Mathematics Paper 1 Questions and Answers - BSJE Mock Exams 2023

• This paper contains two sections: Section I and Section II. Attempt all the questions in section I and strictly any five questions from Section 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.
• Answer the questions in English

                                                                               SECTION I:  (50 MARKS)

                                                                Attempt all questions in the spaces provided

1. An irregular 6 sided polygon has two of its interior angles equal to 2x each, three angles equal to x each and one side equal to 20°
   Calculate the value of x              (3marks)
2. The diagonals of a parallelogram are 20cm and 28.8cm. The acute angle between the diagonals is 62°. Calculate the area of the parallelogram.           (3marks)
3. A mobile phone seller gets a commission of shs 250 on every mobile phone that he sells. In a given month he got 33,000 shillings.
   1. How many phones did he sell that month                               (1mark)
   2. If this commission is 2% what is the sale price of each mobile phone.               (2marks)
4. Given that Sin x = ³/₄, without using tables or calculators, find.   
   1. Cos x^o                             (2marks)
   2. Tan (90-x)^o                     (1mark)
5. Solve the simultaneous equation  (4marks)
   2log y = log2 + log t
         2^y = 4^t                      
6. Evaluate  without using tables or a calculator   (3marks)
   100^-1.5 x 32^0.2            
7. The angle subtended by the major arc at the center of the circle O is twice the angle substended by the minor arc at the center. If the radius of the circle is 3.5cm. Find the length of the minor arc.                                                             (3marks)
8. Two trains T₁ and T₂ travelling in the opposite directions on a parallel tracks are just beginning to pass one another. Train T₁ is 72m long and is travelling at 108km/hr. Train T₂ is 78m long and is travelling at 72km/hr. Find the time in seconds the two trains take to completely pass one another                                                                    (3marks)
9. The angle of depression of a chick from the hawk on top of  a vertical tree 8m tall is 28°.  On seeing the hawk, the chick moves directly towards the base of the tree to a point P such that the angle of elevation of the hawk from P is 32^o. Calculate the distance moved by the chick.           (4marks)
10. Find a 2 x 2 matrix m such that                  (3marks)
    
11. Given that a = 1.2, b = 0.02 and c = 0.2 Express ac + b in the form m/n where m and n are integers              (3mark)
12. The line passing through the point P(−1,3w) and Q(w,3) is parallel to another line whose equation is 2y − 3x = 9 .
    Write down the co-ordinates of P and Q     (3marks)
13. A certain volume of solution has a mass of 2.2kg with density of 0.8g/cm³. Calculate the volume of the solution in liters   (3marks)         
14. The graph below shows frequency densities for the masses of same 200 students selected from a class. Use it to answer the questions that follows. 
                      
    1. Complete the frequency distribution table below.                                           (2 marks)
       

       Mass in kg
       frequency

    2. State the modal frequency                                     (1mark)
15. Calculate the area bounded  by the curve y = x², the line y= −1 and the lines x = 0 and x = 3 using trapezoidal rule with 7 ordinates  (correct to 3 d p)                                (3marks)
16. Mutai imports rice from United States at the initial cost of 500 US dollars per tonnes. He then pays 20% of this amount as shopping costs and 10% of the same amount as custom duty. When the rice reaches Mombasa he has to pay 6% of the initial cost to transport it to Nairobi. Given that on the day of this transaction the exchange rate was 1 us dollar = kshs 76.60. calculate the total cost of importing one tonne of rice up to Nairobi in Kenya shillings.(3mks)                     
    
                                                                                 SECTION II (50 MARKS)
                                                                 Answer only 5 questions from this section.
17.  
    1. On the Cartesian plane given below, draw the quadrilateral ABCD with vertices A( 6,6), B(2,2) C(4,−6) and D(8,0).             (1mrk)
       
