← ASI (Ministerial)
Past Questions
There is a rope of length 20m and you want to cut it at every 2m long. How many times do you need to cut the given rope?
A)
8 times
B)
9 times
C)
10 times
D)
11 times
To solve this problem, we first determine how many pieces the rope will be divided into. Total length = 20m; Length of each piece = 2m. Number of pieces = 20 / 2 = 10 pieces. In a linear object like a rope, the number of cuts required to create 'n' pieces is always 'n - 1'. This is because the final cut creates two pieces simultaneously (the 9th piece and the 10th piece). Therefore, to get 10 pieces, you need 10 - 1 = 9 cuts.
Breakdown of options:
- a. 8 times: This would result in only 9 pieces.
- b. 9 times: Correct. 9 cuts create 10 pieces of 2m each.
- c. 10 times: This is a common mistake where the number of pieces is confused with the number of cuts; 10 cuts would result in 11 pieces.
- d. 11 times: This would result in 12 pieces.
If x + 1/x = 5, then the value of x^2 + 1/x^2 is
A)
21
B)
23
C)
25
D)
22
To find the value of x² + 1/x², we use the algebraic identity (a + b)² = a² + b² + 2ab.
1. Given: x + 1/x = 5
2. Square both sides of the equation:
(x + 1/x)² = 5²
3. Expand the left side using the identity (a + b)² = a² + 2ab + b²:
x² + 2(x)(1/x) + (1/x)² = 25
4. Simplify the middle term: since x * (1/x) = 1, the term 2(x)(1/x) becomes 2(1) = 2.
x² + 2 + 1/x² = 25
5. Isolate x² + 1/x² by subtracting 2 from both sides:
x² + 1/x² = 25 - 2
x² + 1/x² = 23
Therefore, the correct option is (b) 23. Options (a) 21, (c) 25, and (d) 22 are incorrect as they do not follow from the algebraic expansion of the given expression.
A can do a piece of work in 5days, B in 4 days and A, B and C together in 2 days. In what time can C alone do it?
A)
23 days
B)
22 days
C)
20 days
D)
18 days
To solve this problem, we use the concept of work rates (work done per day).
1. **Identify the work rates of A and B:**
- A can complete the work in 5 days, so A's rate = 1/5 of the work per day.
- B can complete the work in 4 days, so B's rate = 1/4 of the work per day.
2. **Identify the combined work rate of A, B, and C:**
- Together, they complete the work in 2 days, so their combined rate (A + B + C) = 1/2 of the work per day.
3. **Calculate the work rate of C:**
- Let C's rate be 'x'.
- Equation: Rate of A + Rate of B + Rate of C = Combined Rate
- 1/5 + 1/4 + x = 1/2
4. **Solve for x:**
- Find a common denominator for 1/5 and 1/4, which is 20.
- (4/20) + (5/20) + x = 1/2
- 9/20 + x = 1/2
- x = 1/2 - 9/20
- x = 10/20 - 9/20
- x = 1/20
5. **Determine the time for C alone:**
- Since C's rate is 1/20 of the work per day, C will take 20 days to complete the work alone.
Comparing this result to the options:
- Option (a) 23 days is incorrect.
- Option (b) 22 days is incorrect.
- Option (c) 20 days is correct.
- Option (d) 18 days is incorrect.
The value of 1/3 ÷ 1/2 × 3/4 ÷ 1/20 × 7/6 is
A)
7/10
B)
7/20
C)
7/40
D)
None of these
To solve the expression 1/3 ÷ 1/2 × 3/4 ÷ 1/20 × 7/6, we follow the BODMAS/PEMDAS rule, which dictates that division and multiplication are performed from left to right.
Step 1: Handle the first division (1/3 ÷ 1/2).
Dividing by a fraction is the same as multiplying by its reciprocal:
1/3 × 2/1 = 2/3
Step 2: Multiply the result by 3/4.
(2/3) × (3/4) = (2 × 3) / (3 × 4) = 6/12 = 1/2
Step 3: Handle the next division (1/2 ÷ 1/20).
Again, multiply by the reciprocal:
1/2 × 20/1 = 20/2 = 10
Step 4: Multiply the result by 7/6.
