A dealer first sells a dining table for ₹X, making a 20% profit. Later the marked price of the same table is raised to ₹1.8X and a festive discount of 30% is given on it. What is the new profit percentage?
A.56%
B.51.2%
C.42%
D.70%
Solution
Profit after new discount
A 20% profit means CP : SP = 5 : 6, so take CP = 5 and X = 6.
New marked price = 1.8X = 1.8 × 6 = 10.8; after 30% discount, SP = 10.8 × = 7.56.
Profit = 7.56 − 5 = 2.56 on a cost of 5.
Profit% = × 100 = 51.2% — option (b).
53
What is the compound interest on a sum of ₹31,250 at 18% p.a. in 2 years, if the interest is compounded 8-monthly?
A.₹12,654
B.₹12,664
C.₹12,754
D.₹13,654
Solution
CI compounded 8-monthly
One period of 8 months means the rate becomes 18 × = 12% per period.
Two years contain 3 such periods.
CI = 31,250[(1 + )³ − 1] = 31,250 × .
= ₹12,654 — option (a).
54
A contractor employs 20 men to complete a work in 28 days, working 7 hours per day. After seven days he adds six more men but reduces the working hours to five per day from the eighth day onwards. If all men work with the same efficiency, how many more days will be required to finish the remaining work?
A.24
B.19
C.27
D.23
Solution
Men, days and hours
Total work = 20 × 28 × 7 = 3920 man-hours.
Done in 7 days = 20 × 7 × 7 = 980, so remaining = 2940.
New daily output = 26 men × 5 hours = 130 man-hours.
2940 ÷ 130 ≈ 22.6, i.e. 23 more days — option (d).
55
A thief and a police officer are 800 metres apart. If they run in the same direction, the officer catches the thief in 16 minutes. If they run towards each other, they meet in 4 minutes. What is the speed of the thief in metres per minute?
A.50 m/min
B.120 m/min
C.75 m/min
D.125 m/min
Solution
Chase and meet
Same direction: relative speed x − y = = 50.
Opposite directions: x + y = = 200.
Subtracting: 2y = 150.
y = 75 m/min — option (c).
56
What type of number is obtained when a rational number is added to an irrational number?
A.Always Rational
B.Always Irrational
C.Always Integer
D.Sometimes Rational, Sometimes Irrational
Solution
Rational + irrational
Suppose a rational r added to an irrational i gave a rational s. Then i = s − r would be the difference of two rationals, hence rational — a contradiction.
So the sum can NEVER be rational: + 7 = 8.414… stays irrational.
A cancelling example like + (−) = 0 does not apply here, because − is itself irrational, not rational.
Always irrational — option (b).
57
Three metallic spheres with radii 4 cm, 2 cm and 7 cm respectively are melted together and recast into a single solid sphere. What is the radius of the new sphere?
A.12.5 cm
B.9.5 cm
C.6.5 cm
D.7.5 cm
Solution
Spheres recast
Volumes add, so πR³ = π(4³ + 2³ + 7³).
R³ = 64 + 8 + 343 = 415.
Taking the cube root: R = ∛415 ≈ 7.46.
≈ 7.5 cm — option (d).
58
A car runs 27.6 km using 3.75 litres of petrol. What is the mileage in km per litre (rounded to 2 decimals)?
A.7.98
B.7.66
C.7.54
D.7.36
Solution
Mileage
Mileage = .
= .
= 7.36.
Hence option (d).
59
In a three-way joint venture the partners share profits in the ratio 5 : 6 : 3. If the venture earns ₹2,80,000 in the first year, how much should the strategist (second partner) receive?
A.₹1,40,000
B.₹1,65,000
C.₹1,20,000
D.₹1,55,000
Solution
Profit sharing
Total parts = 5 + 6 + 3 = 14.
The strategist gets 6 parts.
Share = × 2,80,000.
= ₹1,20,000 — option (c).
60
A wire is cut into three pieces in the ratio 3 : 4 : 8. If the length of the longest piece is 40 cm, what is the total length of the wire?
A.40 cm
B.50 cm
C.75 cm
D.85 cm
Solution
Wire in ratio
The longest piece is 8 parts = 40 cm, so 1 part = 5 cm.
Total parts = 3 + 4 + 8 = 15.
Total length = 15 × 5.
= 75 cm — option (c).
61
In an urban project, triangular plots with side ratios 8 : 15 : 17 are allocated. If the perimeter is 120 m, find the area.