       Draw the image A¹B¹C¹D¹ of ABCD under an enlargement scale factor ½, center origin. State the coordinates of A¹B¹C¹D¹                                                            ( 3mrks)
    2. Describe  the transformation that maps A¹B¹C¹D¹ onto the given image A¹¹B¹¹C¹¹D¹¹              (2mark)
    3. Rotate A¹¹B¹¹C¹¹D¹¹ about center (−2,−1) through a positive quarter turn to get A¹¹¹B¹¹¹C¹¹¹ D¹¹¹. State the co-ordinates of A¹¹¹B1¹¹C¹¹¹D¹¹¹                             (3marks)
    4. State a pair of quadrilaterals that are oppositely congruent.    (1mark )
18. On a certain day, Mwema bought plates worth Kshs 1200. On another day, Mrs. Mwema spent the same amount of money but bought the plates at a discount of 20% per plate.
    1. If Mwema bought a plate at Kshs x, write down a simplified expression for the total number of plates bought by the two people          (3marks)
    2. If Mrs. Mwema bought 6 plates more that her husband, find how much each spent on a plate.                                 (5marks)
    3. Find the total number of plates bought by the family               (2marks)
19.  Using a ruler and a pair of compasses only, construct triangle ABC such that  lines AB=5.5cm, BC=4.8cm and AC =6.8cm, Construct circle passing  through vertices A, B and C                                                (6 marks)
    1. Measure the radius of the circle                        (1mark)
    2. Measure the angle substended at the center of the circle by chord AC (1mark)
    3. Hence calculate the area of the triangle   AOC                         (2marks)
20. A particle P moves in a straight line such that t seconds after passing a fixed point  Q , its velocity  is given by the equation
    2t² − 10t + 12
    Find
    1. The values of t when P is instantaneously at rest.                             (2marks)
    2. An expression for the distance moved by P after t seconds.                         (2marks)
    3. The total distance travelled by P in the first 3 seconds after passing point Q                                     (2marks)
    4. The distance of P from Q when acceleration is zero.                      (4marks)
21. The figure below shows a model of a pillar to be constructed at the Canterbury. The model consists of a circular base of diameter 21cm and a uniform hexagon stand of sides 6cm and height 20cm.
                                                                               
    1. Calculate the cross-sectional area of the hexagon to 2dp.                          (3marks)
    2. Calculate the total volume of the model to 2dp.                      (3marks)
    3. If the height of the red pillar is 52m and the constructor uses two bags of cement for every 500m³ of the construction. Calculate the least number of bags of cement required                      (4marks)
22. The diagram below shows a triangle OPQ in which QN: NP=1:2, OT: TN=3:2 and M is the mid-point of OQ.
                                                                   
    1. Given that OP = P and OQ = q. Express the following vectors inters of p and q 
       1. PQ              (1mark)
       2. ON               (2marks)
       3. PT                 (2marks)
       4. PM               (1mark)
    2.  
       1. Show that points P,T and M are  collinear                       (3marks)
       2. Determine the ratio MT ; TP                    ( 1mark)
23. The diagram below shows a triangle ABC Circumscribed in a circle with AB = 12cm BC = 12cm, BC = 15cm and AC = 14cm
                                                                       
    Calculate to 4 significant figures 
    1. The angle ACB             (3marks)
    2. The radius of the circle             (3marks)
    3. The area  of the shaded region                           (4marks)
24. If Nick gives a quarter of the money he owns to Tom, Tom will have twice as    much as Nick. If Tom gives q shillings to Nick, then Nick will have thrice as much as Tom. Taking the initial amount owned by Nick and Tom to be x and y respectively;
    1. Express y and q in terms of x            (7marks)
    2. Given that Nick’s initial amount was Kshs 40,000. calculate the value of q               (1mark)
    3. The initial amount owned by Tom           (2marks)                             

