Mathematics Paper 2 Questions and Answers - Mangu High School Mock Exams 2023

• Answer All questions in Section I  and only Five questions  from section II
• All answers and working must be written on the question paper in the spaces provided below each
• Show all the steps in your calculations giving answers at each stage in the spaces provided below each
• Marks may be given for correct working even if the answer is wrong
• Candidates should answer questions in English.

1.  Factorise x² – y², hence evaluate 3282² − 3272²               (3mks)
2. Find cos x − Sin x, if tan x = ¾  and 90°≤ x ≤360º                (3mks)
3. Expand  [1-2x]⁶   up to the fourth term. Hence use your expansion to evaluate (1.02) ⁶  to four decimal places. (4mks)
4. The average of the first and fourth terms of a GP is 140. Given that the first term is 64. Find the common ratio.                (3mks)
5. Make b the subject of the formula.                (3mks)
   
   
   
   
6. Two variables P and Q are such that P varies partly as Q and partly as the square root of Q. Determine the equation connecting P and Q. When Q=16, P=500 and when Q = 25, P = 800 (4mks)
7. Calculate the interest on sh 10,000 invested for 1 ½ years at 12 % p.a. Compounded semi-annually.                (3 mks)
8. Given that x=2i+j-2k, y= -3i+4j-k and z =5i + 3j+2k and that P= 3x-y+2z, find the magnitude of vector p to 3 significant figure         (4mks)
9. Eighteen labourers dig a ditch 80m long in 5 days. How long will it take 24 labourers to dig a ditch 64 m long? (3mks).
10. The expression 1+ ^x/₂ is taken as an approximation for √(1+x) Find the percentage error in doing so if x = 0.44                (3mks)
11. The matrices are  and  such that AB = A + B Find a, b, and c.             (3mks)
12. Simplify (3mks)
    
    
           2x² – x−1
              x² – 1  
     
13. On map of scale 1:25000 a forest has an area of 20cm². What is the actual area in Km²                 (3mks)
14. In the figure below, DC = 6cm, AB = 5cm. Determine BC if DC is a  tangent. (3mks).
    
    
    
    
15. Evaluate without using logarithm tables.           (3mks)
    
     3 log ₁₀ 2 + log ₁₀ 750 – log ₁₀ 6             
16. A bag contains 10 balls of which 3 are  red, 5 are white and 2 green. Another bag contains 12 balls of which 4 are red, 3 are white and 5 are green. A bag is chosen at random and a ball picked at random from the bag. Find the probability that the ball so chosen is red.                   (4mks).
    SECTION II (50 MARKS)
    
17. Income tax is charged on annual income at the rates shown below
    

    Taxable Income K£	Rate (shs per K£)
    1 – 1500  1501 – 3000  3001 – 4500  4501 – 6000  6001 – 7500  7501 – 9000  9001 – 12000  Over 12000	2  3  5  7  9  10 12 13

     A certain headmaster earns a monthly salary of Ksh. 8570.. He is entitled to  tax relief of Kshs. 150 per month.

    1. How much tax does he pay in a year.                (6 mks)

    2. From the headmaster’s salary the following deductions are also made every month;

       • W.C.P.S 2% of gross salary

          
       • N.H.I.F Kshs. 1200
          
       • House rent, water and furniture charges Kshs. 246 per month.
         Calculate the headmaster’s net salary.                    (4 mks) 
18.  
    1.  
       1. Taking the radius of the earth, R = 6370 km and p = ²²/₇ calculate the shorter distance between the two cities P (60°N , 29°W) and  Q (60°N, 31°E) along the parallel of latitude.           (3mks)
       2. If it is 1200Hrs at P, what is the local time at Q.         (3mks)
    2. An aeroplane flew due South from a point A (60°N, 45°E) to a point B. The distance covered by the aeroplane was 800km. Determine the position of B.          (4mks).
19. Triangle  PQR whose vertices are p(2,2), Q(5,3) and R(4,1) is mapped onto triangle P'Q'R' by a transformation whose matrix is      
    1. On the grid draw PQR and P¹Q¹R¹.                  (4mks)
    2. The triangle P¹Q¹R¹ is mapped onto triangle P¹¹Q¹¹R¹¹ whose vertices are P¹¹(-2,-2), Q¹¹(-5,-3) and R¹¹ (-4,-1)           (2mks)
       1. Find the matrix of transformation which maps triangle P¹Q¹R¹ onto P¹¹Q¹¹R¹¹              (2mks)
       2. Draw the image P¹¹Q¹¹R¹¹  on the same grid and describe the transformation that maps PQR onto P¹¹Q¹¹R¹¹   (2mks)
    3. Find a single matrix of transformation which will map P¹Q¹R¹ on to P¹¹Q¹¹R¹¹ .(2mks) 
        
20.  
    1. Complete the table for y = Sin x + 2 Cos x.                                 (2mks)
       

