Arithmetic, number systems, percentages, ratios, profit and loss, time and work, algebra, geometry, mensuration, and trigonometry.
The minimum value of the expression 2x2 + 5x + 5 is
5
15/8
0
Correct answer
15/8
Why this answer is correct
For any quadratic equation of form (ax2 + bx + c),
If a > 0, then the minimum value of equation = c - b2/4a
Given, equation ⇒ (2x2 + 5x + 5),
Here, a = 2 > 0
Also, b = 5 and c = 5,
⇒ Minimum value of equation = 5 - (25/8) = 15/8
If H is the harmonic mean of P and Q, then the value of H/P + H/Q is
1
2
\(\frac{{P + Q}}{{PQ}}\)
\(\frac{{PQ}}{{P+Q}}\)
Correct answer
2
Why this answer is correct
Concept:
If H is the Harmonic mean of numbers a and b and is given by
\({\rm{H}} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\)
Calculation:
Given that,
H is the harmonic mean between P and Q.
\(⇒\ \frac{2}{H}=\frac{1}{p}+\frac{1}{Q}\)
⇒ H/P + H/Q = 2
Additional Information AM, GM, Formulas:
If A is the arithmetic mean of numbers a and b and is given by
⇔ \({\rm{A}} = \frac{{{\rm{a;}} + {\rm{;b}}}}{2}\)
If G is the geometric mean of the numbers a and b and is given by
⇔ \({\rm{G}} = \sqrt {{\rm{ab}}} \)
Relation between AM, GM and HM
1. G2 = AH 2. AM ≥ GM ≥ HM
The sum of all possible products taken two at a time out of the numbers ± 1, ± 2, ±3, ± 4 is
0
-30
30
35
Correct answer
-30
Why this answer is correct
Given numbers, 1, - 1, 2, - 2, 3, - 3, 4, - 4
Let the sum of all their possible products be ‘x’,
As we know, (a + b)2 = a2 + b2 + 2ab
Also, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
Hence, we can write,
⇒ Square of sum of ‘n’ numbers = Sum of their squares + 2(Sum of all possible products taken two at a time)
Now,
Square of sum of given numbers = (± 1 ± 2 ± 3 ± 4)2 = 0
Sum of their squares = 1 + 1 + 4 + 4 + 9 + 9 + 16 + 16 = 60
⇒ 0 = 60 + 2x
⇒ x = - 60/2 = - 30
∴ Required sum = - 30
The remainder when 3x3 - 2x2y - 13xy2 + 10y3 is divided by (x - 2y) is equal to
Zero
y
y - 5
y + 3
Correct answer
Zero
Why this answer is correct
According to remainder theorem, when any polynomial f(x) is divided by (x - a), the remainder obtained is given as f(a),
Let f(x) = 3x3 - 2x2y - 13xy2 + 10y3
Putting x = 2y,
⇒ f(2y) = 3(8y3) - 2y(4y2) - 13y2(2y) + 10y3 = 24y3 - 8y3 - 26y3 + 10y3 = 0
∴ Required remainder = 0
If ab + bc + ca = 0, then the value of \(\frac{{\left( {{b^2} - ca} \right)\left( {{c^2} - ab} \right) + \left( {{a^2} - bc} \right)\left( {{c^2} - ab} \right) + \left( {{a^2} - bc} \right)\left( {{b^2} - ca} \right)}}{{\left( {{a^2} - bc} \right)\left( {{b^2} - ca} \right)\left( {{c^2} - ab;} \right)}}\) is
-1
0
1
2
Correct answer
0
Why this answer is correct
Given, ab + bc + ca = 0
⇒ (a2 - bc) = a2 + ab + ca = a(a + b + c)
⇒ (b2 - ca) = b2 + ab + bc = b(a + b + c)
⇒ (c2 - ab) = c2 + bc + ca = c(a + b + c)
Hence,
\(\frac{{\left( {{b^2} - ca} \right)\left( {{c^2} - ab} \right) + \left( {{a^2} - bc} \right)\left( {{c^2} - ab} \right) + \left( {{a^2} - bc} \right)\left( {{b^2} - ca} \right)}}{{\left( {{a^2} - bc} \right)\left( {{b^2} - ca} \right)\left( {{c^2} - ab;} \right)}}\)
\(\Rightarrow \frac{1}{{\left( {{a^2} - bc} \right)}} + \frac{1}{{\left( {{b^2} - ca} \right)}} + \frac{1}{{\left( {{c^2} - ab;} \right)}}\)
\(\Rightarrow \frac{1}{{a\left( {a + b + c} \right)}} + \frac{1}{{b\left( {a + b + c} \right)}} + \frac{1}{{c\left( {a + b + c;} \right)}}\)
\(\Rightarrow \frac{{bc + ac + ab}}{{abc\left( {a + b + c} \right)}}\)
⇒ 0
What is the principal amount which earns Rs. 210 as compound interest for the second year at 5% per annum?
