INSTRUCTIONS TO THE CANDIDATES
 Write your name and index number in the spaces provided above
 This paper contains two sections; Section A and Section B
 All workings and answers must be written on the question paper in the spaces provided below each question.
 Marks may be given for correct working even if the answer is wrong.
 Show all the steps in your calculations, giving your answers at each stage in the spaces below each question.
 This paper consists of 15 printed pages.
 Candidates should check carefully to ascertain that all the pages are printed as indicated and no questions are missing.
Section A. (50mks)
Answer all the questions in this section in the spaces provided.
 Find the value of x that satisfies the equation. (3mks)
log (2x11) log2= log3  log x  The base and the height of a right angled triangle were measured as 6.4cm and 3.5cm respectively. Determine to 1 decimal place the percentage error in calculating the area of the triangle. (3mks)
 The figure below shows a quadrilateral ABCD in which AB = 8cm, DC =12cm <BAD= 45º <CBD= 90º and <BCD =30º
 The length of BD. (1mk)
 The size of angle ADB . (2mks)
 Simply the expression^{√48}/_{√5+√3} leaving the answer in the form a√b+c where a, b and c are integers. (3mks)

 Expand (1+x)^{7} up to the 4th term (1mk)
 Use the expansion in part (a) above to find the approximate value of (0.94)^{7} to 3 decimal places. (2mks)
 A variable P varies directly as t^{3} and inversely as the square root of S. When t=2 and S = 9 P= 16. Determine the equation connecting P, t and S hence find P when S = 36 and t = 3. (3mks)
 Given that Find the values of x for which AB is a singular matrix. (4mks)
 Use completely the square method to solve
3x^{2}+8x6=0 Correct to 3 significant figures. (3mks)  In the figure below the tangent ST meets chord Vu. Produced at T. chord SW passes through the centre O of the circle and intersect chord Vu at x. Line ST = 12cm and uT= 8cm
 Calculate the length of chord Vu. (1mk)
 If wx = 3cm and Vx:xu = 2:3 . find Sx
 Make n the subject of the formula.
^{r}/_{P} = ^{M}/_{√n1} (2mks)  The equation of a circle is given by x^{2}+4x+y^{2}2y4=0. Determine the centre and radius of the circle. (3mks)
 The 5th term of an AP is 82 and the 12th term is 103.
Find The first term and the common difference. (2mks)
 The sum of the first 21 terms. (2mks)
 Use matrices to solve the simultaneous equation. (3mks)
2x+3y=39
5x+2y=81  Solve the following pair of simultaneous inequalities and illustrate the value on a number line. (3mks)
3x1 >4
2x+1 ≤7  A triangle has sides 10cm, 7cm and 9cm. find
 The area. (2mks)
 The size of angle <BAC. (2mks)
 A plot of land is valued at sh 1,250,000 due to increase in demand its appreciates at the rate of 6% every six months. What will be its value after 3 ½ years. (3mks)
Section B (50mks)
Answer any five questions from this section on the spaces provided.  In a mixed school there are 420 boys and 350 girls. The probability that a girl passes her exams in the school is ^{4}/_{7} while that of a boy passing is ^{5}/_{8}. The probability of a girl being made a prefect is ^{2}/_{11} while that of a boy is ^{1}/_{8}.
 Find the probability that a student picked at random.
 Is a boy and passes the exam and is not a perfect. (3mks)
 Is a girl, a prefect and passes the exam. (3mks)
 Is not as prefect and passes the exam. (4mks)
 Find the probability that a student picked at random.
