Mathematics Paper 1 Questions and Answers - Form 3 End Term 3 Exams 2023

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Mathematics Paper 2 Questions and Answers - Form 3 End Term 3 Exams 2023

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INSTRUCTIONS TO THE CANDIDATES

This paper contains two sections; Section I and Section II.
Answer all the questions in section I and only five questions from Section II
All necessary workings and answers must be written on the question paper in the spaces provided below each question.
Show all the steps in your calculations, giving your answers at each stage in the spaces below each question
Marks may be given 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 1 (50 MARKS)

Answer all questions in the spaces provided.

Without using tables or calculators, evaluate; (3 marks)
36 – 8 x (−4) – 15 ÷ (−3)
3 x (−3) + (−8) (6−(−2))
Evaluate ³/₈ of 7³/₅ -¹/₃ (1¹/₄ + 3¹/₃) x 2²/₅ (3 marks)
Use the reciprocal, square and square root tables to evaluate to 4 significant figures the expression. (4 mark)
√¹/_24.56 + 4.346²
Solve for x in the equation (3 marks)
2^−3x = (1/8)⁻²
Solve the inequality−3x + 2 < x + 6 < 17− 2x and write down the integral values that satisfy the inequality. (3 marks)
A bus left Nairobi at 2245hrs on Sunday traveling towards Kisumu. If it took 7 hrs 15 minutes to reach the destination, at what time did it reach the destination? (2 marks)
Determine the centre of enlargement if P (−6, −3) is the image of P (4, 2) under enlargement scale factor −4. (3 marks)
A line L1 runs in such a way that it is always equidistant from points P and Q. Given that P(−2, 3) and Q (4, −5), find its equation in the form ax + by + c = 0. (4 marks)
Construct trapezium PQRS where PQ = 10cm, QR = 5cm PS = 4cm, ∠PQR = 37½° and PQ is parallel to RS. Measure RS (4 marks)
Solve for x in the equation
1 + Sin2x = Cos 60° for 0° ≤ x ≤ 360°
Calculate the area of the shaded part to 1 decimal place. radius of the circle is 5cm.
An irregular polygon has one of its interior angles = 2000. The remaining angles are of equal sizes of 132.5° each. Find the number of sides of the polygon. (3 marks)
A shamba is in the shape of a parallelogram with the lengths of the adjacent sides being 12cm and 15cm.If the area of the parallelogram is 72cm², find the angle between these two sides. (3 marks)
Mr. Oluoch keeps sheep and goats. The number of sheep exceeds the number of goats by 6. During an outbreak of a disease, ¼ of the goats and 1/3 of the sheep died. If he lost a total of 30 animals, how many animals did he have altogether? (3 marks)
Given that log4 = 0.6021 and log 6 =0.7782, without using mathematical tables or a calculator, evaluate log 0.096. (3 marks)
An American tourist converted US dollars 500,000 into Kenya shillings on his arrival to Kenya when the bank rates were as follows:
Buying Selling
In Kenya Shillings) In Kenya Shillings)
1 US Dollar 77.24 77.4
1 Sterling Pound 121.93 122.27
While in Kenya, he spent Kshs. 2,645,000. He later converted the remaining amount into sterling pounds. Calculate the amount he received to the nearest sterling pounds. (3 marks)
SECTION II: 50 MARKS
Answer any five questions in the spaces provided.
Mr. Otieno is a car dealer who specializes in Mercedes Benz and BMW cars. He is given a commission of 5% on Mercedes Benz sold from all sales above sh.5 million and 6% on BMW on sales above 2 million. A Mercedes Benz costs 2.5 million and a BMW costs 1 million. If in the month of December he earned a commission of sh.500, 000 from the sales of Mercedes Benz and sh.480, 000 from BMW sales;
1. Calculate the number of cars of each type that he sold. (5 marks)
2. In the month of January, the number of Mercedes Benz sold rose by 75% and that of BMW sold dropped by 25%. Calculate the percentage increase in his earning. (5 marks)
The distance between two towns A and B is 760 km. A minibus left town A at 8:15a.m and traveled towards B at an average speed of 90km/h. A matatu left B at 10:35a.m and on the same day and travelled towards A at an average speed of 110km/h.
1. 1. How far from A did they meet? (4mks)
  2. At what time did they meet? (2mks)\
2. A motorist starts from his home at 10:30a.m on the same day and traveled at an average speed of 100km/h. He arrived at B at the same time as the minibus. Calculate the distance from B to his home. (4mks)
1. Find the inverse of matrix (2 marks)
2. Two neighbouring schools decided to buy calculators for their students. The first school bought 120 type A calculator while the second school bought 130 type B calculators. Both spent a total of Kshs.298, 000 on the calculators. The following week, the two schools realized that there was still a shortage and ordered for more. The first school ordered for 150 type A calculators while second school ordered for 200 type B calculators and both spent a total of Ksh. 410,000.
  1. Form a matrix equation to represent the above information. (1 mark)
  2. Use matrix method to find the price of each type of calculator bought by the two schools. (4 marks)
  3. If on type A calculators bought, a discount of 10% is given and on type B calculators a discount of 5% is given, find the difference in their discounts. (3 marks)
Triangle ABC has vertices A(1, 2), B(2, 3) and C(4, 1) while triangle A¹B¹C¹ has vertices A1(1, -2), B1(2, -3) and C1(4, -1)
1. Draw the two triangles on the cartesian plane.
2. Describe fully a single transformation that maps triangle ABC onto A¹B¹C¹.
3. On the same axes, draw triangle A¹¹B¹¹C¹¹ the image of ABC under a reflection on the line y = x and write down the coordinates of triangle A¹¹B¹¹C¹¹.
4. Draw triangle A¹¹¹B¹¹¹C¹¹¹ such that it can be mapped onto triangle ABC by a negative quarter turnabout the origin and write down its coordinates.
The figure below shows conical container with hemispherical cap of radius 10cm. Water is in the conical part up to where the radius of the surface of water is 6cm. Find;
1. The height of water level in the conical part. (2mks)
2. The volume of water in the container. (2mks)
3. Volume of the conical part not in contact with water. (3mks)
4. The container was inverted and water poured into the hemisphere cap. Calculate the height of water level in the cap. (3mks)
The diagram below shows a histogram representing marks obtained in a test.

