Arithmetic, number systems, percentages, ratios, profit and loss, time and work, algebra, geometry, mensuration, and trigonometry.
What is the square of 129?
16541
16641
16741
16841
Correct answer
16641
Why this answer is correct
Given:
Value: 129
Formula Used:
Square of a number = Number × Number
Calculation:
⇒ 129 × 129
⇒ 16641
∴ The correct answer is 16641.
A perfect square between 50 and 70 is:
62
63
64
65
Correct answer
64
Why this answer is correct
Given:
A perfect square between 50 and 70.
Formula Used:
A perfect square is a number that can be expressed as n2, where n is an integer.
Calculation:
⇒ Checking perfect squares between 50 and 70:
⇒ 72 = 49 (not in range).
⇒ 82 = 64 (in range).
⇒ 92 = 81 (not in range).
∴ The correct answer is 64.
A and B started a business with investments of ₹5000 and ₹4000, respectively. After 6 months, A added ₹1000 and B added ₹x. If the profit ratio is 11:18, find x.
₹9000
₹10000
₹12000
₹14000
Correct answer
₹10000
Why this answer is correct
Given:
A invests 5000, adds 1000 after 6 months
B invests 4000, adds x after 6 months
Profit ratio = 11 : 18
Formula Used:
Profit ∝ Investment × Time
Calculation:
⇒ A's capital = 5000 × 6 + 6000 × 6 = 30000 + 36000 = 66000
⇒ B's capital = 4000 × 6 + (4000 + x) × 6 = 24000 + (4000 + x) × 6
⇒ = 24000 + 24000 + 6x = 48000 + 6x
⇒ 66000 ÷ (48000 + 6x) = 11 ÷ 18
⇒ 66000 × 18 = 11 × (48000 + 6x)
⇒ 1188000 = 528000 + 66x
⇒ 1188000 - 528000 = 66x
⇒ 660000 = 66x
⇒ x = 10000
∴ The correct answer is 10000.
A started a business with ₹30,000. B joined after 5 months with ₹40,000. At the end of the year, the total profit was ₹80,000. What is the share of A?
₹40000
₹42000
₹45000
₹49000
Correct answer
₹45000
Why this answer is correct
Given:
Investment by A = ₹30,000
Investment by B = ₹40,000
B joined after 5 months.
Total profit = ₹80,000
Formula Used:
Profit share ratio = (Investment × Time)
Individual profit = (Individual ratio ÷ Total ratio) × Total profit
Calculation:
⇒ Time for A = 12 months
⇒ Time for B = 12 − 5 = 7 months
⇒ Profit share ratio for A = ₹30,000 × 12 = ₹360,000
⇒ Profit share ratio for B = ₹40,000 × 7 = ₹280,000
⇒ Total ratio = ₹360,000 + ₹280,000 = ₹640,000
⇒ Share of A = (₹360,000 ÷ ₹640,000) × ₹80,000
⇒ Share of A = (9 ÷ 16) × ₹80,000
⇒ Share of A = ₹45,000
∴ The correct answer is ₹45,000.
A, B, and C start a business. A invests ₹5000 for 12 months, B ₹4000 for 9 months, and C ₹3000 for 6 months. If the total profit is ₹380000, find A's profit share.
₹250000
₹200000
₹260000
₹270000
Correct answer
₹200000
Why this answer is correct
Given:
A's investment = ₹5000 for 12 months
B's investment = ₹4000 for 9 months
C's investment = ₹3000 for 6 months
Total profit = ₹380000
Formula Used:
Profit share ∝ Investment × Time
A's profit share = (A's investment × A's time) ÷ Total ratio × Total profit
Calculation:
⇒ A's ratio = ₹5000 × 12 = ₹60000
⇒ B's ratio = ₹4000 × 9 = ₹36000
⇒ C's ratio = ₹3000 × 6 = ₹18000
⇒ Total ratio = ₹60000 + ₹36000 + ₹18000 = ₹114000
⇒ A's profit share = (₹60000 ÷ ₹114000) × ₹380000
⇒ A's profit share = (10 ÷ 19) × ₹380000
⇒ A's profit share = ₹200000
∴ The correct answer is ₹200000.