       X	0	30	60	90	120	150	180	210	240	270	300
       SinX	0			1.0		0.5		-0.5			−0.87
       2CosY	2			0		−1.73		-1.73			1.0
       Y	2			1.0		−1.23		-2.23			0.13

        
    2. Draw the graph of y = Sin x + 2 cos x.                                               (3mks) 
    3. Solve sinx + 2 cos x = 0 using the graph.                                           (2mks) 
    4. Find the range of values of x for which y ≤ −0.5                                 (3mks). 
21. A bag contains 3 red, 5 white and 4 blue balls. Two balls are picked without replacement. Determine the probability of picking.
    1. 2 red balls                    (2mks) 
        
    2. Only one red ball                     (2mks) 
    3. At least a white ball                    (2mks) 
    4. Balls of same colour.                     (2mks) 
    5. Two white balls                     (2mks) 
22.  
    1. Draw the graph of the function                      (2mks)
       
       y = 10+3x – x² for –2<x <5
    2. Use of the trapezoidal rule with 5 stripes, find the area under the curve from x = −1 to x = 4                     (4mks) 
        
    3. Find the actual area under the curve from x = – 1 to x = 4.                                                                         (2mks)
        
    4. Find the percentage error introduced by the approximation.                                                                       (2mks)
23. The figure below is a cuboid ABCDEFGH such that AB = 8cm, BC  = 6cm and CF 5cm.
    
    Determine
    1. the length
       1. AC                    (2mks)
       2. AF                   (2mks)
    2. The angle AF makes with the plane ABCD. (3mks)
    3. The angle AEFB makes with the base ABCD. (3mks)
24. A manager wishes to hire two types of machine. He considers the following facts.
                                                                  Machine A              Machine B
    Floor space                                                 2m²                         3m²
    Number of men required to operate             4                              3
    He has a maximum of 24m² of floor space and a maximum of 36 men available. In addition he is not allowed to hire more machines of type B than of type A.
    1. If he hires x machines of type A and y machines of type B, write down all the inequalities that satisfy the above conditions. (3mks)
    2. Represent the inequalities on the grid and shade the unwanted region.                            (3mks)
    3. If the profit from machine A is sh. 4 per hour and that from using B is kshs8 per hour. What number of machines of each type should the manager choose to give the maximum profit?                            (4mks

