**Section 1**(50 marks)

Answer all the questions in this section in the spaces provided.

- Use logarithms, correct to 4 decimal places, to evaluate. (4 marks)

- Three grades A,B and C of rice were mixed in the ratio 3:4:5. The cost per kg of each of the grades A,B and C were Ksh 120, Ksh 90 and Ksh 60 respectively.

Calculate:- The cost of one kg of the mixture; (2 marks)
- The selling price of 5 kg of the mixture given that the mixture was sold at 8% profit. (2 marks)

- Make s the subject of the formula.

- Solve the inequalities 2x-5 > -11 and 3+2x ≤ 13, giving the answer as a combined inequality. (3 marks)
- List the integral values of x that satisfy the combined inequality in (a) above. (1 marks)

- In the figure below, ABCD is a cyclic quadrilateral, point O is the center of the circle. Angle ABO=30° and angle ADO=40°

Calculate the size of angle BCD. (2 marks) - The ages in years of five boys are 7,8,9,10 and 11 while those of five girls are 4,5,6,7 and 8. A boy and a girl are picked at random and the sum of their ages recorded.
- Draw a probability space to show all the possible outcomes. (1 mark)
- Find the probability that the sum of their ages is at least 17 years. (1 mark)

- The vertices of a triangle are A(1,2),B(3,5) and C(4,1). The coordinates of C’ the image of C under a translation vector 1, are (6,-2).
- Determine the translation vector T. (1 mark)
- Find the coordinates of A’ and B’ under translation vector T. (2 marks)

- Write sin 45
^{0}in the form^{1}/_{√a}where a is a positive integer .Hence simplify . Leaving the answer in surd form. (3 marks) - The radius of a spherical ball is measured as 7 cm , correct to the nearest centimeter. Determine, to 2 decimal places , the percentage error in calculating the surface area of the ball. (4 marks)
- In the figure below , lines NA and NB represent tangents to a circle at points A and B. Use a pair of compasses and ruler only to construct the circle. (2 marks)

- Measure the radius of the circle. (1 mark)

- In the figure below , lines NA and NB represent tangents to a circle at points A and B. Use a pair of compasses and ruler only to construct the circle. (2 marks)
- Expand and simplify the expression. (a+1⁄2)
^{4 }+ (a-1⁄2)^{4}(3 marks) - The figure below represents a scale drawing of a rectangular piece of land ,RSTU. RS=9cm and ST=7cm.

An electric post P, is to be erected inside the piece of land . On the scale drawing, shade the possible region in which P would lie such that PU=PT and PS ≤ 7cm. (3 marks) - Vector
**OP**=6**i**+**j**and**OQ**=-2**i**+ 5**j**. A point N divides**PQ**internally in the ratio 3:1 Find**PN**in terms of**i**and**j**. (3 marks) - A point M (60
^{0}N,18^{0}E) is on the surface of the earth. Another point N is situated at a distance of 630 nautical miles east of M.

Find:- the longitude difference between M an N; (2 marks)
- The position of N. (1 mark)

- The equation of a circle center (a,b) is
*x*^{2 }+*y*^{2 }- 6*x*- 10*y*+ 30 = 0. Find value of a and b. (3 marks) - The table below shows values
*x*and*y*for the function y = sin 3x^{0}in the range 0^{0}≤ x ≤ 150°.- On the grid provided, draw the graph of y = 2 sin 3x. (2 marks)
- From the graph determine the period. (1 mark)

**SECTION II**(50 marks)

Answer only**five**questions in this section in the spaces provided. - The cash price of a laptop was ksh 60000. On hire purchase terms, a deposit if ksh 7500 was paid followed by 11 monthly installments of ksh 6000 each.
- Calculate;
- The cost of a laptop in a hire purchase terms; (2 marks)
- The percentage increase of hire purchase price compared to the cash price. (2 marks)

- An institution was offered a 5% discount when purchasing 25 such laptops on each terms. Calculate the amount of money paid by the institution. (2 marks)
- Two other institutions X and Y bought 25 laptops each. Institution X bought the laptops on hire purchase terms. Institution Y bought the laptops on cash terms with no discount by securing a loan from a bank. The bank charged 12% p.a compound interest for two years.

