Which of the following numbers is both a Rational Number and a Terminating Decimal?
A.
B.
C.
D.π
Solution
Terminating decimal
A fraction terminates only when the denominator, in lowest terms, has just 2s and 5s as prime factors.
= 1.4, which terminates ✓.
= 0.666… repeats, while and π are irrational.
Hence option (b).
53
Amit walks at 10 km/h for 12 km and at 12 km/h for 16 km. What is his average speed?
A.9.4 km/h
B.12 km/h
C.10 km/h
D.11.05 km/h
Solution
Average speed
Total distance = 12 + 16 = 28 km.
Time = + = 1.2 + 1.333… hours.
Average = .
≈ 11.05 km/h — option (d).
54
The volume of a sphere is 288π cm³. If the radius of the sphere is tripled, by what percentage does the volume increase?
A.800%
B.2600%
C.900%
D.2700%
Solution
Volume and radius
Volume varies as the CUBE of the radius.
Tripling the radius multiplies the volume by 3³ = 27, so old : new = 1 : 27.
Increase = 27 − 1 = 26 parts on the original 1 part.
× 100 = 2600% — option (b).
55
A rectangular plot has a length of 15.75 m and a width of 8.2 m. What is its area?
A.129.15 m²
B.128.9 m²
C.180.2 m²
D.181.5 m²
Solution
Rectangle area
Area = length × breadth.
= 15.75 × 8.2.
= 129.15.
= 129.15 m² — option (a).
56
A recipe requires 0.85 litres of milk for 15 servings. How many servings can be made with 4.25 litres of milk?
A.78
B.70
C.72
D.75
Solution
Unitary method
= 5, so we have five times the milk.
Servings scale in the same proportion.
15 × 5.
= 75 — option (d).
57
D and E start a business with equal investments. After 6 months D adds ₹15,000 more. At the end of 1 year the total profit is ₹35,000 and E's share is ₹10,000. What was their initial investment?
A.₹6,000
B.₹4,000
C.₹5,000
D.₹7,000
Solution
Partnership
Let each invest ₹x. D contributes 6x + 6(x + 15,000) = 12x + 90,000 and E contributes 12x.
Profit ratio D : E = 25,000 : 10,000 = 5 : 2, so = .
24x + 1,80,000 = 60x ⇒ 36x = 1,80,000.
x = ₹5,000 — option (c).
58
If x : y = 3 : 7, then what is (2x + 3y) : (x + y)?
A.27 : 10
B.7 : 19
C.16 : 7
D.7 : 16
Solution
Ratio expression
Take x = 3 and y = 7.
2x + 3y = 6 + 21 = 27.
x + y = 3 + 7 = 10.
= 27 : 10 — option (a).
59
A number is such that if you square it and then subtract five times the number, the result is 150. What is the number?
A.10
B.15
C.20
D.25
Solution
Quadratic
Let the number be x: x² − 5x = 150.
x(x − 5) = 150.
Try x = 15: 15 × 10 = 150 ✓.
Hence option (b).
60
A triangle has vertices at A(4, 6), B(14, 2) and C(8, 12). What is its area?
A person bought a cycle at a discount of 25% on the marked price. He spent ₹510 on repairs and sold it for ₹5,760, gaining 20%. What was the marked price?
A.₹5,500
B.₹6,000
C.₹5,600
D.₹5,720
Solution
Discount, repair, profit
A 20% gain means his total cost = 5,760 × = ₹4,800.
That cost includes ₹510 of repairs, so he paid 4,800 − 510 = ₹4,290 for the cycle.
₹4,290 is 75% of the marked price.
MP = × 100 = ₹5,720 — option (d).
62
A mixture contains 60% water. If 20 litres of milk is added to the 80-litre mixture, what is the new percentage of water?
A.40%
B.48%
C.45%
D.50%
Solution
Mixture percentage
Water = 80 × = 48 L and milk = 32 L.
Adding 20 L of milk makes the milk 52 L, while water stays 48 L.
New total = 48 + 52 = 100 L.
× 100 = 48% — option (b).
63
If 20 boys can build a house in 15 days, in how many days can 15 boys complete the same work?
