Mathematics Paper 2 Form 3 End Term 2 Exams 2021 with Marking Schemes
SECTION I (50MARKS)
Answer all questions in this section
- Evaluate using logarithms. [4 Marks]
- A rectangular card measures 5.3 cm by 2.5cm.Find
- The absolute Error in the area of the card. [2 Marks]
- The Percentage Error in the area of the card [2 Marks]
- The length of a room is 4m longer than its width. Find the length of the room if its area is 32m2. [3 Marks
- If 20 Men can lay 36m of a pipe in 8 hours. How long would 25 Men take to lay the next 54m of the pipe? [2 Marks]
- Expand (2 + x)5 in ascending powers of x up to the term in x3. Hence, approximate the value of (2.03)5 to 4s.f. (4 marks)
- Simplify by rationalizing the denominator; [2 Marks]
3
2√3 − √2 - A scientific calculator is marked at sh. 1560. Under hire purchase it is available for a downpayment of sh. 200 and six monthly instalments of sh. 250 each. Calculate;
- The Hire purchase price. [2 Marks]
- The extra amount paid out over the cash price. [1 Mark]
- Solve the equation; [3 Marks]
log (2x − 10) − 2 log 8 = 2 + log (9 − 2x) - The Equation of a circle is given by x2 + y2 − 6x + 4y − 3 = 0 . Determine the center and the radius of the circle. [3 Marks]
- Make x the subject of the formula in the equation. [3 marks]
- In the figure below, BT is a tangent to the circle to the circle at B. AXCT and BXD are straight lines. AX=6cm, CT=8cm, BX=4.8cm and XD=5cm.
Find the length of;- XC [2 Marks]
- BT [2 Marks]
- Find the value of x if the matrix is a singular matrix. [3 Marks]
- The first term of an arithmetic sequence is −7 and the common difference is 4.
- List the first 6 terms of the sequence [2 Marks]
- Determine the sum of the first 30 terms of the sequence [2 Marks]
- The coordinates of points A and B are (2,5) and (8, −7) respectively. Find the
- Coordinates of M Which Divides AB in the Ratio 1:2 [2 Marks]
- Magnitude of AB [2 Marks]
- Tap A Fills a tank in 6 hours, tap B fills it in 8 hours and tap C empties it in 10 hours.Starting with an empty tank and all the three taps are opened at the same time, how long will it take to fill the tank. [3 Marks]
- Grade X of Tobacco Costs Sh.81.50 per Kg and grade Y cost sh 109 per Kilogram. In what ratio must the two grades be mixed in order to make a profit of 20% when the mixture sells at sh. 112.80 per kg. [3 Marks]
SECTION II: (50MARKS)
Answer any 5 questions from this section
- The figure below shows triangle OAB in which M divides OA in the ratio 2: 3 and N divides OB .in the ratio 4:1 AN and BM intersect at X.
- Given that OA = a and OB = b, express in terms of a and b: (4mks)
- AN
- BM
- If AX = sAN and BX = tBM, where s and t are constants, write two expressions for OX in terms of.a, b s and t. Find the value of s and t. Hence write OX in terms of a and b (6mks)
- Given that OA = a and OB = b, express in terms of a and b: (4mks)
- Kamau, Njoroge and Kariuki are practicing archery. The probability for Kamau hitting the target is 2/5 , that of Njoroge hitting the target is 1/4 and that of Kariuki hitting the target is 3/7.
Find the probability that in one attempt;- Only one hits the target (2mks)
- All three hit the target (2mks)
- None of them hits the target (2mks)
- Two hit the target (2mks)
- At least one hits the target (2mks)
-
- A matrix T is given by . Find T-1 [2 Marks]
- Wanjiku bought 20 bags of maize and 25 bags of beans at a total cost of sh. 77,000. If she had bought 30 bags of maize and 20 bags of beans, she would have spent sh. 7,000 more.
- Form a matrix equation from this information. [1 Mark]
- Determine the cost of a bag of maize and a bag of beans. [3 Marks
- She sold all the maize and beans at a profit of 10% on a bag of maize and 12½ % on a bag of beans. Calculate the total percentage profit. [4 Marks]
- At the beginning of the year 2000, Kanyora bought two houses, one in Thika and the other in Nakuru each at 1,240,000. The value of the house in Thika appreciated at a rate of 12% p.a.
