**Instructions To Candidates **

- This paper has two sections: Section 1 and Section II
- Answer all questions in section I and any three questions in section II
- All answers and working must be written on the question paper in the spaces provided below each question.
- Show all the steps in your calculations, giving your answer at each stage in the space below each question.
- Marks may be awarded for correct working even if the answer is wrong.
- Non-Programmable silent electronic calculators and KNEC Mathematical Tables may be used except where stated otherwise.

**SECTION I (40MKS) (Answer all questions from this section)**

- Use logarithms to evaluate (3mks)

4.73×22.41

82.3 - Solve for x in log
_{3}81 = x (3mks) - Use tables of cubes and reciprocals to evaluate (4mks)
^{2}√0.498 + 0.1

0.0351 - When a number is divided by 8, 9 and 6 the remainders are 7, 8 and 5 respectively. Find the number. (3mks)
- A line with gradient -3 passes through (3, k) and (k, 8). Find the value of k and hence the equation of the line, where a, b and c are constants. (4mks)
- In a fundraising committee of 45 people, the ratio of men to women is 7: 2. Find the number of women required to join the committee so that the ratio of men to women is changed to 5: 4. (3mks).
- The marked price of a car in a dealer’s shop was Ksh. 450 000. Simiyu bought the car at 7% discount. The dealer still made a profit of 13%. Calculate the amount of money the dealer had paid for the car to the nearest thousands. (4mks)
- The size of an interior angle of a regular polygon is 3x0 while that of exterior is (x-20)0. Find the number of sides of the polygon. (3mks)
- The GCD and LCM of three numbers are 3 and 1008 respectively. If two of the numbers are 48 and 72, find the least possible value of the third number. (3mks)
- A straight line through A(2, 1) and B(4, m) is perpendicular to the line whose equation is 3y = 5 − 2x. Determine the value of m. (3mks)
- Two similar solids have surface areas of 48cm
^{2}and 108cm^{2}respectively. Find the volume of the smaller solid if the bigger one has a volume of 162cm^{3}. (3mks) - Given that cos(x-20)° = sin(2x+32)° and that x is an acute angle, find tan(x-4)° (4mks)

**SECTION II (30MKS) **

**(Answer any 3 questions from this section)**

- The coordinates of a triangle ABC are A(1, 1) B(3, 1) and C (1, 3).
- Plot the triangle ABC. (1 mark)
- Triangle ABC undergoes a translation vector Obtain the image of A' B' C ' under the transformation, write the coordinates of A' B' C'. (2marks)
- A' B' C' undergoes a reflection along the line X = 0, obtain the coordinates and plot on the graph points A" B" C", under the transformation

(2 marks) - The triangle A" B" C" , undergoes an enlargement scale factor -1, centre origin. Obtain the coordinates of the image A'" B"' C"'. (2 marks)
- The triangle A"' B"' C"' undergoes a rotation centre (1, −2) angle 1200. Obtain the coordinates of the image Aiv Biv Civ. (2 marks)
- Which triangles are directly congruent. (1 mark)

- A country bus left town A at 11.45 am and travelled towards town B at an average speed of 60km/hr. A matatu left town B at 1.15 pm on the same day and travelled towards town A along the same road at an average speed of 90km/hr. The distance between the two towns is 540 km. Determine
- The time of the day the two vehicles met. (4marks)
- How far from town A they met. (2marks)
- How far from town B the bus was when the matatu reached town A (4marks)

- The table below shows the mass to the nearest gram, of 101 mango seeds in a research station.

**Mass (gram)**10-14 15-19 20-24 25-29 30-34 35-39 **Frequency**2 14 33 35 14 3 - State the modal class. (1mark)
- Calculate to 2 decimal places:
- The mean mass (4marks)
- The difference between the median mass and the mean mass. (5marks)

- A helicopter is stationed at an airport H on a bearing of 060° and 800km from another airport P. A third airport J is on a bearing of 140° and 120km from H.
- Using a scale of 1cm represents 100km;
- Show the relative positions of P, H and J (3mks)
- Determine the distance between P and J (2mks)
- State the bearing of P from J (2mks)

- A jet flying at a speed of 103km/h left J towards P. The helicopter at H also took off towards P at the same time. Find the speed at which the helicopter will fly so as to arrive at P 12 minutes later than the jet. (3mks)

- Using a scale of 1cm represents 100km;
- Given that y = 2x
^{2}+ 3x − 7 for −4 ≤ x ≤ 3- Complete the table below (2mks)

x −4 −3 −2 −1 0 1 2 3 2x2 32 18 2 18 3x −9 −3 3 6 −7 −7 −7 −7 −7 −7 −7 −7 y 4 −5 −7 7 - Draw the graph y = 2x
^{2}+ 3x − 7 for −4 ≤ x ≤ 3 (3mks) - Use the graph to find the roots of the equation
- 2x
^{2}+ 3x − 7 = 0 (2mks) - 2x
^{2}+ 4x − 9 = 0 (3mks)

