## SECTION I (50 marks)

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

- Solve for n
^{6n}/_{n-1}=^{25}/_{n}(3 marks) - A family used two-fifths of its monthly income on school fees. Three-quarters of the remaining amount was used on family upkeep while the rest was invested. The family invested Ksh 13 500 monthly.Calculate the amount of money the family used on school fees every month. (4 marks)
- Solve for x in the equation.

5^{2x-1}-25^{x}= 500 (3 marks) - Kipkoech and Tanui began a 5000m race together at the starting line. Kipkoech and Tanui took 72 seconds and 80 seconds respectively to run a 400 m lap. The two athletes were together again at the starting line after some time.
- Determine the number of laps that Tanui had to run to complete the race after they were together. (3 marks)

- Simplify

18ax-(3a-4x)(3a+4x) (3mrks)

3a-8x - In the quadrilateral ABCD, AD = CD = 6 cm and BA BC= 12 cm. Angle ADC = 300°.

Calculate, correct to 2 decimal places, the area of the quadrilateral ABCD. (4 marks) - A watch loses 8 seconds every hour. It was set to read the correct time at 1100 h on Sunday. Determine the time, in a 12-hour system, the watch will show on the following Thursday when the correct time is 0500 h. (3 marks)
- A lorry left town A for town B and maintained an average speed of 50 km/h. A car left town A for town B 42 minutes later and maintained an average speed of 80 km/h. At the time the car arrived in town B, the lorry had 25 km to cover to town B. Determine the distance between town A and B. (3 marks)
- Port L is 120 km on a bearing of $30°W from port K. A ship left port K at 1000 h and sailed at a speed of 40 km/h along the bearing of S60°E. Using scale drawing, determine the bearing of the ship from port L at 1400 h. (4 marks)
- The image of P(-2, 5) under a translation T is P'(2, 2). Q'(9, -5) is the image of Q under the same translation T. Determine the coordinates of Q. (3 marks)
- A Kenyan bank bought and sold United Arab Emirates (UAE) dirhams on two different dates as shown below
**buying(Ksh)****Selling (Ksh)**1s ^{t}August 20211 UAE dirham 28.40 28.90 16 ^{th}August 20211UAE dirham 28.00 28.40

During her stay in UAE, she spent 3 520 UAE dirhams. She arrived back to Kenya on16th August 2021. On the same day she converted the remaining amount of money to Kenya shillings at the same bank.

Calculate the amount of money in Kenya shillings that she received from the bank. (3 marks) - An electric post erected vertically is 20m from point P on the same level ground. The angle of elevation of the top, T, of the post from P is 30°. Given that S is the mid point of the post, calculate, correct to I decimal place, the angle of elevation of S from P. (3 marks)
- Given that determine the values of u,v and w. (3 marks)
- The capacities of two similar containers are 54 ml and 250 ml respectively. The difference in the heights of the two containers is 4 cm.

Calculate the height of the larger container. (3 marks) - The table below shows the mean marks in a mathematics test of two classes.
Class Number of students Mean mark X 43 65 Y 45 62 - The base, ABCDEF, of a right pyramid is a regular hexagon of side 2.5 cm. Point V is the vertex of the pyramid and the length of the slanting edges is 4 cm.Draw a labelled net of the pyramid. (3 marks)

**SECTION 11 (50 marks)**

Answer only five questions in this section in the spaces provided.

- A contractor hired Wema and Tatu to transport 144 tonnes of stones to building sites A and B. To transport 48 tonnes of stones for a distance of 28 km, the contractor paid Ksh 24000.
- Wema transported 96 tonnes of stones to site A, a distance of 49 km.
- Calculate the amount of money that was paid to Wema.(2 marks)
- For every 8 tonnes of stones Wema transported to site A, he spent Ksh 3000.Calculate the profit Wema made. (3 marks)

- Tatu transported the remaining 48 tonnes of stones to site B, a distance of 84 km. If Tatu made 44% profit, calculate the amount of money Tatu spent to transport the stones. (3 marks)
- Determine the ratio of the profit made by Wema to that made by Tatu. (2 marks)

- Wema transported 96 tonnes of stones to site A, a distance of 49 km.
- A shot put is spherical and has mass of 7.26kg. It is made of a metal with a density of 6.93 g/cm3. (Take =π=
^{22}/_{7})- Determine the radius of the shot put, correct to I decimal place. (3 marks)
- A bucket is in the shape of a frustum of a cone. The base radius of the bucket is 7 cm. The bucket contains water to a height of 15 cm. The radius of the surface of the water is 10.5 cm.
- Find the volume of the water in the bucket.(3 marks)
- The shot put ball is completely submerged in the water in the bucket.

