Study the following table and answer the given question.
The table shows the expenditure of a family over the given years.
| Year | Expenditure in ₹ | ||||
| Food | Clothing | Education | Rent | Others | |
| 2011 | 2000 | 1200 | 3300 | 2000 | 4000 |
| 2012 | 2500 | 1100 | 3200 | 2300 | 4100 |
What is the total expenditure (in ₹) on food by the family for the year 2011 and 2012?
The problem requires us to calculate the total expenditure on food for the years 2011 and 2012 from the given table. We need to identify the expenditure on food for each year and then sum them up.
Total expenditure on Food \( = \) Expenditure on Food (2011) \( + \) Expenditure on Food (2012)
Total expenditure on Food \( = \) ₹2000 \( + \) ₹2500
Total expenditure on Food \( = \) ₹4500
Therefore, the total expenditure on food by the family for the years 2011 and 2012 is ₹4500.
C) 4,500 — This is the correct sum of food expenditures for 2011 (₹2000) and 2012 (₹2500).
To find the number of sides of a regular polygon given the number of diagonals, we use the formula for the number of diagonals in an n-sided polygon. We will set up an equation and solve for the number of sides, \(n\).
Recall the formula for the number of diagonals:
The number of diagonals, \(D\), in a polygon with \(n\) sides is given by the formula:
Substitute the given number of diagonals into the formula:
We are given that the polygon has 65 diagonals. So, \(D = 65\).
Solve the equation for \(n\):
First, multiply both sides by 2:
Rearrange the equation into a standard quadratic form \(an^2 + bn + c = 0\):
\[ n^2 - 3n - 130 = 0 \]Factor the quadratic equation:
We need two numbers that multiply to -130 and add to -3. These numbers are -13 and 10.
Find the possible values for \(n\):
\[ n - 13 = 0 \implies n = 13 \] \[ n + 10 = 0 \implies n = -10 \]Choose the valid number of sides:
Since the number of sides of a polygon cannot be negative, we discard \(n = -10\).
Therefore, the number of sides of the polygon is \(n = 13\).
C) 13
Find the volume of a sphere whose diameter is 84 m. [Use π = \( \frac{22}{7} \)]
To find the volume of a sphere, we first need to determine its radius from the given diameter. Then, we use the standard formula for the volume of a sphere, substituting the radius and the given value of \(\pi\).
Find the radius (r) from the diameter (d):
The diameter of the sphere is given as \(84 \text{ m}\).
The radius is half of the diameter:
\[ r = \frac{d}{2} = \frac{84 \text{ m}}{2} = 42 \text{ m} \]Use the formula for the volume of a sphere:
The formula for the volume (\(V\)) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]Substitute the values and calculate:
Given \(\pi = \frac{22}{7}\) and \(r = 42 \text{ m}\).
\[ V = \frac{4}{3} \times \frac{22}{7} \times (42)^3 \]\[ V = \frac{4}{3} \times \frac{22}{7} \times 42 \times 42 \times 42 \]Simplify the expression:
Cancel out common factors:
\[ V = 4 \times 22 \times \frac{42}{3} \times \frac{42}{7} \times 42 \]\[ V = 4 \times 22 \times 14 \times 6 \times 42 \]Perform the multiplication:
\[ V = 88 \times 14 \times 6 \times 42 \]\[ V = 88 \times 84 \times 42 \]\[ V = 7392 \times 42 \]\[ V = 310464 \text{ m}^3 \]B) 310,464 m³ is the correct volume calculated using the given diameter and the formula for the volume of a sphere.
The question asks about the monthly stipend amount for candidates under the Prime Minister's Internship Scheme, launched in October 2024. This is a factual question related to a government scheme, requiring knowledge of its specific provisions.
Correct Option: B) ₹5,000
The Prime Minister's Internship Scheme, launched in October 2024, aims to provide practical work experience to young graduates. A key feature of this scheme is the financial support offered to interns. The stipulated monthly stipend for the full 12-month internship period is ₹5,000.
To distribute a total amount in a given ratio, first find the sum of the ratio parts. Then, determine the fraction of the total amount that each person receives by dividing their respective ratio part by the sum of the ratio parts. Finally, multiply this fraction by the total amount to find each person's share.
Identify the total amount and the ratio:
Total amount to be distributed = ₹2,400
Ratio between A and B = 7 : 5
Calculate the sum of the ratio parts:
Sum of ratio parts \( = 7 + 5 = 12 \)
Determine B's share as a fraction of the total:
B's ratio part is 5.
