INSTRUCTIONS TO CANDIDATES
- This paper consists of TWO sections. Section A and Section B.
- Answer ALL the questions in section A and only FIVE questions from Section B.
- Show all the steps in your calculations, giving your answers at each stage
- Marks may be given for correct working even if the answer is wrong.
- Non-programmable silent calculators and KNEC mathematical tables may be used except where stated otherwise.
Questions
SECTION A
- Make x the subject of the formula. (3mks)
- Simplify the following by rationalizing the denominator. (3mks)
- A quantity P is partly constant and partly varies inversely as square of t. p =6 when t=6 and p=18 when t=3. Find t when p=11. (3mks)
- Solve for x in the equation; (3mks)
Log8( x + 6) –log8(x – 3) = 2/3 - In the figure below QT is a tangent to a circle at Q. PXRT and QXS are straight lines.
PX =6cm, RT=8cm, QX=4.8cm and XS= 5cm.
|
Find the length of;- XR (2mks)
- QT (2mks)
- Solve for x and y in the simultaneous equation below. (3mks)
xy + 6 = 0
x – 2y = 7 - Solve for x. (3mks)
2x2 + x – 36 = 0 - Expand (1+2x)7 up to the term in x3, hence use the expansion to estimate the value of (1.02)7 correct to four decimal places. (3mks)
- Find the value of y for which is a singular matrix. (3mks)
-
- Find the inverse of the matrix . (1mk)
- Hence solve the simultaneous equation using the matrix method. (3mks)
4x+3y =6
3x+5y =5
- An item that costs sh. 24,000 cash can be bought on hire purchase. A customer pays sh.6,000 as deposit and then makes 6 monthly installments of sh.3,500 each. Calculate the monthly rate of compound interest, giving your answer to 1 d.p. (3mks)
- Barasa shared sh.360,000 among his children Simiyu, Wasike and Nekesa in the ratio 1:3:5 respectively. How much did each receive? (3mks)
- In the arithmetic series 1+4+7+10+…find the sum of the first 100 terms. (3mks)
- If = , and , find 3a – 2b + c (3mks)
- Make x the subject (3mks)
- The figure below shows a triangle xyz in which x=13.4cm, z=5cm and ∠xyz= 57.7º. Find length y. (3mks)
SECTION B
-
- Complete the table below for the function y=2x2+3x-5
x -4 -3 -2 -1 0 1 2 2x2 0 3x -12 18 -5 -5 -3 6 y - On the grid provided draw the graph of y=2x2+3x-5 for -4≤x≤2. (4mks)
- Use your graph to state the roots of:
- 2x2 + 3x –5 = 0 (1mk)
- 2x2 + 6x – 2 = 0 (3mks)
- Complete the table below for the function y=2x2+3x-5
- A trader bought 8 cows and 12 goats for a total of ksh.294,000. If he had bought 1 more cow and 3 more goats he would have spent ksh.337,500.
- Form two equations to represent the above information. (2mks)
- Use matrix method to determine the cost of a cow and that of a goat. (3mks)
- The trader sold the animals he had bought making a profit of 40% per cow and 45% per goat.
- Calculate the total amount of money he received. (3mks)
- Determine his profit in Kenyan shillings. (2mks)
- A group of young men decided to raise ksh.480,000 to start a business. Before actual payment was made four members pulled out and each of the remaining had to pay an additional ksh.20,000 write an expression in terms of p for;
-
- Original contribution of each member. (1mk)
- Contribution after withdrawal of four members. (1mk)
- Form an equation in p and hence determine the number of initial members. (5mks)
- Three men Kamau, James and Hassan shared shs.480,000 such that Kamau:James is 3:2 and James:Hassan is 4:2. Find how much each got. (3mks)
-
- The probability that boys goes to school by bus is 1/3 and by matatu is 1/2. if he uses a bus the probability that he is late to school is 1/5 and if he uses a matatu the probability of being late is 3/10. If he uses other means of transport the probability of being late is 1/20. What is the probability that:
- He will be late to school?
- he will not be late to school?
- he will be late to school if he does not use a matatu?
- he neither uses a bus nor matatu but arrives to school early?
