Mathematics Paper 2 Questions and Answers - Lanjet Joint Mock Exams 2023

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English Paper 1 Questions and Answers - Lanjet Joint Mock Exams 2023

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INSTRUCTION TO CANDIDATES:

This paper consists of TWO sections: Section I and Section II.
Answer ALL the questions in Section I and any five questions from Section II.
Show all the steps in your calculation, giving your answer at each stage in the spaces provided 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.
Candidates should answer the questions in English.

SECTION A (50 MARKS)
Answer all questions in this section in the spaces provided

A piece of wire 10cm long is to be cut into two parts. If the parts are bent to form a square and a circle respectively, find the length each of the two parts, and hence find the radius of the circle formed, if the sum of their areas is minimum. (4 marks)
In 2022, the cost of processing a bag of wheat was Ksh 250, and this was divided between electricity costs and labour in the ratio 2:3. In the year 2023, the cost of electricity doubled, while the cost of labour increased by x%. Calculate the value of x given that the cost of processing a bag of wheat in 2023 is 425. (2 marks)
A triangle PQR is such that the equation of line PQ is 3y − 2x − 11 =0, while that of line QR is 2y − 3 = −3x. Find the coordinates of the point of intersection of the two lines. (3 marks)
Three quantities b, n and m are such that b varies directly as the product of m and n, and inversely as the square root of the difference of n² and m. Form an equation connecting b, n and m. Express n in terms of b, m and k, where k is the constant of proportionality. (3 marks)
The locus of points 3 cm away from a point (m,m) passes through the point (2,5). Find the value of m. (3 marks)
Using a ruler and pair of compasses only, construct triangle MNP in which MN = 5cm, NP = 4.4cm and angle PNM = 45°. Draw a circle passing through all the vertices of the triangle and measure its radius. (3 marks)
The deviations from the mean of a set of data are given as −4, −1, x², 1, x, and −2. Find the value of x, hence find the standard deviation of the data, given that x is positive. (3 marks)
Two parallel chords of length 4.6cm and 6.8cm are on the same side of a circle of radius r cm. calculate the radius of the circle, correct to four significant figures, given that distance between the two chords is 2cm. (3 marks)
A piece of land is in the shape of a quadrilateral ABCD such that side AB = AD = 50m, < DAB = 1000, < ABC = 87° and < ADC = 79°. Determine the length of barbed wire that would go around the land to the nearest metre. (4 marks).
OABC is parallelogram in which OA = a and OC = c. Y is the mid-point of OC, and AY meets OB at X. X divides line OB and AY in the ratio 3:4 and 1:2 respectively. Find OB in terms of a and c. (3 marks)
Simplify without using tables or calculators: (3 marks)
sinsin 480° − cos 765°
tantan 2250 − cos⁡(−3300)
Use the first four terms of the expansion (2a+b)⁸ to find the value of 2.05⁸ correct to 4 significant figures. (3 marks)
Bazu owns a total more than 10 buses and matatus combined. There are four more matatus than buses, and the buses are not more than 8 in number. By taking x to represent the number of buses, write down all the inequalities for the information and hence state all possible solutions of x. (3 marks)
Two similar metallic cylinders have masses of 64g and 125g respectively. The smaller cylinder has a volume of 21.56cm³. The larger cylinder is molded into a sphere of radius r. Find the value of r correct to 3 decimal places. (3marks)
A ship is observed from the top of a cliff 200 m high in a direction N180E at an angle of depression of 15°. Thirty minutes later, the same ship is observed in a direction S72°E at an angle of depression of 10°. Find the distance travelled in the 30 minutes assuming that the ship travelled along a straight line. (4 marks)
A right pyramid VABCD has its vertex at V and a rectangular base ABCD. AB = 8cm, BC = 6cm and all the slant edges of the pyramid are 10cm. Find the angle between plane VBC and plane VAD correct to two decimal places. (3 marks)

SECTION B (50 MARKS)
Answer only FIVE questions in this section in the spaces provided

The table below shows the income tax rate for a certain year

Income in Ksh. p.m	Tax rate(%)
1 – 11180	10
11181 – 21714	15
21715 – 32248	20
32249 – 42781	25
42782 and above	30

In a certain month, Mr Zakayo paid a net tax of Ksh 26416. He was entitled to a monthly personal relief of Ksh 1648. He also paid insurance premiums amounting to 18200 and was entitled to an insurance relief at 15% of premiums paid, subject to a maximum of Ksh 2000 per month. He also lived in a company house for which he paid a nominal rent of 5200 per month. Other taxable allowances amounted to sh. 22400.