Calculate the new height of the water in the bucket.(4 marks)

- A triangle ABC is right angled at point A. The vertices of the triangle are A(1,-2), B(5, 4) and C(m, n).

The equation of line BC is 5y-x= 15.- Determine:
- the equation of line AC in the form ar+by+c=0, where a, b and c are integers.(4 marks)
- the coordinates of point C.(3 marks)

- line passes through point A and is parallel to line BC. Determine the x-intercept of the line.(3 marks)

- Determine:
- In the figure below, line AB = 10cm and is part of a trapezium ABCD. Point X is such that angle BAX= 45°

.- Using a ruler and a pair of compasses only:
- locate point D on line AX such that AD: DX=3:1. (3 marks)
- complete trapezium ABCD such that line DC is parallel to line AB and angle ABC = 67.5°. (3 marks)
- draw a perpendicular line from D to meet AB at E. Measure DE. (2 marks)

- Calculate the area of the trapezium ABCD. (2 marks)

- Using a ruler and a pair of compasses only:
- The amount of money, in Kenya shillings, spent on airtime by a group of 30 people in a period of an hour was recorded as shown below.
- Complete the frequency distribution table below.

(2 marks) - On the grid below, draw a histogram to represent the data.

(3marks) - Using the histogramto determine ;
- the mediam amount of money spent on airtime bye 30 peope. (3marks)
- the number of people who spent more than Ksh 41.50 on airtime over that period . (2marks)

- Complete the frequency distribution table below.

## MARKING SCHEME

**SECTION 1 (50mark**)

**answer all the queastions in the spaces provided **

- Solve for
^{6n}/_{n-1}=^{25}/_{n}

n(6n)=25(n-1)

6n²-25n+25=0^{m1}

6n²-15n-10n+25=0

(3n-5)(2n-5)=0

3n-5=0 ⇒n=^{5}/3 = 1^{2}/_{3}or 2n-5=0⇒ n=^{5}/_{2}=2^{1}/_{2} - Afamily used two-fifths of its monthly income on school fees. Three-quaters of the remaining amount was used on family upkeep while the rest was invested . The family invested Ksh13500 monthly .

calculate the amount of money the family used on school fees every month . (4marks)

Fees=^{2}/_{5}, rem=^{5}/_{5}-^{2}/_{5}=^{3}/_{5}

upkeep=^{3}/_{4}×^{3}/_{5}=^{9}/_{20}, rem=^{3}/_{5}-^{9}/_{20}=^{3}/_{20}

invest. =^{3}/_{20}=13500

=^{2}/_{5}×13500×^{20}/_{3}

=ksh36,000 - Solve for x in the equation

5^{2x - 1 }- 25^{x}=500

5^{2x-1}-5^{2x}=500

5^{2x},5^{1}- 5^{2x }=500

let 5^{2x}be k

∴^{1}/_{5 }k - k =500^{-4}/_{5}k = 500

k =500×^{5}/_{4}= -62

5^{2x}= −625

5^{2x}= −5

x is indeterminate - Kipkoech and Tanui began a 5000m race together at the starting line .kipkoech and Tanui took 72 seconds and 80seconds respectively to to run a 400m lap .The two athletes were together againat the starting line after some time. Determine the number of laps thatTanui had to run to complete the race after they were together.

Time taken to be together again ;

L.C.M =2^{4}×3²×5=720sec

No. of laps made by Tanui=^{720}/_{80}=9

Remaining laps =^{5000}/_{400 }=9 =3^{1}/_{2 } - Simplify 18ax−(3a−4a)(3a+4x)

3a−8x

(3a−4x)(3a+4x) =9a² −16²

18ax−(9a−16x²) = 16x²+18ax −9a²

3a−8x 3a−8x

=16x² +24ax -6ax -9a²

3a-8x

=(2x+3a)(8x-3a)

-1(8x-3a)

= -2x-3a - In the equadrilateral ABCD, AD =CD=6cm and BA=BC=12cm. Angle ADC =300°

Calculate ,correct to 2 decimal places, the area of the quadrilateral ABCD.