 The table below shows some values of the curves.
y=2 sinx and y=3 cosx Complete the table for the values of y=2 sinx and y=3 cosx correct to 1 decimal place. (2mks)
xº 0º 30º 60º 90º 120º 150º 180º 210º 240º 270º 300º 330º 360º y=2 sinx 0 1 2 1 0 1 1.7 0 y=3 cos3 3 1.5 0 2.6 1.5 3  On the grid below draw the graph of y=3 cosx and y=2 sinx for 0º ≤x ≤360º (5mks)
 Use the graph to find the values of x when 3 cos32 sinx=0 (2mks)
 Find the difference in amplitude of y=3 cosx and y=2 cosx (1mk)
 Complete the table for the values of y=2 sinx and y=3 cosx correct to 1 decimal place. (2mks)
 In the figure below , OP = P ,OQ = ⏟q PQ∶ QR =1:1 and OQ: QS = 3:1
 Determine, in terms of P and q
 PQ (1mk)
 RS (2mks)
 If RS:ST = 1:K and OP:PT = 1:n
Determine. ST in terms of P, q and K. (2mks)
 The values of K and n. (5mks)
 Determine, in terms of P and q
 Using a ruler and a compass only construct.
 Triangle PQR such that PQ = 6cm , <PQR = 60º and <RPQ = 45º (4mks)
 Locate the point A in the triangle when is equidistant from all the three sides of triangle PQR. (3mks)
 Find the distance of A from the sides of the triangles. (1mk)
 Drop a perpendicular height to PQ and Measure its height. (2mks)
 The figure below represents a cuboids EFGHJKLM in which EF= 40cm FG=9cm and GM= 30cm. N is the midpoint of LM.
Calculate correct to 3 significant figures The length of GL. (2mks)
 The length of FJ. (3mks)
 The angle between EM and the plane EFGH. (3mks)
 The angle between the planes EFGH and ENH. (2mks)
 The table below shows income tax rates for a certain year.
Monthly income in Kshs Tax rate in each shillings 1 – 9400 10% 9401 – 18000 10% 18001 – 26600 20% 26601 – 35600 25% 35601 –and above 30%  Calculate
 His taxable income per month. (2mks)
 The amount of tax he paid in a month. (5mks)
 Opunyi’s salary included a medical allowance of Shs 8000. He contributed 6% of his basic salary to a sacco. Calculate his net pay. (3mks)
 Calculate
 The masses of 100 patients in a hospital were distributed as shown in the table below.
Mass (Kg) 09 1019 2029 3039 4049 5059 6069 7079 8089 9099 Frequency 3 7 8 9 12 18 25 10 6 2  State the modal class. (1mk)
 Calculate
 The mean mass of the patients. (3mks)
 The standard deviation of the distribution. (3mks)
 Find the interquartile range for the data. (3mks)
 OABC is a parallelogram with vertices O (0, 0) A (2, 0) B (3, 2) and C (1, 2).
O^{1} A^{1} B^{1} C^{1} is the image of OABC under transformation matrix Find the coordinates of O^{1} A^{1} B^{1} C^{1}. (2mks)