10 × 7/6 = 70/6
Step 5: Simplify the final fraction.
70/6 = 35/3
Comparing the result (35/3 or 11.67) with the given options:
a. 7/10 (0.7)
b. 7/20 (0.35)
c. 7/40 (0.175)
Since 35/3 does not match any of the provided options, the correct choice is 'None of these'.
Find the amount on ₹ 6250 for 1 Year at 8% per annum compounded half yearly.
A)
₹ 6760
B)
₹ 6770
C)
₹ 6750
D)
None of these
Based on established principles in Quantitative Aptitude, the correct and most appropriate choice is '₹ 6760' (Option A).
An inlet pipe can fill a cistern in 5 hours which an outlet pipe can empty it in 6 hours. If both the two pipes are turned on together simultaneously in the empty cistern, when will the cistern be filled completely?
A)
11 hours
B)
22 hours
C)
24 hours
D)
30 hours
Step-by-step solution:
1. **Identify the rates of the pipes**:
- The inlet pipe fills the cistern in 5 hours, so its filling rate is 1/5 of the cistern per hour.
- The outlet pipe empties the cistern in 6 hours, so its emptying rate is 1/6 of the cistern per hour.
2. **Calculate the net rate when both are open**:
Since the inlet pipe adds water and the outlet pipe removes water, we subtract the outlet rate from the inlet rate:
Net Rate = (1/5) - (1/6)
3. **Find a common denominator**:
The least common multiple (LCM) of 5 and 6 is 30.
Net Rate = (6/30) - (5/30) = 1/30 of the cistern per hour.
4. **Determine the total time**:
The time taken to fill the cistern is the reciprocal of the net rate:
Time = 1 / (1/30) = 30 hours.
Comparing this to the options:
- a. 11 hours: Incorrect (this is the sum of 5+6).
- b. 22 hours: Incorrect.
- c. 24 hours: Incorrect.
- d. 30 hours: Correct.
In a group of 850 persons 600 can speak English, 340 can speak Hindi, find the number of persons that can speak English only.
A)
420
B)
510
C)
570
D)
610
To find the number of persons who speak English only, we use the principles of Set Theory.
1. **Identify the given values:**
- Total number of persons (n(E ∪ H)) = 850
- Persons who can speak English (n(E)) = 600
- Persons who can speak Hindi (n(H)) = 340
2. **Find the number of persons who speak both languages (n(E ∩ H)):**
Using the formula: n(E ∪ H) = n(E) + n(H) - n(E ∩ H)
850 = 600 + 340 - n(E ∩ H)
850 = 940 - n(E ∩ H)
n(E ∩ H) = 940 - 850
n(E ∩ H) = 90
So, 90 people speak both English and Hindi.
3. **Calculate the number of persons who speak English only:**
The number of people who speak English only is the total number of English speakers minus those who speak both languages.
English only = n(E) - n(E ∩ H)
English only = 600 - 90
English only = 510
Therefore, 510 persons can speak English only. Option (b) is correct.
A copper wire of length 44cm is bent to form a circle. The same wire is also bent to form a square. Which of the two will have more area and by how much?
A)
square, by 33 cm²
B)
circle, by 33 cm²
C)
square, by 121 cm²
D)
circle, by 154 cm²
Step-by-step calculation:
1. **Case 1: The wire is bent into a circle.**
- The length of the wire is the circumference of the circle: $C = 44$ cm.
- Formula for circumference: $C = 2\pi r$
- $44 = 2 \times \frac{22}{7} \times r$
- $44 = \frac{44}{7} \times r$
- $r = 7$ cm.
- Area of the circle: $A_{circle} = \pi r^2 = \frac{22}{7} \times 7^2 = 22 \times 7 = 154$ cm².
2. **Case 2: The wire is bent into a square.**
- The length of the wire is the perimeter of the square: $P = 44$ cm.
- Formula for perimeter: $P = 4 \times side$
- $44 = 4 \times side$
- $side = 11$ cm.
- Area of the square: $A_{square} = side^2 = 11^2 = 121$ cm².