Fraction of the total amount B receives \( = \frac{5}{12} \)
Calculate B's actual share:
B's share \( = \text{Fraction of total} \times \text{Total amount} \)
B's share \( = \frac{5}{12} \times 2400 \)
B's share \( = 5 \times \frac{2400}{12} \)
B's share \( = 5 \times 200 \)
B's share \( = 1000 \)
So, B's share is ₹1,000.
Correct Option: C) ₹1,000
The problem describes a pattern based on the English alphabetical order. We need to identify the relationship between the given pairs of letter groups and then apply the same logic to find the corresponding group for "PSQT".
B) UXVY is the correct option because applying a +5 shift to each letter of PSQT results in UXVY.
This problem can be solved using Venn diagrams or by applying the rules of syllogism. We will draw Venn diagrams for the given statements and then check which conclusions logically follow.
Statement 1: No dogs are pigs.
This means the set of 'Dogs' and the set of 'Pigs' are mutually exclusive. There is no overlap between them.
We can represent this as two separate circles: one for Dogs and one for Pigs, with no intersection.
Statement 2: All pigs are stones.
This means the set of 'Pigs' is entirely contained within the set of 'Stones'.
We can represent this by drawing the 'Pigs' circle completely inside the 'Stones' circle.
Combine the diagrams:
Draw a circle for 'Stones'. Inside this circle, draw a smaller circle for 'Pigs'. Separate from both of these, draw another circle for 'Dogs'.
Evaluate Conclusion (I): Some stones are pigs.
From our combined diagram, since all pigs are stones, it logically follows that some part of the stones must be pigs (specifically, the part that constitutes all pigs). If the set of Pigs is entirely inside the set of Stones, then there must be some stones that are pigs. This conclusion is true.
Evaluate Conclusion (II): No stones are dogs.
We know that 'No dogs are pigs'. We also know that 'All pigs are stones'. This means the 'Pigs' circle is inside the 'Stones' circle. The 'Dogs' circle is separate from the 'Pigs' circle.
However, the 'Dogs' circle could potentially overlap with the 'Stones' circle (the part of stones that are not pigs) or it could be entirely separate from the 'Stones' circle. The statements do not provide enough information to definitively say that 'No stones are dogs'. There is a possibility that some stones (those that are not pigs) could be dogs. Therefore, this conclusion does not logically follow.
Correct Option: B) Only conclusion (I) follows
If a sum amounts to 2\( \frac{1}{4} \) times the sum after 2 years on a certain rate of compound interest, compounded annually, then the rate of interest per annum is:
This problem involves compound interest. We are given the principal amount, the final amount after a certain period, and the time duration. We need to find the annual rate of interest. The formula for compound interest will be used to establish a relationship between these variables and then solve for the rate.
Let the principal sum be \(P\). The problem states that the sum amounts to \(2\frac{1}{4}\) times the sum. So, the Amount (\(A\)) after 2 years is:
\[ A = 2\frac{1}{4} P = \frac{9}{4} P \]The time period (\(n\)) is 2 years.
Let the rate of interest per annum be \(R\).
The formula for compound interest, compounded annually, is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]Substitute the given values into the formula:
\[ \frac{9}{4} P = P \left(1 + \frac{R}{100}\right)^2 \]Divide both sides by \(P\):
\[ \frac{9}{4} = \left(1 + \frac{R}{100}\right)^2 \]Take the square root of both sides:
\[ \sqrt{\frac{9}{4}} = 1 + \frac{R}{100} \]\[ \frac{3}{2} = 1 + \frac{R}{100} \]Subtract 1 from both sides:
\[ \frac{3}{2} - 1 = \frac{R}{100} \]\[ \frac{3-2}{2} = \frac{R}{100} \]\[ \frac{1}{2} = \frac{R}{100} \]Solve for \(R\):
\[ R = \frac{100}{2} \]\[ R = 50 \]Therefore, the rate of interest per annum is 50%.
B) 50% — As calculated, the rate of interest is 50% per annum.
The question asks about the Indian state where a 1,300-year-old stupa was discovered by the Archaeological Survey of India (ASI) in February 2023. This is a factual question related to recent archaeological findings in India. Recalling current affairs and significant discoveries is key to answering this.
Correct Option: A) Odisha
The 1,300-year-old stupa was indeed discovered by the Archaeological Survey of India (ASI) in February 2023 in the state of Odisha. Specifically, it was found during a conservation project near the Jagannath Temple in Parala, Jajpur district. This discovery is significant for understanding the Buddhist heritage of the region.
The question asks for the duration for which the Union Cabinet approved a one-time budgetary support of ₹150 crore for the International Big Cat Alliance (IBCA).
Correct Option: C) The Union Cabinet approved a one-time budgetary support of ₹150 crore for the International Big Cat Alliance (IBCA) for a period of 5 years.