MARKING SCHEME

1.	(x- y) ( x+y) ( 3282 – 3272) ( 3282 + 3272)       65540	M1 M2 A1
2.	Tan x = is positive 3rd quadrant Then sin x = -3/5      h =  √( 42 + 32) =  √ 25 = 5  Sin  x = -3                5   Cos x – sin x = - 4   − -3    =  -3                             5         5       5                                              =  -1                                                    5	B1 M1   A1	Identification of the hypotenuse  Cao  Accept  (−0.2)
3.	16 + 6(- ½  x ) + 15(- ½  x )2 + 20(- ½  x )3  = 1 – 3x + 15/4x2  - 5 x3 X = -0.04 1-3 (−0.04) +  15/4(−0.04)2  − 5/4( −0.04)3 = 1 + 0.12 + 0.006 + 0.000616 = 1.12616 = 1.1262	M1 M1 M1 A1	For ✓ simplification  For ✓ substitution of x
4.	a + ar3 = 140      2 64 + 64 r3 = 140      2 64 + 64 r3 = 280 64 → r3 = 280 – 64 64 r3 = 216 → r = √ 216                                   64 r =  3        2	M1 M1 A1
5.	a2 =   b2d2              b2 – d a2b2 - a2d = b2d2 a2b2 - b2d2 = a2d b2(a2 - d2) = a2d b2 =     a2d               a2 - d2 =  √( a2d )        (a2d2)	M1   M1   A1	✓ sq on both sides
6.	P = aQ + √Q P = 16a + 4b ( 500 = 16a + 4b) (800 = 25a + 5b) 2500 = 80a + 20b 3200 = 100a + 20b -700 = -20a 35 = a Then b = -15 Equation connecting P and Q p = 35Q – 15√Q	M1   M1 M1   A1	For ✓ equation For ✓ formation of simultaneous equations For ✓ values of both a and b
7.	1000 (1+ 6/100)3 1000 x 1.063 Ksh 11910.16 Ksh 11910	M1  A1
8.	4.562 x 0.38 = 1.73356  4 √1.73356      = 1.14745÷ 0.82                          = 1.3993                        = 1.4	M1 M1   A1
9.	18 x 64 x 5      24 x 80  6  x 64    8 x 16  3 days	M1 M1   A1	For ✓ simplification
10	True value = √ (1 + n )= 1.44 = 1.2 Approx. value  1 + n  = 1 +  44  = 1.22       2             2 = 1.22 – 1.2   = 0.02 1.2 x 100 =  1.2 = 1.67 %	M1 M1   A1
11.	3a + 0 = 3  + a    3b  + 0  = b 3a = 3 + a   →   a =  3                                   2 3b + 0 = b 2b = 0 B = 0 0 + 4c = 4 + c 3c = 4 C=  4         3	M1 M1   A1	For matrix equation   For forming of simultaneous equation   For values of a, b and c ( correct)
12.	2x2 – 2x  + x -1     ( x + 1 ) ( x – 1) 2x ( x – 1 ) + 1 ( x- 1)   ( x + 1 ) (n- 1 ) =   ( 2x + 1)        ( x + 1 ) = 2x + 1     x + 1	M1 M1   A1
13.	1    cm = 25000cm 2   1cm = 250m 3   1cm = 0.25 1cm2 = 0.0625 20cm2  = 20 x 0.0625              = 1.25/ cm2	M1 M1   A1
		3
14.	AB . BC = DC -2      5: BC = 36          BC =  36                      5             = 7.2 cm	M1 M1   A1
		3
15.	Log108 + Log10750 - Log106 Log10 Log10 ( 8 x 750)                              6 = Log101000 = 3	M1 M1   A1
		3
16.	=   1    +   3       6        20        =   20 + 18               120 = 38      =   19    120          60	M1   M1   M1   A1
		4
17.	Taxable income 115 x 8570 = 9855.50                           100 9855.50 x 12 p.a          20 = 5913.30                                               Tax 1 – 1500                        →     150 1501 – 3000                  →     225 3001 – 4500                  →     375 4501 – 5913.30             →    494.30                                              1244.30     -                                                  90.00  K £ 1154.30 pa . or  Ksh 1923.83 per month Total Decuctions   2      x  9855.5 100 = 197.11 ( wcps)               +      20.00    246.00               + Tax per month 1923.83                          2386.94 Net salary 9855.50 – 2386.94 Ksh 7468.65	M1   A1   M1   M1   M1   A1   M1   A1   M1   A1
		10
18	i)   θ   ( 2∏R cos θ)     360 =   60  x 2 x 22 x 6370 cos 60     360          7 = 1/3 x 22 x 910 x 0.5 = 3336.7 ii) Time (4 x 60) hrs                  60               4 hrs. Local time 1200 + 4                   = 1600hrs b)  θ  x 2πR = 800     360    =   θ     x 2 x πx 6370 = 800       360   θ =  800 x 360          2 x π x 6370    = 7.196° ∠ (60 – 7.196) = 52.80° ( 52.8°N 45°E)	M1   A1   B1   M1   M1   A1   B1
19.
20.	(a)  X 30  60  120  180  240  270   Sin X  0.5 0.87  0.87  0  -0.87  -1.0  2 cos X  1.73  1.0  -1.0  -2.0  -1.0  0   Y 2.23  1.87  -0.13  -2  -1.87 -1.0   (b)   c) x = 114 ± 3° and x = 294 ± 3° d) line thro y = − 1.5	B2  S1  P1 C1 B2 L1 B2	For all 6 values of y✓  B1 for at least 4 ✓ Appropriate scale use ✓ plotting  ✓ curve   Points identified and stated B1 only stated ✓line Points identified and stated  B1 only stated
21.	P ( RR) = 3/12 x 2/11             =1/22 P(IR) = RW or RB or WR or BR 15/132 + 12/132 + 15/132 + 12/132               = 9/22 p( At least white Ball ) = P(RW) + P(WR) + P(WW) + P ( WB) + P(B) 15/132 + 9/132 + 20/132 + 20/132 + 20/132 = 84/132  or  7/11 P(RR or WW or BB) = 6/ 132 + 20/132 + 12/132 = 19/66	B2   M1   A1   M1   A1   M1   A1   M1   A1	✓✓ prob, tree  Or equivalent 0.04545 Or equivalent 0.4091  Or equivalent 0.6364  Or equivalent 0.2424
22	X -2 -1 0 1 2 3 4 5 Y 0 6 10 12 12 10 6 0	B2       S1   P1   C1         M1   A1 M1 M1 M1 A1	8 values ✓   B1 at least 6 ✓   Appropriate scale use   ✓ Plotting   ✓ Curve
		10
23.	AC2 = 82 + 62 = 100 AC  = 10cm AF2  = 102 + 52  = 125   AF = 11.18cm Tan x = 5/11 = 0.5        x = 26.52° Tan x =  5/6  = 0.8333 x = 39.7°	M1 A1 M1 A1 B1 M1 A1 B1 M1 A1	Sketch   Sketch
		10
24.	(c) P profit function object we function P = 4x + 3y Max profit at point (6,4) P = 4(6) + 4,88)(4)    = 56 Hence he should here 6 medium of type A and 4 machine of type B	Inequalities x≥O y≥O 4x + 3y ≤ 36 4x + 3y = 24 y = n For  shading of x≥o and y ≥ o 2x + 3y ≤ 36 y ≤ n	B1   B1   B1   B1 - shading and line   B1 - shading and line drawn   B1 - for shading and line drawn   B1   B1   B1
			10.