- The bearing of towns P and Q on a horizontal ground from a tower are 050 and 142 respectively. The angle of elevation of the top of the lower from town P is 34 . Given that P is 200m from the top of the tower and Q is 120m from the base of the tower determine:
- The height of the tower (3mks)
- The angle of elevation of the top of the lower from Q (3mks)
- The distance between the two towns P and Q (4mks)
- The relationship between two variables S and T is given by the equation S=KTn where K and n are constant
T 2 3 4 5 6 7 S 12.8 28.8 51.2 80.8 115.2 156.8 - Write down the linear equation relating to S and T (1mk)
- Complete the table above for the linear equation relating to S and T(to one decimal place) (2mks)
- Draw a suitable straight line graph to represent the data (3mks)
- Use your graph to determine the value of K and n (2mks)
- Find the value of S when T =3.5 (2mks)
-
- The current price of a vehicle is shs 500,000. If the vehicle depreciates at a rate of 15% p.a . Find the number of years it will take for its value to fall to shs 180,000. (4mks)
- The cash price of a cooker is shs 9,000. A customer bought the cooker by paying 15 monthly installments of shs 950 each. Calculate:
- the carrying charge (3mks)
- the rate of interest (3mks)
- The table below shows the income tax rates in a certain year.
Total income in K£ per annum Rate in shs per round 1 - 3,900 2 3,901 - 7,800 3 7,801 - 11,700 4 11,701 - 15,600 5 15,601 - 19,500 7 over 19,500 7.5
Mrs Musau earned a basic salary of ksh 18,600 per month and allowances amounting to ksh.7, 800 per month. She claimed a personal relief of ksh 1,080 per month. Calculate;- Total taxable income in k£ p.a. (2mks)
-
- The tax payable in ksh per month without relief. (4mks)
- The tax payable in ksh per month after relief. (2mks)
- Mrs Musau’s net monthly income. (2mks)
MARKING SCHEME
SECTION A
-
P2= x+2w
4x+3R
P2(4x+3R)= x+2w (4x +3R)
4x+3R
P24x+3P2R=x+2w
4P2x − x = 2w − 3P2R
x(4P2 −1)= 2w − 3P2R
4P2 − 1 4P2-1
x = 2w−3P2R
4P2−1 - 8 (4 + 2√3)
4 − 2√3 (4 + 2√3)
8 × 2 (2 + √3)
16 −12
16 (2 + √3)
48 (2 + √3)
16 + 8√3 - p = a + b/t2
6 = a + b/36
216 = 36a + b
216 = 36a + b
18 = a + b/9
162 = 9a + b
36a + b = 216
– 9a + b = 162
27a + 0 = 54
27a = 54
27 27
a=2
9a + b = 162
18 + b = 162
b = 162 – 18
b = 144
P = 2 + 144/t2
11 = 2 + 144/t2
9 = 144/t2
9t2 = 144
9 9
t = √144/9
√t2 = √16
= +4
=12/3
= +4 - Log8 ( x + 6 ) – log8 ( x – 3 ) = 2/3
Log8 x + 6 = Log8 4
x - 3
x +6 = 4
x – 3 1
x + 6 = 4x – 12
18 = 3x
3 3
x = 6 -
- XR.PX = QX.XS
6X = 4.8 × 5
X = 4.8 × 5
6
X = 0.8 × 5
XR = 4 cm - QT2 = PT.RT
QT2 = 18 × 8
QT2=144
QT=+12 cm
- XR.PX = QX.XS
- xy + 6 = 0
x − 2y = 7
x = 7 + 2y
y(7 + 2y) + 6 = 0
7y + 2y2 + 6 = 0
s = 7
p = 12
f = 4 × 3
2y2 + 4y + 3y + 6 = 0
2y(y + 2) + 3(y + 2) = 0
(2y + 3)(y + 2) = 0
2y = –3
y can either be = –2 or –3/2
x = 7 + 2y
x = 7 + 2( –2)
x = 3
x = 7 + 2(–3/2)
x = 7 + –3
x = 4 -
2x2 + x – 36 = 0
s = 1
p = 72
f = 9, –8
2x2 + 9x – 8x – 36 = 0
x (2x + 9) – 4(2x + 9) = 0
(x – 4)(2x + 9) = 0
2x = –9
x can either be x = –4.5 or x = 4 - (1 + 2x)7
1 7 21 35 35 17 16 15 14 13 (2x)0 (2x)1 (2x)2 (2x)3
1 + 14x + 84x2 + 280x3
1 + 14x + 84x2 + 280x3+........