A.380 m²
B.320 m²
C.400 m²
D.540 m²
Solution
Right triangle area
Total parts = 8 + 15 + 17 = 40, and 40 parts = 120 m ⇒ 1 part = 3 m.
Sides are 24 m, 45 m and 51 m; since 8² + 15² = 17², it is a right triangle.
Area = × 24 × 45.
= 540 m² — option (d).
62
A student's marks in English were 40% more than in Sanskrit. If he scored 80 in Sanskrit, how many marks did he score in English?
A.82
B.108
C.104
D.112
Solution
Percentage more
English = Sanskrit + 40% of Sanskrit.
= 80 × .
= 80 × 1.4.
= 112 — option (d).
63
7 women and 6 men can complete a work in 8 days, while 5 women and 9 men complete it in 9 days. In how many days will 9 women complete it?
A.13 days
B.8 days
C.16.5 days
D.14 days
Solution
Men and women work
(7W + 6M) × 8 = (5W + 9M) × 9 ⇒ 56W + 48M = 45W + 81M.
11W = 33M ⇒ M : W = 1 : 3, so take W = 3 and M = 1.
Total work = (7×3 + 6) × 8 = 216 units; 9 women give 9 × 3 = 27 units a day.
216 ÷ 27 = 8 days — option (b).
64
The average of a group of 8 consecutive even numbers is 43. What is the sum of the smallest and largest numbers in the group?
A.79
B.86
C.83
D.85
Solution
Consecutive even numbers
In an evenly spaced set, the average equals the average of the first and last terms.
So = 43.
Smallest + largest = 43 × 2.
= 86 — option (b).
65
A prism with a square base of side 6 cm is constructed in layers. Its height follows an arithmetic progression, increasing from 6 cm in the first layer to 24 cm in the fourth layer. What is the total volume of the prism?
A.1800 cm³
B.1950 cm³
C.1210 cm³
D.2160 cm³
Solution
Prism with AP heights
Base area = 6² = 36 cm².
The four heights in AP from 6 to 24 are 6, 12, 18, 24 — total 60 cm.
Volume = 36 × 60.
= 2160 cm³ — option (d).
66
A vessel has 90 litres of a solution with sugar and soda in the ratio 7 : 2. If 18 litres of the solution are replaced with soda, what is the new ratio?
A.20 : 17
B.28 : 17
C.31 : 15
D.13 : 23
Solution
Replacement in mixture
Initially sugar = 70 L and soda = 20 L.
The 18 L removed keeps the same 7 : 2 ratio — 14 L sugar and 4 L soda.
Left: sugar 56 L, soda 16 L; adding 18 L soda gives soda = 34 L.
56 : 34 = 28 : 17 — option (b).
Find the solution to the system: y = 7x − 4 and y = −2x + 14.
A.(2, 10)
B.(17, 2)
C.(1.5, 3.5)
D.(1.5, −4.5)
Solution
Simultaneous equations
Equate the two expressions: 7x − 4 = −2x + 14.
9x = 18 ⇒ x = 2.
y = 7(2) − 4 = 10.
= (2, 10) — option (a).
69
When a transversal intersects two parallel lines, one of the alternate interior angles is 135°. What is the measure of the angle that lies opposite to it on the alternate interior side?
A.75°
B.80°
C.160°
D.135°
Solution
Alternate interior angles
When a transversal cuts parallel lines, alternate interior angles are EQUAL.
One of them is 135°.
So the other is also 135°.
Hence option (d).
70
If △PQR ∼ △MNO and PQ = 8 cm, QR = 10 cm, MN = 16 cm, what is the length of NO?
A.20 cm
B.8 cm
C.4 cm
D.12 cm
Solution
Similar triangles
In similar triangles corresponding sides are proportional: = .
= .
NO = 10 × 2.
= 20 cm — option (a).
71
If x = 4 + , find x − .
A.
B.2
C.
D.
Solution
Surd — x − 1/x
Rationalise: × = = 4 − .
So x − = (4 + ) − (4 − ).
= 2.
Hence option (b).
72
A circle has a radius of 17 cm and a chord measuring 30 cm. What is the distance from the centre of the circle to the chord?
A.6 cm
B.8 cm
C.10 cm
D.12 cm
Solution
Chord distance
The perpendicular from the centre bisects the chord, so half-chord = 15 cm.
d² = 17² − 15² = 289 − 225.
d² = 64.
d = 8 cm — option (b).