 On the grid provided draw OABC and O^{1} A^{1} B^{1} C^{1} (2mks)
 Find O^{11} A^{11} B^{11} C^{11} the image of O^{1} A^{1} B^{1 }C^{1} under the transformation matrix (2mks)
 On the same grid draw O^{11} A^{11} B^{11} C^{11} . (1mk)
 Find the single matrix that map O^{11} A^{11} B^{11} C^{11} onto OABC. (3mks)
MARKING SCHEME
 log(2x11) log2= log3 logx
^{2x11}/_{2} = ^{3}/_{x}
x(2x11) = 6
2x^{2} – 11x – 6 = 0
2x^{2} – 12x + x – 6 = 0
2x(x6) + 1(x – 6) = 0
2x + 1 = 0 x 6= 0
2x = 1 x =6
X=  ½
M_{1} for ^{2x11}/_{2} = ^{3}/_{3}M_{1} for attempting to solve quadritic equation
A_{1} for both values of x  Actual area
= ½̸ ×64^{32} ×3.5 =11.2
Limits 6.45 3.55
6.35 3.45
Max area ½ × 6.45 ×3.55 =11.45
Min area ½ × 6.35 ×3.45= 10.93
A.E. A  max min 11.4510.95=0.25
2 2
% Error = ^{0.25}/_{1.2} ×100
= =2.2%
M_{1} for limits
M_{1} for ½ ×6.35 ×3.45
A_{1} for % error
BD = 12
sin30 sin90
BD = 12 × sin30 = 6
sin90
8 = 6
SinADB sin45
SinADB= 8 × sin45
6
= 0.942
ADB = sin^{1}0.942
= 70.39
M_{1} for expression to find BD
M_{1} for SinADB= 8 × sin45
6
A_{1} for % correct Angle √48 × √5√3
√5+√3 √5 √3
√240√144 = 4√1512
52 3
= ^{4}/_{3} √15  4
M_{1} for multipying by conjugate
M_{1} for 52
A_{1} for % C.A 
 (1+ x)^{7}1^{7}xº + x^{1} – x^{2 }+ x^{3}
1 4 6 4
= 1 + 4x + 6x^{2} + 4x^{2}  (0.94)^{7} = (1 + 0.06)^{7} x = 0.06
1 + 4 (0.06) + 6( 0.06^{2}) + 4 (0.06^{3})
1 + 0.24 + 0.0036 =1.244
B_{1} for corrrect expansion
M_{1} for x= 0.06
A_{1} for correct value
 (1+ x)^{7}1^{7}xº + x^{1} – x^{2 }+ x^{3}
 P = Kt^{3}
√5
16 = K2^{3} √9
16 × 3= 8 K 16 × 3=K
8 8 8
K=6 K= 6
P = 6t^{3}
√5
P = 6 ×3^{3 }= 6 × 27 = 27
√36 6
P = 6 ×3^{3 }= 6 × 27 = 27
√36 6
M_{1} for P = 6 × 3^{3 } √3
A_{1} correct value of P.
(3+3x)(2x+2) = 6( x+7)
6x+6+6x^{2}+ 6x = 6x+42
6x^{2}+ 12x – 6x + 6 – 42 = 0
6x^{2} – 6x – 36 = 0
x^{2} – x – 6 = 0
x^{2}– 3x + 2x – 6 = 0
x( x 3) + 2( x3 ) = 0
x+ 2 = 0 x – 3 = 0
x = 2 x = 3
M_{1} for ( 3+3x)( 2x+2) =6 ( x+7)
M_{1} for Attempting to solve quadratic equation
A_{1} for x = 2
A_{1} for x = 3 3x^{2}+8x6=0
x^{2} + ^{8}/_{8} x – ^{6}/_{3}=0
x^{2} + ^{8}/_{3} x+K=2+K
K=(^{8}/_{3} ÷2)^{2}= (^{8}/_{3} ×½ = (^{4}/_{3})^{2}= ^{16}/_{9}
x^{2}+ ^{8}/_{3} x + ^{16}/_{9}= 2+ ( 16 )/(9 )
( x+ ^{4}/_{3})^{2 }=3^{7}/_{9}
x+^{4}/_{3}= √3^{7}/_{9 }x= 1.9441.333
x=1.9441.333 x= 3.277
x=0.611
M_{1} for x^{2}+ ^{8}/_{3} x + ^{16}/_{9}= 2+ ( 16 )/(9 )
M_{1} for x+^{4}/_{3}= √3^{7}/_{9 }A_{1} for both values of x 
 TS^{2 }=8(8+x)
144 = 64 + 8x
80 = 8
10 = x
Vu = 10cm  vx = 4cm (^{2}/_{5} ×10^{2})
