Aggie’s earnings were as follows:

• Basic salary – Ksh. 38,000  per month
• House allowance – Ksh. 14,000  per month
• Transport allowance – Ksh. 8,500  per month
• Medical allowance – Ksh. 3,300  per month

In that year, income tax was charged as shown in the table below

Aggie was entitled for a tax relief of Ksh. 18,000 for that year.

(a) Calculate

i.  Aggie’s taxable income in K£ per annum.                                                      (2 mks)

ii. Aggie’s pay as you earn.                                               (4 mks)

(b) Aggie is deducted the following monthly:

• Provident fund – Ksh. 1,000

• Loan Repayment – Ksh. 6,250

• Union dues – Ksh. 500

• NHIF – Ksh. 1,000

• Cooperative shares – Ksh. 3,000

Calculate her net monthly salary in Ksh.                                 (4 mks)

A cargo plane leaves an airport P(60⁰N, 20⁰E) and flies due west to airport Q(60⁰N, 25⁰W). 100 minutes later, it then flies due South for 5400 nautical miles to airport R.

(a) Find the position of airport R.                                                          (2 mks)

(b) Calculate the total distance that the plane travelled between airports P and Q via R in kilometres. Use the radius of the earth R=6370 km and  = ²²/₇ .          (3 mks)

(c) If the average speed of the plane was 600 knots, calculate the total time that the plane was airborne.              (3 mks)

(d) If the plane arrived at Q on Sunday 0215 hours. At what time and day did it leave airport P?         (2 mks)

The ratio of Ali’s wages to Ben’s earnings is 5:3. If Ali’s earnings increase by 12%, his new earning becomes Ksh. 5,040. Find the corresponding percentage change in Ben’s earnings if the sum of their new earnings is Ksh. 8,145.                 (4 mks)

A triangle T  with vertices A(2, 3) , B(5, 3)  and C(4, 1)  is mapped onto triangle T'  whose vertices are A'(-4, 3) , B'(-1, 3)  and C'(x, y)  by a transformation M .

Find:

(a) i. The transformation matrix represented by M .                                       (3 mks)

ii. The coordinates of C'                                                                (2 mks)

(b) Triangle T''  is the image of T'  under a reflection in the line y  -  x  =  0 . Find the coordinates of T'' .       (3 mks)

(c) Determine a single matrix of transformation that maps triangle T  onto T'' .      (2 mks)

Below is a line segment XY . Using a ruler and a pair of compasses only construct the locus of a variable point P  above XY  such that angle XPY  60⁰  and the area of DXPY>8.75  cm².           (4 mks)

      _A   .............................................................  _B

The figure below shows a circle centre O, with two parallel chords AB and CD. Chord CD is produced to E and EF is a tangent to the curve at F. The radius of the circle is 5 cm,AB=8cm   and chords AB and CD are 5 cm apart.

Calculate correct to 1 decimal place:

(a) The length of the chord CD.                                                            (2 mks)

(b) The length FE if DE = ³/₂CD .                                                            (2 mks)

Ujenzi Quarries Limited is contracted to supply ballast for the construction of a health centre. The construction is estimated to use 144 tonnes of ballast. The firm intends to use two trucks; Fuso and Faw to transport this ballast. A Fuso truck can carry 8000 kg of ballast while the Faw truck can carry 12,000 kg of ballast per trip. The Fuso truck should make less than 9 trips and the Faw truck should make at most twice the number of trips made by the Fuso truck. The total number of trips should be at least 10 trips. By letting x  and y  to represent the Fuso and Faw trucks respectively, write down all the inequalities to represent the above information.            (4 marks)

The period of the function below is 540⁰

(a) Find the value of k                                            (1 mk)

(b) Determine the maximum displacement of the curve over the x-axis.  (1 mk)

An auditorium is designed so that the next row of seats has d  less seats than the row behind it. The first row has a  seats while the tenth row has 435 seats. There are 25 rows of seats in all. The entire auditorium contains 9000 seats. Find the values of a  and d .              (3 mks)

Find the values of a  and b  in the expression below without using a calculator or mathematical tables. (3 mks)