A circular iron plate of radius 24 cm is heated, and its radius increases by 1.8%. What is the approximate percentage increase in its area?
1.8%
2.2%
3.6%
4.3%
Correct answer
3.6%
Why this answer is correct
Shortcut Trick
For a 2D shape, if the linear dimension (radius) increases by x%, the area increases by [2x + (x2/100)]%.
Given x = 1.8%.
Net Increase = 1.8 + 1.8 + (1.8 × 1.8 / 100)
Net Increase = 3.6 + 0.0324 = 3.6324% ≈ 3.6%.
∴ The correct answer is 3.6%.
Alternate Method
Given: Initial radius (r) = 24 cm, Percentage increase in radius = 1.8%.
Formula: Area of Circle (A) = πr2.
⇒ Let the initial radius be r.
⇒ New radius (r') = r + 1.8% of r = 1.018r.
⇒ Initial Area (A1) = πr2.
⇒ New Area (A2) = π(1.018r)2 = 1.036324 × πr2.
⇒ Percentage Increase in Area = [(A2 − A1) / A1] × 100
⇒ Percentage Increase = [(1.036324πr2 − πr2) / πr2] × 100
⇒ Percentage Increase = (1.036324 − 1) × 100 = 3.6324%.
⇒ Approximate percentage increase = 3.6%.
∴ The correct answer is 3.6%.
Additional Information
Successive Percentage Change
If a value changes by a% and then by b%, the net percentage change is given by (a + b + ab/100)%. For area calculation, a = b.
Linear vs Area vs Volume Change
If linear dimensions change by k%, the Perimeter changes by k%, Area changes by approximately 2k%, and Volume changes by approximately 3k% (for small values of k).
Binomial Approximation
For very small percentage changes x, (1 + x)n ≈ 1 + nx. In this case, (1 + 0.018)2 ≈ 1 + 2 × 0.018 = 1.036.
If x + y = 6 and x2 + y2 = 20, then find the value of (x4 + x2 y2 + y4).
412
336
408
372
Correct answer
336
Why this answer is correct
Given:
x + y = 6
x2 + y2 = 20
Formula Used:
(x + y)2 = x2 + y2 + 2xy
x4 + y4 = (x2 + y2)2 - 2x2y2
Calculation:
⇒ (x + y)2 = 36 = 20 + 2xy
⇒ 2xy = 16 ⇒ xy = 8
⇒ x2y2 = 82 = 64
⇒ x4 + y4 = 202 - 2×64 = 400 - 128 = 272
⇒ x4 + x2y2 + y4 = 272 + 64 = 336
∴ The correct answer is 336.
If (6m - 5n)2 = 7 and 30mn = 60, then find the value of (6m + 5n)2 + 16mn.
235
207
279
244
Correct answer
279
Why this answer is correct
Shortcut Trick
Use the algebraic identity (a + b)2 = (a − b)2 + 4ab, where a = 6m and b = 5n.
Given (6m − 5n)2 = 7 and 4ab = 4(30mn) = 4 × 60 = 240.
⇒ (6m + 5n)2 = 7 + 240 = 247.
From 30mn = 60, find 16mn = (60 ÷ 30) × 16 = 32.
Required value = 247 + 32 = 279.
∴ The correct answer is 279.
Alternate Method
Given:
(6m − 5n)2 = 7
30mn = 60
Formula Used:
(a + b)2 = a2 + b2 + 2ab
(a − b)2 = a2 + b2 − 2ab
Calculations:
⇒ (6m − 5n)2 = 36m2 + 25n2 − 60mn = 7
⇒ 36m2 + 25n2 − 60 = 7 (as 60mn = 2 × 30mn = 2 × 60 = 120, wait let us use direct substitution)
⇒ 36m2 + 25n2 − 2(30mn) = 7
⇒ 36m2 + 25n2 − 2(60) = 7
⇒ 36m2 + 25n2 = 7 + 120 = 127
⇒ (6m + 5n)2 = 36m2 + 25n2 + 2(30mn)
⇒ (6m + 5n)2 = 127 + 2(60) = 127 + 120 = 247
⇒ mn = 60 ÷ 30 = 2
⇒ 16mn = 16 × 2 = 32
⇒ Required Expression = 247 + 32 = 279
∴ The correct answer is 279.
