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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 | 7 | 5 | 5 | 6 | 7 | 7 | 8 | 8 | 8 | 9 | 10 | 84 |

Score |

Question 1 Code: 9709/11/M/J/12/2, Topic: Series

Find the coefficient of $x^{6}$ in the expansion of $\displaystyle\left(2 x^{3}-\frac{1}{x^{2}}\right)^{7}$. $[4]$

Question 2 Code: 9709/11/M/J/15/3, Topic: Series

$\text{(i)}$ Find the first three terms, in ascending powers of $x$, in the expansion of

$\text{(a)}$ $(1-x)^{6}$ $[2]$

$\text{(b)}$ $(1+2 x)^{6}$. $[2]$

$\text{(ii)}$ Hence find the coefficient of $x^{2}$ in the expansion of $[(1-x)(1+2 x)]^{6}$. $[3]$

Question 3 Code: 9709/11/M/J/20/3, Topic: Series

Each year the selling price of a diamond necklace increases by $5 \%$ of the price the year before. The selling price of the necklace in the year 2000 was $\$ 36000$.

$\text{(a)}$ Write down an expression for the selling price of the necklace $n$ years later and hence find the selling price in 2008. $[3]$

$\text{(b)}$ The company that makes the necklace only sells one each year. Find the total amount of money obtained in the ten-year period starting in the year 2000. $[2]$

Question 4 Code: 9709/11/M/J/14/4, Topic: Differentiation

A curve has equation $\displaystyle y=\frac{4}{(3 x+1)^{2}}$. Find the equation of the tangent to curve at the point where the line $x=-1$ intersects the curve. $[5]$

Question 5 Code: 9709/12/M/J/18/4, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=a+b \cos x$ for $0 \leqslant x \leqslant 2 \pi$. It is given that $\mathrm{f}\left(\frac{1}{3} \pi\right)=5$ and $\mathrm{f}(\pi)=11$.

$\text{(i)}$ Find the values of the constants $a$ and $b$. $[3]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $\mathrm{f}(x)=k$ has no solution. $[3]$

Question 6 Code: 9709/13/M/J/10/5, Topic: Differentiation, Integration

The equation of a curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{\sqrt{(} 3 x-2)}.$ Given that the curve passes through the point $P(2,11)$, find

$\text{(i)}$ the equation of the normal to the curve at $P$, $[3]$

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

Question 7 Code: 9709/13/M/J/12/6, Topic: Series

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

$\text{(i)}$ Find the common difference of the progression. $[2]$

The first term, the ninth term and the $n$th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.

$\text{(ii)}$ Find the common ratio of the geometric progression and the value of $n$. $[5]$

Question 8 Code: 9709/13/M/J/13/7, Topic: Coordinate geometry

The diagram shows three points $A(2,14), B(14,6)$ and $C(7,2).$ The point $X$ lies on $A B$, and $C X$ is perpendicular to $A B$. Find, by calculation,

$\text{(i)}$ the coordinates of $X$, $[6]$

$\text{(ii)}$ the ratio $A X: X B$. $[2]$

Question 9 Code: 9709/13/M/J/14/8, Topic: Quadratics

$\text{(i)}$ Express $2 x^{2}-10 x+8$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants, and use your answer to state the minimum value of $2 x^{2}-10 x+8$. $[4]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $2 x^{2}-10 x+8=k x$ has no real roots. $[4]$

Question 10 Code: 9709/13/M/J/15/9, Topic: Series

$\text{(a)}$ The first term of an arithmetic progression is $-2222$ and the common difference is 17. Find the value of the first positive term. $[3]$

$\text{(b)}$ The first term of a geometric progression is $\sqrt{3}$ and the second term is $2 \cos \theta$, where $0 < \theta < \pi$. Find the set of values of $\theta$ for which the progression is convergent. $[5]$

Question 11 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 12 Code: 9709/12/M/J/15/10, Topic: Integration, Differentiation

The equation of a curve is $\displaystyle y=\frac{4}{2 x-1}$.

$\text{(i)}$ Find, showing all necessary working, the volume obtained when the region bounded by the curve, the $x$-axis and the lines $x=1$ and $x=2$ is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

$\text{(ii)}$ Given that the line $2 y=x+c$ is a normal to the curve, find the possible values of the constant $c$. $[6]$