                                                                           MARKING SCHEME 

S/NO
1.	Sum of interior angles = (6 − 2)180 = 720°                               2(2x) + 3(x) + 20 = 720°                                          4x+3x+20 = 720                                                      7x = 700                                                        x = 100	B1  M1      A1
		03
2.	A = ½ x 14.4 x 10 x 10 x Sin 62°                                                         63.57cm2                      63.57 x 4                      254.28cm2	M1   M1 A1
		03
3.	Let No of phones be x   250 x = 33000   X = 33000           250    X = 132 phone   2% = 250 Kshs   1% = 250              2   100% = 250 x100                  2 Price of 1 mobile phone = 12,500 Kshs	B1         M1   A1
		03
4.	y = √7 Cos x = 7              4 Tan(90 − x) =  7	B1   B1   B1
		03
5.	y2 = y      y2 − y = 0      y (y−1) = 0      y = 0 or 1      when y = 0, 2t = 0      t = 0      and y = 1, 2t = 1       t = ½	B1     M1   A1       B1
		04
6.	100-3/2 x 321/5  (102)−3/2 x (25)1/5   10-3  x 2   0.002	M1   M1   A1
		03
7.	2x + x = 360°  3x = 360     x =120°  Arc length = θ    x 2πr                     360  = 120 x 22/7 x 2 x 3.5     360  = 7.33cm	B1     M1   A1
		03
8.	Relative speed = 108 + 72 = 180km/hr?  Distance = 78 +72 = 150km  T=   150 x 60 x 60          180 x 1000          = 3 seconds	B1   M1   A1
		03
9.	Tan 28 = 8/x                  x = 15.046m                  Tan 32 = 8/y                  y = 12.803                  Distance = 15.046 − 12.803                      = 2.243m	M1   M1   M1 A1
		04
10.		M1             M 1 Use of inverse        A1
		03
11.	ac + b   1.2(0.2) + 0.02   = 0.26   =  26      100   = 13      50	M1   A1     B1
		03
12.	2y = 3x +9    y = 3/2x+ 9/2   m1 = m2= 3/2   3 − 3w  = 3   W + 1      2  6 − 6w = 3w + 3  w =1/3  P(−1,1) and Q(1/3 ,3)	B1   M1           A1
		03
13.	Density = 0.8g/cm3   = 0.8   ÷       1            1000     1000,000   = 0.8  x 1,000,000     1000   = 800kg/m3   r = 2.2 x 1000 litres        800      2.75 litres	B1 M1   A1
		03
14.	f.d = f  x k         c.s   1st bar 2 = f  x1, f = 20                    10     2end bar 14 = f   x 1, f = 70                          5     3rd bar 5 = f    x 1,f = 30                    10     4th bar 10 = f  x1, −f = 50                       5   5th bar 1 = f  x1, f =1 0                    10    Mass in kg  41-50  51-55  56-65  66-70 71-80              Frequency   20  70  50  50 10 Modal frequency = 70	B1   B1   B1
		03
15.	H = 3 − 0  = 0.5            6    X  0  0.5   1   1.5   2   2.5   3   Y = x2  0  0.25   1   2.25   4   6.25   9   New y  1 1.25   2   3.25   5   7.25   10                    Area = 0.5   (1 +10) +2(1.25 + 2 + 3.25 + 5 + 7.25)                2           = 0.25(11 +3.75)           = 12.125sq units	B1 all values correct       M1       A1
		03
16.	Shipping cost + custom duty + transport   = 20 +10 + 5 = 35%      135  x  500      100  = 675 Us dollars.    1US. Dollar = 76.60kshs    675 dollars =?    675 x 76.60           1  = 51,705 kshs	M1               M1   A1
		03
	SECTION II
17.