Additional Information
Square of Sum and Difference Relation
The relationship between the square of a sum and a difference is given by (a + b)2 − (a − b)2 = 4ab.
Sum of Squares Identity
Adding the two basic identities results in (a + b)2 + (a − b)2 = 2(a2 + b2).
BODMAS Rule
Always follow the order of Brackets, Orders (powers), Division, Multiplication, Addition, and Subtraction for simplification.
If sec x - tan x = 1/√ 3, then the value of sec x × tan x is:
2/3
4/√ 3
2/√ 3
4/3
Correct answer
2/3
Why this answer is correct
Given:
secx - tanx = 1 ÷ √3
Formula Used:
(secx - tanx)(secx + tanx) = 1
Calculation:
⇒ secx + tanx = 1 ÷ (1 ÷ √3) = √3
⇒ secx = ((√3 + 1 ÷ √3) ÷ 2)
⇒ tanx = ((√3 - 1 ÷ √3) ÷ 2)
⇒ secx × tanx = ((√3 + 1 ÷ √3)(√3 - 1 ÷ √3)) ÷ 4
⇒ = (3 - 1 ÷ 3) ÷ 4
⇒ = (8 ÷ 3) ÷ 4
⇒ = 2 ÷ 3
∴ The correct answer is 2 ÷ 3.
If x = 7sin A + 24cos A and y = 7cos A - 24sin A, then what is the value of 3(x2 + y2)?
1760
2120
1875
1290
Correct answer
1875
Why this answer is correct
Given:
x = 7sinA + 24cosA
y = 7cosA - 24sinA
Formula Used:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
sin2A + cos2A = 1
Calculation:
⇒ x2 = (7sinA + 24cosA)2 = 49sin2A + 336sinAcosA + 576cos2A
⇒ y2 = (7cosA - 24sinA)2 = 49cos2A - 336sinAcosA + 576sin2A
⇒ x2 + y2 = (49 + 576)(sin2A + cos2A)
⇒ = 625 × 1 = 625
⇒ 3(x2 + y2) = 3 × 625 = 1875
∴ The correct answer is 1875.
A man pulls a rope tied to a flagpole at ground level. The rope is 20 meters long and makes an angle of 60° with the ground. How high up the flagpole is the point where the rope is tied? Also, find the horizontal distance between the man and the base of the flagpole.
9.55 meters and 15 meters
11.45 meters and 16 meters
17.32 meters and 10 meters
12.88 meters and 19 meters
Correct answer
17.32 meters and 10 meters
Why this answer is correct
Given:
Rope length = 20 m
Angle with ground = 60°
Formula Used:
Height = Hypotenuse × sinθ
Base = Hypotenuse × cosθ
Calculation:
⇒ Height = 20 × sin60°
⇒ Height = 20 × (√3 ÷ 2)
⇒ Height = 10√3 ≈ 17.32 m
⇒ Base = 20 × cos60°
⇒ Base = 20 × (1 ÷ 2)
⇒ Base = 10 m
∴ The correct answer is 17.32 m and 10 m.
An amount of Rs. 'z' is invested in two different schemes—one offers simple interest at 13% p.a. for 4 years, and the other offers compound interest at 12% p.a. compounded annually for 2 years. If the total amount from the simple interest scheme exceeds the compound interest scheme by Rs. 78, then find the value of z.
