Mathematics Paper 2 Questions and Answers - Wahundura Boys Mock Examination 2023

INSTRUCTIONS 

• The paper contains two sections: Section I and II.
• 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.

QUESTIONS

1. Use logarithms to evaluate (4mks)
   
2. Express the following in surd form and simplify by rationalizing the denominator (3mks)
   ¹/ _{(cos60°− sin45°)}
3. Make q the subject of the formula (3mks)
   
4. Without using logarithms tables or calculator evaluate (3mks)
   log₁₀⁡96 + 3⁄4 log₁₀⁡625 - log_10⁡12
5. The volume V of a cylinder varies jointly as its height, (h) and the square of its radius, (r), calculate the percentage increase in its volume V, when radius increases by 5% and height, h increases by 10%. (3 mks)
6. Chords AB and CD in the figure below intersects externally at Q. If AB=5cm BQ=6CM and DQ=4cm, calculate the length of chord CD ( 2mks)
   
7. Jane can do a piece of work in 4 days while Mary can do the same piece of work in 7 days. Mary and Jane did the job together for two days before Jane fell sick. Mary was left to complete the job. How long did it take to do the job (3mks)
8. Solve the equation below in the range θ ≤ x ≤ 180° (3mks)
   2 sin (3x+10) = -1.674
9. Expand and simplify (3x –y)⁴ hence use the first three terms of the expansion to approximate the value of (6 – 0.2)⁴ (3mks)
10. The prefects body of a certain school consists of 7 boys and 5 girls. Three prefects are to be chosen at random to represent the school at a certain function at Nairobi. Find the probability that the chosen prefects are boys. (2mks)
11. The sides of a triangle were measured and recorded as 8.4 cm, 10.5 cm and 15.32 cm .Calculate the percentage error in it’s perimeter 2d.p. (3marks)
12. A(3,2) and B(7,4) are points on the circumference of a circle. Given that chord
    AB passes through the centre of the circle determine the equation of the circle. (4marks)
13. Use matrix method to determine the co-ordinates of the point of intersection of the two lines. (3mks)
    3x - 2y =13
    2y + x + 1=0
14. The sum of two numbers is 9. The sum of the square of the number is 41. Find the numbers (4 Marks)
15. Draw a line DF=4.6cm. Construct the locus of a point K above DF such that the angle DKF=90°, (3mks)
16. A right pyramid has a rectangular base of 12 cm by 16cm. its slanting lengths are 26 cm. Determine:
    1. The length of AC (1 mk)
       
    2. The angle AV makes with the base ABCD. (2 mks)

SECTION II (Answer only five questions)

17.  
    1. An arithmetic progression is such that the first term is -5, the last is 135 and the sum of the progression is 975. Calculate:
       1. The number of terms in the series (4mks)
       2. The common difference of the progression (2mks)
    2. The sum of the first three terms of a geometric progression is 27 and first term is 36. Determine the common ration and the value of the fourth term (4 marks)
18. A′(−6,0), B′(−2,−3) and C′(−2,0) are the vertices of the image of triangle ABC under a transformation described by the matrix ?=M=
    1. Determine the coordinates of triangle ABC (3 marks)
    2.  
       1. On the same grid, draw triangles ABC, A′B′C′ (2 marks)
          
       2. Describe fully the transformation M (1 mark)
    3. Triangle A′′B′′C′′ is the image of triangle A′B′C′ such that A′′(0,6), B′′(6,2) and C′′(0,2)
       1. Draw triangle A′′B′′C′′on the same axes (1 mark)
       2. Find a single matrix of transformation that maps triangle ABC onto triangle A′′B′′C′′ (3 marks)
19. In the triangle PQR below, L and M are points on PQ and QR respectively such that PL : LQ = 1 : 3 and Qm : mR = 1 : 2. Pm and RL intersect at X. Given that PQ = (b ) ̃and PR = c ̃
    
    Q L m XP R
    1. Express the following vectors in terms of (b ) ̃and c ̃ .
       1. QR (1 mark)
       2. Pm (1 mark)
       3. RL (1 mark)
    2. By taking PX = kPm and RX = hRL where h and k are constants. Find two expressions of PX in terms of h, k, b and c. Hence determine the values of the constants h and k. (6 marks)
    3. Determine the ratio LX : XR. (1 mark)
20. The curve given by the equation y = x² +1 is defined by the values in the table below.
    1. Complete the table by filling in the missing values.
       

