\text{(a)}$ The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20000. $[4]$

$\text{(b)}$ A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression. $[4]$

Question 9 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 10 Code: 9709/12/M/J/17/10, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3 \tan \left(\frac{1}{2} x\right)-2$, for $-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)+4=0$, giving your answer correct to 1 decimal place. $[3]$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and find the domain of $\mathrm{f}^{-1}$. $[5]$

$\text{(iii)}$ Sketch, on the same diagram, the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$. $[3]$

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: 9709/13/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. $[6]$