Area of ΔADC=^{1}/_{2}×6×6sin60=9√3cm²

LSF=^{12}/_{6}=2 ⇒ A.sf =2²=4

Area of ΔABC=4×9√3 = 36√3cm²

A.of ABCD =36√3 −9√3 =27√3cm²

=46.77cm² - A watch loses 8 seconds every hour . it was set to read the correct time at 1100h on sunday . Determine the time 12-hours system , the watch will show on the following thusday when the correct time is 0500 (3marks)

1100h sun →1100h wed =24×3 = 72hrs

1100h wed →0500hthus =18h

Total =18+72=90h

time lost =90×8 =12mins

60

Time on thur=5:00a.m −12min

=4:48a.m - A lorry left town A for town B and maintained an average speed of 50 km/h. A car left town A for town B 42 minutes later and maintained an average speed of 80 km/h. At the time the car arrived in town B, the lorry had 25 km to cover to town B. Determine the distance between town A and B. (3 marks)

A←–——————→B

Time taken by car =^{x}/_{80}h

Distance by lory =50×^{42}/_{60}+50×(^{x}/_{80}) =(x−25)

0.625x+35=x−25

(1−0.625)x =35+20

x=160km - Port L is 120 km on a bearing of $30°W from port K. A ship left port K at 1000 h and sailed at a speed of 40 km/h along the bearing of S60°E. Using scale drawing, determine the bearing of the ship from port L at 1400 h. (4 marks)
- The image of P(-2, 5) under a translation T is P'(2, 2). Q'(9, -5) is the image of Q under the same translation T. Determine the coordinates of Q. (3 marks)
- A Kenyan bank bought and sold United Arab Emirates (UAE) dirhams on two different dates as shown below
**buying(Ksh)****Selling (Ksh)**1s ^{t}August 20211 UAE dirham 28.40 28.90 16 ^{th}August 20211UAE dirham 28.00 28.40

During her stay in UAE, she spent 3 520 UAE dirhams. She arrived back to Kenya on16th August 2021. On the same day she converted the remaining amount of money to Kenya shillings at the same bank.

Calculate the amount of money in Kenya shillings that she received from the bank. (3 marks)

Dirhams received =^{130050}/_{28.90 }or4500Dirhams

Remainder after expended ; =4500 -3520 or 980 Dirhams

ksh received =980×28.00

= ksh. 29440 - An electric post erected vertically is 20m from point P on the same level ground. The angle of elevation of the top, T, of the post from P is 30°. Given that S is the mid point of the post, calculate, correct to I decimal place, the angle of elevation of S from P. (3 marks)

Rt=20tan30° or 11.55m

Rs^{1}/_{2 }×20tan30°

=10tan30° or o.5774m

θ = tan^{-1 }(^{10tan30°}/_{20 }) or tan^{-1 }(^{5.774}/_{20 })

16.1° - Given that determine the values of u,v and w. (3 marks)
- The capacities of two similar containers are 54 ml and 250 ml respectively. The difference in the heights of the two containers is 4 cm.

Calculate the height of the larger container. (3 marks)

VSF =^{250}/_{54}=^{125}/_{27 }

LSF =(^{125}/_{27})^{1/3}=(^{5}/_{3}) =^{5}/_{3 }

∴^{5}/_{3}=^{X+4}/_{X }

5X =3X+12 ⇒X=6

H. of the container = 6+4 =10 cm - The table below shows the mean marks in a mathematics test of two classes
Class Number of students Mean X 43 65 Y 45 62

Calculate , correct to 2 decimal places,the mean of the classes.

Mean =(43×65)+(45×62)

(43+45)

=63.47 - The base, ABCDEF, of a right pyramid is a regular hexagon of side 2.5 cm. Point V is the vertex of the pyramid and the length of the slanting edges is 4 cm.Draw a labelled net of the pyramid. (3 marks)

**SECTION 2(50marks)**

**Answer only five questions in this section in the spaces provided **

- A contractor hired Wema and Tatu to transport 144 tonnes of stones to building sites A and B. To transport 48 tonnes of stones for a distance of 28 km, the contractor paid Ksh 24000.
- Wema transported 96 tonnes of stones to site A, a distance of 49 km.
- Calculate the amount of money that was paid to wema .
^{49}/_{28 }×^{96}/_{48 }×2400

ksh. 8400 - For every 8stonnes Wema transported to site A ,he spent ksh3000.calculate the profit Wema made. (3marks)

Expenses =^{96}/_{8}×3000 =3600

profit =8400 -36000

=ksh48000

- Calculate the amount of money that was paid to wema .
- Tatu transported the remaining 48 tonnes of stones to site B, a distance of 84 km. If Tatu made 44% profit, calculate the amount of money Tatu spent to transport the stones. (3 marks)

Paid ;^{84}/_{28 }×^{48}/_{48 }×24000 =ksh.72000

Expenses =^{100-44}/_{100 }×72000 = ksh.40,320 - Determine the ratio of the profit made by wema to that made by tatu.
^{44}/_{100 }×72,000 =ksh 31,68