Calculate Mr Zakayo’s monthly gross tax (2 marks)
Calculate Mr Zakayo’s total taxable income in that month (4 marks)
Calculate Mr Zakayo’s net monthly income. (4 marks)

The equation of a curve is given as y = x(x − 2 )
(x−1)⁻¹.
1. Sketch the curve. (7 marks)
2. Determine the exact area bounded by the curve and the x-axis. (3 marks)
Water flows through a cylindrical pipe of diameter 2.8cm at a speed of 25m/min.
1. Calculate the volume of water delivered by the pipe per minute in litres. (3 marks)
2. A cylindrical storage tank of depth 2.5m is filled by water from this pipe and at the same rate of flow. Water begins flowing into the empty storage tank at 8:00am and is full at 2:00pm. Calculate the area of the cross section of this tank in m². (4 marks)
3. A family consumes the capacity of this tank in one month. The cost of water is sh. 150 per 1000 litres plus a fixed maintenance charge of sh. 200. Calculate this family’s water bill for a month. (3 marks)
The second, 4^th and 7^th terms of an AP form the first three consecutive terms of a GP.
1. Find the common ratio of the G.P (5 marks)
2. The first term of the GP if the common difference of the AP is 3 (2 marks)
3. The seventh and the fifth terms of the G.P form the first two consecutive terms of an AP. Find the sum of the first 20 terms of the AP formed. (3 marks)
Use a ruler and a pair of compasses only for all constructions in this question.
1. Construct parallelogram ABCD such that <ABC = 67.5°, BA = 7.8cm and CB = 5.6cm. (3 marks)
2. Construct a perpendicular from A to meet DC at E. Measure AE, hence find the area of the parallelogram. (2 marks)
3. On the same side of AB as C and D, construct the locus of points P such that < APB = 45°(2 marks)
4. Identify by shading the unwanted region, the locus of points R such that AR > 2.8 cm, AR≤ RB and <ARB ≥ 45°. (3 marks)
In a form 4 chemistry classes consisting of both boys and girls, ²/₅ of the students are girls. ²/₃ of the boys and ⁷/₁₀ of the girls are right handed. The probability that a right-handed student breaks a conical flask in any practical session is ³/₇ and the corresponding probability of a left-handed student is ⁴/₉, independent of the student’s gender.
1. Draw a tree diagram to represent the above information (2 marks)
2. Determine the probability that a student chosen at random from the class is left handed and does not break a conical flask during a practical session in simplest form.(3 marks)
3. Determine the probability that a conical flask is broken in any chemistry practical session in simplest form. (3 marks)
4. Determine the probability that a conical flask is not broken by a right-handed student in the simplest form. (2 marks)
A farmer wishes to grow two crops, maize and beans. He has 70 acres of land available for this purpose. He has 240 man-days of labour available to work out the land and he can spend up to sh. 180,000. The requirements of the crops are as follows:
Maize Beans
Minimum number of acres to be sown 10 20
Man – day per acre 2 4
Cost per acre in sh. 3,000 2,000
Profit per acre in sh. 15,000 10,000
1. If x and y represent the number of acres to be used for maize and beans respectively, write down in their simplest form, all the inequalities which x and y must satisfy. (2 marks)
2. Represent the inequalities on the grid provided (5 marks)
3. Profit per acre of maize and beans is sh. 15,000 and 10,000 respectively. Draw a suitable search line on the graph and use it to find the number of acres to be used for maize and beans so as to give maximum profit. (2 marks)
4. Find the maximum profit. (1 mark)
The table below gives values of P with corresponding values of G.

