A certain sum amounts to ₹1,800 in two years and to ₹2,160 in three years at compound interest, compounded annually. Find the rate of interest.
A.18%
B.16%
C.25%
D.20%
Solution
CI from two amounts
The third-year amount is the second-year amount grown by one year’s interest.
Interest of the third year = 2,160 − 1,800 = ₹360.
Rate = × 100.
= 20% — option (d).
53
Evaluate the following statements to determine if the information provided is sufficient to answer the question.
Question: If Sachin, Naveen and Ashish work together, how many days will they take to complete a project?
Statement I: Naveen and Ashish together can complete the project in 10 days.
Statement II: Sachin and Ashish together can complete the project in 15 days.
A.Statement I alone provides enough information, but Statement II alone does not
B.Statement II alone provides enough information, but Statement I alone does not
C.Both Statements combined are needed, as neither on its own is sufficient
D.Even when combined, the information in both Statements is not enough
Solution
Data sufficiency
Let the daily rates be S, N and A. Statement I gives N + A = and Statement II gives S + A = .
Adding them gives S + N + 2A, not S + N + A — the rate of Ashish is counted twice.
With two equations and three unknowns, S + N + A cannot be found.
Not sufficient even together — option (d).
54
A 100-litre container of pure milk undergoes a 10-litre removal and 10-litre water replacement process, repeated 3 times. Approximately how much original milk remains?
A donor deposits ₹10,00,000 in a trust that earns 8% simple interest annually. The interest is used to distribute three annual awards. If the second and third awards are ₹30,000 and ₹15,000 respectively, what is the value of the first award?
A.₹45,000
B.₹20,000
C.₹35,000
D.₹25,000
Solution
SI and awards
Annual interest = = ₹80,000.
The three awards together use up this whole amount.
First award = 80,000 − (30,000 + 15,000).
= ₹35,000 — option (c).
56
The time taken to cover 150 km by car is 2 hours. Find the time taken to cover 125 km by the same car.
A.1 hour 45 min
B.1 hour 40 min
C.1 hour 30 min
D.1 hour 20 min
Solution
Time from distance
Speed = = 75 km/h.
Time = = hours.
hours = 1 hour and of an hour.
= 1 hour 40 minutes — option (b).
57
The surface area of a cuboid is 736 cm² and its dimensions are in the ratio 1 : 3 : 5. Find its volume.
A.960 cm³
B.948 cm³
C.860 cm³
D.528 cm³
Solution
Cuboid from surface area
Let the sides be x, 3x and 5x. Surface area = 2(lb + bh + hl) = 2(3x² + 15x² + 5x²) = 46x².
46x² = 736 ⇒ x² = 16 ⇒ x = 4.
Volume = x × 3x × 5x = 15x³ = 15 × 64.
= 960 cm³ — option (a).
58
A building has 4 square pyramidal turrets, each with base side 5 m and height 7 m. If each needs 1.5 times its volume for structural fill, find the total fill needed.
A.350 m³
B.260 m³
C.280 m³
D.180 m³
Solution
Pyramidal turrets
Volume of one pyramid = × base area × height = × 25 × 7.
Four turrets = 4 × = m³.
Fill needed = 1.5 times this = × .
= 350 m³ — option (a).
59
A man spends 70% of his income. His income increases by 15% and his expenditure by 10%. What is the percentage increase in his savings?
A.26.67%
B.31.67%
C.41.67%
D.51.67%
Solution
Change in savings
Take income = 100, so expenditure = 70 and savings = 30.
New income = 115 and new expenditure = 70 × 1.10 = 77.
New savings = 115 − 77 = 38.
Increase = × 100 ≈ 26.67% — option (a).
60
The average score of a batsman in 20 innings is 43. If his highest score of 115 is excluded, what is the new average?
A.37.89
B.38.89
C.39.21
D.36.89
Solution
Average after removal
Total runs = 20 × 43 = 860.
Removing 115 leaves 860 − 115 = 745 runs in 19 innings.
New average = .
≈ 39.21 — option (c).
61
If the price tag of a microwave is ₹4,500 and the shopkeeper offers a discount of 19%, what is the selling price?