73
A circle has a radius of 14 cm. Two radii are drawn such that the chord connecting their endpoints is 14 cm. What is the area of the minor segment formed?
A.( − 49) sq.cm
B.( − 59) sq.cm
C.( − 49) sq.cm
D.( − 49) sq.cm
Solution
Minor segment area
Radius = chord = 14 cm, so the triangle is equilateral and θ = 60°.
Sector area = × π × 14² = cm².
Triangle area = R² sinθ = × 196 × = 49 cm².
Segment = − 49 — option (a).
Multiply 14.345 by 0.037 and round your answer to 4 decimal places.
A.0.5307
B.0.5370
C.0.5037
D.0.5703
Solution
Decimal multiplication
Multiply without decimals: 14345 × 37 = 5,30,765.
The two numbers together have 3 + 3 = 6 decimal places.
So the product is 0.530765.
Rounded to four places: 0.5307 — option (a).
Finished reading? Now test yourself.
The same 25 questions, but with a timer and the answers hidden — 15 minutes.
Apply componendo and dividendo: = = = .
Squaring: = .
45 + 9x = 80 − 16x ⇒ 25x = 35.
x = — option (b).
A 20% profit means CP : SP = 5 : 6, so take CP = 5 and X = 6.
New marked price = 1.8X = 1.8 × 6 = 10.8; after 30% discount, SP = 10.8 × = 7.56.
Profit = 7.56 − 5 = 2.56 on a cost of 5.
Profit% = × 100 = 51.2% — option (b).
One period of 8 months means the rate becomes 18 × = 12% per period.
Two years contain 3 such periods.
CI = 31,250[(1 + )³ − 1] = 31,250 × .
= ₹12,654 — option (a).
Total work = 20 × 28 × 7 = 3920 man-hours.
Done in 7 days = 20 × 7 × 7 = 980, so remaining = 2940.
New daily output = 26 men × 5 hours = 130 man-hours.
2940 ÷ 130 ≈ 22.6, i.e. 23 more days — option (d).
Same direction: relative speed x − y = = 50.
Opposite directions: x + y = = 200.
Subtracting: 2y = 150.
y = 75 m/min — option (c).
Total parts = 5 + 6 + 3 = 14.
The strategist gets 6 parts.
Share = × 2,80,000.
= ₹1,20,000 — option (c).
The longest piece is 8 parts = 40 cm, so 1 part = 5 cm.
Total parts = 3 + 4 + 8 = 15.
Total length = 15 × 5.
= 75 cm — option (c).
Total parts = 8 + 15 + 17 = 40, and 40 parts = 120 m ⇒ 1 part = 3 m.
Sides are 24 m, 45 m and 51 m; since 8² + 15² = 17², it is a right triangle.
Area = × 24 × 45.
= 540 m² — option (d).
Base area = 6² = 36 cm².
The four heights in AP from 6 to 24 are 6, 12, 18, 24 — total 60 cm.
Volume = 36 × 60.
= 2160 cm³ — option (d).
Initially sugar = 70 L and soda = 20 L.
The 18 L removed keeps the same 7 : 2 ratio — 14 L sugar and 4 L soda.
Left: sugar 56 L, soda 16 L; adding 18 L soda gives soda = 34 L.
56 : 34 = 28 : 17 — option (b).
cosec²A = 1 + cot²A = 1 + .
= .
cosecA = (since 61² = 3721 and 60² = 3600).
Hence option (a).
Equate the two expressions: 7x − 4 = −2x + 14.
9x = 18 ⇒ x = 2.
y = 7(2) − 4 = 10.
= (2, 10) — option (a).
When a transversal cuts parallel lines, alternate interior angles are EQUAL.
One of them is 135°.
So the other is also 135°.
Hence option (d).
In similar triangles corresponding sides are proportional: = .
= .
NO = 10 × 2.
= 20 cm — option (a).
The perpendicular from the centre bisects the chord, so half-chord = 15 cm.
d² = 17² − 15² = 289 − 225.
d² = 64.
d = 8 cm — option (b).
Radius = chord = 14 cm, so the triangle is equilateral and θ = 60°.
Sector area = × π × 14² = cm².
Triangle area = R² sinθ = × 196 × = 49 cm².
Segment = − 49 — option (a).
sinx = gives cosx = (3-4-5 triplet).
sin2x = 2 sinx cosx = 2 × × = .
cos2x = 1 − 2sin²x = 1 − = −.
Sum = = 0.68 — option (a).