Which of the following statements is CORRECT about Whole Numbers?
A.Whole numbers are the same as natural numbers
B.Whole numbers include all positive integers and zero
C.Whole numbers include both positive and negative integers
D.Whole numbers include all fractions and decimals
Solution
Whole numbers
Whole numbers are 0, 1, 2, 3, …
They are the natural numbers together with zero, so (a) is wrong.
They take in no negatives, no fractions and no decimals.
Hence option (b).
53
Aarav travels from his home to the library at a speed of 4.5 km/h and returns at 3 km/h. If the total journey time is 5 hours and 6 minutes, what is the distance between his home and the library?
A.9.33 km
B.9.18 km
C.8.71 km
D.7.62 km
Solution
To and fro journey
Total time = 5 + = hours.
For equal distances each way, distance = × total time.
= × = 1.8 × 5.1.
≈ 9.18 km — option (b).
54
A spherical shell has an outer diameter of 24 cm and is 3 cm thick. What is the volume of the material in the shell? (Use π = 3.14)
A.4,782.48 cm³
B.4,182.48 cm³
C.2,182.48 cm³
D.3,182.58 cm³
Solution
Spherical shell
Outer radius R = 12 cm; thickness 3 cm gives inner radius r = 9 cm.
Volume of material = π(R³ − r³).
= × 3.14 × (1728 − 729) = × 3.14 × 999.
= 4,182.48 cm³ — option (b).
55
A rectangular tank measuring 80 cm long and 40 cm wide contains water with an initial level of 15 cm. If 5 identical metal spheres, each 10 cm in diameter, are gently lowered into the tank, how much does the water level rise? (All spheres are completely submerged.)
A. cm
B. cm
C. cm
D.25π cm
Solution
Rise in water level
Each sphere has r = 5 cm, so its volume = π(125) = cm³; five give cm³.
Base area of the tank = 80 × 40 = 3200 cm².
Rise = = .
= cm — option (c).
56
P, Q and R started a textile business with ₹12,00,000, ₹8,00,000 and ₹10,00,000 respectively. After four months P added ₹2,00,000 and R withdrew ₹1,00,000, while Q kept his investment unchanged. The partnership lasted 12 months. What was the ratio of their profits?
The price of a refrigerator is increased by y% and then decreased by y%. The final price is 20% less than the original price. What is the value of y?
A.25
B.20
C.22
D.20
Solution
Rise then fall
A rise of y% followed by a fall of y% gives a net change of − percent.
So = 20.
y² = 2000.
y = = 20 — option (b).
58
A warehouse floor is 16.5 metres long and 16 metres wide. If 2.5 m × 2.2 m rectangular tiles are used to completely cover the floor, how many tiles will be needed?
A.48
B.68
C.72
D.80
Solution
Tiling a floor
Floor area = 16.5 × 16 = 264 m².
Area of one tile = 2.5 × 2.2 = 5.5 m².
Number of tiles = .
= 48 — option (a).
59
A large advertisement board is made on a 10 m × 6 m rectangle. From this, a 3 m × 2 m rectangular portion is cut out and removed. If painting the remaining area costs ₹250 per square metre, what is the total cost?
A.₹13,240
B.₹13,500
C.₹13,960
D.₹14,200
Solution
Remaining area cost
Board area = 10 × 6 = 60 m².
Cut-out area = 3 × 2 = 6 m².
Remaining = 60 − 6 = 54 m².
Cost = 54 × 250 = ₹13,500 — option (b).
60
A regular hexagonal nut has an outer side of 8 cm and an inner hollow side of 4 cm. What is the approximate area of the material in the cross-section?
A man made a 20% profit on his investment. If he invested ₹10,000 and reinvested the profit, what would his net worth be after the second year at the same rate of return?
A.₹16,000
B.₹16,050
C.₹14,400
D.₹14,150
Solution
Compounded profit
First year: profit = 20% of 10,000 = ₹2,000, so the total becomes ₹12,000.
The profit is reinvested, so the second year runs on ₹12,000.
Second year profit = 20% of 12,000 = ₹2,400.
Total = 12,000 + 2,400 = ₹14,400 — option (c).
62
A vendor offers a 10% discount on the marked price and sells a packet of salty biscuits for ₹180. If the profit earned on this sale is 50%, find the cost price.
A.₹136
B.₹120
C.₹130
D.₹110
Solution
CP from SP
The selling price after the discount is ₹180.
A 50% profit means SP = CP × .
CP = 180 × .
= ₹120 — option (b).