Rs. 287
Rs. 293.67
Rs. 245.5
Rs. 299.88
Correct answer
Rs. 293.67
Why this answer is correct
Given:
Principal = z
SI rate = 13%
SI time = 4 years
CI rate = 12%
CI time = 2 years
Difference = 78
Formula Used:
Amount (SI) = Principal × (1 + Rate × Time ÷ 100)
Amount (CI) = Principal × (1 + Rate ÷ 100)Time
Calculation:
⇒ Amount (SI) = z × (1 + 13 × 4 ÷ 100)
⇒ Amount (SI) = z × (1 + 52 ÷ 100) = z × 1.52
⇒ Amount (CI) = z × (1 + 12 ÷ 100)2
⇒ Amount (CI) = z × (1.12)2
⇒ Amount (CI) = z × 1.2544
⇒ z × 1.52 - z × 1.2544 = 78
⇒ z × (1.52 - 1.2544) = 78
⇒ z × 0.2656 = 78
⇒ z = 78 ÷ 0.2656
⇒ z = 293.67
∴ The correct answer is 293.67.
A sum of Rs. 7,500 is invested in a scheme offering r% compound interest for 2 years. Find the value of 'r' if the difference in the simple interest and compound interest for the same amount at the same rate of interest after 2 years is Rs. 108.
10%
8%
12%
9%
Correct answer
12%
Why this answer is correct
Given:
Principal = 7500
Time = 2 years
Difference (CI - SI) = 108
Formula Used:
Difference for 2 years = P × (r ÷ 100)2
Calculation:
⇒ 108 = 7500 × (r ÷ 100)2
⇒ 108 = 7500 × r2 ÷ 10000
⇒ 108 = (3 ÷ 4) × r2
⇒ r2 = 108 × (4 ÷ 3) = 144
⇒ r = 12
∴ The correct answer is 12%.
Gaurav invested Rs. ‘x’ in a scheme offering 10% compound interest for three years. Prem invested Rs. 18,000 in a scheme offering 15% simple interest for four years. Find the interest earned by Gaurav if the interest earned by Prem is Rs. ‘2x – 1,200’.
Rs. 1,215
Rs. 1,779
Rs. 1,560
Rs. 1,986
Correct answer
Rs. 1,986
Why this answer is correct
Given:
Gaurav invests = x
Rate = 10%
Time = 3 years
Prem invests = 18000
Rate = 15%
Time = 4 years
Prem interest = 2x - 1200
Formula Used:
SI = P × r × t ÷ 100
CI interest = P × [(1 + r ÷ 100)n - 1]
Calculation:
⇒ Prem SI = 18000 × 15 × 4 ÷ 100 = 10800
⇒ 10800 = 2x - 1200
⇒ 2x = 12000
⇒ x = 6000
⇒ CI interest = 6000 × [(1 + 10 ÷ 100)3 - 1]
⇒ = 6000 × [(11 ÷ 10)3 - 1]
⇒ = 6000 × (1331 ÷ 1000 - 1)
⇒ = 6000 × (331 ÷ 1000)
⇒ = 1986
∴ The correct answer is 1986.
The sum of three numbers is 144. If the numbers are in a ratio of 3:4:5, find the smallest number.
36
48
46
38
Correct answer
36
Why this answer is correct
Given:
Sum of three numbers = 144
Ratio of numbers = 3:4:5
Formula Used:
Total ratio = Sum of individual ratios
Value of each number = (Individual ratio ÷ Total ratio) × Sum of three numbers
Calculation:
⇒ Total ratio = 3 + 4 + 5 = 12
⇒ Smallest number = (3 ÷ 12) × 144
⇒ Smallest number = 3 × 12
⇒ Smallest number = 36
∴ The correct answer is 36.
Box A, contains tea worth Rs. 350 per kg and in box B contains tea worth Rs. 450 per kg. If both box A and B are mixed in the ratio 4 : 6 then the price of the mixture per kg is approximately:
400/-
350/-
410/-
440/-
Correct answer
410/-
Why this answer is correct
Given:
Price of A = 350
Price of B = 450
Ratio = 4 : 6
Formula Used:
Mean price = (a×x + b×y) ÷ (x + y)
Calculation:
⇒ Mean price = (350×4 + 450×6) ÷ (4 + 6)
⇒ = (1400 + 2700) ÷ 10
⇒ = 4100 ÷ 10
⇒ = 410
∴ The correct answer is 410.
