Mathematics Paper 2 Questions and Answers - Asumbi Girls Highschool Pre-Mock Exams May - June 2022

Instructions to candidates

• Write your name and index number in the spaces provided above.
• Sign and write the date of examination in the spaces provided above.
• This paper consists of two sections I, II.
• Answer all the questions in section 1 and any five questions from section II.
• All working and answers must be written on the question paper in the spaces provided below each question.
• Show all steps in your calculations, giving answers 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.

QUESTIONS

SECTION 1 (50MARKS)

1. Use logarithms to evaluate (4mks)
   
2. A rectangular card measures 5.3cm by 2.5cm. find
   1. The absolute error in the area of the card. (2mks)
   2. The relative error in the area of the card (2mks)
3. Solve the equation (4mks)
   Sin (2x +10)º=-0.5 for 0º ≤ x≤ 360º
4. In a transformation, an object with an area of 52cm2 is mapped onto an image whose area is 30cm². Given that the matrix of the transformation is  find the value of x (3mks)
5. Simplify leaving your answer in the form of a√b + c where a, b and c are integers. (3mks)
6. A customer deposited sh 14000 in a saving account. Find the accumulated amount after one year if interest was paid at 12% p.a compounded quarterly (3mks)
7. Expand (1+x)⁵, hence use the expansion to estimate (1.04)⁵ correct to 4 decimal place (3mks)
8. Find the centre and the radius of circle whose equation is (3mks)
   x²+4x+y²-5=0
9. Make d the subject of the formula (3mks)
   P=1/2mn^2-gd²
                    n
10. In what proportion should grades of sugar costing sh 45 and sh 50 per kg be mixed in order to produce a blend worth sh 48 per kg (3mks)
11. Simplify the expression (3mks)
    
12. Find the equation of the tangent to the curve (3mks)
    y=2x² at (2, 3)
13. Use matrix method to solve the given simultaneous equation (3mks)
    3x+y=7
    5x+2y=12
14. The sum of n terms of the sequence 3, 9, 15, 21 ... is 7500. Determine the value of n (3mks)
15. The figure below (not drawn to scale) shows a triangle ABC in which AB=6cm, BC=9cm, AC=10cm. calculate the radius of the circle touching the three vertices of the triangle. (3mks)
    
16. The point p (40ºS, 45ºE) and point Q (40ºS, 60ºW) are on the surface of the earth. Calculate the shortest distance along a circle of latitude between the two points. (3mks)
17. The table below shows monthly income tax rates.
    

    Monthly taxable pay K£	Rate of tax sh per K£
    1-342 343-684 685-1026 1027-1368 1369-1710 Over 1710	2 3 4 5 6 7

    A civil servant earns a monthly salary of sh 20 000 and is provided with a house at a normal rent of sh 700 per month.
    1. calculate the civil servant taxable pay in K£ (4mks)
    2. Calculate the total tax (4mks)
    3. If the employee is entitled to a tax relief of sh 600 per month. What is the net tax paid? 2mks)
18. In an agricultural research centre, the length of a sample of 50 maize cobs were measured and recorded as shown in the frequency distribution table below.
    

    Length in cm	Number of cobs
    8-10 11-13 14-16 17-19 20-22 23-25	4 7 11 15 8 5

    Calculate
    1. The mean (2mks)
    2.                    
       1. the variance (5mks)
       2. The standard deviation (3mks)
19. In the diagram shown below O is the centre of the circle, angle RTV=1500,and angle RST=500,
    
    
    1. Calculate the size of
       1. <ORS (2mks)
       2. <USP (1mk)
       3. <PQR (2mks)
    2. Given that RT =7cm and ST=9cm, calculate to 3.s.f
       1. The length of line PR (2mks)
       2. The radius of the circle (3mks)
20. The position of two towns A and B on the earth surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively
    1. Find the difference in longitude between town A and town B (2mks)
    2. Given that the radius of the earth is 6370km calculate the distance between town A and B (4mks)
    3. Another town C is 840km east of town B and on the same latitude as towns A and B. find the longitude of town C (4mks)
21. The distance sm from a fixed point O, covered by a particle after ts is given by the equation
    S=t³-6t²+9t+5
    1. Calculate the gradient to the curve at t=0.5s (3mks)
    2. Determine the values of s at the maximum and minimum turning points of the curve. (4mks)
    3. On the space provided sketch the curve of s=t³-6t²+9t+5 (3mks)
22. In the figure below OQ =a and OB=b. m is the midpoint of OA and AN:NB =2:1
    
    
    1. Express in terms of a and b
       1. BA (1mk)
       2. BN (1mk)
       3. ON (2mks)
    2. Given that BX=hBM and OX=KON determine the values of h and k (6mks)
23.                            
    1. Complete the table below, giving the values correct to 1 d.p (2mks)
       