(1.02)7 → (1 + 0.02)7
(1 + 2x)7
2x = 0.02
2 2
x = 0.01
1 +14(0.01) + 84(0.01)2 + 280(0.01)3
1 + 0.14 + 0.0084 + 0.00028
=1.14868 -
18 – 4y = 0
18b = 4y
4 4
y = 4.5 -
Inverse =-
20 – 9
= 11
× =
=
X = 30/11 – 15/11 = 15/11
Y = –18/11 +20/11 = 2/11
- A = P (1 + r/100)n
27,000 = 24,000 (1 + r/100 )6
(1 + r/100 )6 = 27,000/24,000
n = 6 months
6√(1 + r/100 )6 = 6√1.125
= 1.02 to 2 d.p.
r/100 = 1.02 – 1
r/100= 0.02
r = 21% - 1/9 × 360,000 = 40,000
3/9 × 360,000 = 120,000
5/9 × 360,000 = 200,000 - Sn = n /2( 2a + (n –1) d)
s/100 = 100/2 (2 + 99 × 3)
= 14,950 - 3a + 2b +c
3 + 2 +
3 – 8 + 1 = –4
6 –8 0 14
9 10 –5 –6
P(y) = 5
9- Y2 = X2 +Z2 – 2 × Z cos 57.7º
y2 = 13.42 + 52 – 2 × 13.4 × 5 cos 57.7º
y2 = 179.56 + 25 – 134 × 0.5344
y2 = 204.56 – 134 × 0.5344
y2 = 204.56 –71.6096
y2 = 132.95
y = √132.9504
y= ±11.53
SECTION B
-
x –4 –3 –2 –1 0 1 2 2x2 32 18 8 2 0 2 8 3x –12 –9 –6 3 0 3 6 –5 –5 –5 –5 –5 –5 –5 –5 y 15 4 –3 –6 –5 0 9 -
-
- x = 1 or x = –2.5
y = 2x2 + 3x –5
0 = 2x2 + 3x –5
y = 0
x = 1 or x = –2.5 - 2x2 + 6x – 2 = y
2x2 + 3x –5 = 0
3x + 3 = y
y = 3x + 3
x ¦0 1 –1
y ¦3 6 0
x = –2 or 1.9
- x = 1 or x = –2.5
- 8x + 12y = 294,000
9x + 15y = 337,500
4x + 6y = 147,000
3x + 5y = 112,500 - 4x + 6y = 147,000
3x + 5y = 112,500
(4 6) (x) = (147000)
(3 5) (y) = (112500)
20 – 18 = 2
1 (5 –6) (4 6) (x) = 1 (5 –6)
2 (–3 4) (3 5) (y) = 2 (–3 4)
(1 0) x = 5/2 –6/2) (147000)
(0 1) y = –3/2 2 ) (112500)
367500 + –337500
x (cow)= 30,000
–220500 + 225000
y (goat)= 4,500 -
- 40 × 30000
100
profit = ksh 120,000
45 × 45000
100
= sh 2,025
42,600 × 8 + 12 × 6,525
336,000 + 78,300
=sh 414,300 - 414,300 – 294,000
total profit = 120,300
- 40 × 30000
-
-
-
- 480,000
x - 480,000 + 20,000
x – 4
- 480,000
- 480,000 – 480,000 = 20,000
x x – 4
480,000 (x – 4) – 480,000x = 20,000 (x2 – 4x)
480,000x – 1,920,000 – 480,000x = 20,000x – 80,000x
20,000x2 – 800,000x +1,920,000 = 0
1000 1000 1000 1000
20x2 – 80x – 1920 = 0
4 4 4 4
5x2 – 20x – 480 = 0
x2 – 4x – 96 = 0
4± √ 42 – 4x1x – 96
2
4 ± 20
2
x = 24 = 12
2
-
-
- (1/3 × 1/5) + (1/2 ×3/10) + (1/6 ×1/20)
1/15 + 3/20 + 1/120 = 8 + 18 + 1
120
= 27/120
= 9/40 - (1/3 × 4/5) + (1/2 × 7/10) + (1/6 × 19/20)
= 4/15 + 7/20 +19/20 = 32 + 42 + 19 = 93/120
120
- (1/3 × 1/5) + (1/2 ×3/10) + (1/6 ×1/20)
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