A recipe for a cake calls for 2 cups of milk and 3 cups of flour. With this recipe, a cake was baked using 12 cups of flour. How many cups of milk were required?
7 cups
9 cups
8 cups
10 cups
Correct answer
8 cups
Why this answer is correct
Given:
Milk : Flour = 2 : 3
Flour used = 12
Formula Used:
Direct proportion
Calculation:
⇒ Milk ÷ Flour = 2 ÷ 3
⇒ Milk ÷ 12 = 2 ÷ 3
⇒ Milk = 12 × (2 ÷ 3) = 24 ÷ 3 = 8
∴ The correct answer is 8.
Train 'P' can cross a pole in 8 seconds, while train 'Q' takes 15 seconds to cross train 'P' when they travel in opposite directions on parallel tracks. If the ratio of length of train 'P' to that of train 'Q' is 2:3 and the speed of train 'P' is 54 km/h, then find the speed [in km/h] of train 'Q'.
28 km/h
12 km/h
22 km/h
18 km/h
Correct answer
18 km/h
Why this answer is correct
Given:
Time for P to cross pole = 8 s
Speed of P = 54 km/h
Time to cross each other = 15 s
Length ratio P : Q = 2 : 3
Formula Used:
Distance = Speed × Time
Relative speed = Sum of speeds (opposite direction)
Calculation:
⇒ Speed of P = 54 × 5 ÷ 18 = 15 m/s
⇒ Length of P = 15 × 8 = 120 m
⇒ Length ratio 2 : 3 ⇒ Length of Q = 120 × 3 ÷ 2 = 180 m
⇒ Total distance = 120 + 180 = 300 m
⇒ Relative speed = 300 ÷ 15 = 20 m/s
⇒ Speed of Q = 20 - 15 = 5 m/s
⇒ Speed of Q = 5 × 18 ÷ 5 = 18 km/h
∴ The correct answer is 18 km/h.
A thief is 2,000 metres ahead of a policeman when the policeman starts a chase by running at 20 m/s. If the thief is running at 12 m/s, then find the time taken by the policeman to catch the thief.
210 seconds
280 seconds
250 seconds
220 seconds
Correct answer
250 seconds
Why this answer is correct
Given:
Initial distance = 2000 m
Speed of policeman = 20 m/s
Speed of thief = 12 m/s
Formula Used:
Relative speed = Difference of speeds
Time = Distance ÷ Relative speed
Calculation:
⇒ Relative speed = 20 - 12 = 8 m/s
⇒ Time = 2000 ÷ 8 = 250 s
∴ The correct answer is 250 seconds.
An item undergoes two successive discounts: 12.5% (i.e., 1/8) and then 20%. After those discounts, a tax of 18% is applied on the reduced price, and the final amount the customer pays is ₹927. What was the original price of the item? [Give exact value and a rounded amount to two decimal places]
Rs. 1022.28
Rs. 1028.22
Rs. 1128.22
Rs. 1122.28
Correct answer
Rs. 1122.28
Why this answer is correct
Given:
First discount = 12.5%
Second discount = 20%
Tax = 18%
Final price = ₹927
Formula used:
Net multiplier = (1 − discount) × (1 + tax)
Calculations:
After 12.5% discount
⇒ Remaining = 7/8
After 20% discount
⇒ Remaining = 4/5
Total after discount
⇒ 7/8 × 4/5 = 28/40 = 7/10
After 18% tax
⇒ 7/10 × 118/100 = 826/1000 = 0.826
Let original price = x
⇒ x × 0.826 = 927
⇒ x = 927 ÷ 0.826
⇒ x = 1122.28 (approx)
∴ Original price ≈ ₹1122.28
Solving previous year questions by topic helps identify recurring concepts, question patterns, and frequently asked sub-topics. Attempt each question, review the step-by-step explanation, and note down key formulas or facts for quick revision before the exam.