       		X0	0	40	80	120	160	200	240
       2sin (x+20)0		0.7			2.0		0.0		-2.0
       √3 cos x		1.7	1.3			-0.9		-1.6

    2. On the grid provided, using the same scale and axes, draw the graphs of y=2sin (x+20)º and y=√3 cos x for 0≤x≤ 240º (5mks)
    3. Use the graphs drawn in (b) above to determine:
       The values of x for which 2 sin (x+20)º =√3 cos x (2mks)
       The difference in the amplitudes of y=2sin(x+20) and y=√3 cox x (1mk)
24. The probabilities that a husband and wife will be a live 25 years from now are 0.7 and 0.9 respectively. Find the probability that in 25 years time;
    1. Both will be a live (2mks)
    2. Neither will be a live (3mks)
    3. One will be a live (2mks)
    4. At least one will be a live (3mks)

MARKING SCHEME

SECTION 1 (50MARKS)

1. Use logarithms to evaluate (4mks)
   
   
   

   No	std	log
   0.8932	8.932 x 10-1	1.9509 x 2
   582.3	5.823 x 102	2.7651      3
   09.36	0.935 x 101	1.8410

2. A rectangular card measures 5.3cm by 2.5cm. find
   1. The absolute error in the area of the card. (2mks)
      max V 5.35   min 5.25      max 2.55   min  2.45
      max A -  min A = M1
               2
      13.3875 - 12.8625
                 2
      = 0.2626%
   2. The relative error in the area of the card (2mks)
3. Solve the equation (4mks)
   Sin (2x +10)º=-0.5 for 0º ≤ x≤ 360º
   2x + 10 = sin-1-0.5
   = -30º
   2x + 10 = 210º, 330º, 570º, 690º
   x = 100º, 160º, 280º, 335º
4. In a transformation, an object with an area of 52cm2 is mapped onto an image whose area is 30cm². Given that the matrix of the transformation is  find the value of x (3mks)
   ASF = 30/52
   = 15/26 = 4x - 2x + 2
   15/26 = 2x + 2
   x = -37/26
5. Simplify leaving your answer in the form of a√b + c where a, b and c are integers. (3mks)
   
6. A customer deposited sh 14000 in a saving account. Find the accumulated amount after one year if interest was paid at 12% p.a compounded quarterly (3mks)
   A = 14000 (1 + 12/100 x 1/4)⁴
   = 14000(1.03)⁴
   = 15757
7. Expand (1+x)⁵, hence use the expansion to estimate (1.04)⁵ correct to 4 decimal place (3mks) r = 3
8. Find the centre and the radius of circle whose equation is (3mks)
   x²+4x+y²-5=0
9. Make d the subject of the formula (3mks)
   P=1/2mn^2-gd²
                    n
   2pn - mn³ = 2gd²
   d² = mn³ - 2pn
                2q
10. In what proportion should grades of sugar costing sh 45 and sh 50 per kg be mixed in order to produce a blend worth sh 48 per kg (3mks)
    let a:b = ratio
    45a + 50b = 48
        a + b
    45a - 48a = 48b - 50b
    a:b = 2:3
11. Simplify the expression (3mks)
    
    N:(4m - 3n)(4m + 3n)
    D: 4m³ - 4mn + 3mn - 3n²
    4m(m - n) + 3n(m - n)
    (4m + 3n)(m - n) 
    = 4m - 3n
        m - n
12. Find the equation of the tangent to the curve (3mks)
    y=2x² at (2, 3)
    dy/dx = 4x
    at x = 2
    m = 2(4)
    = 8
    (x,y)(2,3) 
    m = 8
    y - 3 = 8
    x - 2    1
    y - 3 = 8 x -16
    y = 8x + 15
13. Use matrix method to solve the given simultaneous equation (3mks)
    3x+y=7
    5x+2y=12
    
14. The sum of n terms of the sequence 3, 9, 15, 21 ... is 7500. Determine the value of n (3mks)
    n/2(2a(n - 1)d) = 7500
    n/2(6 + (n - 1)6) = 7500
    6n + 6n² = 15000
    6n² = 15000
    n² = 2500
    n = 50
15. The figure below (not drawn to scale) shows a triangle ABC in which AB=6cm, BC=9cm, AC=10cm. calculate the radius of the circle touching the three vertices of the triangle. (3mks)
    
16. The point p (40ºS, 45ºE) and point Q (40ºS, 60ºW) are on the surface of the earth. Calculate the shortest distance along a circle of latitude between the two points. (3mks)
    Ò = 105º
    = 105 x 2 x 22 x 6370 cos 40
       360           7
    = 8946 km
17. The table below shows monthly income tax rates.
    