Xu= 10cm – 4 = 6
4×6=3 × x x = 4 x 6^{2 }=8cm
3
Sx = 8cm
M_{1} for 144=64+8x
A_{1} for C.A
M_{1} for vx or xu
A_{1} for 8cm
 TS^{2 }=8(8+x)
 ^{r}/_{p} = ^{m}/_{√n1}^{r2}/_{p2} = (m^{2})/(n1 )
r^{2} (n1 )= m^{2}p^{2}
r^{2}n  r^{2} =m^{2}p^{2}
r^{2} n =m^{2}p^{2}^{ }+r^{2} r^{2} r^{2}n=m^{2}p^{2}^{ }+r^{2} r^{2}M_{1} for squaring
A_{1} for C.A  x^{2}+4x+K+y^{2}2y+K=4
K= (^{4}/_{2})^{2}=4 K= (^{2}/_{2})^{2}=1
x^{2}+4x+4 +y^{2}2y+1=4+4+1
(x+2)^{2} + (y1)^{2}= a
(xa)^{2} +(yb )^{2}= r^{2}a= 2 b=+1 r=3
M_{1 }for competing the square
A_{1} for centre
A_{1} for radius 
 5th = a+4d=82
12th= a+11d=103
7=21
d=3
a+4d=82
a+12=82
a=8212
= 70  = ^{n}/_{2}( 2a+(n1)d
= 2½ (2×70+20 ×3)
= 2½ ((140+60)
= 2100
M_{1} for 7d= 21 (or equivalent)
A_{1} for term 1
M_{1} for substitution
A_{1} for C.A
 5th = a+4d=82
x=^{2}/_{11}×39+ ^{3}/_{11}×81=15
y= ^{5}/_{11}×39 +^{ 2}/_{11}×81=3
M_{1} for determinant
A_{1} for correct values of x and y 3x1 > 4 2x+1≤7
3x >4+1 2x≤6
3x > 3 x ≤3
x> 1
 1 <x ≤3
M_{1 }for 3x > 3
A_{1} for x ≤3 and x >1
B_{1} for number line with correct arrow  Area =P=10+7+9
= 26
S= 13
A = √(13(1310)( 137)(139))
=30.5
½ ×10 ×7 ×sinBAC=30.5
= sinBAC= ^{30.5}/_{35}
= 0.8714
BAC=60.6
M_{1} for for substitution of a, b c and S in the formula
A_{1 }Correct Area
M_{1} for sinθ= ^{30.5}/_{35}A_{1} for correct angle  A= P( 1+ ^{r}/_{100} )^{n}
= 1250000 (1+ ^{6}/_{100})^{7}
=1250000 × 1.06^{7}=1879537.82
M_{1 }for substitution
M_{1} for (1+ ^{6}/_{100})^{7}A_{1} for Amount 
 B and passes and not prefect
^{6}/_{11} × ^{5}/_{8 }× ^{7}/_{8 }= ^{105}/_{352}  G and prefect and pass
^{5}/_{11} × ^{2}/_{11} × ^{4}/_{7}= ^{40}/_{847}  B or G
B NP passes or G NP passes
= ^{6}/_{11} × ^{5}/_{8 }× ^{7}/_{8} + ^{5}/_{11} × ^{9}/_{11} × ^{4}/_{7}= ^{105}/_{352} + ^{180}/_{847}
= 0.5108
M_{1} for 7/(8 )
M_{1} for ^{6}/_{11} × ^{5}/_{8 }× ^{7}/_{8}
A_{1} for C.A
M_{1} for ^{5}/_{11} × ^{2}/_{11} × ^{4}/_{7}
A_{1} for C.A
M_{1} for ^{6}/_{11} × ^{5}/_{8 }× ^{7}/_{8} + ^{5}/_{11} × ^{9}/_{11} × ^{4}/_{7}
A_{1} for ^{105}/_{352}
A_{1} for ^{180}/_{847}
A_{1} for 0.5108
 B and passes and not prefect



 PQ=PO+OQ
=  P+q  RS = RQ+Q S
= PQ+ ^{1}/_{3} OQ
= P q+ ^{1}/_{3} q
P  ^{2}/_{3} q
 PQ=PO+OQ
 ^{ }
 ^{ST}/_{RS }= ^{K}/_{T}
ST=KRS
K=(P ^{2}/_{3}q)
=KP  ^{2}/_{3} qK  Expressing RT in two ways
RT = RS + ST
= P ^{2}/_{3} q+KP ^{2}/_{3}Kq
=(1+K)P +(^{}^{2}/_{3} ^{2}/_{3} K)^{q}
RT=RP+PT
= 2PQ + nOP
= 2(q p) + nP
= (2+ n)p – 2q
(1 +K) P + (^{2}/_{3} ^{2}/_{3}K)^{q }=(2+n )p2q
^{2}/_{3 } ^{2}/_{3} K= 2 1 + K = 2+ n