- Calculate the value of the house in Thika after 9 years to the nearest shilling. [2 Marks
- After n years, the value of the house in Thika was 2,741,245 while the value of the house in Nakuru was 2,917,231.
- Find n [4 Marks]
- Find the annual rate of appreciation of the house in Nakuru. [4 Marks]
- The table below shows income tax rates.
Taxable income
K£ per monthRate in shs. per K£ 1 - 325 2 326 - 650 3 651 - 975 4 976 - 1300 5 1301 - 1625 6 over 1626 7 - Calculate
- Wafula’s taxable income in k£ per month. [2 Marks]
- Gross tax. [3 Marks]
- Net Tax [2 Marks]
- His net income per month has the following deductions
Health insurance fund – sh. 150
Loan interest – sh. 200
Service charge – sh. 200
Sacco loan – sh. 2,500
Calculate his net income per month. [3 Marks]
- Calculate
-
- P varies jointly as Q and the square of R. P = 18 when Q = 9 and R = 15. Find R when P=32 and Q=81. [5 Marks]
- A varies Directly as B and inversely as the square root of C. Find the percentage change in A When B is decreased by 10% and C increased by 21%. [5 Marks]
-
- The first term of an arithmetic progression is 2. The sum of the first 8 terms of the AP is 240.
- Find the common difference of the AP. [2 Marks]
- Given that the sum of the first n terms of the AP is 1,560. Find n [2 Marks]
- The 3rd, 5th and 8th terms of another AP from the first three terms of a G.P. If the common difference of the AP is 3. Find.
- The first term of G. P [4 Marks]
- The sum of the first 9 terms of the G.P to 4 s.f. [2 Marks]
- The first term of an arithmetic progression is 2. The sum of the first 8 terms of the AP is 240.
-
- Complete the table below for the function Y=2x2 + 4x − 3 [2 Marks]
x −4 −3 −2 −1 0 1 2 2x2 32 1 0 8 4x −8 −12 −8 4 8 −3 −3 −3 −3 −3 −3 −3 −3 y 21 −3 - On the grid provided, draw the graph of the function y=2x2 + 4x − 3for −4 ≤ x ≤2 [3 Marks]
- Use your graph to solve the roots of the quadratic equations.
- 2x2 + x − 5 = 0 [2 Marks]
- 2x2+3x − 2 = 0 [2 Marks]
- x2 + 4x − 3 = 0 (1 mark)
- Complete the table below for the function Y=2x2 + 4x − 3 [2 Marks]
Marking Scheme
-
-
-
Minimum Actual Maximum 5.25 5.3 5.35 2.45 2.5 2.55
Max. Area = 5.35 × 2.55
= 13.46 cm2
Min. Area = 5.25 × 2.45
= 12.8625 cm2
Absolute Error = max − min
2
= 13.6425 − 12.8025 = 0.78 ✓1
2 - % error = Absolute error × 100 ✓1
Actual area
= 0.78 × 100
5.3 × 2.5
= 5.8868 % ✓1
-
-
x(x+4) = 32
x2 + 4x − 32 = 0
x2 + 8x − 4x − 32 = 0
x (x+4) − 4(x+8) = 0
(x − 4)(x+8) = 0
x = 4 or x = -8
∴ x =4 since length can't be negative
length = x + 4 = 4+4 = 8m - Men Length Hours
20 36 8
25 54 ?