       X	0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0	5.5	6.0
       Y	1.0		2.0		5.0		10.0		17.0		26.0		37.0

    2. Sketch the curve for y = x² + 1 for 0 ≤ x ≤ 6
       
    3. Use the mid-ordinate rule with 6 ordinates to find the area of the region bounded by the curve y = x² + 1, the axis the lines x = 0 and x = 6 (2mk)
    4. Use method of integration to find the exact value of the area of the region in (C) above. (2mk)
    5. Calculate the percentage error involved in using the mid- ordinate rule to find the area. (2mk)
21. The table below shows the taxation rates for income earned.
    

    Income in Ksh  p.m	Tax rates %
    1 – 9680 9681 – 18800 18801 – 27920 27921 – 37040 Excess over 37041	10 15 20 25 30

    
    In that year, Mr. Hamisi paid a net tax of KSh. 5,512 per month. He gets a house allowance of KShs. 10,000, medical allowance of KShs. 2400 and acting allowance of KShs. 2820 per month. He was entitled to a monthly personal relief of KShs. 1162. He has a life insurance policy for which he pays a monthly premium of KSh. 1,500 and claims a relief at a rate of 10% of the premium paid per month. The following deductions also made every month.
    1. N.H.I.F. KSh. 320
    2. Co-operative society shares KSh. 6000
    3. Union dues KSh. 200
       1. Calculate Mr. Hamisi’s monthly basic salary in KSh. (7 Marks)
       2. Calculate his net monthly salary. (3 Marks)
22. The table below gives the marks obtained in a mathematics test by 47 students.
    1.  
       1. Fill in the missing class (1mark)
       2. State the modal class (1 mark)
    2. Draw a cumulative frequency curve to represent the above data. (3 marks)
       
    3. Use your graph to estimate
       1. Median (1 mark)
       2. Lower quartile (1 mark)
       3. Upper quartile (1 mark)
    4. Calculate the quartile deviation (2 marks)
23. An aircraft leaves town P (30oS, 17°E) and moves directly towards Q (60°N, 17°E). It then moved at an average speed of 300 knots for 8 hours Westwards to town R. Determine
    1. The distance PQ in nautical miles. (2 marks)
    2. The position of town R. (4 marks)
    3. The local time at R if local time at Q is 3.12p.m (2 marks)
    4. The total distance moved from P to R in kilometers. (Take 1nm = 1.853km) (2 marks)
24. A certain uniform supplier is required to supply two types of shirts: one for girls labeled G and the other for boys labeled B. The total number of shirts must not be more than 400. He has to supply more of type G than of type B. However the number of type G shirts must not be more than 300 and the number of type B shirts must not be less than 80. By taking x to be the number of type G shirts and y the number of type B shirts.
    1. Write down in terms of x and y all the inequalities representing the information above. (4 marks)
    2. On the grid provided draw the inequalities and shade the unwanted regions. (4 marks)
       
    3. Given that type G costs Ksh500 per shirt and type B costs Ksh. 300 per shirt
       1. Use the graph in (b) above to determine the number of shirts of each type that should be made to maximize profit. (1 mark)
       2. Determine the maximum profit (1 mark)