A.25
B.10
C.20
D.15
Solution
Men and days
M₁D₁ = M₂D₂ since the work is the same.
20 × 15 = 15 × D.
300 = 15D.
D = 20 days — option (c).
64
40 men can do a piece of work in 90 days. Find the percentage of the work done by 80 men in 15 days.
A.35.32%
B.31.21%
C.40%
D.33.33%
Solution
Percentage of work
Total work = 40 × 90 = 3600 man-days.
Work done = 80 × 15 = 1200 man-days.
× 100.
= 33.33% — option (d).
65
The base of a prism is a rectangle whose length is twice its width. The perimeter of the base is 54 cm. The prism's height decreases in an arithmetic progression from 17 cm in the first layer to 11 cm in the fourth layer. What is the total volume of the prism?
A.9000 cm³
B.9500 cm³
C.9060 cm³
D.9072 cm³
Solution
Prism with AP heights
l = 2b and 2(l + b) = 54 ⇒ 3b = 27 ⇒ b = 9 cm, l = 18 cm; base area = 162 cm².
The four heights form an AP from 17 to 11, so their sum = (17 + 11) = 56 cm.
Volume = base area × total height = 162 × 56.
= 9072 cm³ — option (d).
66
A merchant wants to mix two types of tea priced at ₹400/kg and ₹340/kg. In what ratio should they be mixed to sell the mixture at ₹370/kg without profit or loss?
A.1 : 1
B.2 : 1
C.3 : 2
D.1 : 2
Solution
Alligation
With no profit or loss the mean cost is ₹370 per kg.
Alligation: (400 − 370) : (370 − 340) = 30 : 30.
That is 1 : 1.
Hence option (a).
67
The average marks of 40 students in a class is 72. If the marks of one student were wrongly entered as 92 instead of 72, what is the CORRECT average?
A.71.5
B.71.67
C.71
D.70.35
Solution
Corrected average
The total was overstated by 92 − 72 = 20 marks.
Spread over 40 students, the average is overstated by = 0.5.
Correct average = 72 − 0.5.
= 71.5 — option (a).
68
If sinθ + cosθ = , find the value of sin²θ + cos²θ + 2sinθcosθ.
A.1
B.
C.
D.
Solution
Identity
Notice that sin²θ + cos²θ + 2sinθcosθ = (sinθ + cosθ)².
So we simply square the given value.
()².
= — option (c).
69
Find the point where y = 10x − 20 crosses the x-axis.
A.(4, 0)
B.(0, −10)
C.(−4, 0)
D.(2, 0)
Solution
x-intercept
On the x-axis the y-coordinate is 0.
0 = 10x − 20.
10x = 20 ⇒ x = 2.
The point is (2, 0) — option (d).
70
The sum of the measures of 8 interior angles of a 9-sided polygon is 1,120°. What is the measure of the 9th exterior angle?
Two triangles, △ABC and △XYZ, have AB = XY, AC = XZ and ∠BAC = ∠YXZ. By what rule are they congruent?
A.SSS
B.SAS
C.ASA
D.AAS
Solution
SAS congruence
Two pairs of sides are equal: AB = XY and AC = XZ.
The angle given, ∠BAC, lies BETWEEN those two sides.
Two sides and the included angle equal means side-angle-side.
Congruent by SAS — option (b).
72
Simplify:
A.2
B.
C.3
D.
Solution
Nested surds
16 = 11 + 5 and = × , so 16 ± 2 = ( ± )².
The two square roots become ( + ) and ( − ).
Their sum = 2, and the denominator = 2.
= — option (b).
73
A chord subtends an angle of 140° at the centre of a circle. What is the measure of the angle subtended by the same chord at any point on the remaining part of the circle?
A.35°
B.70°
C.140°
D.105°
Solution
Angle at circumference
The angle at the centre is twice the angle at any point on the remaining circumference.
So the angle at the circle = .
= 70°.
Hence option (b).
74
A circle is inscribed in a regular hexagon. What is the angle subtended by one side of the hexagon at the centre?
A.36°
B.54°
C.60°
D.108°
Solution
Hexagon central angle
A regular hexagon has six equal sides, so the centre is divided into six equal angles.
Angle = .
= .