Rs. 2000
Rs. 3200
Rs. 4000
Rs. 4800
Correct answer
Rs. 4000
Why this answer is correct
Given:
Rate = 5% p.a.
Compound interest for second year = Rs. 210
Assumption:
Let the principal amount be Rs. ‘P’
Formula:
Compound interest after ‘n’ years = Principal × [(1 + Rate/100)n – 1]
Calculation:
CI after 2 years – CI after 1 year = Rs. 210
⇒ [P × {(1 + 5/100)2 – 1}] – [P × {(1 + 5/100) – 1}] = 210
⇒ P × (1.052 – 1 – 0.05) = 210
⇒ P × (1.1025 – 1 – 0.05) = 210
⇒ P = 210/0.0525
⇒ P = Rs. 4000
∴ Required principal amount = Rs. 4000
In an examination, 50% of the candidates failed in English, 40% failed in Hindi and 15% failed in both the subjects. The percentage of candidates who passed in both English and Hindi is
20%
25%
60%
75%
Correct answer
25%
Why this answer is correct
Short trick∶
Required percentage = 100% – (50 + 40 – 15)% = 25%
Detailed solution∶
Let the total number of students appearing in examination be ‘x’
No. of students that failed in English = 50% of x = 0.5x
No. of students that failed in Hindi = 40% of x = 0.4x
No. of students that failed in both = 15% of x = 0.15x
⇒ No. of students that either failed in English or Hindi or both = 0.5x + 0.4x – 0.15x = 0.75x
⇒ No. of students that passed both in English & Hindi = x – 0.75x = 0.25x
∴ Percentage of students that passed both in English & Hindi = (0.25x/x) × 100 = 25%
A train 100 m long passes a platform 100 m long in 10 seconds. The speed of the train is
36 kmph
45 kmph
54 kmph
72 kmph
Correct answer
72 kmph
Why this answer is correct
Given:
Length of train = 100 m
Length of platform = 100 m
Time to cross the platform = 10 sec
Formula used:
Speed = Distance/Time
To convert m/sec into km/hr, multiply the number by 18/5
Calculation:
Distance covered by train in passing the platform
⇒ Length of train + Length of platform
⇒ 100 + 100 = 200 m
Using the above formula
⇒ Speed of train = 200/10 = 20 m/sec.
But 1 m/s = 18/5 km/hr
⇒ Speed of train = 20 × 18/5 = 72 kmph.
∴ The speed of the train is 72 kmph.
A cyclist covers his first 20 km at an average speed of 40 kmph, another 10 km at an average speed of 10 kmph and the last 30 km at an average speed of 40 kmph. Then the average speed of the entire journey is
20 kmph
26.67 kmph
28.24 kmph
30 kmph
Correct answer
26.67 kmph
Why this answer is correct
Topper’s approach∶
Average speed = \(\frac{{20 + 10 + 30}}{{\frac{{20}}{{40}} + \frac{{10}}{{10}} + \frac{{30}}{{40}}}} = 26.67\) kmph
Detailed solution∶
∵ Average speed = Total distance covered/Total time taken
Time taken in covering first 20 km = 20/40 = 0.5 hrs.
Time taken in covering next 10 km = 10/10 = 1 hr.
Time taken in covering last 30 km = 30/40 = 0.75 hrs.
⇒ Total time taken = 0.5 + 1 + 0.75 = 2.25 hrs.
⇒ Total distance covered = 20 + 10 + 30 = 60 km
∴ Average speed of entire journey = 60/2.25 = 26.67 kmph.
In a race of 1000 m, A beats B by 150 m, while in another race of 3000 m, C beats D by 400 m. Speed of B is equal to that of D. (Assume that A, B, C and D run with uniform speed in all the events). If A and C participate in a race of 6000 m, then which one of the following is correct?
A beats C by 250 m
C beats A by 250 m
A beats C by 115.38 m
C beats A by 115.38 m
Correct answer
A beats C by 115.38 m
Why this answer is correct
Suppose the speed of B = speed of D = ‘x’ m/sec.
In the race of 1000 m, A beats B by 150 m,
⇒ Distance covered by B = 1000 – 150 = 850 m
∵ Time taken by both participants will be equal,
⇒ Time taken = 1000/(Speed of A) = 850/x
⇒ Speed of A = (20x/17) m/sec.