K=2 1+ 2 = 2+ n
1 = n
B_{1} for PQ
M_{1} for P q + ^{1}/_{3}q
A_{1} for P ^{2}/_{3}q
M_{1} for K(P^{2}/_{3}q)
A_{1} for correct vector
M_{1} for P ^{2}/_{3} q+KP ^{2}/_{3} Kq
M_{1} for (1 +K) P + (^{2}/_{3} ^{2}/_{3}K)^{q}M_{1} for equating the two expression
A_{1} for K
A_{1} for n
 ^{ST}/_{RS }= ^{K}/_{T}



 GL^{2}= 9^{2} + 30^{2}
GL=31.3  FJ^{2 }=FH^{2}+ HJ^{2}
FH^{2 }= 40+ 9^{2}
FH =HI
FJ^{2}= 41^{2 }+30^{2}
FJ^{2}= 50.8  EM and EFGH
EM projection = EG
<GEM
Tan θ = ^{ 30}/_{41}=0.75
Q = 36.87
M_{1} for 9^{2} )+ 30^{2}A_{1} for correct lenght
M_{1} for FH^{2}=40+ 9^{2}M_{1} for FJ^{2}= 41^{2 }+30^{2}A_{1} for 50.8
M_{1} for Tan θ= ^{30}/_{41}M_{1} for projution
A_{1} for Q
M_{1} for Tan θ
A_{1 }for angle
 GL^{2}= 9^{2} + 30^{2}
 Taxable income
= 35600 + 3200
= 38800 1st 9400 ×^{10}/_{100}=940
Next 8600× ^{15} /_{100}×8600=1290
Next 8600× ^{20}/_{100}×8600=1720
Next 9000 × ^{25}/_{100}×9000=2250
Next 3200 × ^{30}/_{100} ×3200=960
(Total tax = & 71 60
less relief 1172 )
Shs 5988  Basic salary = taxable income – allowances
= 38800 – 8000
Sacco = ^{6}/_{100 }×30800 = 1848
Net salary = T.1 – (PAYE + Sacco
= 38800 – (5988 + 1848)
= 38800 – 7836
= Ksh 30964
M_{1} for addition
A_{1} for 38800
M_{1} for first slab
M_{1} for 2nd and 3rd slabs
M_{1} for last 2 slabs
A_{1} for gross tax
A_{1} for Net tax
M_{1} for 1848
M_{1} for substration
A_{1} for C.A
 1st 9400 ×^{10}/_{100}=940

Mass X F Fx dxx d^{2} Fd^{2} x^{2} Fx^{2} 09 4.5 3 13.5 20.25 60.73 1019 14.5 7 101.5 210.25 1471.75 2029 24.5. 8 196 600.25 4802 3039 34.5 9 310.5 1190.25 10712.25 4049 44.5 12 534 1980.25 23769 5059 54.5 18 981 2970.25 53464.5 6069 64.5 25 1612.5 4160.25 104006.25 7079 74.5 10 745 5550.25 55502.5 8089 84.5 6 507 7140.25 42841.5 9099 94.5 2 189 8930.25 17860.5 100 5190  Modal class
= 60 – 69 
 (Mean =^{ƸFx}/_{ƸF} = ^{5190}/_{100 }= 51.9
 standard deviation
√ƸFx^{2}/ƸF (ƸFx/ƸF)^{2}= ^{314485}/_{100} 51.9^{2}= 3144.86 – 2696.61
=√448.24
=21.17± 0.1
B_{1} for Modal class
M_{1} for ƸFx
M_{1} for ƸF
A_{1} for Mean
M_{1} for ^{314485}/_{100} 51.9^{2}M_{1} for Square root
A_{1} for C.A
 Modal class
M_{1} for finding cordinates
A_{1} for cordinates
B_{1} for OABC drawn
B_{1} for O^{1} A^{1} B^{1} C^{1} drawn
M_{1} for Attempting to find cordinates
of O^{11} A^{11} B^{11 }C^{11}A_{1} coordinates of O^{11} A^{11} B^{11} C^{11}B_{1} for O^{11 }A^{11} B^{11} C^{11} drawn
M_{1} for Attempting to find matrix
A_{1} for a= ½
A_{1} for matrix
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