63
The average monthly salary of 10 employees in a company's IT department is ₹45,000. If two new employees join, taking the total to 12, and the average becomes ₹46,500, what is the average salary of the two new employees?
A.₹50,000
B.₹54,000
C.₹57,000
D.₹52,000
Solution
Average of new joiners
Old total = 10 × 45,000 = ₹4,50,000.
New total = 12 × 46,500 = ₹5,58,000.
The two new salaries add up to 5,58,000 − 4,50,000 = ₹1,08,000.
Average = = ₹54,000 — option (b).
64
A triangular storage prism has a base in the shape of a right triangle whose side lengths are in the ratio 3 : 4 : 5. If the perimeter of the base is 54 cm and the height of the prism is 8 cm, find the lateral surface area of the prism.
A.360 cm²
B.300 cm²
C.480 cm²
D.432 cm²
Solution
Lateral surface area
For any prism, lateral surface area = perimeter of base × height.
The perimeter is given directly as 54 cm.
LSA = 54 × 8.
= 432 cm² — option (d).
65
The average marks of 12 students is 85. When one student is removed, the average of the remaining 11 becomes 82. What were the marks of the removed student?
A.115
B.118
C.120
D.110
Solution
Removed student
Total of 12 students = 12 × 85 = 1020.
Total of 11 students = 11 × 82 = 902.
Removed student’s marks = 1020 − 902.
= 118 — option (b).
66
A distributor marks a batch of 220 identical mobile chargers at 45% above cost price and offers a 15% discount to retailers. A retailer buys all the chargers and spends ₹14,790. What is the cost price per charger for the wholesaler?
A.₹54.5
B.₹57.5
C.₹56.5
D.₹55.5
Solution
Markup and discount
Take CP = 100 units ⇒ MP = 145; after a 15% discount the SP = 145 × 0.85 = 123.25 units.
123.25 units = ₹14,790, so 100 units = = ₹12,000 — the total cost of all 220.
Cost per charger = .
≈ ₹54.5 — option (a).
67
If cotA = , then what is the value of cosecA?
A.
B.
C.
D.1
Solution
cosecA from cotA
cotA = = , so base = 4 and perpendicular = 3.
Hypotenuse = = 5.
cosecA = .
= — option (c).
68
If 1 + cot²θ = cosec²θ, then what is the value of 3cosec²30° − 3cot²30°?
A.5
B.1
C.3
D.6
Solution
Trig identity
The identity gives cosec²θ − cot²θ = 1 for every angle.
Take 3 common: 3(cosec²30° − cot²30°).
= 3 × 1.
= 3 — option (c).
69
A sector of a circle has a radius of 20 cm and a central angle of radians. What is the area of the sector?
A.141.4π cm²
B.171.4π cm²
C.145.5π cm²
D.175.8π cm²
Solution
Sector area
Area of a sector = r²θ, with θ in radians.
= × 400 × .
= .
≈ 171.4π cm² — option (b).
70
If sec(90° − θ) = y and θ is an acute angle, then what is the value of sinθ in terms of y?
A.y
B.
C.
D.Undefined
Solution
Complementary ratio
For complementary angles, sec(90° − θ) = cosecθ.
So cosecθ = y.
But cosecθ = .
Therefore sinθ = — option (c).
71
The radii of two circles are 8 cm and 3 cm. The centres of the two circles are 12 cm apart. What is the length of a transverse common tangent?
A. cm
B. cm
C. cm
D. cm
Solution
Transverse common tangent
Transverse common tangent = .
= .
= .
= cm — option (c).
72
In a circle, a chord AB subtends an angle of 40° at point C on the circumference. What angle does it subtend at point D on the circumference, on the same side as C?
A.40°
B.80°
C.140°
D.150°
Solution
Angles in the same segment
Angles in the same segment of a circle are equal.
C and D lie on the same side of the chord, so both are in the same segment.
Hence ∠ADB = ∠ACB.
= 40° — option (a).
73
A chord of length 24 cm is drawn in a circle of radius 13 cm. What is the distance of the chord from the centre of the circle?
A.5 cm
B.10 cm
C.8 cm
D.11 cm
Solution
Distance of a chord
The perpendicular from the centre bisects the chord, so half-chord = 12 cm.
Distance = .
= = .
= 5 cm — option (a).
74
In a circle, the chord PQ subtends an angle of 100° at the centre O. What is the measure of the angle PRQ subtended by the same chord PQ at a point R on the remaining part of the circle?
A.50°
B.60°
C.90°
D.120°
Solution
Angle at circumference
The angle at the centre is twice the angle at any point on the remaining part of the circumference.
∠POQ = 2 × ∠PRQ.
