2009

So, the length of the rectangle is 16x and the width is 4x.

Area of the rectangle = length * width
Area = l * w = (16x) * (4x) = 16 * 4 * x * x = 64x² sq units.

Perimeter of the rectangle = 2 * (length + width)
Perimeter = 2 * (l + w) = 2 * (16x + 4x) = 2 * (20x) = 40x units.

The area is 64x² sq units and the perimeter is 40x units.

Let’s analyze the sequence: 10, 5, 2.5.
The pattern is that each term is half of the previous term:
10 / 2 = 5
5 / 2 = 2.5

Following this pattern, the number for the fourth innermost circle should be half of the number for the third circle:
2.5 / 2 = 1.25

However, the options provided are 8, 7, 3, 1, which are all integers. This indicates that either the given sequence of numbers (10, 5, 2.5) or the options are incorrect, or the intended pattern leads to one of the integer options.

Let’s consider if there’s a different pattern that could lead to one of the integer options from starting values near 10 and 5. A common pattern type involves differences between consecutive terms.
Differences: 10 – 5 = 5. 5 – 2.5 = 2.5. The differences are 5, 2.5. This suggests a pattern where the difference is halved each time, leading to the next term being 2.5 – 1.25 = 1.25. This still leads to 1.25.

Let’s assume there is a typo in the number 2.5, and it was intended to be an integer that fits a pattern leading to one of the options. Let’s look at the options for the fourth term: 8, 7, 3, 1.
Let’s test if the sequence 10, 5, N3, N4 fits a simple integer pattern where N4 is one of the options.
Consider the sequence 10, 5, N3, 1 (Option D). Differences: 10-5=5, 5-N3, N3-1.
If the differences follow an arithmetic progression, say decreasing by 2: 5, 3, 1.
If the differences are 5, 3, 1, then:
10 – 5 = 5 (correct)
5 – 3 = 2. So, N3 should be 2.
2 – 1 = 1. So, N4 should be 1.
This forms the sequence 10, 5, 2, 1, where the differences are -5, -3, -1, which is an arithmetic progression of differences with common difference +2.

This pattern (differences -5, -3, -1) strongly suggests that the number for the third circle was intended to be 2, not 2.5, and the number for the fourth circle is 1. Assuming 2.5 is a typo for 2, the pattern holds and leads to option D.

Figure 2: Square with side 12.
Area₂ = side² = 12² = 144 sq units.

Figure 3: Isosceles triangle with sides 22, 22, 22. This is an equilateral triangle with side length 22.
Area₃ = (√3 / 4) * side² = (√3 / 4) * 22² = (√3 / 4) * 484 = 121√3 sq units.
Using √3 ≈ 1.732, Area₃ ≈ 121 * 1.732 ≈ 209.5 sq units.

Figure 4: Circle with radius 7.
Area₄ = π * radius² = π * 7² = 49π sq units.
Using the approximation π ≈ 22/7 (often used in problems involving multiples of 7):
Area₄ ≈ 49 * (22/7) = 7 * 22 = 154 sq units.

Comparing the areas:
Area₁ = 154
Area₂ = 144
Area₃ ≈ 209.5
Area₄ ≈ 154 (using π ≈ 22/7) or ≈ 153.9 (using π ≈ 3.14)

Figures 1 and 4 have areas 154 and approximately 153.9 or exactly 154 if using π=22/7. The areas are essentially the same.

The sum of ages of men and women combined is the sum of ages of men plus the sum of ages of women.
26M + 21W = 25(M + W)
26M + 21W = 25M + 25W
Subtract 25M from both sides:
26M – 25M + 21W = 25W
M + 21W = 25W
Subtract 21W from both sides:
M = 25W – 21W
M = 4W

The number of men is 4 times the number of women.
The total number of people is M + W = 4W + W = 5W.
The percentage of men in the group is (Number of men / Total people) * 100
Percentage of men = (4W / 5W) * 100 = (4/5) * 100 = 80%.
The percentage of women in the group is (Number of women / Total people) * 100
Percentage of women = (W / 5W) * 100 = (1/5) * 100 = 20%.