Calculate how much more money institution Y paid than institution X. (4 marks)

- Calculate;
- The first, fifth and seventh terms of an arithmetic Progression (AP) correspond to the first three consecutive terms of a decreasing Geometric Progression (GP).The first term of each progression is 64, the common difference of AP is
*d*and the common ratio of the G.P is*r*.- Write two equations involving
*d*and*r*. (2 marks) - Find the values of
*d*and*r*. (4 marks)

- Write two equations involving
- Find the sum of the first 10 terms of
- The arithmetic Progression.(AP); (2 marks)
- The Geometric Progression (GP). (2 marks)

- The vertices of a rectangle are A(-1,-1), B(-4.-1),C(-4,-3) and D(-1,-3).
- On the grid provided, draw the rectangle and its image A’B’C’D’ under a transformation whose matrix is (
^{-2}_{0}^{0}_{-2}). (4 marks)

- A’’B’’C’’D’’ is the image of A’B’C’D’ under a transformation matrix

P = (^{1/2}_{1}^{1}_{1/2})- Determine the coordinates of A’’,B’’,C’’ and D’’. (2 marks)
- On the same grid draw the quadrilateral A’’B’’C’’D’’. (1 mark)

- Find the area of A’’B’’C’’D’’. (3 marks)

- On the grid provided, draw the rectangle and its image A’B’C’D’ under a transformation whose matrix is (
- A parent has two children whose age difference is 5 years. Twice the sum of the ages of the two children is equal to the age of the parent.
- Taking
*x*to be the age of the elder child , write an expression for,- The age of the younger child; (1 mark)
- The age of the parent. (1 mark)

- In twenty years time, the product of the children’s ages will be 15 times the age of their parent. (2 mark)
- Form an equation in x and hence determine the present possible ages of the elder child. (4 marks)
- Find the present possible ages of the parent. (2 marks)

- Determine the possible ages of the younger child in 20 years time. (2 marks)

- Taking
- The table below shows values of
*x*and some values of*y*for the curve*y*=*x*^{3 }+ 2*x*^{2}- 3*x*- 4 for -3≤*x*≤2.- Complete the table by filling in the missing values
*y*, correct to 1 decimal place. (2 marks) - On the grid provided, draw the graph of
*y*=*x*^{3}+ 2*x*^{2}- 3*x*- 4.

Use the scale: 1cm represents 0.5 units on*x*-axis.

1 cm represents 1 unit in y-axis. (3 marks) - Use the graph to:
- Solve the equation
*x*^{3}+ 2*x*^{2}- 3*x*- 4 = 6; (3 marks) - Estimate the coordinates of the turning points of the curve. ( 2 marks)

- Solve the equation

- Complete the table by filling in the missing values
- The figure below represents a rectangular based pyramid VABCD. AB = 12 cm and AD = 16cm. Point O is vertically below V and VA =26 cm.

Calculate;- The height,VO, of the pyramid. (4 marks)
- The angle between the edge VA and the plane ABCD; (3 marks)
- The angle between the planes VAB and ABCD.(3 marks)

- The cost C, of producing n items varies partly as n and partly as the inverse of n . To produce two items it costs ksh 135 and to produce three items it cost ksh 140.

Find:- The constants of proportionality and hence write the equation connecting C and n; (5 marks)
- The cost of producing 10 items; (2 marks)
- The number of item produced at a cost of ksh 756. (3 marks)

- A building contractor has two lorries, P and Q used to transport at least 42 tonnes of sand to a building site. Lorry P carries 4 tonnes of sand per trip while lorry Q carries 6 tonnes of sand per trip. Lorry P uses 2 litres of fuel per trip while lorry Q uses 4 litres of fuel per trip. The two lorries are to use less than 32 litres of fuel. The number of trips made by lorry P should be less than 3 times the number of trips made by lorry Q .Lorry P should make more than 4 trips.
- Taking
*x*to represent the number of trips made by lorry P and*y*to represent the number of trips made by lorry Q, write the inequalities that represent the above information. (4 marks) - On the grid provided, draw the inequalities and shade the unwanted regions. (4 marks)
- Use the graph drawn in (b) above to determine the number of trips made by lorry P and by lorry Q to deliver the greatest amount of sand. (2 marks)

- Taking