MARKING SCHEME

1. Use logarithms to evaluate (4mks)
   
   0.2098 - 0.3406 = ^-1.8692
   13 × ^-1.8692 = ^-1.9564
   log of ^-1.9564
   9.044 × 10^-1 = 0.9044
2. Express the following in surd form and simplify by rationalizing the denominator (3mks)
   1/ (cos60°-sin45°)
   1/ 1⁄2 - 1⁄  √2 
   1/ √2-2/ 2√2  = 2√2/ √2-2
   2√2 (√2+2)/ (√2-2)(√2+2)
   =4+4√2/ -2  =-2(1+2√2)
3. Make q the subject of the formula (3mks)
   p³q=nq−m
   p³q−nq=−m
   q(p³−n)=−m
   q=  -m   or q=  m     
        p³-n           n-p³
4. Without using logarithms tables or calculator evaluate (3mks)
   log₁₀⁡96 + 3⁄4 log_10⁡625 - log_10⁡12
   log₁₀ 96 + log₁₀ 6253/₄ -log₁₀ 12
   log₁₀96+log₁₀(5⁄4)3/4-log₁₀12
   log₁₀96+log₁₀125-log₁₀12
   log₁₀ 96×125/ ₁₂
   log_1o1000=3
5. The volume V of a cylinder varies jointly as its height, (h) and the square of its radius, (r), calculate the percentage increase in its volume V, when radius increases by 5% and height, h increases by 10%. (3 mks)
   Vαhr2
   V=Khr2
   V1=Kh1(r1)2
   h1=1.1h
   r1=1.05r
   V1KX1.1h×(1.05r)2
   V1=1.2128Hhr2
   butKhr2=V
   V1=1.2128V
   (1.2128V-V)/_V ×100
   =21.28%
6. Chords AB and CD in the figure below intersects externally at Q. If AB=5cm BQ=6CM and DQ=4cm, calculate the length of chord CD ( 2mks)
   AQ.BQ=CG.DG   let cd=x
   11×6 =(x+4)4
   66=4x+16
   50=4x
   x=12.5
   CD=12.5
7. Jane can do a piece of work in 4 days while Mary can do the same piece of work in 7 days. Mary and Jane did the job together for two days before Jane fell sick. Mary was left to complete the job. How long did it take to do the job (3mks)
   both in a day 1/₄ +1/₇ =11/₂₈
   both in two days 11/₂₈×2=11/₁₄
   remaining job 1-11/₁₄=3/14
   1/₇-1day
   3/₁₄-?
   3/₁₄×⁷/₁=3/₂=1 1/₂ days
8. Solve the equation below in the range θ ≤ x ≤ 180° (3mks)
   2 sin (3x+10) = -1.674
   sin(3x+10°)=0.8370
   sin(3x+10°)=236.82°,303.18°
   3x+10°=236.82°
   x=75.6067°
   3x+10°=303.18°
   x=97.7267°
9. Expand and simplify (3x –y)4 hence use the first three terms of the expansion to approximate the value of (6 – 0.2)4 (3mks)
   81x4-108x3y+54x2y2-12xy3+y4
   3x=6
   x=2
   y=0.2
   81(2)4-108(2)3(0.2)+54(2)2(0.2)2
   1296-172.8+8.64
   =1131.84
10. The prefects body of a certain school consists of 7 boys and 5 girls. Three prefects are to be chosen at random to represent the school at a certain function at Nairobi. Find the probability that the chosen prefects are boys. (2mks)
    
    p(BBB)=⁷/₁₂× ⁶/₁₁ × ⁵/₁₀=⁷/₄₄
11. The sides of a triangle were measured and recorded as 8.4 cm, 10.5 cm and 15.32 cm .Calculate the percentage error in it’s perimeter 2d.p. (3marks)
    max P= 8.45+10.55+15.325=34.325
    minP=8.35+10.45+15.315=34.115
    actualp=8.4+10.5+15.32=34.22
    absolute error =  34.325−34.115
                                                   ₂
                               = 0.105
    %error = 0.105 × 100
                    34.22
    =0.3068%
12. A(3,2) and B(7,4) are points on the circumference of a circle. Given that chord
    AB passes through the centre of the circle determine the equation of the circle. (4marks)
    center(3+7, 2+4 = 5,3)
                  2       2
    r=
    r=√5
    (x-5)2+(y-3)2=(√5)2
    x2-10x+25+y2-6y+9=5
    x2+y2-10x-6y+29=0
13. Use matrix method to determine the co-ordinates of the point of intersection of the two lines. (3mks)
    3x - 2y =13
    2y + x + 1=0
    
14. The sum of two numbers is 9. The sum of the square of the number is 41. Find the numbers (4 Marks)
    x + y = 9...(i)
    x2 + y2 = 41...(ii)
    from eqn i x = 9 - y
    solution in (ii)
    (9-y)² + y²=41
    81-81+y²+y²=41
    40-18y+2y2=0
    y²-9y+20=0
    y²-5y-4y+20=0
    y(y-5)-4(y-5)=0
    (y-4)(y-5)=0
    y=4  or y=5
    x=5  or x=4
15. Draw a line DF=4.6cm. Construct the locus of a point K above DF such that the angle DKF=90°, (3mks)
    
16. A right pyramid has a rectangular base of 12 cm by 16cm. its slanting lengths are 26 cm. Determine:
    1. The length of AC (1 mk)
         =13.42
    2. The angle AV makes with the base ABCD. (2 mks)
       cosθ = 6.71  =0.2581
                     26
         θ=75.04°

SECTION II (Answer only five questions)