= 25/20 × 54/36 × 8 ✓1
= 15 hours ✓1 - (2+x)5
Co-efficients 1 5 10 10 5 1 Expansion 25x0 24x1 23x2 22x3 21x4 20x5 Combined 32 80x 80x2 40x3 10x4 x5
(2+x)5 = 2.035
2+x = 2.03
x = 2.03 − 2
x = 0.03
∴ 2.035 = 32 + 80 (0.03) + 80 (0.03)2 + 40 (0.03)3
= 32 + 2.4 + 0.072 + 0.00108
= 34.47308 ≈ 34.47 ✓1 -
-
- = 200 + (2500 × 6)
= sh. 1700 - 1700 − 1560
= sh. 140
- = 200 + (2500 × 6)
- log (2x − 10) = log (100 × (9 − 2x) )
2x − 10 = 900 − 200x
64 1
2x − 10 = 57600 − 12800x
12802x = 57610
x = 57610
12802
= 4.5001
≈ 4.5 - (x2 − 6x + ___) + (y2 + 4y + ___) = 3 + ___+ ___
x2 − 6x + 6/2 + y2 + 4y + 4/2 = 3 + (6/2) + (4/2)2
(x − 3)2 + (y+2)2 = 16
= centre (3, −2)
= Radius = √16
= 4 units -
-
- 6 (XC) = 4.8 × 5
XC = 4.8 × 5
6
= 4 cm - BT2 = AT.CT
= 18 × 8
BT = √144
= 12 cm
- 6 (XC) = 4.8 × 5
- For singular matrix det = 0
∴ {x(x − 3)} − (4 × 1) = 0
x2 − 3x − 4 = 0
x2 + x − 4x − 4 = 0
x(x+1) − 4(x+1) = 0
(x−4) (x+1) = 0
x = −1 or 4 -
- 1st term = −7
2nd term = −7 + 4 = −3
3rd term = −3 + 4 = 1
4th term = 1 + 4 = 5
5th term = 5 + 4 = 9
6th term = 9 + 4 = 13 - S30 = 30/2{2(−7) + (30 − 1)4}
= 15 {−14+116}
= 1530
- 1st term = −7
-
- Tap A ⇒ 6 hours
in 1hr = 1/6 of the tank
Tap B ⇒ 8 hours
in 1hr = 1/8 of the tank
in 1hr both = 1/6 + 1/8
Tap C ⇒ 10 hours to empty
in 1hr = 1/10
All three in hour = 7 − 1 = 35 − 12 = 23
24 10 120 120
23/20 = 1 hr
1 = ?
1 × 1 × 120/23
= 5.22 hours
= 5 hrs 13 min - Grade X Grade Y Mixture
B.P sh 81.50 sh. 109
Ratio x : y
Total 81.50x 109y 81.50x + 109y
120/100(81.50x + 109y) = 112.80
x + y
97.8x + 130.8y = 112.80x + 112.80y
97.8x − 112.80x = 112.80y − 130.8y
−15x = −18y
x = 18 ⇒ x:y = 6:5
y 15 -
-
- AN = AO + ON
= −a + 4/5b - BM = BO + OM
= −b + 2/5a
- AN = AO + ON
- OX = OA + AX
= a + sAN
= a + s (4/5b − a)
= a − sa + 4/5sb
= a(1 − s) + 4/5sb
Also:
OX = OB + BX
= b + t BM
= b + t(2/5a − b)
= b − tb +2/5ta
= b (1 − t) + 2/5ta
a(1 − s) + 4/5sb = b (1 − t) + 2/5ta
1 − s = 2/5ta
t = 5/2 − 5/2s ------- (i)
4/5s = (1 − t) --------(ii)
4/5s = 1 − (5/2 − 5/2s)
4/5s = 1 − 5/2 + 5/2s
4/5s = − 3/2 + 5/2s
3/2 = 17/10s
s = 15/17
t = 5/2 − 5/2s
t = 5/2 − 5/2 × 15/17
t = 5/17
∴ OX = b(1 − 5/17) + (2/5 × 5/17)a
= 12/17b + 2/17a
-
-
- (2/5 × 3/4 × 4/7) + (3/5 × 1/4 × 4/7) + (3/5 × 3/4 × 3/7)
= 6/35 + 3/35 + 27/140
= 9/20 - P(HHH) = 2/5 × 1/4 × 3/7
= 3/70 - P('H'H'H) = 3/5 × 3/4 × 4/7
= 9/35 - (2/5 × 1/4 × 4/7) + (2/5 × 3/4 × 3/7) + (3/5 + 1/4 + 3/7)
= 2/35 + 9/70 + 9/140 = 1/4 - P(1 or 2 or 3) = 1 − (P(none hits))
= 1 − 9/35
= 26/35
- (2/5 × 3/4 × 4/7) + (3/5 × 1/4 × 4/7) + (3/5 × 3/4 × 3/7)
-
-
-
- 4 5 15400
20m + 25b = 77000
30m + 20b = 84000
