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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 | 4 | 5 | 5 | 6 | 7 | 5 | 8 | 8 | 7 | 8 | 8 | 75 |

Score |

Question 1 Code: 9709/12/M/J/10/2, Topic: Integration

The diagram shows part of the curve $y=\displaystyle \frac{a}{x}$, where $a$ is a positive constant. Given that the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis is $24 \pi$, find the value of $a$. $[4]$

Question 2 Code: 9709/11/M/J/18/2, Topic: Differentiation

A point is moving along the curve $\displaystyle y=2 x+\frac{5}{x}$ in such a way that the $x$-coordinate is increasing at a constant rate of $0.02$ units per second. Find the rate of change of the $y$-coordinate when $x=1$. $[4]$

Question 3 Code: 9709/12/M/J/18/2, Topic: Quadratics

The equation of a curve is $y=x^{2}-6 x+k$, where $k$ is a constant.

$\text{(i)}$ Find the set of values of $k$ for which the whole of the curve lies above the $x$-axis. $[2]$

$\text{(ii)}$ Find the value of $k$ for which the line $y+2 x=7$ is a tangent to the curve. $[3]$

Question 4 Code: 9709/12/M/J/18/3, Topic: Series

A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by $2 \%$ of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained $8000 \mathrm{~kg}$ of salt.

$\text{(i)}$ Find the amount of salt obtained in the 12 th week after the change. $[3]$

$\text{(ii)}$ Find the total amount of salt obtained in the first 12 weeks after the change. $[2]$

Question 5 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 6 Code: 9709/11/M/J/10/5, Topic: Quadratics

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=2 \sin ^{2} x-3 \cos ^{2} x$ for $0 \leqslant x \leqslant \pi$.

$\text{(i)}$ Express $\mathrm{f}(x)$ in the form $a+b \cos ^{2} x$, stating the values of $a$ and $b$. $[2]$

$\text{(ii)}$ State the greatest and least values of $\mathrm{f}(x)$. $[2]$

$\text{(iii)}$ Solve the equation $f(x)+1=0$. $[3]$

Question 7 Code: 9709/12/M/J/19/5, Topic: Circular measure

The diagram shows a semicircle with diameter $A B$, centre $O$ and radius $r$. The point $C$ lies on the circumference and angle $A O C=\theta$ radians. The perimeter of sector $B O C$ is twice the perimeter of sector $A O C$. Find the value of $\theta$ correct to 2 significant figures. $[5]$

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: 9709/11/M/J/16/8, Topic: Coordinate geometry

A curve has equation $\displaystyle y=3 x-\frac{4}{x}$ and passes through the points $A(1,-1)$ and $B(4,11)$. At each of the points $C$ and $D$ on the curve, the tangent is parallel to $A B$. Find the equation of the perpendicular bisector of $C D$. $[7]$

Question 11 Code: 9709/12/M/J/19/9, Topic: Coordinate geometry

The curve $C_{1}$ has equation $y=x^{2}-4 x+7$. The curve $C_{2}$ has equation $y^{2}=4 x+k$, where $k$ is a constant. The tangent to $C_{1}$ at the point where $x=3$ is also the tangent to $C_{2}$ at the point $P$. Find the value of $k$ and the coordinates of $P$. $[8]$

Question 12 Code: 9709/11/M/J/21/10, Topic: Coordinate geometry

The equation of a circle is $x^{2}+y^{2}-4 x+6 y-77=0$.

$\text{(a)}$ Find the $x$-coordinates of the points $A$ and $B$ where the circle intersects the $x$-axis. $[2]$

$\text{(b)}$ Find the point of intersection of the tangents to the circle at A and B. $[6]$