A.₹3,545
B.₹3,645
C.₹4,345
D.₹4,200
Solution
Discount
After a 19% discount the buyer pays 81% of the marked price.
SP = 4,500 × .
= 45 × 81.
= ₹3,645 — option (b).
62
The radii of three concentric circles are in arithmetic progression. If the innermost radius is 5 cm and the outermost is 15 cm, what is the radius of the middle circle?
A.12 cm
B.11 cm
C.10 cm
D.14 cm
Solution
AP of radii
In an AP the middle term is the average of the other two.
Middle radius = .
= .
= 10 cm — option (c).
63
If the height of a cylinder is equal to its diameter and the radius is 6 cm, find its volume.
If tanP = cotQ and both P and Q are acute, what is P + Q?
A.60°
B.90°
C.120°
D.180°
Solution
tan = cot
cotQ = tan(90° − Q).
So tanP = tan(90° − Q), giving P = 90° − Q.
Therefore P + Q = 90°.
Hence option (b).
68
The measures of the interior angles of a pentagon are y, y + 5, y + 10, y + 15 and y + 20. What is the measure of the smallest angle?
A.88°
B.98°
C.108°
D.118°
Solution
Angles of a pentagon
Sum of interior angles of a pentagon = (5 − 2) × 180° = 540°.
y + (y+5) + (y+10) + (y+15) + (y+20) = 540°.
5y + 50 = 540 ⇒ 5y = 490.
y = 98°, the smallest angle — option (b).
69
In triangle PQR an altitude PS is drawn from P to side QR. Given the coordinates P(3, 4), Q(1, 1) and R(5, 3), what is the slope of the line representing the altitude PS?
A.2
B.−2
C.−
D.
Solution
Slope of an altitude
Slope of QR = = = .
PS is perpendicular to QR, and for perpendicular lines m₁m₂ = −1.
So the slope of PS = −1 ÷ .
= −2 — option (b).
70
Triangle ABC has vertices A(0, 0), B(5, 0) and C(0, 12). Triangle XYZ has vertices X(1, 1), Y(13, 1) and Z(1, 6). Are these two triangles congruent?
A.Yes, by SSS
B.Yes, by SAS
C.Yes, by ASA
D.No, they are not congruent
Solution
Congruence by SSS
For △ABC: AB = 5, CA = 12 and BC = = 13.
For △XYZ: XY = 12, XZ = 5 and YZ = = 13.
Both have sides 5, 12 and 13 — all three pairs match.
Congruent by SSS — option (a).
71
Simplify: 2( − )
A.6
B.4
C.5
D.10
Solution
Surds
= 3 and = 2.
So the bracket becomes 3 − 2 = .
2 × = 2 × 5.
= 10 — option (d).
72
If a chord subtends an angle of 100° at the centre, the angle it subtends at the circumference on the same side is:
A.50°
B.37.5°
C.45°
D.60°
Solution
Angle at circumference
The angle at the centre is twice the angle at the circumference for the same arc.
So the angle at the circumference = × 100°.
= 50°.
Hence option (a).
73
What is the area of the segment formed by a chord in a circle of radius 4 cm, if the angle subtended at the centre is 90°?
A.4π − 8
B.8π − 4
C.16π − 8
D.4π − 4
Solution
Area of a segment
Area of segment = area of sector − area of the triangle.
Sector = × π × 4² = × 16π = 4π cm².
Triangle = r² sin90° = × 16 × 1 = 8 cm².
Segment = (4π − 8) cm² — option (a).
74
If cosA = , then what is sin²A + cos²A − 2 sinA cosA?
A.1
B.0
C.
D.
Solution
Perfect square identity
The expression is (sinA − cosA)².
With cosA = , the (3, 4, 5) triplet gives sinA = .
sinA − cosA = − = −.
(−)² = — option (c).
75
Kuldeep can complete a task in 30 days and Pankaj can complete the same task in 60 days. How long will it take them to complete the task together?
A.11 days
B.13 days
C.15 days
D.20 days
Solution
Work together
Take the work as LCM(30, 60) = 60 units.