6 4 16800 -
- 4 5 15400
- B.P = sh. 77,000
Maize
S.P = 1600 × 110
100
= sh. 1760
Beans
S.P = 1800 × 112.5
100
S.P = (20 × 1760) + (25 × 2025)
= sh. 85, 825
Profit = sh. 85825 − 77,000
= sh. 8,825
% Profit = 8,825 × 100 %
77,000
= 0.115 × 100 %
= 11.5 %
-
-
- A = P (1 + r/100)n
= 1,240,000 (1 + 12/100)9
= 1,240,000 × 2.773
= sh. 3,438,618 -
- A = P (1 + r/100)n
2,741,245 = 1,240,000 (1 + 12/100)n
2.211 1
2,741,245 = 1,240,000
1,240,00 1,240,000
2.211 = (1.12)n
log 2.211 = log 1.12 n
log 2.211 = n log 1.12
log 1.12 log 1.12
= 0.3445 = n
0.0492
n= 7 years - 2,917,231 = 1,2400,000 (1 + r/100)7
2,917,231 = 1,2400,000(1 + r/100)7
1,2400,000 1,2400,000
2.3526 = (1 + r/100)7
7√(2.3526) = 1 + r/100
1.13 − 1 = r/100
100 × 0.13 = r/100 × 100
r = 13 % p.a
- A = P (1 + r/100)n
- A = P (1 + r/100)n
-
-
- T.I = B.S + T.A
= 30,500 + 6000 + 4000 + 3000
20
= K£ 2175 - Gross tax
325 × 2 = sh. 650
325 × 3 = sh. 975
325 × 4 = sh. 1300
325 × 5 = sh. 1625
325 × 6= sh. 1950
550 × 7 = sh. 3850
Gross tax = Ksh. 10,350 - net tax = Gross tax − relief
= 10350 − 1056
= Ksh. 9294
- T.I = B.S + T.A
- Total dedcutions = 9294 + 150 + (200 × 2) + 2500
= Shs. 12,344
Net Income = Sh. 43500 − 12,344
= Shs. 31,156
-
-
- P ∝ QR2
P = kQR2
18 = k × 9 × (15)2
18 = 225 × 9 × k
18 = 2025k
2025 2025
k = 2/225
∴ P = 2/225 QR2
32 = 2/225 × 81 × R2
225 × 32 = 162R2 × 225
162 225 162
444/9 = R2
R = √444/9
= 62/3 - A ∝ B/√C
A = kB/√C
A = k × 0.9B
√(1.21C)
A = 0.9kB
1.1√C
= 0.818kB
√C
B = 100%
= 90%
= 90 × 13 = 0.9B
100
C = 100%
= 121%
121 × C = 1.21C
100
Δ in A: 0.818kB − kB
√C √C
= k B(0.818 − 1)
√C
% Δ in A = k B/√C(0.818 − 1) × 100 %
kB/√C
= − 0.1818 × 100%
A decreases by 18.18%
- P ∝ QR2
-
-
- S8 = n/2 (2a + (n−1)d)
240 = 8/2 ((2×2) + (8 −1)d)
240 = 4 (4+7d)
240 = 16 + 28d
28d = 224
28 28
d = 8 - 2 × 1560 = n/2 ((2×2) + (n − 1)8) × 2
3120 = n (4+8n − 8)
3120 = 4n + 8n2 − 8n
4 4 4 4
780 = n + 2n2 − 2n
780 = 2n2 − n
2n2 − n − 780 = 0
n = −(−1) ± √(1+ 4×2×780)
4
= 80/2
n = 40 times
- S8 = n/2 (2a + (n−1)d)
-
- a+2d, a+4d, a+2d ....G.P
a+4d = a+7d
a+2d a+4d
a2 + 8ad + 16d2 = a2 + 9ad +14d2
16d2 − 14d2 = 9ad − 8ad
2d2 = ad
d d
a = 2d
but d=3
∴ a = 6 (first term A.P) - C.R = 18/12 = 3/2
Sn = a (rn − 1
r − 1
S9 = 12(1.59 − 1)
1.5 − 1
= 898.6
- a+2d, a+4d, a+2d ....G.P
-
-
-
x −4 −3 −2 −1 0 1 2 2x2 32 18 8 1 0 2 8 4x −8 −12 −8 −4 0 4 8 −3 −3 −3 −3 −3 −3 −3 −3 y 21 3 −3 −6 −3 3 13 -
-
- y = 2x2 + 4x −3
−0 = 2x2 + x −5
y = 3x + 2
x 0 1 y 2 5 - y = 2x2 + 4x −3
−0 = 2x2 + 3x −2
y = x − 1
x 0 3 y −1 2 - x = 0.6 or -2.6 ± 0.1
- y = 2x2 + 4x −3
-
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