18	Plates bought by Mwema  = 1200                                                 x Plates bought by Mrs. Mwema = 1200                                                       0.8x Total 1200 + 1200            x         0.8x 1200 − 1200 = 6   0.8x      x  1200x − 1200(0.8x) = 6(0.8x2) 1200x − 960x = 4.8x2 240x = 4.8x2 4.8x2 − 240x = 0 x(4.8x − 240) = 0 x = 0 or 4.8x − 240 = 0 x = 0 or 50   x = 50 Mwema – shs 50 per plate Mrs. Mwema – kshs 40 per plate 1200   + 1200     50        40 = 24 + 30 = 54	B1   B1   B1     M1      M1    M1     A1     B1    M1   A1
		10
19.
20.	2t2 − 10t +12 = 0 t2 − 5t + 6 = 0  t2 + 3t − 2t + 6 = 0 t(t − 3) −2(t − 3) = 0 (t - 2)(t - 3) = 0  t = 2 or 3 seconds S= ᶘr dt S= ᶘ2t2 − 10t +12 S= 2/3t3 −5t2 +12t + c When t = 0, s = 0, c = 0 S = 2/3t3 – st2 +12t When t = 3, S = 2/3(3)3 − 5(3)2 +12(3)    = 18 − 45 +36    = 9m A = dr/dt A = 4t − 10 4t − 10 = 0 4t = 10 t = 2.5seconds S = 2/3(2.5)3 −5(2.5)2 +12(2.5)    = 9.167m	M1         A1         M1         M1   A1     M1     A1 M1   A1
		10
21.	Angle at the center = 360 = 60                                     6 Cross sectional area  = ½ x 6 x 6 Sin 60 x 6    = 93.53 cm2   volume of model Hexagonal = 93.53 x 20                   = 1870.60cm3   Circular part = 3.142 x 10.52 x 6                      = 2078.43 Total volume = 1870.60 + 2078.43                      = 3949.03cm3 L.S.F = 5200 = 200                26 r.s.f =  200 = 8000 000              1           1 Volume of the pillar = 8000,000 x 3949.03                                             1000000                                     = 31592.24m3 If 500m3 = 2bags 31592.24 =? 31592.24 x 2                               500                  = 126.37bags Least No of bags = 127 bags	B1   M1     A1   M1         M1       A1           M1   M1       M1      A1
		10
22.	Q = −p + q or q – p ON = q + 1/3(p - q)       = 1/3p + 2/3q PT = − p + 3/5(1/3p + 2/3q)      = −4/5 (p + 2/5q) Pm = −p + 1/2q   Tm = 3/5(−1/3p − 2/3q) + ½2q       = − 1/5p − 2/5q + ½q       = − 1/5p  + 1/10q PT = − 4/5p + 2/5q −1/5p + 1/10q = K(-4/5p + 2/5q)   K = ¼ Tm = ¼ PT Tm //PT, T is common point hence PT and m are collinear Ratio of mT;Tp = 1:4	B1   M1   A1   M1   A1   B1       M1         M1     A1 B1
		10
23.	a2 + b2  − 2ab Cos c = C2 152 + 142 − 2 x 14 x 15 Cos c = 122  Cos C = 152 +142 − 122 = 0.6595                           2 x14 x15 < C = Cos-1 0.6595 < C = 48.74o      12           = 2r Sin 48.74o r =    6             Sin 48.74 r = 7.982 Area of sector = 97.48 x 3.142  x7.9822                                  360               = 54.21cm2 Area of triangle ABC = ½  x 7.9822 Sin 97.48 = 31.59cm2 Area of shaded region 54.21 − 31.59 22.62cm2	M1         M1   A1   M1     M1   A1 M1         M1     M1   A1
		10
24.	Nicks money = x Tom’s initial money = y When Nick  gives  ¼ x to Tom than  Tom has y + ¼x Nick ¼x y + ¼x = 2(3/4x) y = 6/4x  − ¼x y = 5/4x Tom gives q shillings to Nick then Tom’s money = y − q Nick = x + q (x + q) =3(y − q) x + q = 3y − 3q) x + q = 3y − 3q but y = 5/4x 4q = 3(5/4x) − x q = 11/16x   Value or q                  = 11/16 x 40, 000                  = 27,500 Kshs  Initial amount owned by Tom y = 5/4 x 40 000 = 50,000 Kshs	B1   M1 A1          B1   M1   M1 A1   B1 M1 A1
		10