Kuldeep does = 2 units a day and Pankaj = 1 unit a day.
Together = 3 units a day.
Time = = 20 days — option (d).
Finished reading? Now test yourself.
The same 25 questions, but with a timer and the answers hidden — 15 minutes.
= 79, since 79² = 6241.
= 35.5, since 35.5² = 1260.25.
79 − 35.5 = 43.5.
= 7.25 — option (c).
The third-year amount is the second-year amount grown by one year’s interest.
Interest of the third year = 2,160 − 1,800 = ₹360.
Rate = × 100.
= 20% — option (d).
Let the daily rates be S, N and A. Statement I gives N + A = and Statement II gives S + A = .
Adding them gives S + N + 2A, not S + N + A — the rate of Ashish is counted twice.
With two equations and three unknowns, S + N + A cannot be found.
Not sufficient even together — option (d).
Final quantity = initial × (1 − )ⁿ.
= 100 × (1 − )³ = 100 × ()³.
= 100 × = .
= 72 litres — option (d).
Annual interest = = ₹80,000.
The three awards together use up this whole amount.
First award = 80,000 − (30,000 + 15,000).
= ₹35,000 — option (c).
Let the sides be x, 3x and 5x. Surface area = 2(lb + bh + hl) = 2(3x² + 15x² + 5x²) = 46x².
46x² = 736 ⇒ x² = 16 ⇒ x = 4.
Volume = x × 3x × 5x = 15x³ = 15 × 64.
= 960 cm³ — option (a).
Volume of one pyramid = × base area × height = × 25 × 7.
Four turrets = 4 × = m³.
Fill needed = 1.5 times this = × .
= 350 m³ — option (a).
Take income = 100, so expenditure = 70 and savings = 30.
New income = 115 and new expenditure = 70 × 1.10 = 77.
New savings = 115 − 77 = 38.
Increase = × 100 ≈ 26.67% — option (a).
Total runs = 20 × 43 = 860.
Removing 115 leaves 860 − 115 = 745 runs in 19 innings.
New average = .
≈ 39.21 — option (c).
After a 19% discount the buyer pays 81% of the marked price.
SP = 4,500 × .
= 45 × 81.
= ₹3,645 — option (b).
In an AP the middle term is the average of the other two.
Middle radius = .
= .
= 10 cm — option (c).
Height = diameter = 2r = 12 cm.
Volume = πr²h.
= π × 36 × 12.
= 432π cm³ — option (a).
Volume of a prism = base area × height.
= 35 × 8.
= 280.
= 280 cm³ — option (c).
cotA = = , so base = 3 and perpendicular = 1.
Hypotenuse = = .
cosA = .
= — option (d).
Sum of interior angles of a pentagon = (5 − 2) × 180° = 540°.
y + (y+5) + (y+10) + (y+15) + (y+20) = 540°.
5y + 50 = 540 ⇒ 5y = 490.
y = 98°, the smallest angle — option (b).
Slope of QR = = = .
PS is perpendicular to QR, and for perpendicular lines m₁m₂ = −1.
So the slope of PS = −1 ÷ .
= −2 — option (b).
For △ABC: AB = 5, CA = 12 and BC = = 13.
For △XYZ: XY = 12, XZ = 5 and YZ = = 13.
Both have sides 5, 12 and 13 — all three pairs match.
Congruent by SSS — option (a).
The angle at the centre is twice the angle at the circumference for the same arc.
So the angle at the circumference = × 100°.
= 50°.
Hence option (a).
Area of segment = area of sector − area of the triangle.
Sector = × π × 4² = × 16π = 4π cm².
Triangle = r² sin90° = × 16 × 1 = 8 cm².
Segment = (4π − 8) cm² — option (a).
The expression is (sinA − cosA)².
With cosA = , the (3, 4, 5) triplet gives sinA = .
sinA − cosA = − = −.
(−)² = — option (c).
Take the work as LCM(30, 60) = 60 units.
Kuldeep does = 2 units a day and Pankaj = 1 unit a day.
Together = 3 units a day.
Time = = 20 days — option (d).