17.  
    1. An arithmetic progression is such that the first term is -5, the last is 135 and the sum of the progression is 975. Calculate:
       1. The number of terms in the series (4mks)
          a=-5
          l=135
          sn=ⁿ/ ₂(a+l)
          975=ⁿ/ ₂(-5+135)
          975=¹³⁰/₂n
          n=975ײ/₁₃₀=15
       2. The common difference of the progression (2mks)
          n^th=a+(n-1)d
          135=-5+14d
          140=14d
          d=10
    2. The sum of the first three terms of a geometric progression is 27 and first term is 36. Determine the common ration and the value of the fourth term (4 marks)
       a+ar+ar²=27
       36+36r+36r²=27
       divided by 9 all through
       4r²+4r+1=0
       4r²+2r+2r+1=0
       2r(2r+1)+1(2r+1)=0
       (2r+1)(2r+1)=0
       2r+1=0
       2r=-1
       r=-1 /₂
       4th=ar³
       36(-1⁄2)³
       =-4 1⁄2
18. A′(−6,0), B′(−2,−3) and C′(−2,0) are the vertices of the image of triangle ABC under a transformation described by the matrix ?=m=
    1. Determine the coordinates of triangle ABC (3 marks)
       let the coordinates
       A(m,n) B(o,p) C(q,r)
       
       n=0     A= (-6,0)
       m=6
       p=-3    B=(-8,-3)
       o=-8
       r=0      C=(-2,0)
       q=-2
    2.  
       1. On the same grid, draw triangles ABC, A′B′C′ (2 marks)
          
       2. Describe fully the transformation M (1 mark)
          shear
          x-axis invariant
          pointB(-8,-3)mapped onto B1(-2,-3)
    3. Triangle A′′B′′C′′ is the image of triangle A′B′C′ such that A′′(0,6), B′′(6,2) and C′′(0,2)
       1. Draw triangle A′′B′′C′′on the same axes (1 mark)
       2. Find a single matrix of transformation that maps triangle ABC onto triangle A′′B′′C′′ (3 marks)
           
          
19. In the triangle PQR below, L and M are points on PQ and QR respectively such that PL : LQ = 1 : 3 and Qm : mR = 1 : 2. Pm and RL intersect at X. Given that PQ = (b ) ̃and PR = c ̃
    Q L m XP R
    1. Express the following vectors in terms of (b ) ̃and c ̃ .
       1. QR (1 mark)
          QR=QP+PR
          QR=-b+c
          QR=c-b
       2. Pm (1 mark)
          PM=MQ+QM
          but QM=1⁄3 QR
          PM=b+1⁄3 (-b+c)
          =b-1⁄3 b+1⁄3 c
          =2⁄3 b+1⁄3 c
       3. RL (1 mark)
          RL=RP+Pl  but PL+1⁄4 PQ
          -c+1⁄4 b
    2. By taking PX = kPm and RX = hRL where h and k are constants. Find two expressions of PX in terms of h, k, b and c. Hence determine the values of the constants h and k. (6 marks)
       PX=K(2⁄3 b+1⁄3 c)
       PX=2⁄3 Kb+1⁄3 Kc.....i
       PX=PR+RX
       =c+h(-c+1⁄4 b)
       c-hc+1⁄4 hb
       (1-h)c+1⁄4 hb.....ii
       2⁄3 kb=1⁄4 hb
       K=3⁄8 h
       1⁄3 Kc=(1-h)c
       1⁄3(3⁄8 h)=1-h   1⁄8 h=1-h
       h=8⁄9
       K=3⁄8 h =3⁄8×8⁄9=1⁄3
    3. Determine the ratio LX : XR. (1 mark)
       1⁄9 : 8⁄9=1:8
20. The curve given by the equation y = x² +1 is defined by the values in the table below.
    1. Complete the table by filling in the missing values.
       

       X	0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0	5.5	6.0
       Y	1.0	1.25	2.0	3.25	5.0	7.25	10.0	13.25	17.0	21.25	26.0	31.25	37.0

    2. Sketch the curve for y = x² + 1 for 0 ≤ x ≤ 6
       
    3. Use the mid-ordinate rule with 6 ordinates to find the area of the region bounded by the curve y = x² + 1, the axis the lines x = 0 and x = 6 (2mk)
       1(1.25+3.25+7.25+13.25+21.25+31.25)
       =77.5 square units
    4. Use method of integration to find the exact value of the area of the region in (C) above. (2mk)
       
       =78 sq units
    5. Calculate the percentage error involved in using the mid- ordinate rule to find the area. (2mk)
       78 − 77.5 = 0.5
       0.5 × 100 = 0.6410%
       78
21. The table below shows the taxation rates for income earned.
    