3. **Comparison:**
- Area of circle = 154 cm²
- Area of square = 121 cm²
- Difference = $154 - 121 = 33$ cm².
Conclusion: The circle has a larger area than the square by 33 cm². Therefore, option (b) is correct.
3√2 + 8√2 - 18√2 = ?
A)
5√2
B)
3√2
C)
2√2
D)
4√3
Based on established principles in the relevant subject area, the correct and most appropriate choice is '5√2' (Option A).
Find the sum of the AP 2, 7, 12, …………………to 10 terms
A)
224
B)
245
C)
345
D)
425
To find the sum of an Arithmetic Progression (AP), we use the formula: Sₙ = (n/2) [2a + (n - 1)d]
1. **Identify the given values:**
- First term (a) = 2
- Common difference (d) = Second term - First term = 7 - 2 = 5
- Number of terms (n) = 10
2. **Substitute the values into the formula:**
S₁₀ = (10/2) [2(2) + (10 - 1)5]
S₁₀ = 5 [4 + (9)5]
S₁₀ = 5 [4 + 45]
S₁₀ = 5 [49]
S₁₀ = 245
Therefore, the sum of the first 10 terms of the AP is 245.
- Option (a) 224 is incorrect.
- Option (c) 345 is incorrect.
- Option (d) 425 is incorrect.
What should be subtracted from -5/7 to get -1?
A)
2/7
B)
5/7
C)
3/7
D)
None of these
To find the value that should be subtracted from -5/7 to get -1, we can set up an algebraic equation. Let the required number be 'x'.
Equation: (-5/7) - x = -1
Step 1: Rearrange the equation to solve for x. Move x to the right side and -1 to the left side:
-5/7 + 1 = x
Step 2: To add these values, find a common denominator. Since the denominator of the first term is 7, we write 1 as 7/7:
-5/7 + 7/7 = x
Step 3: Perform the addition of the numerators:
(-5 + 7) / 7 = x
2/7 = x
Verification: (-5/7) - (2/7) = (-5 - 2) / 7 = -7/7 = -1. The result is correct.
Comparing this result to the options:
- Option (a) 2/7 matches our result.
- Option (b) 5/7 is incorrect as (-5/7) - (5/7) = -10/7.
- Option (c) 3/7 is incorrect as (-5/7) - (3/7) = -8/7.
The average age of 25 students is 10 years. If the teacher’s age is included, the average age becomes 11 years. The age of the teacher is
A)
25 years
B)
30 years
C)
32 years
D)
36 years
To find the age of the teacher, we use the concept of the sum of ages based on the average formula: Sum = Average × Number of people.
Step 1: Calculate the total age of the 25 students.
Total age of students = 25 students × 10 years/student = 250 years.
Step 2: Calculate the total age of the group after the teacher is included.
When the teacher is added, the total number of people becomes 26 (25 students + 1 teacher), and the new average is 11 years.
Total age of (students + teacher) = 26 people × 11 years/person = 286 years.
Step 3: Find the teacher's age by subtracting the students' total age from the combined total age.
Teacher's age = Total age of 26 people - Total age of 25 students
Teacher's age = 286 - 250 = 36 years.
Alternative Shortcut Method:
The teacher's age = New Average + (Original Number of Students × Increase in Average)
Teacher's age = 11 + (25 × (11 - 10)) = 11 + 25 = 36 years.
Therefore, option (d) is correct. Options (a), (b), and (c) are incorrect as they do not satisfy the mathematical requirement of increasing the average of 25 people by 1 year.
A loan of ₹ 60,000 is to be cleared in three equal instalments of ₹ 22,000 each. The total interest charged is
A)
₹ 3,000
B)
₹ 4,000
C)
₹ 5,000
D)
₹ 6,000
To find the total interest charged on a loan, we use the basic formula: Total Interest = (Total Amount Paid) - (Principal Loan Amount).
1. **Identify the Principal Loan Amount:** The loan amount is given as ₹ 60,000.
2. **Calculate the Total Amount Paid:** The loan is cleared in three equal instalments of ₹ 22,000 each.
Total Amount Paid = 3 instalments × ₹ 22,000 per instalment = ₹ 66,000.