= 60° — option (c).
75
If sinC = , then find cotC in simplest form.
A.
B.
C.3
D.
Solution
cotC from sinC
Take perpendicular = 1 and hypotenuse = .
Base = = = 3.
cotC = .
= = 3 — option (c).
Finished reading? Now test yourself.
The same 25 questions, but with a timer and the answers hidden — 15 minutes.
Amount = P(1 + )ᵗ.
= 4,000(1 + )⁵ = 4,000 × ()⁵.
= 4,000 × .
= ₹9,953.2 — option (c).
Total distance = 12 + 16 = 28 km.
Time = + = 1.2 + 1.333… hours.
Average = .
≈ 11.05 km/h — option (d).
Volume varies as the CUBE of the radius.
Tripling the radius multiplies the volume by 3³ = 27, so old : new = 1 : 27.
Increase = 27 − 1 = 26 parts on the original 1 part.
× 100 = 2600% — option (b).
= 5, so we have five times the milk.
Servings scale in the same proportion.
15 × 5.
= 75 — option (d).
Let each invest ₹x. D contributes 6x + 6(x + 15,000) = 12x + 90,000 and E contributes 12x.
Profit ratio D : E = 25,000 : 10,000 = 5 : 2, so = .
24x + 1,80,000 = 60x ⇒ 36x = 1,80,000.
x = ₹5,000 — option (c).
Take x = 3 and y = 7.
2x + 3y = 6 + 21 = 27.
x + y = 3 + 7 = 10.
= 27 : 10 — option (a).
Area = |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
= |4(2 − 12) + 14(12 − 6) + 8(6 − 2)|.
= |−40 + 84 + 32| = × 76.
= 38 square units — option (b).
A 20% gain means his total cost = 5,760 × = ₹4,800.
That cost includes ₹510 of repairs, so he paid 4,800 − 510 = ₹4,290 for the cycle.
₹4,290 is 75% of the marked price.
MP = × 100 = ₹5,720 — option (d).
Water = 80 × = 48 L and milk = 32 L.
Adding 20 L of milk makes the milk 52 L, while water stays 48 L.
New total = 48 + 52 = 100 L.
× 100 = 48% — option (b).
Total work = 40 × 90 = 3600 man-days.
Work done = 80 × 15 = 1200 man-days.
× 100.
= 33.33% — option (d).
l = 2b and 2(l + b) = 54 ⇒ 3b = 27 ⇒ b = 9 cm, l = 18 cm; base area = 162 cm².
The four heights form an AP from 17 to 11, so their sum = (17 + 11) = 56 cm.
Volume = base area × total height = 162 × 56.
= 9072 cm³ — option (d).
With no profit or loss the mean cost is ₹370 per kg.
Alligation: (400 − 370) : (370 − 340) = 30 : 30.
That is 1 : 1.
Hence option (a).
The total was overstated by 92 − 72 = 20 marks.
Spread over 40 students, the average is overstated by = 0.5.
Correct average = 72 − 0.5.
= 71.5 — option (a).
On the x-axis the y-coordinate is 0.
0 = 10x − 20.
10x = 20 ⇒ x = 2.
The point is (2, 0) — option (d).
Sum of interior angles = (n − 2) × 180° = 7 × 180° = 1260°.
So the 9th interior angle = 1260° − 1120° = 140°.
Interior + exterior = 180°.
Exterior = 180° − 140° = 40° — option (c).
Two pairs of sides are equal: AB = XY and AC = XZ.
The angle given, ∠BAC, lies BETWEEN those two sides.
Two sides and the included angle equal means side-angle-side.
Congruent by SAS — option (b).
16 = 11 + 5 and = × , so 16 ± 2 = ( ± )².
The two square roots become ( + ) and ( − ).
Their sum = 2, and the denominator = 2.
= — option (b).
The angle at the centre is twice the angle at any point on the remaining circumference.
So the angle at the circle = .
= 70°.
Hence option (b).
A regular hexagon has six equal sides, so the centre is divided into six equal angles.
Angle = .
= .
= 60° — option (c).
Take perpendicular = 1 and hypotenuse = .
Base = = = 3.
cotC = .
= = 3 — option (c).