    Income in Ksh  p.m	Tax rates %
    1 – 9680 9681 – 18800 18801 – 27920 27921 – 37040 Excess over 37041	10 15 20 25 30

    In that year, Mr. Hamisi paid a net tax of KSh. 5,512 per month. He gets a house allowance of KShs. 10,000, medical allowance of KShs. 2400 and acting allowance of KShs. 2820 per month. He was entitled to a monthly personal relief of KShs. 1162. He has a life insurance policy for which he pays a monthly premium of KSh. 1,500 and claims a relief at a rate of 10% of the premium paid per month. The following deductions also made every month.
    1. N.H.I.F. KSh. 320
    2. Co-operative society shares KSh. 6000
    3. Union dues KSh. 200
       1. Calculate Mr. Hamisi’s monthly basic salary in KSh. (7 Marks)
          relief 10⁄100×1500=150
          1162+150=1312
          gross tax 5512+1312=6824
          1st slab 9680×10⁄100=968
          2nd slab 9120×15⁄100=1368
          3rd slab 9120×20⁄100=1824
          4thslab 9120×25⁄100=2280
          68-6440=384
          30⁄100 x-11112=384
          30⁄100 x=11496
          x=38320
          taxable income=38320
          B/S=taxable income-allowances
          38320-(10000+2400+2820)
          38320-15220
          =sh23100
       2. Calculate his net monthly salary. (3 Marks)
          total deductions
          320+6000+200+1500+5512
          =13532
          net salary=38320-13532
          =sh24,748
22. The table below gives the marks obtained in a mathematics test by 47 students.
    1.  
       1. Fill in the missing class (1mark)
          56-60
          51-55
       2. State the modal class (1 mark)
    2. Draw a cumulative frequency curve to represent the above data. (3 marks)
       
    3. Use your graph to estimate
       1. Median (1 mark)
          47⁄2   =23.5
          median =55.5
       2. Lower quartile (1 mark)
          1⁄4 × 47 =11.75
          Q1 =51.0
       3. Upper quartile (1 mark)
          3⁄4 × 47 =36.25
          Q3 =60.0
    4. Calculate the quartile deviation (2 marks)
       Q3 − Q1
             2
       60.0− 51.0
               2
       =4.5
23. An aircraft leaves town P (30oS, 17°E) and moves directly towards Q (60°N, 17°E). It then moved at an average speed of 300 knots for 8 hours Westwards to town R. Determine
    1. The distance PQ in nautical miles. (2 marks)
       90×60=5400nm
    2. The position of town R. (4 marks)
       d=300×8=2400nm
       60×lat diff×cos60=2400nm
       lat diff =2400/ _60coc60
       80°
       80°-17°=63°
       R(60°N63°W)
    3. The local time at R if local time at Q is 3.12p.m (2 marks)
       1°=4min
       80°?
       80×⁴/ ₁ =320min
       =5hrs 20min
       1512−520=0952
       =9:54am
    4. The total distance moved from P to R in kilometers. (Take 1nm = 1.853km) (2 marks)
       5400+2400=7800nm
       1nm=1.853
       7800=?
       7800×1.853
       =14,453.4km
24. A certain uniform supplier is required to supply two types of shirts: one for girls labeled G and the other for boys labeled B. The total number of shirts must not be more than 400. He has to supply more of type G than of type B. However the number of type G shirts must not be more than 300 and the number of type B shirts must not be less than 80. By taking x to be the number of type G shirts and y the number of type B shirts.
    1. Write down in terms of x and y all the inequalities representing the information above. (4 marks)
       x+y≤400
       x>y
       x≤300
       y≥80
    2. On the grid provided draw the inequalities and shade the unwanted regions. (4 marks)
       
    3. Given that type G costs Ksh500 per shirt and type B costs Ksh. 300 per shirt
       1. Use the graph in (b) above to determine the number of shirts of each type that should be made to maximize profit. (1 mark)
          (300,100)          500(300)+300(100)=180000
          (300,80)            500(300)+300(80)=174000
          (290,110)          500(240)+300(110)=178000
          300 shirts
          100 blouses
       2. Determine the maximum profit (1 mark)
          500(300) +300(100) =180000