100° = 2 × ∠PRQ.
∠PRQ = 50° — option (a).
75
A line is drawn from a point P, 15 cm away from the centre O of a circle of radius 12 cm. If this line is a tangent to the circle, what is the length of the tangent from P to the point of contact?
A.3 cm
B.9 cm
C.12 cm
D.10 cm
Solution
Tangent length
The radius meets the tangent at right angles, giving a right triangle.
Tangent = .
= = .
= 9 cm — option (b).
Finished reading? Now test yourself.
The same 25 questions, but with a timer and the answers hidden — 15 minutes.
Amount = P(1 + )ᵗ.
= 10,000 × (1.05)³.
= 10,000 × 1.157625.
= ₹11,576.25 — option (a).
Total time = 5 + = hours.
For equal distances each way, distance = × total time.
= × = 1.8 × 5.1.
≈ 9.18 km — option (b).
Outer radius R = 12 cm; thickness 3 cm gives inner radius r = 9 cm.
Volume of material = π(R³ − r³).
= × 3.14 × (1728 − 729) = × 3.14 × 999.
= 4,182.48 cm³ — option (b).
Each sphere has r = 5 cm, so its volume = π(125) = cm³; five give cm³.
Base area of the tank = 80 × 40 = 3200 cm².
Rise = = .
= cm — option (c).
P: 12,00,000 × 4 + 14,00,000 × 8 = 48 + 112 = 160 lakh-months.
Q: 8,00,000 × 12 = 96 lakh-months.
R: 10,00,000 × 4 + 9,00,000 × 8 = 40 + 72 = 112 lakh-months.
160 : 96 : 112 = 10 : 6 : 7 — option (d).
A rise of y% followed by a fall of y% gives a net change of − percent.
So = 20.
y² = 2000.
y = = 20 — option (b).
Floor area = 16.5 × 16 = 264 m².
Area of one tile = 2.5 × 2.2 = 5.5 m².
Number of tiles = .
= 48 — option (a).
Board area = 10 × 6 = 60 m².
Cut-out area = 3 × 2 = 6 m².
Remaining = 60 − 6 = 54 m².
Cost = 54 × 250 = ₹13,500 — option (b).
Outer hexagon = × 8² = 96 cm².
Inner hexagon = × 4² = 24 cm².
Material = 96 − 24 = 72 = 72 × 1.732.
≈ 124.7 cm² — option (a).
First year: profit = 20% of 10,000 = ₹2,000, so the total becomes ₹12,000.
The profit is reinvested, so the second year runs on ₹12,000.
Second year profit = 20% of 12,000 = ₹2,400.
Total = 12,000 + 2,400 = ₹14,400 — option (c).
The selling price after the discount is ₹180.
A 50% profit means SP = CP × .
CP = 180 × .
= ₹120 — option (b).
Old total = 10 × 45,000 = ₹4,50,000.
New total = 12 × 46,500 = ₹5,58,000.
The two new salaries add up to 5,58,000 − 4,50,000 = ₹1,08,000.
Average = = ₹54,000 — option (b).
For any prism, lateral surface area = perimeter of base × height.
The perimeter is given directly as 54 cm.
LSA = 54 × 8.
= 432 cm² — option (d).
Total of 12 students = 12 × 85 = 1020.
Total of 11 students = 11 × 82 = 902.
Removed student’s marks = 1020 − 902.
= 118 — option (b).
Take CP = 100 units ⇒ MP = 145; after a 15% discount the SP = 145 × 0.85 = 123.25 units.
123.25 units = ₹14,790, so 100 units = = ₹12,000 — the total cost of all 220.
Cost per charger = .
≈ ₹54.5 — option (a).
cotA = = , so base = 4 and perpendicular = 3.
Hypotenuse = = 5.
cosecA = .
= — option (c).
Area of a sector = r²θ, with θ in radians.
= × 400 × .
= .
≈ 171.4π cm² — option (b).
Transverse common tangent = .
= .
= .
= cm — option (c).
Angles in the same segment of a circle are equal.
C and D lie on the same side of the chord, so both are in the same segment.
Hence ∠ADB = ∠ACB.
= 40° — option (a).
The perpendicular from the centre bisects the chord, so half-chord = 12 cm.
Distance = .
= = .
= 5 cm — option (a).
The angle at the centre is twice the angle at any point on the remaining part of the circumference.
∠POQ = 2 × ∠PRQ.
100° = 2 × ∠PRQ.
∠PRQ = 50° — option (a).
The radius meets the tangent at right angles, giving a right triangle.
Tangent = .
= = .
= 9 cm — option (b).