If the frequency of the first class is 20.
1. Prepare a frequency distribution table for the data. (4 marks)
2. Calculate the mean mark (3 marks)
3. Calculate the median mark (3 marks)
In a triangle ABC, M is a point on BC such that BM: MC = 3:2. A point N divides AB externally in the ratio 3:2. If AB = a and AC = c
1. Find in terms of a and c
  1. AM (2mks)
  2. AN (1mk)
  3. NM (1mk)
2. If NM produced meets AC at L, determine the ratios AL: LC. (6mks)
Given that y = 7 + 3x - x²
1. Complete the below. (3 marks)
  
  x −3 −2 −1 0 1 2 3 4 5 6
  
  y −11 7 7 −11
2. Using a suitable scale, draw the graph of y = 7 + 3x - x². (3 marks)
3. On the same graph, draw the straight line y = 4 - x. (2 marks)
4. Use your graph to solve the equation x² - 4x - 3 = 0 (2 marks)

MARKING SCHEME

SECTION 1 (50 MARKS)

Answer all questions in the spaces provided.

Without using tables or calculators, evaluate; (3 marks)
36 – 8 x (−4) – 15 ÷ (−3)
3 x (−3) + (−8) (6−(−2)
Evaluate ³/₈ of 7³/₅ -¹/₃ (1¹/₄ + 3¹/₃) x 2²/₅ (3 marks)
Use the reciprocal, square and square root tables to evaluate to 4 significant figures the expression. (4 mark)
√¹/_24.56 + 4.346²
Solve for x in the equation (3 marks)
2^−3x = (1/8)⁻²
Solve the inequality−3x + 2 < x + 6 < 17− 2x and write down the integral values that satisfy the inequality. (3 marks)
A bus left Nairobi at 2245hrs on Sunday traveling towards Kisumu. If it took 7 hrs 15 minutes to reach the destination, at what time did it reach the destination? (2 marks)
Determine the centre of enlargement if P (−6, −3) is the image of P (4, 2) under enlargement scale factor −4. (3 marks)
A line L1 runs in such a way that it is always equidistant from points P and Q. Given that P(−2, 3) and Q (4, −5), find its equation in the form ax + by + c = 0. (4 marks)
Construct trapezium PQRS where PQ = 10cm, QR = 5cm PS = 4cm, ∠PQR = 37½° and PQ is parallel to RS. Measure RS (4 marks)
Solve for x in the equation
1 + Sin2x = Cos 60° for 0° ≤ x ≤ 360°
Calculate the area of the shaded part to 1 decimal place. radius of the circle is 5cm.
An irregular polygon has one of its interior angles = 2000. The remaining angles are of equal sizes of 132.5° each. Find the number of sides of the polygon. (3 marks)
A shamba is in the shape of a parallelogram with the lengths of the adjacent sides being 12cm and 15cm.If the area of the parallelogram is 72cm², find the angle between these two sides. (3 marks)
Mr. Oluoch keeps sheep and goats. The number of sheep exceeds the number of goats by 6. During an outbreak of a disease, ¼ of the goats and 1/3 of the sheep died. If he lost a total of 30 animals, how many animals did he have altogether? (3 marks)
Given that log4 = 0.6021 and log 6 =0.7782, without using mathematical tables or a calculator, evaluate log 0.096. (3 marks)
An American tourist converted US dollars 500,000 into Kenya shillings on his arrival to Kenya when the bank rates were as follows:
Buying Selling
In Kenya Shillings) In Kenya Shillings)
1 US Dollar 77.24 77.4
1 Sterling Pound 121.93 122.27
While in Kenya, he spent Kshs. 2,645,000. He later converted the remaining amount into sterling pounds. Calculate the amount he received to the nearest sterling pounds. (3 marks)