3. **Calculate the Total Interest:**
Total Interest = ₹ 66,000 - ₹ 60,000 = ₹ 6,000.
Therefore, the total interest charged is ₹ 6,000, which corresponds to option (d). Options (a), (b), and (c) are incorrect as they do not match the difference between the total repayment and the principal.
The smallest number by which 396 must be multiplied so that the product becomes a perfect square is
A)
11
B)
22
C)
3
D)
2
To find the smallest number by which 396 must be multiplied to become a perfect square, we must first find its prime factorization.
1. Prime factorization of 396:
- 396 ÷ 2 = 198
- 198 ÷ 2 = 99
- 99 ÷ 3 = 33
- 33 ÷ 3 = 11
- 11 ÷ 11 = 1
So, 396 = 2 × 2 × 3 × 3 × 11 = 2² × 3² × 11¹.
2. For a number to be a perfect square, all the exponents of its prime factors must be even.
- The prime factor 2 has an exponent of 2 (even).
- The prime factor 3 has an exponent of 2 (even).
- The prime factor 11 has an exponent of 1 (odd).
3. To make the exponent of 11 even, we must multiply 396 by another 11.
- 396 × 11 = 4356
- √4356 = 66 (which is a perfect square).
Therefore, the smallest number to multiply by is 11.
Verification of other options:
- Multiplying by 22 (2 × 11) would make it a square, but it is not the smallest.
- Multiplying by 3 or 2 would leave the factor 11 unpaired, so the result would not be a perfect square.
The HCF and LCM of two numbers are 4 and 24 respectively. If one of the numbers is 8, what is the other number?
A)
6
B)
8
C)
10
D)
12
To solve this problem, we use a fundamental property of number theory regarding the relationship between two numbers and their Highest Common Factor (HCF) and Least Common Multiple (LCM).
1. **The Formula**: For any two positive integers 'a' and 'b', the product of the numbers is equal to the product of their HCF and LCM.
Formula: a × b = HCF(a, b) × LCM(a, b)
2. **Given Values**:
- HCF = 4
- LCM = 24
- One number (a) = 8
- Other number (b) = ?
3. **Calculation**:
Substitute the known values into the formula:
8 × b = 4 × 24
8b = 96
b = 96 / 8
b = 12
4. **Verification**:
- The numbers are 8 and 12.
- Factors of 8: 1, 2, 4, 8
- Factors of 12: 1, 2, 3, 4, 6, 12
- HCF(8, 12) = 4 (Correct)
- Multiples of 8: 8, 16, 24, 32...
- Multiples of 12: 12, 24, 36...
- LCM(8, 12) = 24 (Correct)
Therefore, the other number is 12, which corresponds to option (d).
The electrical energy meter of a household reads 5800 units. After one month, it reads 6100 units. How much the house owner has to pay for one month’s bill if the cost of 1unit of electrical energy is ₹ 2?
A)
₹ 300
B)
₹ 400
C)
₹ 500
D)
₹ 600
To find the total bill for the month, we follow these steps:
1. **Calculate the total units consumed**: The units consumed are found by subtracting the initial meter reading from the final meter reading.
Units consumed = Final Reading - Initial Reading
Units consumed = 6100 units - 5800 units = 300 units.
2. **Calculate the total cost**: The total cost is the product of the units consumed and the cost per unit.
Total Cost = Units consumed × Cost per unit
Total Cost = 300 units × ₹ 2/unit = ₹ 600.
Therefore, the house owner has to pay ₹ 600 for one month's bill.
Comparing this to the options:
- Option (a) ₹ 300 is only the number of units consumed, not the cost.
- Option (b) ₹ 400 is incorrect.
- Option (c) ₹ 500 is incorrect.
- Option (d) ₹ 600 is the correct calculation.
Find the 8th term from the end of the AP 7, 10, 13, …………………,184
A)
153
B)
161
C)
163
D)
172
To find the n-th term from the end of an Arithmetic Progression (AP), we can use the formula:
Term from end = l - (n - 1)d
Given data:
1. First term (a) = 7
2. Common difference (d) = 10 - 7 = 3
3. Last term (l) = 184
4. Position from the end (n) = 8
Applying the values to the formula:
8th term from end = 184 - (8 - 1) * 3
8th term from end = 184 - (7 * 3)
8th term from end = 184 - 21
8th term from end = 163
Alternatively, if we reverse the AP, the new first term becomes 184 and the common difference becomes -3.