Similarly, in race of 3000 m, C beats D by 400 m,
⇒ Distance covered by D = 3000 – 400 = 2600 m
⇒ Time taken = 3000/(Speed of C) = 2600/x
⇒ Speed of C = (15x/13) m/sec.
Now, in the race of 6000 m,
∵ Speed of A > Speed of C
This means that A will win the race,
⇒ Time taken by A to complete the race = 6000/(20x/17) = (5100/x) sec.
⇒ Distance covered by C in the same time = (15x/13) × (5100/x) = 5884.62 m
∴ In race of 6000 m, A beats C by = 6000 – 5884.62 = 115.38 m
The sum of ages of a father, a mother, a son Sonu and daughters Savita and Sonia is 96 years. Sonu is the youngest member of the family. The year Sonu was born, the sum of the ages of all the members of the family was 66 years. If the father’s age now is 6 times that of Sonu’s present age, then 12 years hence, the father’s age will be
44 years
45 years
46 years
48 years
Correct answer
48 years
Why this answer is correct
Given:
Sum of current ages of Father, Mother, Sonu, Savita & Sonia = 96 years
Concept used:
We will be using the fundamental concept of ratio.
Calculation:
Let Sonu’s current age be ‘x’ years
When Sonu was born, i.e., ‘x’ years ago,
Sum of ages of Father, Mother, Savita & Sonia = 66 years
⇒ 66 = 96 – 5x
⇒ x = 30/5 = 6 years
⇒ Sonu’s current age = 6 years
⇒ Father’s current age = 6 × 6 = 36 years
∴ Father’s age after 12 years = 36 + 12 = 48 years
‘A’ is thrice as good a workman as ‘B’ and takes 10 days less to do a piece of work than ‘B’ takes. The number of days taken by ‘B’ alone to finish the work is
12
15
20
30
Correct answer
15
Why this answer is correct
Let ‘B’ takes ‘x’ days to finish the work alone
⇒ No. of days A will take to do the work alone = (x – 10)
Now,
Work efficiency of A = 3 × Work efficiency of B
⇒ 1/(x – 10) = 3 × 1/x
⇒ x = 3x – 30
⇒ 2x = 30
⇒ x = 30/2 = 15
∴ B will alone finish the work in 15 days
Out of 85 children playing badminton or table tennis or both, the total number of girls in the group is 70% of the total number of boys in the group. The number of boys playing only badminton is 50% of the number of boys and the total number of boys playing badminton is 60% of the total number of boys. The number of children playing only table tennis is 40% of the total number of children and a total of 12 children play badminton and table tennis both. The number of girls playing only badminton is
14
16
17
35
Correct answer
14
Why this answer is correct
Total no. of children = 85
Given, total no. of girls = 70% of total no. of boys
⇒ 85 – Total no. of boys = 0.7 × Total no. of boys
⇒ Total no. of boys = 85/1.7 = 50
Now,
No. of children playing only table tennis = 40% of 85 = 34
No. of children playing badminton & table tennis both = 12
⇒ No. of children playing only badminton = 85 – 34 – 12 = 39
∵ No. of boys playing only badminton = 50% of No. of boys = 0.5 × 50 = 25
∴ No. of girls playing only badminton = 39 – 25 = 14
A person bought two articles X and Y from a departmental store. The sum of prices before the sales tax was Rs. 130. There was no sales tax on the article X and 9% sales tax on the article Y. The total amount the person paid, including the sales tax, was Rs. 136.75. What was the price of the article Y before sales tax?
Rs. 75
Rs. 85
Rs. 122
Rs. 125
Correct answer
Rs. 75
Why this answer is correct
Given,
Price of X + Price of Y = Rs. 130 ----(1)
Price of X + Price of Y + 9% of Price of Y = Rs. 136.75 ----(2)
Subtracting eq.
(1) from (2), we get,
⇒ 9/100 × Price of Y = 136.75 – 130 = 6.75
⇒ Price of Y = 100/9 × 6.75 = Rs. 75
∴ Price of article Y before sales tax = Rs. 75
According to Mr. Sharma’s will, half of his property goes to his wife and the rest is equally divided between his two sons, Ravi and Raj.
