# Mathematics Questions and Answers - Form 3 Term 1 Opener Exams 2023

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)

1. Use logarithms to evaluate (3mks)
4.73×22.41
82.3
2. Solve for x in log381 = x (3mks)
3. Use tables of cubes and reciprocals to evaluate (4mks)
2√0.498 +   0.1
0.0351
4. When a number is divided by 8, 9 and 6 the remainders are 7, 8 and 5 respectively. Find the number.    (3mks)
5. 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)
6. 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).
7. 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)
8. 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)
9. 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)
10. 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)
11. Two similar solids have surface areas of 48cm2 and 108cm2 respectively. Find the volume of the smaller solid if the bigger one has a volume of 162cm3. (3mks)
12. 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)

1. The coordinates of a triangle ABC are A(1, 1) B(3, 1) and C (1, 3).
1. Plot the triangle ABC. (1 mark)
2. Triangle ABC undergoes a translation vector Obtain the image of A' B' C ' under the transformation, write the coordinates of A' B' C'. (2marks)
3. 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)
4. The triangle A" B" C" , undergoes an enlargement scale factor -1, centre origin. Obtain the coordinates of the image A'" B"' C"'. (2 marks)
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)
6. Which triangles are directly congruent. (1 mark)
2. 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
1. The time of the day the two vehicles met. (4marks)
2. How far from town A they met. (2marks)
3. How far from town B the bus was when the matatu reached town A (4marks)
3. 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
1. State the modal class. (1mark)
2. Calculate to 2 decimal places:
1. The mean mass (4marks)
2. The difference between the median mass and the mean mass. (5marks)
4. 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.
1. Using a scale of 1cm represents 100km;
1. Show the relative positions of P, H and J (3mks)
2. Determine the distance between P and J (2mks)
3. State the bearing of P from J (2mks)
2. 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)
5. Given that y = 2x2 + 3x − 7 for −4 ≤ x ≤ 3
1. 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
2. Draw the graph y = 2x2 + 3x − 7 for −4 ≤ x ≤ 3 (3mks)
3. Use the graph to find the roots of the equation
1. 2x2 + 3x − 7 = 0 (2mks)
2. 2x2 + 4x − 9 = 0 (3mks) ## MARKING SCHEME

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

1. Use logarithms to evaluate (3mks)
4.73×22.41
82.3 2. Solve for x in log381 = x (3mks) 3x = 81
3x = 34
x = 4
3. 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
4. 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 28 × 32 = 72
N = 72 − 1
= 71
5. 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
6. 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    =
10+x      4
50 + 5x = 140
5x = 90
x = 90/5
x = 18 women
7. 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
8. 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
9. 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
= 24 × 32 × 7
1st No: 48 = 24 × 3
2nd No: 72 = 22 × 32
3rd No: = 3 × 7
= 21
or
3rd No: = 32 × 7
= 63
Least possible #
= 21
10. 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 = −2/3x + 5/3
For lines, m1m2 = −1
m2−2/3
m1 = 3/2
m − 1 = 3/2
4 − 2
2m − 2 = 6
2m = 8
m = 4
11. Two similar solids have surface areas of 48cm2 and 108cm2 respectively. Find the volume of the smaller solid if the bigger one has a volume of 162cm3. (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 = 48cm3
12. 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)

1. The coordinates of a triangle ABC are A(1, 1) B(3, 1) and C (1, 3).
1. Plot the triangle ABC. (1 mark) 2. Triangle ABC undergoes a translation vector Obtain the image of A' B' C ' under the transformation, write the coordinates of A' B' C'. (2marks) 3. 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)
4. 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)
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)
Aiv(0.9, 0.4)      Biv(0, 2.1)          Civ(2.6, 1.3)
6. Which triangles are directly congruent. (1 mark)
ABC and A'B'C'
A'''B'''C''' and AivBivCiv
2. 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
1. 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
2. How far from town A they met. (2marks)
Bus distance: from 1.15pm
= 60km/h × 3 hours
= 180km
Distance from A
90 + 180 = 270km
3. 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
3. 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
1. State the modal class. (1mark)
25 - 29
2. Calculate to 2 decimal places:
1. The mean mass (4marks)
 Mass Midpoint x f c.f fx 10-14 15-19 20-24 25-29 30-34 35-39 12  17 22 27 32 37 2 14 33 35 14  3 2  16 49 84 98 101 24 238 726 945 448 111 Σf = 101 Σfx = 2492
Mean, x̄ = Σfx
Σf
= 2492
101
= 24.67g
2. 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
4. 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.
1. Using a scale of 1cm represents 100km;
1. Show the relative positions of P, H and J (3mks) 2. Determine the distance between P and J (2mks)
• 8.7 ± 0.1 = 870km
3. State the bearing of P from J (2mks)
• 267°
2. 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
839/60
= 800 × 20/173
= 92.5 km/h
5. Given that y = 2x2 + 3x − 7 for −4 ≤ x ≤ 3
1. 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
2. Draw the graph y = 2x2 + 3x − 7 for −4 ≤ x ≤ 3 (3mks) 3. Use the graph to find the roots of the equation
1. 2x2 + 3x − 7 = 0 (2mks)
y = 2x2 + 3x − 7
0 = 2x2 + 3x − 7
y = 0
x = −2.60 or x = 1.25
2. 2x2 + 4x − 9 = 0 (3mks)
y = 2x2 + 3x − 7
−0 = 2x2 + 4x − 9
y =         −x + 2 x = −3.2  or x = 1.3

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