W:T =48,000:31680

=50:33

- Wema transported 96 tonnes of stones to site A, a distance of 49 km.
- A shot put is spherical and has mass of 7.26kg. It is made of a metal with a density of 6.93 g/cm3. (Take =π=
^{22}/_{7})- Determine the radius of the shot put, correct to I decimal place. (3 marks)

Vol; =^{4}/_{3 }×^{22}/_{7 }×r³ =77.26×^{1000}/_{6.93 }

r³ =^{7.26×1000×21}/_{6.93×4×22 }=250

r =3√250 =6.3cm - A bucket is in the shape of a frustum of a cone. The base radius of the bucket is 7 cm. The bucket contains water to a height of 15 cm. The radius of the surface of the water is 10.5 cm.
- Find the volume of the water in the bucket.(3 marks)
^{10.5}/_{7 }=^{h+15}/_{h }⇒h =30cm

H=45cm

v =^{1}/_{3 }×^{22}/_{7 }(10.5²×45 -7²×30)

=3657.5cm³ - The shot put ball is completely submerged in the water in the bucket.

Calculate the new height of the water in the bucket.(4 marks)

New vol, =3657.5+7.26×^{1000}/_{6.93 }

=^{197615}/_{42 }cm³ or 4705.1190476cm³

Vsf =(^{45+x}/_{45}) =^{4705.1190476}/_{3657.5 }x =3.941

New weight = 15+3.441

=18.941cm

- Find the volume of the water in the bucket.(3 marks)

- Determine the radius of the shot put, correct to I decimal place. (3 marks)
- A triangle ABC is right angled at point A. The vertices of the triangle are A(1,-2), B(5, 4) and C(m, n).

The equation of line BC is 5y-x= 15.- Determine:
- the equation of line AC in the form ar+by+c=0, where a, b and c are integers.(4 marks)

5y= x+15⇒y =^{1}/_{5 }x+3

Grad of AB =^{4-(-2)}/_{5-1 }=^{6}/_{4 }=^{3}/_{2 }

Grad of AC = -^{1}/_{3}= −^{2}/_{3 }^{y−(−2)}/_{x−1 }=^{−2}/_{3 }

⇒-2x-3y-4=0 or 2x+3y+4=0 - the coordinates of poin C

- the equation of line AC in the form ar+by+c=0, where a, b and c are integers.(4 marks)
- A line passes through poin A and is parallel to line BC.Determine thr x-intercept of the line

BC ; y =^{1}/_{5 }x+3 ⇒m_{1 }=^{1}/_{5 }

m_{1 }=m_{2 }=^{1}/_{5 }^{y-(-2)}/_{x-1 }=^{1}/_{5 }

y=2 =^{1}/_{5}x -^{1}/_{5 }

y=^{1}/_{5 }x -2^{1}/_{5 }=0

x=11

x intercept=11

- Determine:
- In the figure below, line AB = 10cm and is part of a trapezium ABCD. Point X is such that angle BAX= 45°
- Using a ruler and a pair of compasses only:
- locate point D on line AX such that AD: DX=3:1. (3 marks)
- complete trapezium ABCD such that line DC is parallel to line AB and angle ABC = 67.5°. (3 marks)
- draw a perpendicular line from D to meet AB at E. Measure DE. (2 marks)

DE =4.0cm

- Calculate the area of the trapezium ABCD.

A =^{1}/_{2}×DE (AB+CD)

=^{1}/_{2 }×4 (10+4)

= 28cm²

- Using a ruler and a pair of compasses only:
- The amount of money , in kenya shillings , spent on airtime by a group of 30 people in the period of an hour was recorded as shown .
- Complete the frequency distribution table below .
AMOUNT (KSH) 20 -24 24 -29 30 -34 35 -44 45 -59 FREQUENCY 5 7 9 6 3 - On the grid provided below, draw a histogram to represent the data
- Use the histogram to determine ;
- the medium amount of money spent on airtime by the 30 people .

A=5×1.0 +5×1.4 +1.8x =^{1}/_{2}×30

18x =15-12 =3

x=1^{2}/_{3 }or 1.667

median =29.5+1.667

=31.167

=31.17 - the number of people who spent more than ksh 31.50 on airtime over the period (2marks)

5×1.0+5×1.4+5×1.8+7×0.6

= 25.2

=26 people

- the medium amount of money spent on airtime by the 30 people .

- Complete the frequency distribution table below .

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