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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 | 4 | 5 | 5 | 7 | 4 | 7 | 7 | 8 | 8 | 11 | 10 | 79 |

Score |

Question 1 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 2 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 3 Code: 9709/11/M/J/13/3, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $8 \mathrm{~cm}$. Angle $B O A$ is $\alpha$ radians. $O A C$ is a semicircle with diameter $O A$. The area of the semicircle $O A C$ is twice the area of the sector $O A B$.

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

$\text{(ii)}$ Find the perimeter of the complete figure in terms of $\pi$. $[2]$

Question 4 Code: 9709/12/M/J/19/4, Topic: Trigonometry

Angle $x$ is such that $\sin x=a+b$ and $\cos x=a-b$, where $a$ and $b$ are constants.

$\text{(i)}$ Show that $a^{2}+b^{2}$ has a constant value for all values of $x$. $[3]$

$\text{(ii)}$ In the case where $\tan x=2$, express $a$ in terms of $b$. $[2]$

Question 5 Code: 9709/11/M/J/17/5, Topic: Coordinate geometry

The equation of a curve is $y=2 \cos x$.

$\text{(i)}$ Sketch the graph of $y=2 \cos x$ for $-\pi \leqslant x \leqslant \pi$, stating the coordinates of the point of intersection with the $y$-axis. $[2]$

Points $P$ and $Q$ lie on the curve and have $x$-coordinates of $\frac{1}{3} \pi$ and $\pi$ respectively.

$\text{(ii)}$ Find the length of $P Q$ correct to 1 decimal place. $[2]$

The line through $P$ and $Q$ meets the $x$-axis at $H(h, 0)$ and the $y$-axis at $K(0, k)$.

$\text{(iii)}$ Show that $h=\frac{5}{9} \pi$ and find the value of $k$. $[3]$

Question 6 Code: 9709/11/M/J/21/5, Topic: Series

The fifth, sixth and seventh terms of a geometric progression are $8 k,-12$ and $2 k$ respectively. Given that $k$ is negative, find the sum to infinity of the progression. $[4]$

Question 7 Code: 9709/11/M/J/12/6, Topic: Vectors

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are such that $\mathbf{u}=\left(\begin{array}{c}p^{2} \\ -2 \\ 6\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}2 \\ p-1 \\ 2 p+1\end{array}\right)$, where $p$ is a constant.

$\text{(i)}$ Find the values of $p$ for which $\mathbf{u}$ is perpendicular to $\mathbf{v}$. $[3]$

$\text{(ii)}$ For the case where $p=1$, find the angle between the directions of $\mathbf{u}$ and $\mathbf{v}$. $[4]$

Question 8 Code: 9709/11/M/J/15/6, Topic: Coordinate geometry

The line with gradient $-2$ passing through the point $P(3 t, 2 t)$ intersects the $x$-axis at $A$ and the $y$-axis at $B$.

$\text{(i)}$ Find the area of triangle $A O B$ in terms of $t$. $[3]$

The line through $P$ perpendicular to $A B$ intersects the $x$-axis at $C$.

$\text{(ii)}$ Show that the mid-point of $P C$ lies on the line $y=x$. $[4]$

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: 9709/12/M/J/20/8, Topic: Integration, Coordinate geometry

The diagram shows part of the curve $\displaystyle y=\frac{6}{x}$. The points $(1,6)$ and $(3,2)$ lie on the curve. The shaded region is bounded by the curve and the lines $y=2$ and $x=1$.

$\text{(a)}$ Find the volume generated when the shaded region is rotated through $360^{\circ}$ about the $y$-axis. $[5]$

$\text{(b)}$ The tangent to the curve at a point $X$ is parallel to the line $y+2 x=0$. Show that $X$ lies on the line $y=2 x$. $[3]$

Question 11 Code: 9709/11/M/J/15/10, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{8}{\sqrt{(} 3 x+4)}.$ The curve intersects the $y$-axis at $A(0,4).$ The normal to the curve at $A$ intersects the line $x=4$ at the point $B$.

$\text{(i)}$ Find the coordinates of $B$. $[5]$

$\text{(ii)}$ Show, with all necessary working, that the areas of the regions marked $P$ and $Q$ are equal. $[6]$

Question 12 Code: 9709/13/M/J/15/11, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $r$. The point $C$ on $O B$ is such that angle $A C O$ is a right angle. Angle $A O B$ is $\alpha$ radians and is such that $A C$ divides the sector into two regions of equal area.

$\text{(i)}$ Show that $\sin \alpha \cos \alpha=\frac{1}{2} \alpha$. $[4]$

It is given that the solution of the equation in part $\text{(i)}$ is $\alpha=0.9477$, correct to 4 decimal places.

$\text{(ii)}$ Find the ratio

perimeter of region $O A C$ : perimeter of region $A C B$,

giving your answer in the form $k: 1$, where $k$ is given correct to 1 decimal place. $[5]$

$\text{(iii)}$ Find angle $A O B$ in degrees. $[1]$