Length of pipe that would contain the water from the full tank
\(= \frac{18480}{1000 \times \pi \times 0.014 \times 0.014} = 30000m \)Water takes 1s to flow through 2m of the pipe
Thus for 30000m it would take, \(\frac{30000}{2} = 15000s\)
\(=\frac{15000}{3600}\text{hours}\) \(=4 \frac{1}{6} \text{hours}\)2. The first term of a GP is 2. The common ratio of the GP is also 2. The product of the last two terms of the GP is 512. Determine the number of terms in the GP
\begin{align*} ar^{n-1}ar^{n-2} =512\\ a^{2}r^{(n-1)+(n-2)}=512\\ 2^{2}\times2^{(2n-3)}=512\\ 2^{(2n-3)}=128\\ 2^{(2n-3)}=2^{7}\\ 2n-3=7\\ 2n=7+3\\ 2n=10\\ n=5\\ \end{align*}3. The expression \(ax^{2}-30+9\) is a perfect square , where \(a\) is a constant. Find the value of \(a\)
\begin{align*} ax^{2}-30+9 \\ 4ac =b^{2}\\ 4\times a\times9=(-30)^{2}\\ 36a=900\\ a=\frac{900}{36}\\ a=25 \end{align*}4. Make \(x\) the subject of the formula
\begin{align*} y=\frac{bx}{\sqrt{cx^{2}-a}} \end{align*} \begin{align*} y=\frac{bx}{\sqrt{cx^{2}-a}}\\ y({\sqrt{cx^{2}-a}})=bx\\ y^{2}(cx^{2}-a)=b^{2}x^{2}\\ cx^{2}y^{2}-ay^{2}=b^{2}x^{2}\\ cx^{2}y^{2}-b^{2}x^{2}=ay^{2}\\ x^{2}(cy^{2}- b^{2})=ay^{2}\\ x^{2}=\frac{ay^{2}}{(cy^{2}- b^{2})}\\ x= \pm \sqrt{\frac{ay^{2}}{(cy^{2}- b^{2})}} \end{align*}6. Four quantities P,Q,R and S are such that P varies directly as the square root of Q and inversely as the square of the difference of R and S. Quantity Q is increased by 44% while quantity S and R are each decreased by 10%
The cash price of a gas cooker is Ksh 20 000. A customer bought the cooker on hire purchase terms by paying a deposit of Ksh 10 000 followed by 18 equal monthly instalments of Ksh 900 each. Annual interest, compounded quarterly, was charged on the balance for the period of 18 months. Determine, correct to 1 decimal place, the rate of interest per annum. (4 marks)
\(Balance = 20000-10000 \) \(=10000\) \( Amount = 18 \times 900\) \(=16200\) \(A= P(1+\frac{\frac{r}{4}}{100})^6\) \( 16200=10000(1+\frac{\frac{r}{4}}{100})^6 \\ \) \( \sqrt[6]{1.62}= 1+\frac{\frac{r}{4}}{100}\\ \) \( 1.083725192 - 1 = \frac{\frac{r}{4}}{100}\) \(0.083725 =\frac{\frac{r}{4}}{100}\\ \) \(8.3725 =\frac{r}{4} \\ \) \(r=33.49%=33.5\)11. A point Q is 2000nm to the west of a point P (\(40^\text{o}N ,155^\text{o}W\)). Find the longitude of Q to the nearest degree. 3mks
Let \(\theta = \) longitude difference between P and Q
\(\theta \times 60\times \cos \alpha =2000nm \)
\(\theta \times 60 \cos 40^\text{o} =2000nm \\ \) \(\theta = 43.51^{o} \\ \)Point P has gone into the the eastern side of the globe by 198.51-180
\( =15.49^{o} \\ \) Longitude of point P \(=180-(15.49) \) \(= 161.49^\text{o}E\)10. The equation of a trigonometric wave is \(y=4sin(ax-70)^\text{o}\). The wave has a period of \(180^\text{o}\)
(a) Determine the value of \(a\)
\begin{align*} \frac{360}{a}=180\\ a=2\\ \end{align*}(a) Deduce the phase angle of the wave
\begin{align*} \text{Phase angle }=+70\\ \end{align*}14. The position vectors of point P,Q and R are \(OP=6i-2j+3k\), \(OQ = 12i-5j+6k\) and \(OR = 8i-3j+4k\). Show that \(P\), \(Q\) and \(R\) are collinear.
Think of a straight line with three points on it
\(PQ=\left(\begin{array}{c}12\\ -5\\6 \end{array}\right)-\left(\begin{array}{c}6\\ -2\\ 3\end{array}\right)\\ =\left(\begin{array}{c}6\\ -3\\3\end{array}\right) \\ \) \(PR=\left(\begin{array}{c}8\\ -3\\4 \end{array}\right)-\left(\begin{array}{c}6\\ -2\\3\end{array}\right) \\ =\left(\begin{array}{c}2\\ -1\\1 \end{array}\right) \\ \)If PQ and PR are parallel, then PQ=kPR
\(\left(\begin{array}{c}6\\ -3\\3\end{array}\right) =k\left(\begin{array}{c}2\\ -1\\1\end{array}\right)\\ k=3\\ \therefore PQ=3PR\)P is common point. Points P,Q and R are collinear
15. In a transformation, an object of area \(x cm^2 \) is mapped onto an image whose area is \(13xcm^2\).Given that the matrix of transformation is
\( \left(\begin{array}{c}x& 7\\ x-1&3x\end{array}\right) \\ \).Find the possible values of \(x\). \(3mks\)
Determinant \(= \text{A.S.F} \\ \) \(3x^2-7(x-1)=\frac{13x}{x} \\ \) \(3x^2-7x+7=13 \\ \) \(3x^2-7x+7-13=0 \\ \) \(3x^2-7x-6=0\) \(3x^2-9x+2x-6=0 \\ \) \(3x(x-3)+2(x-3)=0 \\ \) \((3x+2)(x-3)=0\) \(x=3 \\or\\ 3x=-2 \\ \) \(x=\frac{-2}{3} \\ \)7. The figure below represents a prism ABCDEFGH of length 6 cm. The cross section BCFG of the prism is a trapezium in which GF = 11 cm, BC =8 cm, BG= 5 cm and \(\angle{GFC} = \angle{BCF} = 90^{o} \)
Calculate correct to 1 decimal place the angle between the line FA and the plane GFEH.
3d-geometry
\((XF)^2=\sqrt(8^2+6^2) \\ \)
\(=\sqrt(100) \\ \)
XF=10
\((AX)^2=5^2-3^2=16 \\ \)AX=4cm
\(\tan \theta = \frac{4}{10}\) \(\theta =\tan^{-1}0.4=21.8^{o}\)