The 8th term of this reversed AP is: a + (n-1)d = 184 + (8-1)(-3) = 184 - 21 = 163.
Comparing this result with the options:
(a) 153: Incorrect
(b) 161: Incorrect
(c) 163: Correct
(d) 172: Incorrect
The mean of 10 numbers is 25. If 5 is added to each number, what will be new mean?
A)
25
B)
30
C)
35
D)
40
Step-by-step solution:
1. **Understand the given data**: The mean of 10 numbers is 25. Let the numbers be $x_1, x_2, ..., x_{10}$.
2. **Formula for Mean**: $\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}$. Therefore, $\text{Sum} = \text{Mean} \times \text{Number of observations} = 25 \times 10 = 250$.
3. **Apply the change**: If 5 is added to each of the 10 numbers, the total sum increases by $5 \times 10 = 50$.
4. **Calculate new sum**: $\text{New Sum} = 250 + 50 = 300$.
5. **Calculate new mean**: $\text{New Mean} = \frac{300}{10} = 30$.
**Mathematical Property**: There is a fundamental property in statistics stating that if a constant 'k' is added to every observation in a data set, the mean of the new data set will be the original mean plus 'k'.
New Mean = Original Mean + Constant
New Mean = $25 + 5 = 30$.
Therefore, option (b) is correct. Options (a), (c), and (d) are incorrect as they do not follow this mathematical property.
Find the median of 4, 7, 5, 2, 13, 9, 11, 16
A)
6
B)
7
C)
8
D)
9
To find the median of a data set, follow these steps:
1. **Arrange the data in ascending order:**
The given numbers are: 4, 7, 5, 2, 13, 9, 11, 16.
Sorted order: 2, 4, 5, 7, 9, 11, 13, 16.
2. **Determine the number of observations (n):**
There are n = 8 observations.
3. **Identify the median position:**
Since n is even (8), the median is the average of the two middle terms. The middle terms are the (n/2)th and the (n/2 + 1)th terms.
- 4th term: 7
- 5th term: 9
4. **Calculate the average of the middle terms:**
Median = (7 + 9) / 2
Median = 16 / 2
Median = 8
Therefore, the median of the given data set is 8. Option (c) is correct, while options (a), (b), and (d) are incorrect as they do not represent the average of the two central values of the sorted set.
6 bowls cost ₹ 180. What would be the cost of 20 such bowls?
A)
₹ 900
B)
₹ 600
C)
₹ 700
D)
None of these
This problem is solved using the Unitary Method, which involves finding the value of a single unit first and then multiplying it by the required number of units.
Step 1: Find the cost of one bowl.
Cost of 6 bowls = ₹ 180
Cost of 1 bowl = 180 / 6 = ₹ 30
Step 2: Find the cost of 20 bowls.
Cost of 20 bowls = 20 × Cost of 1 bowl
Cost of 20 bowls = 20 × 30 = ₹ 600
Verification of options:
- Option (a) ₹ 900 is incorrect (this would be the cost of 30 bowls).
- Option (b) ₹ 600 is correct.
- Option (c) ₹ 700 is incorrect.
- Option (d) is incorrect because a correct option is available.
The class mark of the class 120 – 140 is
A)
120
B)
125
C)
130
D)
135
In statistics, the 'class mark' (also known as the mid-value or class midpoint) of a class interval is the average of the lower limit and the upper limit of that class.
Step-by-step calculation:
1. Identify the lower limit (L) = 120
2. Identify the upper limit (U) = 140
3. Apply the formula for Class Mark:
Class Mark = (Lower Limit + Upper Limit) / 2
4. Substitute the values:
Class Mark = (120 + 140) / 2
Class Mark = 260 / 2
Class Mark = 130
Therefore, option (c) 130 is the correct answer. Options (a), (b), and (d) are incorrect as they do not represent the arithmetic mean of the given interval boundaries.