- 2x

- Complete the table below (2mks)

## MARKING SCHEME

**SECTION I (40MKS) (Answer all questions from this section)**

- Use logarithms to evaluate (3mks)

4.73×22.41

82.3 - Solve for x in log
_{3}81 = x (3mks)**3x = 81**

3x = 3^{4}

x = 4 - Use tables of cubes and reciprocals to evaluate (4mks)
^{2}√0.498 + 0.1

0.0351**(49.8 ×**^{1}/_{100})^{½}+ 1

0.351

7.0569 × 1 + 1 × 1

10 3.51 10^{−1}

0.70569 + 0.2849 × 10

0.70569 + 2.849

= 3.55469 - When a number is divided by 8, 9 and 6 the remainders are 7, 8 and 5 respectively. Find the number. (3mks)
**Let the number be N**^{N}/_{8}, rem = 7^{N}/_{9}, rem = 8^{N}/_{6}, rem = 5

N is given by the L.C.M of 8, 9, and 6 and subtracting 1 from it

2^{8}× 3^{2}= 72

N = 72 − 1

= 71 - A line with gradient -3 passes through (3, k) and (k, 8). Find the value of k and hence the equation of the line, where a, b and c are constants. (4mks)
**8 − k = −3**

k − 3 1

8 − k = −3k + 9

3k - k = 9 − 8

2k = 1

k = ½ or 0.5

y − ½ = − 3

x − 3 1

y − ½ = −3x + 9

2y − 1 = − 6x + 18

2y = −6x + 19

y = −3x +^{19}/_{2} - In a fundraising committee of 45 people, the ratio of men to women is 7: 2. Find the number of women required to join the committee so that the ratio of men to women is changed to 5: 4. (3mks).
**Let # of wmen joining be x**

Initial # of men

=^{7}/_{9}× 45 = 35

Initial # of women

=^{ 2}/_{9}× 45 = 10

After x women joined, the ratio changed to 5:4

35 = 5

10+x 4

50 + 5x = 140

5x = 90

x =^{90}/_{5}

x = 18 women - The marked price of a car in a dealer’s shop was Ksh. 450 000. Simiyu bought the car at 7% discount. The dealer still made a profit of 13%. Calculate the amount of money the dealer had paid for the car to the nearest thousands. (4mks)
^{93}/_{100}× 450000

= 418500/=

113% = 418500

100% = ?

100 × 418500

113

= Sh. 370000 - The size of an interior angle of a regular polygon is 3x0 while that of exterior is (x-20)0. Find the number of sides of the polygon. (3mks)
**3x° + (x − 20°) = 180°**

4x = 180 + 20

4x = 200°

x = 50°

size of exterior angle

= 50° − 20° = 30°

No. of sides of polygon - The GCD and LCM of three numbers are 3 and 1008 respectively. If two of the numbers are 48 and 72, find the least possible value of the third number. (3mks)
**G.C.D = 3**

L.C.M = 1008

= 2^{4}× 3^{2}× 7

1^{st}No: 48 = 2^{4}× 3

2^{nd}No: 72 = 2^{2}× 3^{2}

3^{rd}No: = 3 × 7

= 21

or

3^{rd}No: = 3^{2}× 7

= 63

Least possible #

= 21 - A straight line through A(2, 1) and B(4, m) is perpendicular to the line whose equation is 3y = 5 − 2x. Determine the value of m. (3mks)
**3y − 5 − 2x**

y =^{−2}/_{3}x +^{ 5}/_{3}

For lines, m_{1}m_{2}= −1

m_{2}=^{−2}/_{3}

m_{1}=^{3}/_{2}

m − 1 =^{3}/_{2}

4 − 2

2m − 2 = 6

2m = 8

m = 4 - Two similar solids have surface areas of 48cm
^{2}and 108cm^{2}respectively. Find the volume of the smaller solid if the bigger one has a volume of 162cm^{3}. (3mks)**A.S.F =**^{108}/_{48}=^{9}/_{4}

L.S.F = √(^{9}/_{4}) =^{3}/_{2}

V.S.F = (L.S.F)^{3}

=(3/2)^{3}

=^{27}/_{8}^{27}/_{8}=^{162}/_{x}

x = 162 × 8

27

x = 48cm^{3} - Given that cos(x-20)° = sin(2x+32)° and that x is an acute angle, find tan(x-4)° (4mks)

(x − 20)° + (2x + 32)° = 90

3x = 78

x =^{78}/_{3}= 26°

tan (x − 4)° = tan(26 − 4)°

= tan 22°

= 0.4040

**SECTION II (30 MKS) **

**(Answer any 3 questions from this section)**

- The coordinates of a triangle ABC are A(1, 1) B(3, 1) and C (1, 3).
- Plot the triangle ABC. (1 mark)
- Triangle ABC undergoes a translation vector Obtain the image of A' B' C ' under the transformation, write the coordinates of A' B' C'. (2marks)
- A' B' C' undergoes a reflection along the line X = 0, obtain the coordinates and plot on the graph points A" B" C", under the transformation