P 90.1 223.3 371.2 693.3 4450.1 11000

G 60 105 147 226 780 1500
1. Using the data from the table above, draw a suitable straight line graph given that P and G are connected by the equation
  log P = G + log K (5 marks)
2. Use the graph to determine:
  1. The values of k and n (3 marks)
  2. The value of P when G is 113 (1 mark)
  3. The value of G when P is 400 (1 mark)

MARKING SCHEME

INSTRUCTION TO CANDIDATES:

This paper consists of TWO sections: Section I and Section II.
Answer ALL the questions in Section I and any five questions from Section II.
Show all the steps in your calculation, giving your answer at each stage in the spaces provided 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.
Candidates should answer the questions in English.

SECTION A (50 MARKS)
Answer all questions in this section in the spaces provided

A piece of wire 10cm long is to be cut into two parts. If the parts are bent to form a square and a circle respectively, find the length each of the two parts, and hence find the radius of the circle formed, if the sum of their areas is minimum. (4 marks)
In 2022, the cost of processing a bag of wheat was Ksh 250, and this was divided between electricity costs and labour in the ratio 2:3. In the year 2023, the cost of electricity doubled, while the cost of labour increased by x%. Calculate the value of x given that the cost of processing a bag of wheat in 2023 is 425. (2 marks)
A triangle PQR is such that the equation of line PQ is 3y − 2x − 11 =0, while that of line QR is 2y − 3 = −3x. Find the coordinates of the point of intersection of the two lines. (3 marks)
Three quantities b, n and m are such that b varies directly as the product of m and n, and inversely as the square root of the difference of n² and m. Form an equation connecting b, n and m. Express n in terms of b, m and k, where k is the constant of proportionality.
(3 marks)
The locus of points 3 cm away from a point (m,m) passes through the point (2,5). Find the value of m. (3 marks)
Using a ruler and pair of compasses only, construct triangle MNP in which MN = 5cm, NP = 4.4cm and angle PNM = 45°. Draw a circle passing through all the vertices of the triangle and measure its radius. (3 marks)
The deviations from the mean of a set of data are given as −4, −1, x², 1, x, and −2. Find the value of x, hence find the standard deviation of the data, given that x is positive. (3 marks)
Two parallel chords of length 4.6cm and 6.8cm are on the same side of a circle of radius r cm. calculate the radius of the circle, correct to four significant figures, given that distance between the two chords is 2cm. (3 marks)
A piece of land is in the shape of a quadrilateral ABCD such that side AB = AD = 50m, < DAB = 1000, < ABC = 87° and < ADC = 79°. Determine the length of barbed wire that would go around the land to the nearest metre. (4 marks).
OABC is parallelogram in which OA = a and OC = c. Y is the mid-point of OC, and AY meets OB at X. X divides line OB and AY in the ratio 3:4 and 1:2 respectively. Find OB in terms of a and c. (3 marks)
Simplify without using tables or calculators: (3 marks)
sinsin 480° − cos 765°
tantan 2250 − cos⁡(−3300)
Use the first four terms of the expansion (2a+b)⁸ to find the value of 2.05⁸ correct to 4 significant figures. (3 marks)
Bazu owns a total more than 10 buses and matatus combined. There are four more matatus than buses, and the buses are not more than 8 in number. By taking x to represent the number of buses, write down all the inequalities for the information and hence state all possible solutions of x. (3 marks)
Two similar metallic cylinders have masses of 64g and 125g respectively. The smaller cylinder has a volume of 21.56cm³. The larger cylinder is molded into a sphere of radius r. Find the value of r correct to 3 decimal places. (3marks)
A ship is observed from the top of a cliff 200 m high in a direction N180E at an angle of depression of 15°. Thirty minutes later, the same ship is observed in a direction S72°E at an angle of depression of 10°. Find the distance travelled in the 30 minutes assuming that the ship travelled along a straight line. (4 marks)
A right pyramid VABCD has its vertex at V and a rectangular base ABCD. AB = 8cm, BC = 6cm and all the slant edges of the pyramid are 10cm. Find the angle between plane VBC and plane VAD correct to two decimal places. (3 marks)

SECTION B (50 MARKS)
Answer only FIVE questions in this section in the spaces provided