The value of ∛2744 is
A)
7
B)
12
C)
14
D)
16
To find the cube root of 2744 (∛2744), we can use the prime factorization method:
1. Divide 2744 by the smallest prime number, 2:
2744 ÷ 2 = 1372
1372 ÷ 2 = 686
686 ÷ 2 = 343
2. Now, 343 is not divisible by 2, 3, or 5. We test 7:
343 ÷ 7 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1
3. The prime factorization of 2744 is: 2 × 2 × 2 × 7 × 7 × 7, which can be written as 2³ × 7³.
4. The cube root is found by taking one factor from each triplet:
∛(2³ × 7³) = 2 × 7 = 14.
Verification of options:
- a. 7³ = 343 (Incorrect)
- b. 12³ = 1728 (Incorrect)
- c. 14³ = 14 × 14 × 14 = 196 × 14 = 2744 (Correct)
- d. 16³ = 4096 (Incorrect)
Which term of the AP 72, 68, 64, 60, ………. is 0?
A)
15th term
B)
17th term
C)
19th term
D)
None of these
To find which term of the Arithmetic Progression (AP) is 0, we use the formula for the nth term of an AP:
a_n = a + (n - 1)d
Given values:
1. First term (a) = 72
2. Common difference (d) = Second term - First term = 68 - 72 = -4
3. The nth term (a_n) = 0
Substituting these values into the formula:
0 = 72 + (n - 1)(-4)
Now, solve for n:
0 = 72 - 4n + 4
0 = 76 - 4n
4n = 76
n = 76 / 4
n = 19
Therefore, the 19th term of the AP is 0.
Verification of options:
- Option (a) 15th term: 72 + (14)(-4) = 72 - 56 = 16
- Option (b) 17th term: 72 + (16)(-4) = 72 - 64 = 8
- Option (c) 19th term: 72 + (18)(-4) = 72 - 72 = 0 (Correct)
If 3^(2x+1) ÷ 9 = 27, the value of x is
A)
1
B)
-1
C)
0
D)
2
To solve the equation 3^(2x+1) ÷ 9 = 27, we need to express all terms as powers of the same base, which is 3.
Step 1: Express 9 and 27 as powers of 3.
9 = 3^2
27 = 3^3
Step 2: Substitute these values back into the equation:
3^(2x+1) ÷ 3^2 = 3^3
Step 3: Apply the law of exponents for division (a^m ÷ a^n = a^(m-n)):
3^((2x+1) - 2) = 3^3
3^(2x - 1) = 3^3
Step 4: Since the bases are the same, we can equate the exponents:
2x - 1 = 3
Step 5: Solve for x:
2x = 3 + 1
2x = 4
x = 4 / 2
x = 2
Verification:
If x = 2, then 3^(2(2)+1) ÷ 9 = 3^5 ÷ 9 = 243 ÷ 9 = 27. The result is correct.
Comparing with options:
a. 1: 3^(2+1) ÷ 9 = 27 ÷ 9 = 3 (Incorrect)
b. -1: 3^(-2+1) ÷ 9 = 3^-1 ÷ 9 = (1/3) ÷ 9 = 1/27 (Incorrect)
c. 0: 3^(0+1) ÷ 9 = 3 ÷ 9 = 1/3 (Incorrect)
d. 2: Correct.
The average of three numbers is 7 and the average of the first two numbers is 4. The third number is
A)
13
B)
14
C)
15
D)
16
Step-by-step solution:
1. Find the total sum of the three numbers: Since the average of three numbers is 7, the sum is calculated as Average × Count. Therefore, Sum of 3 numbers = 7 × 3 = 21.
2. Find the total sum of the first two numbers: Since the average of the first two numbers is 4, the sum is 4 × 2 = 8.
3. Find the third number: The third number is the difference between the total sum of the three numbers and the sum of the first two numbers. Third number = 21 - 8 = 13.
Verification of options:
- Option (a) 13: Matches our calculation.
- Option (b) 14: Incorrect; would result in a total sum of 22.
- Option (c) 15: Incorrect; would result in a total sum of 23.
- Option (d) 16: Incorrect; would result in a total sum of 24.
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