(2 marks)**A''(−3,3) B''(−5,3) C"(−3, 5)** - The triangle A" B" C" , undergoes an enlargement scale factor -1, centre origin. Obtain the coordinates of the image A'" B"' C"'. (2 marks)
**A'''(3,−3) B'''(5, −3) C'''(−3, 5)** - The triangle A"' B"' C"' undergoes a rotation centre (1, −2) angle 1200. Obtain the coordinates of the image Aiv Biv Civ. (2 marks)
**A**^{iv}(0.9, 0.4) B^{iv}(0, 2.1) C^{iv}(2.6, 1.3) - Which triangles are directly congruent. (1 mark)
**ABC and A'B'C'**

A'''B'''C''' and A^{iv}B^{iv}C^{iv}

- Plot the triangle ABC. (1 mark)
- A country bus left town A at 11.45 am and travelled towards town B at an average speed of 60km/hr. A matatu left town B at 1.15 pm on the same day and travelled towards town A along the same road at an average speed of 90km/hr. The distance between the two towns is 540 km. Determine
- The time of the day the two vehicles met. (4marks)
**1315**

−1145

1.30

Distance travelled by bus:

60 × 1.5 = 90km

Distnace left = 540 − 90

= 450km

R.speed = 90 + 60 = 150km/h

Time taken to meet:

450 = 3hrs

150

Time of the day of meeting:

1.15

+ 3.00

4.15pm - How far from town A they met. (2marks)
**Bus distance: from 1.15pm**

= 60km/h × 3 hours

= 180km

Distance from A

90 + 180 = 270km - How far from town B the bus was when the matatu reached town A (4marks)
**Time taken by matatu**

540km = 6 hours

90km/h

Distance travelled by bus from 1.15pm

= 450km

Distance covered in 6hrs:

60kmk/h × 6 = 360km

Distance of Bus from B when matatu reached A

450km − 360km

= 90km

- The time of the day the two vehicles met. (4marks)
- The table below shows the mass to the nearest gram, of 101 mango seeds in a research station.

**Mass (gram)**10-14 15-19 20-24 25-29 30-34 35-39 **Frequency**2 14 33 35 14 3 - State the modal class. (1mark)

25 - 29 - Calculate to 2 decimal places:
- The mean mass (4marks)

Mass Midpoint x f c.f fx 10-14

15-19

20-24

25-29

30-34

35-3912

17

22

27

32

372

14

33

35

14

32

16

49

84

98

10124

238

726

945

448

111Σf = 101 Σfx = 2492 **Mean, x̄ = Σfx**

Σf

= 2492

101

= 24.67g - The difference between the median mass and the mean mass. (5marks)
**Median = 24.5 + (**^{2}/_{35})5

= 24.5 + 0.2857

= 24.7857g

≅ 24.76g

Difference = 24.79 − 24.67

= 0.12g

- The mean mass (4marks)

- State the modal class. (1mark)
- A helicopter is stationed at an airport H on a bearing of 060° and 800km from another airport P. A third airport J is on a bearing of 140° and 120km from H.
- Using a scale of 1cm represents 100km;
- Show the relative positions of P, H and J (3mks)
- Determine the distance between P and J (2mks)
**8.7 ± 0.1 = 870km**

- State the bearing of P from J (2mks)

**267°**

- Show the relative positions of P, H and J (3mks)
- A jet flying at a speed of 103km/h left J towards P. The helicopter at H also took off towards P at the same time. Find the speed at which the helicopter will fly so as to arrive at P 12 minutes later than the jet. (3mks)
**Time taken by jet = Distance = 870km = 8hrs 27min**

speed 103km/h

Time taken by Helicopter = 8h 27min + 12 = 8hr 39min**S =**^{ D}/_{T}

= 800

8^{39}/_{60}

= 800 ×^{20}/_{173}= 92.5 km/h

- Using a scale of 1cm represents 100km;
- Given that y = 2x
^{2}+ 3x − 7 for −4 ≤ x ≤ 3- Complete the table below (2mks)

x −4 −3 −2 −1 0 1 2 3 2x2 32 18 **8****2****0**2 **8**18 3x **−12**−9 **−6**−3 **0**3 6 **9**−7 −7 −7 −7 −7 −7 **−7**−7 −7 y **13**4 −5 **−8**−7 **−2**7 **20** - Draw the graph y = 2x
^{2}+ 3x − 7 for −4 ≤ x ≤ 3 (3mks) - Use the graph to find the roots of the equation
- 2x
^{2}+ 3x − 7 = 0 (2mks)**y = 2x**^{2}+ 3x − 7

0 = 2x^{2}+ 3x − 7

y = 0

x = −2.60 or x = 1.25 - 2x
^{2}+ 4x − 9 = 0 (3mks)**y = 2x**^{2}+ 3x − 7

−0 = 2x^{2}+ 4x − 9

y = −x + 2

x = −3.2 or x = 1.3

- 2x

- Complete the table below (2mks)

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