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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 | 6 | 5 | 7 | 5 | 9 | 7 | 6 | 11 | 9 | 9 | 9 | 9 | 92 |

Score |

Question 1 Code: 9709/12/M/J/17/1, Topic: Series

$\text{(i)}$ Find the coefficient of $x$ in the expansion of $\displaystyle\left(2 x-\frac{1}{x}\right)^{5}$. $[2]$

$\text{(ii)}$ Hence find the coefficient of $x$ in the expansion of $\left(1+3 x^{2}\right)\left(2 x-\frac{1}{x}\right)^{5}$. $[3]$

Question 2 Code: 9709/12/M/J/20/1, Topic: Series

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

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

Question 3 Code: 9709/13/M/J/18/6, Topic: Coordinate geometry

The coordinates of points $A$ and $B$ are $(-3 k-1, k+3)$ and $(k+3,3 k+5)$ respectively, where $k$ is a constant $(k \neq-1)$.

$\text{(i)}$ Find and simplify the gradient of $A B$, showing that it is independent of $k$. $[2]$

$\text{(ii)}$ Find and simplify the equation of the perpendicular bisector of $A B$. $[5]$

Question 4 Code: 9709/11/M/J/21/6, Topic: Quadratics

The equation of a curve is $y=(2 k-3) x^{2}-k x-(k-2)$, where $k$ is a constant. The line $y=3 x-4$ is a tangent to the curve.

Find the value of $k$ $[5]$

Question 5 Code: 9709/11/M/J/13/7, Topic: Quadratics, Differentiation, Coordinate geometry

A curve has equation $y=x^{2}-4 x+4$ and a line has equation $y=m x$, where $m$ is a constant.

$\text{(i)}$ For the case where $m=1$, the curve and the line intersect at the points $A$ and $B$. Find the coordinates of the mid-point of $A B$. $[4]$

$\text{(ii)}$ Find the non-zero value of $m$ for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve. $[5]$

Question 6 Code: 9709/13/M/J/16/7, Topic: Differentiation

The point $P(x, y)$ is moving along the curve $\displaystyle y=x^{2}-\frac{10}{3} x^{\frac{3}{2}}+5 x$ in such a way that the rate of change of $y$ is constant. Find the values of $x$ at the points at which the rate of change of $x$ is equal to half the rate of change of $y$. $[7]$

Question 7 Code: 9709/11/M/J/20/8, Topic: Circular measure

In the diagram, $A B C$ is a semicircle with diameter $A C$, centre $O$ and radius $6 \mathrm{~cm}$. The length of the $\operatorname{arc} A B$ is $15 \mathrm{~cm}$. The point $X$ lies on $A C$ and $B X$ is perpendicular to $A X$.

Find the perimeter of the shaded region $B X C$. $[6]$

Question 8 Code: 9709/11/M/J/10/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x^{2}-12 x+7$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $\mathrm{f}(x)$ in the form $a(x-b)^{2}-c$. $[3]$

$\text{(ii)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(iii)}$ Find the set of values of $x$ for which $f(x)<21$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 2 x+k$ for $x \in \mathbb{R}$.

$\text{(iv)}$ Find the value of the constant $k$ for which the equation $\operatorname{gf}(x)=0$ has two equal roots. $[4]$

Question 9 Code: 9709/11/M/J/16/9, Topic: Series

$\text{(a)}$ The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression. $[3]$

$\text{(b)}$ The first three terms of an arithmetic progression are $2 \sin x, 3 \cos x$ and $(\sin x+2 \cos x)$ respectively, where $x$ is an acute angle.

$\text{(i)}$ Show that $\tan x=\frac{4}{3}$. $[3]$

$\text{(ii)}$ Find the sum of the first twenty terms of the progression. $[3]$

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

The equation of a curve is $y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.

$\text{(i)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $[3]$

$\text{(ii)}$ Find the equation of the tangent to the curve at the point where the curve intersects the $y$-axis. $[3]$

$\text{(iii)}$ Find the set of values of $x$ for which $\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$ is an increasing function of $x$. $[3]$

Question 11 Code: 9709/13/M/J/12/10, Topic: Coordinate geometry, Quadratics

The equation of a line is $2 y+x=k$, where $k$ is a constant, and the equation of a curve is $x y=6$.

$\text{(i)}$ In the case where $k=8$, the line intersects the curve at the points $A$ and $B$. Find the equation of the perpendicular bisector of the line $A B$. $[6]$

$\text{(ii)}$ Find the set of values of $k$ for which the line $2 y+x=k$ intersects the curve $x y=6$ at two distinct points. $[3]$

Question 12 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration

Points $A(2,9)$ and $B(3,0)$ lie on the curve $y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at $A$ intersects the $x$-axis at $C$. Showing all necessary working,$\text{(i)}$ find the equation of the tangent $A C$ and hence find the $x$-coordinate of $C$, $[4]$

$\text{(ii)}$ find the area of the shaded region $A B C$. $[5]$