HIGH
**General Certificate of Education Advanced Level**
**LOWER \(6^{\text{th}}\)**
**PURE MATHEMATICS**
**PAPER 1**
END OF YEAR 2017 SESSION
3 hours
Additional materials:
FRIDAY 17 NOVEMBER 2017
MORNING
Answer Paper Non-programmable calculator
TIME 3 hours
INSTRUCTIONS TO CANDIDATES
Write your name in the spaces provided on the answer sheet/answer booklet.
There is no restriction on the number of questions which you may attempt.
If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to 2 significant figures.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 120.
Questions are printed in the order of their mark allocations and candidates are advised to attempt questions sequentially.
The use of an electronic calculator is expected, where possible.
You are reminded of the need for clear presentations in your answers.
This question paper consists of 4 printed pages.
Copyright: Tarakino N.P. (Trockers) - Chegutu High School, End of Year 2017
© Tarakino N.P. C.H.S. End of Year 2017
[Turn Over
Compiled and Typed by Trockers
Question 1
Given that \(f(x)=2 x^{3}+k x^{2}-32 x+15\), find the value of \(k\) if ( \(x-3\) ) is a factor. Hence,
(i)
factorise \(f(x)\),
(ii)
sketch the graphs of
#### a)
\(y=f(x)\) [show all the coordinates of the turning points],
#### b)
\(y=f(x)-2\) [no need to show the coordinates of the turning points],
#### c)
\(y=f(x-2)\) [no need to show the coordinates of the turning points],
#### d)
\(y=2 f(x)\) [no need to show the coordinates of the turning points].
Question 2
Express \(\sqrt{3} \cos \emptyset+\sin \emptyset\) in the form \(R \cos (\emptyset-\alpha)\), where \(0 \leq \alpha \leq \frac{\pi}{2}\) and \(R>0\). Hence,
(i)
solve the equation \((\sqrt{3} \cos \emptyset+\sin \emptyset)^{2}=2\),
(ii)
state the maximum and minimum values of \(\frac{4}{\sqrt{3} \cos \emptyset+\sin \emptyset+4}\).
Question 3
In the diagram below, \(D O B C\) is a semicircle, center \(O\) and radius 6 cm. \(A C\) is perpendicular to \(D O B\) where \(A B=2 \mathrm{~cm}\).
Calculate
a)
The exact length of \(A C\).
b)
Angle \(C \hat{O} A\) in radians.
c)
The perimeter of the shaded region.
d)
Express the area of the shaded region as a percentage of the area of the semicircle.
Question 4
Expand \((p+q x)^{4}\) up to and including the term in \(x^{2}\).
Given that the first two terms are \(16-\frac{8}{3} x\), find the values of \(p\) and the value of \(q\).
Hence find the third term of the expansion.
Question 5
Sketch the graph of \(y=|\sin x|\) for \(0 \leq x \leq 360\).
Question 6
Verify that the equation \(10 \cos x-x=0\) has a root between \(x=1\) and \(x=2\). Using \(x=\frac{\pi}{2}\) as a first approximation, show that the next approximation given by applying the Newton Rhaphson method once is \(\frac{5 \pi}{11}\).
Question 7
The tangent to the circle \(x^{2}+y^{2}-2 x-6 y+5=0\) at the point \((3,4)\) meets the \(x\)-axis at \(M\). Find the distance of \(M\) from the centre of the circle.
Question 8
Two straight corridors, \(P\) and \(Q\), each of width \(w\), meet at right angles. \(A B C D\) is a rectangular crate of length \(a\) and breadth \(b\). In the position shown in the diagram, the angle between \(D C\) and the wall of corridor \(P\) is \(\theta\). The crate touches the outer walls at \(A\) and \(B\), and touches the inside corner at \(E\), where \(C E=x\).
i.
Show that \(x \cos \theta+b \sin \theta=w\), and find an equation relating \(a, b, x, w\) and \(\theta\).
ii.
By eliminating \(x\) from the equations in part (i), or otherwise, show that
\(\frac{1}{2} a \sin 2 \theta+b=w(\sin \theta+\cos \theta)\).
iii.
Let \(\theta=45^{\circ}-\emptyset\). Show that the equation in part (ii) may be expressed as
$$ a \cos ^{2} \emptyset-(w \sqrt{2}) \cos \emptyset+\left(b-\frac{1}{2} a\right)=0 $$
iv.
Find \(\theta\) in the case where \(a=4, b=1\) and \(w=2\).
Question 9
Show that \(\frac{d}{d x}\left(\frac{1}{2} x-\frac{1}{4} \sin 2 x\right)=\sin ^{2} x\) and deduce that
$$ \int_{0}^{\pi} \sin ^{2} x d x=\frac{1}{2} \pi $$
Question 10
Points \(A\) and \(B\) have position vectors \(2 \boldsymbol{i}-\boldsymbol{j}+\boldsymbol{k}\) and \(\boldsymbol{i}+3 \boldsymbol{j}+\boldsymbol{k}\), respectively.
a)
Given that \(\overrightarrow{O C}=\overrightarrow{A B}\) and \(\overrightarrow{A D}=\overrightarrow{C B}\), find the position vectors of \(C\) and \(D\).
b)
Hence find the angle between \(\overrightarrow{O C}\) and \(\overrightarrow{A D}\).
Question 11
Prove that \(\frac{1-\cos 2 \beta}{1+\cos 2 \beta} \equiv \sec ^{2} \beta-1\).
Question 12
Given that : \(f(x) \rightarrow \frac{x+p}{x-3}(x \neq 3)\), where \(p\) is a constant.
a)
Find the value of \(p\) if \(f(5)=1 \frac{1}{2}\).
b)
Hence
#### (i)
Find \(f^{-1}(x)\) in a similar form.
#### (ii)
State the value of \(x\) for which \(f^{-1}(x)\) is undefined.
Question 13
The diagram above shows a rectangular box with height \(y cm\), length \(3x cm\) and width \(x cm\). Given that the volume is \(972 cm^{3}\), show that \(y=\frac{324}{x^{2}}\).
Hence find the dimensions of the box if the surface area is to be minimised.
Question 14
(a)
An arithmetic progression has 13 terms whose sum is 143. The third term is 5. Find the first term.
(b)
The sum of the first 3 terms of a GP is 7 times the first term. Find the possible values of the common ratio \(r\).
Hence, using the value of the first term you obtained in 14 (a) above, find the sum to infinity of this progression.
Question 15
(a)
Solve the equation \(\frac{d y}{d x}=2 x+5\), given that when \(x=3, y=-1\).
(b)
At time \(t\) minutes after an oven is switched on, its temperature, \(\theta^{\circ} \mathrm{C}\) is given by \(\theta=200-180 e^{-0.1 t}\).
#### (i)
State the value which the oven's temperature approaches after a long time.
#### (ii)
Find the time taken for the oven's temperature to reach \(150^{\circ} \mathrm{C}\).
#### (iii)
Find the rate at which the temperature is increasing at the instant when the temperature reaches \(150^{\circ} \mathrm{C}\).