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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 | 5 | 6 | 7 | 8 | 8 | 8 | 8 | 11 | 9 | 12 | 91 |

Score |

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

$\text{(i)}$ Show that the equation

$$ 3(2 \sin x-\cos x)=2(\sin x-3 \cos x) $$can be written in the form $\tan x=-\frac{3}{4}$. $[2]$

$\text{(ii)}$ Solve the equation $3(2 \sin x-\cos x)=2(\sin x-3 \cos x)$, for $0^{\circ} \leqslant x \leqslant 360^{\circ}$. $[2]$

Question 2 Code: 9709/11/M/J/19/1, Topic: Series

The term independent of $x$ in the expansion of $\displaystyle \left(2 x+\frac{k}{x}\right)^{6}$, where $k$ is a constant, is 540.

$\text{(i)}$ Find the value of $k$. $[3]$

$\text{(ii)}$ For this value of $k$, find the coefficient of $x^{2}$ in the expansion. $[2]$

Question 3 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$ The point $P(2,9)$ lies on the curve.

$\text{(i)}$ A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of $0.05$ units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. $[2]$

$\text{(ii)}$ Find the equation of the curve. $[3]$

Question 4 Code: 9709/12/M/J/11/4, Topic: Differentiation, Coordinate geometry

A curve has equation $\displaystyle y=\frac{4}{3 x-4}$ and $P(2,2)$ is a point on the curve.

$\text{(i)}$ Find the equation of the tangent to the curve at $P$. $[4]$

$\text{(ii)}$ Find the angle that this tangent makes with the $x$-axis. $[2]$

Question 5 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 6 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 7 Code: 9709/11/M/J/18/7, Topic: Vectors

Relative to an origin $O$, the position vectors of the points $A, B$ and $C$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{r} 1 \\ -3 \\ 2 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} -1 \\ 3 \\ 5 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 3 \\ 1 \\ -2 \end{array}\right) $$$\text{(i)}$ Find $\overrightarrow{A C}$. $[1]$

$\text{(ii)}$ The point $M$ is the mid-point of $A C$. Find the unit vector in the direction of $\overrightarrow{O M}$. $[3]$

$\text{(iii)}$ Evaluate $\overrightarrow{A B} \cdot \overrightarrow{A C}$ and hence find angle $B A C$. $[4]$

Question 8 Code: 9709/12/M/J/10/9, Topic: Integration

The diagram shows the curve $y=(x-2)^{2}$ and the line $y+2 x=7$, which intersect at points $A$ and $B$. Find the area of the shaded region. $[8]$

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

The base of a cuboid has sides of length $x \mathrm{~cm}$ and $3 x \mathrm{~cm}$. The volume of the cuboid is $288 \mathrm{~cm}^{3}$.

$\text{(i)}$ Show that the total surface area of the cuboid, $A \mathrm{~cm}^{2}$, is given by $[3]$

$$ A=6 x^{2}+\frac{768}{x}. $$$\text{(ii)}$ Given that $x$ can vary, find the stationary value of $A$ and determine its nature. $[5]$

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: 9709/12/M/J/20/10, Topic: Differentiation

The equation of a curve is $y=54 x-(2 x-7)^{3}$.

$\text{(a)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[4]$

$\text{(b)}$ Find the coordinates of each of the stationary points on the curve. $[3]$

$\text{(c)}$ Determine the nature of each of the stationary points. $[2]$

Question 12 Code: 9709/13/M/J/16/11, Topic: Coordinate geometry

Triangle $A B C$ has vertices at $A(-2,-1), B(4,6)$ and $C(6,-3)$.

$\text{(i)}$ Show that triangle $A B C$ is isosceles and find the exact area of this triangle. $[6]$

$\text{(ii)}$ The point $D$ is the point on $A B$ such that $C D$ is perpendicular to $A B$. Calculate the $x$-coordinate of $D$. $[6]$