    Monthly taxable pay K£	Rate of tax sh per K£
    1-342 343-684 685-1026 1027-1368 1369-1710 Over 1710	2 3 4 5 6 7

    A civil servant earns a monthly salary of sh 20 000 and is provided with a house at a normal rent of sh 700 per month.
    1. calculate the civil servant taxable pay in K£ (4mks)
       20000 + 15/100 x 20000 - 700
       22300/20 = 1115t
    2. Calculate the total tax (4mks)
       ans = 3523
    3. If the employee is entitled to a tax relief of sh 600 per month. What is the net tax paid? 2mks)
        3523
       - 600
       29231
18. In an agricultural research centre, the length of a sample of 50 maize cobs were measured and recorded as shown in the frequency distribution table below.
    

    Length in cm	Number of cobs	x	x2	fx	fx2
    8-10 11-13 14-16 17-19 20-22 23-25	4 7 11 15 8 5	9 12 15 18 21 24	81 144 225 324 441 576	36 84 165 270 168 120	324 1008 2475 4860 3528 2880
    			∑x = 843		∑fx = 15075

    Calculate
    1. The mean (2mks)
       = 16.86
    2.                    
       1. the variance (5mks)
          = 17.25
       2. The standard deviation (3mks)
          = 4.153
19. In the diagram shown below O is the centre of the circle, angle RTV=1500,and angle RST=500,
    
    
    1. Calculate the size of
       1. <ORS (2mks)
          = 40º - base < s isosceles angles are equal
       2. <USP (1mk)
          = 80º - alternate segments theorem
       3. <PQR (2mks)
          = 130º - opposites <s a cycle quadrilateral add up to 180º
    2. Given that RT =7cm and ST=9cm, calculate to 3.s.f
       1. The length of line PR (2mks)
          (x + 7)7 = 9²   7x + 49 = 81   7x = 22
          x = 3.142
       2. The radius of the circle (3mks)
          PR = 7 + 3.142
          = 10.142
          3.142 = 2R
          sin 50
          R = 2.05cm
20. The position of two towns A and B on the earth surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively
    1. Find the difference in longitude between town A and town B (2m
       = 49 + 131
       180= )
    2. Given that the radius of the earth is 6370km calculate the distance between town A and B (4mks)
       = 16196 km
    3. Another town C is 840km east of town B and on the same latitude as towns A and B. find the longitude of town C (4mks)
       = 121.66ºW
21. The distance sm from a fixed point O, covered by a particle after ts is given by the equation
    S=t³-6t²+9t+5
    1. Calculate the gradient to the curve at t=0.5s (3mks)
       ds/dt = 3t² - 12t + 9
       = 3(1/4) - (12)1/2 + 9
       = 3/4 - 6 + 9
       = 15/4
    2. Determine the values of s at the maximum and minimum turning points of the curve. (4mks)
       
    3. On the space provided sketch the curve of s=t³-6t²+9t+5 (3mks)
       intercepts at t = 0 at s = s
       => (0,5)   min (3,0)  max(1,9)
       
22. In the figure below OQ =a and OB=b. m is the midpoint of OA and AN:NB =2:1
    
    
    1. Express in terms of a and b
       1. BA (1mk)
          b + a
       2. BN (1mk)
          1/3b + 1/3a
       3. ON (2mks)
          a + 2/3AB = 1/3a + 2/3b
    2. Given that BX=hBM and OX=KON determine the values of h and k (6mks)
       k = 3/2h
23.                            
    1. Complete the table below, giving the values correct to 1 d.p (2mks)
       

       		X0	0	40	80	120	160	200	240
       2sin (x+20)0		0.7	1		2.0		0.0		-2.0
       √3 cos x		1.7	1.3			-0.9		-1.6

    2. On the grid provided, using the same scale and axes, draw the graphs of y=2sin (x+20)º and y=√3 cos x for 0≤x≤ 240º (5mks)
    3. Use the graphs drawn in (b) above to determine:
       The values of x for which 2 sin (x+20)º =√3 cos x (2mks)
       The difference in the amplitudes of y=2sin(x+20) and y=√3 cox x (1mk)
       
24. The probabilities that a husband and wife will be a live 25 years from now are 0.7 and 0.9 respectively. Find the probability that in 25 years time;
    1. Both will be a live (2mks)
       0.7 x 0.9 = 0.63
    2. Neither will be a live (3mks)
       0.3 x 0.37 = 0.111
    3. One will be a live (2mks)
       (0.7 x 0.37) + (0.9 x 0.3) = 0.529
    4. At least one will be a live (3mks)
       0.529 x 0.63 = 0.33327