Some years later, Ravi dies and leaves half of his property to his widow and rest to his brother Raj. When Raj dies, he leaves half of his property to his widow and remaining to his mother, who is still alive. The mother now owns 88,000 worth of the property. The total worth of the property of Mr. Sharma was
Rs. 1,00,000
Rs. 1,24,000
Rs. 1,28,000
Rs. 1,32,000
Correct answer
Rs. 1,28,000
Why this answer is correct
Let the total worth of the property of Mr. Sharma be Rs. ‘x’
Initial property share of his wife = x/2
Property share of Ravi = Property share of Raj = (1/2) × (x/2) = x/4
After death of Ravi,
Property share of Raj = x/4 + (1/2) × (x/4) = x/4 + x/8 = 3x/8
After death of Raj,
Property share of their mother = x/2 + (1/2) × (3x/8) = x/2 + 3x/16 = 11x/16
But, she owns Rs. 88000 worth of the property,
⇒ 11x/16 = 88000
⇒ x = 16/11 × 88000 = Rs. 128000
∴ Total worth of property of Mr. Sharma = Rs. 128000
X bought 4 bottles of lemon juice and Y bought one bottle of orange juice. Orange juice per bottle costs twice the cost of lemon juice per bottle. Z bought nothing but contributed Rs. 50 for his share of the drink which they mixed together and shared the cost equally. If Z’s Rs. 50 is covered from his share, then what is the cost of one bottle of orange juice?
Rs. 75
Rs. 50
Rs. 46
Rs. 30
Correct answer
Rs. 50
Why this answer is correct
Let the cost of one bottle of lemon juice be Rs. ‘x’
⇒ Cost of one bottle of orange juice = Rs. 2x
X bought 4 bottles of lemon juice, while Y bought 1 bottle of orange juice,
Total cost of drink = 4(x) + 2x = Rs. 6x
Now, the cost was divided equally between X, Y & Z,
⇒ Share of each of them = 6x/3 = Rs. 2x
But, Z contributed Rs. 50 as his share,
⇒ 2x = 50
∴ Cost of 1 bottle of orange juice = Rs. 2x = Rs. 50
Ten years before, the ages of a mother and her daughter were in the ratio 3 :
39 years, 21 years
55 years, 25 years
75 years, 25 years
49 years, 31 years
Correct answer
55 years, 25 years
Why this answer is correct
Let the ages of the mother and daughter ten years ago be ‘3x’ years and ‘x’ years respectively
Ratio of their current ages = (3x + 10)/(x + 10)
Ratio of their ages after ten years = (3x + 20)/(x + 20)
But, ratio of their ages after ten years = 13 ∶ 7
⇒ (3x + 20)/(x + 20) = 13/7
⇒ 21x + 140 = 13x + 260
⇒ 8x = 120
⇒ x = 120/8 = 15
⇒ Current age of mother = (3x + 10) = 45 + 10 = 55 years
⇒ Current age of daughter = (x + 10) = 15 + 10 = 25 years
In a class of 60 boys, there are 45 boys who play chess and 30 boys who play carrom. If every boy of the class plays at least one of the two games, then how many boys play carrom only?
30
20
15
10
Correct answer
15
Why this answer is correct
Total no. of boys in class = 60
No. of boys who play chess = 45
∵ Each boy of class play at least one of the two games,
∴ No. of boys who play carrom only = 60 – 45 = 15
Two equal amounts were borrowed at 5% and 4% simple interest. The total interest after 4 years amounted to Rs. 405. What was the total amount borrowed?
Rs. 1075
Rs. 1100
Rs. 1125
Rs. 1150
Correct answer
Rs. 1125
Why this answer is correct
Let the equal amount borrowed be Rs. ‘P’
Time = 4 years
∵ Simple interest = (Principal × Rate × Time)/100
Simple interest at 5% = (P × 5 × 4)/100 = Rs. 0.2P
Simple interest at 4% = (P × 4 × 4)/100 = Rs. 0.16P
⇒ Total interest after 4 years = 0.2P + 0.16P = Rs. 0.36P
But, total interest = Rs. 405
⇒ 0.36P = 405
⇒ P = 405/0.36 = Rs. 1125
∴ Amount borrowed = Rs. 1125
Note: This question has some error in the official paper. The question asks for the total amount borrowed which should be equal to 2P. The answer is given according to the answer key.
12 men work 8 hours per day and require 10 days to build a wall. If 8 men are available, how many hours per day must they work to finish the work in 8 days?
10 hours
12 hours
15 hours
18 hours
Correct answer
15 hours
Why this answer is correct
12 men build a wall in 10 days working for 8 hours per day
⇒ Total no. of hours 12 men took = 10 × 8 = 80 hours
⇒ No. of hours 1 man will take = 12 × 80 = 960 hours
⇒ No. of hours 8 men will take = 960/8 = 120 hours
But, 8 men have only 8 days to finish the work,
∴ No. of hours per day 8 men must work = 120/8 = 15 hours
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.