The table below shows the income tax rate for a certain year

Income in Ksh. p.m	Tax rate(%)
1 – 11180	10
11181 – 21714	15
21715 – 32248	20
32249 – 42781	25
42782 and above	30

Calculate Mr Zakayo’s monthly gross tax (2 marks)
Calculate Mr Zakayo’s total taxable income in that month (4 marks)
Calculate Mr Zakayo’s net monthly income. (4 marks)

The equation of a curve is given as y = x(x − 2 )
(x−1)⁻¹.
1. Sketch the curve. (7 marks)
2. Determine the exact area bounded by the curve and the x-axis. (3 marks)
Water flows through a cylindrical pipe of diameter 2.8cm at a speed of 25m/min.
1. Calculate the volume of water delivered by the pipe per minute in litres. (3 marks)
2. A cylindrical storage tank of depth 2.5m is filled by water from this pipe and at the same rate of flow. Water begins flowing into the empty storage tank at 8:00am and is full at 2:00pm. Calculate the area of the cross section of this tank in m². (4 marks)
3. A family consumes the capacity of this tank in one month. The cost of water is sh. 150 per 1000 litres plus a fixed maintenance charge of sh. 200. Calculate this family’s water bill for a month. (3 marks)
The second, 4^th and 7^th terms of an AP form the first three consecutive terms of a GP.
1. Find the common ratio of the G.P (5 marks)
2. The first term of the GP if the common difference of the AP is 3 (2 marks)
3. The seventh and the fifth terms of the G.P form the first two consecutive terms of an AP. Find the sum of the first 20 terms of the AP formed. (3 marks)
Use a ruler and a pair of compasses only for all constructions in this question.
1. Construct parallelogram ABCD such that <ABC = 67.5°, BA = 7.8cm and CB = 5.6cm. (3 marks)
2. Construct a perpendicular from A to meet DC at E. Measure AE, hence find the area of the parallelogram. (2 marks)
3. On the same side of AB as C and D, construct the locus of points P such that < APB = 45°(2 marks)
4. Identify by shading the unwanted region, the locus of points R such that AR > 2.8 cm, AR≤ RB and <ARB ≥ 45°. (3 marks)
In a form 4 chemistry classes consisting of both boys and girls, ²/₅ of the students are girls. ²/₃ of the boys and ⁷/₁₀ of the girls are right handed. The probability that a right-handed student breaks a conical flask in any practical session is ³/₇ and the corresponding probability of a left-handed student is ⁴/₉, independent of the student’s gender.
1. Draw a tree diagram to represent the above information (2 marks)
2. Determine the probability that a student chosen at random from the class is left handed and does not break a conical flask during a practical session in simplest form.(3 marks)
3. Determine the probability that a conical flask is broken in any chemistry practical session in simplest form. (3 marks)
4. Determine the probability that a conical flask is not broken by a right-handed student in the simplest form. (2 marks)
A farmer wishes to grow two crops, maize and beans. He has 70 acres of land available for this purpose. He has 240 man-days of labour available to work out the land and he can spend up to sh. 180,000. The requirements of the crops are as follows:
Maize Beans
Minimum number of acres to be sown 10 20
Man – day per acre 2 4
Cost per acre in sh. 3,000 2,000
Profit per acre in sh. 15,000 10,000
1. If x and y represent the number of acres to be used for maize and beans respectively, write down in their simplest form, all the inequalities which x and y must satisfy. (2 marks)
2. Represent the inequalities on the grid provided (5 marks)
  * scale (1cm rep 5 acres)
3. Profit per acre of maize and beans is sh. 15,000 and 10,000 respectively. Draw a suitable search line on the graph and use it to find the number of acres to be used for maize and beans so as to give maximum profit. (2 marks)
4. Find the maximum profit. (1 mark)
The table below gives values of P with corresponding values of G.
1. Using the data from the table above, draw a suitable straight line graph given that P and G are connected by the equation
  log P = G + log K (5 marks)
2. Use the graph to determine:
  1. The values of k and n (3 marks)
  2. The value of P when G is 113 (1 mark)
  3. The value of G when P is 400 (1 mark)