SECTION II: 50 MARKS
Answer any five questions in the spaces provided.
Mr. Otieno is a car dealer who specializes in Mercedes Benz and BMW cars. He is given a commission of 5% on Mercedes Benz sold from all sales above sh.5 million and 6% on BMW on sales above 2 million. A Mercedes Benz costs 2.5 million and a BMW costs 1 million. If in the month of December he earned a commission of sh.500, 000 from the sales of Mercedes Benz and sh.480, 000 from BMW sales;
1. Calculate the number of cars of each type that he sold. (5 marks)
2. In the month of January, the number of Mercedes Benz sold rose by 75% and that of BMW sold dropped by 25%. Calculate the percentage increase in his earning. (5 marks)
The distance between two towns A and B is 760 km. A minibus left town A at 8:15a.m and traveled towards B at an average speed of 90km/h. A matatu left B at 10:35a.m and on the same day and travelled towards A at an average speed of 110km/h.
1. 1. How far from A did they meet? (4mks)
  2. At what time did they meet? (2mks)
2. A motorist starts from his home at 10:30a.m on the same day and traveled at an average speed of 100km/h. He arrived at B at the same time as the minibus. Calculate the distance from B to his home. (4mks)
1. Find the inverse of matrix (2 marks)
2. Two neighbouring schools decided to buy calculators for their students. The first school bought 120 type A calculator while the second school bought 130 type B calculators. Both spent a total of Kshs.298, 000 on the calculators. The following week, the two schools realized that there was still a shortage and ordered for more. The first school ordered for 150 type A calculators while second school ordered for 200 type B calculators and both spent a total of Ksh. 410,000.
  1. Form a matrix equation to represent the above information. (1 mark)
  2. Use matrix method to find the price of each type of calculator bought by the two schools. (4 marks)
  3. If on type A calculators bought, a discount of 10% is given and on type B calculators a discount of 5% is given, find the difference in their discounts. (3 marks)
Triangle ABC has vertices A(1, 2), B(2, 3) and C(4, 1) while triangle A1B1C1 has vertices A1(1, -2), B1(2, -3) and C1(4, -1)
1. Draw the two triangles on the cartesian plane.
2. Describe fully a single transformation that maps triangle ABC onto A1B1C1.
3. On the same axes, draw triangle A¹¹B¹¹C¹¹ the image of ABC under a reflection on the line y = x and write down the coordinates of triangle A¹¹B¹¹C¹¹.
4. Draw triangle A¹¹¹B¹¹¹C¹¹¹ such that it can be mapped onto triangle ABC by a negative quarter turnabout the origin and write down its coordinates.
The figure below shows conical container with hemispherical cap of radius 10cm. Water is in the conical part up to where the radius of the surface of water is 6cm. Find;
1. The height of water level in the conical part. (2mks)
2. The volume of water in the container. (2mks)
3. Volume of the conical part not in contact with water. (3mks)
4. The container was inverted and water poured into the hemisphere cap. Calculate the height of water level in the cap. (3mks)
The diagram below shows a histogram representing marks obtained in a test.

If the frequency of the first class is 20.
1. Prepare a frequency distribution table for the data. (4 marks)
2. Calculate the mean mark (3 marks)
3. Calculate the median mark (3 marks)
In a triangle ABC, M is a point on BC such that BM: MC = 3:2. A point N divides AB externally in the ratio 3:2. If AB = a and AC = c
1. Find in terms of a and c
2. If NM produced meets AC at L, determine the ratios AL: LC. (6mks)
Given that y = 7 + 3x - x²
1. Complete the below. (3 marks)
  
  x −3 −2 −1 0 1 2 3 4 5 6
  
  y −11 −3 3 7 9 9 7 3 −3 −11
2. Using a suitable scale, draw the graph of y = 7 + 3x - x². (3 marks)
3. On the same graph, draw the straight line y = 4 - x. (2 marks)
4. Use your graph to solve the equation x² - 4x - 3 = 0 (2 marks)