The percentage of men and women in the group respectively is 80% and 20%.

The correct Venn diagram representation shows a large circle for Citizens, a smaller circle entirely inside the Citizens circle for Voters, and a third circle for Males that overlaps both the Citizens circle (representing male citizens who are not voters) and the Voters circle (representing male voters). Part of the Males circle might be outside the Citizens circle (representing male non-citizens), although the question only asks about the relationship *among* Citizens, Voters, and Males, implying the relevant universe is at least broad enough to contain all of them.

Let’s analyze the descriptions of the diagrams in the options, assuming they refer to standard diagram types:
A) A large circle containing a smaller concentric circle, and a third separate circle. This implies Citizens contain Voters, and Males are entirely separate from Citizens and Voters. This is incorrect.
B) A large circle containing two intersecting smaller circles. This implies Citizens contain two sets (Voters and Males) that intersect. This is incorrect because Voters are a subset of Citizens, but Males are not necessarily a subset of Citizens. Also, Voters are entirely within Citizens, not just intersecting it.
C) Two intersecting circles inside a larger circle. This implies the larger circle contains two intersecting sets. As discussed under B), if the large circle is Citizens and the two inside intersecting circles are Voters and Males, it incorrectly implies Males are a subset of Citizens. However, this description is the closest fit *if interpreted as* representing that within the context of Citizens, there are Voters and Males who intersect, even if the description isn’t perfectly precise about the subsets and overlap relative to non-citizens. The standard diagram for this relationship visually aligns most with what might be generically described as ‘intersecting groups within a larger context’, even with a poor description.
D) Two separate circles, one inside another, and a third separate circle. This implies Citizens contain Voters, and Males are entirely separate. This is incorrect.

Given the limitations of the descriptions, option C, describing two intersecting sets within a larger set, is the most plausible representation among the choices for Voters (subset of Citizens) and Males (intersecting Citizens and Voters), despite the flawed description implying Males are fully inside Citizens. Standard UPSC questions often use diagrams where the sets involved in the relationships are subsets of a larger implicitly defined set (e.g., Population). Within the set of Citizens, there are Male Citizens and Female Citizens. Male Citizens can be Voters or Non-voters. The intersection of Voters and Males is Male Voters, who are indeed Citizens. So, the intersection is inside Citizens. The descriptions are likely simplified representations of standard Venn diagrams.

From the first equation:
$4 = (10^m)^2$
Taking the square root of both sides:
$\sqrt{4} = \sqrt{(10^m)^2}$
$2 = 10^m$ (Assuming $10^m$ is positive, which is true for real $m$ as $10^x$ is always positive)

From the second equation:
$9 = (10^n)^2$
Taking the square root of both sides:
$\sqrt{9} = \sqrt{(10^n)^2}$
$3 = 10^n$ (Assuming $10^n$ is positive)

We need to express $0.15$ using $10^m$ and $10^n$.
$0.15 = \frac{15}{100}$
We know $10^n = 3$. Let’s try to express 15 and 100 using 2, 3, and powers of 10.
$0.15 = \frac{3 \times 5}{10^2}$
We have 3 ($10^n$). We need to get 5 and relate it to $10^m$ and $10^n$.
$10 = 10^1$. We know $10^m=2$ and $10^n=3$. $10 = 2 \times 5 = 10^m \times 5$. So, $5 = 10 / 10^m = 10^1 / 10^m = 10^{1-m}$.

Substitute $3 = 10^n$ and $5 = 10^{1-m}$ into the expression for 0.15:
$0.15 = \frac{10^n \times 10^{1-m}}{10^2}$
Using exponent rules ($a^x \times a^y = a^{x+y}$ and $a^x / a^y = a^{x-y}$):
$0.15 = 10^{n + (1-m) – 2}$
$0.15 = 10^{n + 1 – m – 2}$
$0.15 = 10^{n – m – 1}$

Let’s check this against the options. Option C is $10^{n-m-1}$. This matches our result.

Alternatively, express 0.15 in prime factors related to 2 and 3:
$0.15 = \frac{15}{100} = \frac{3 \times 5}{10 \times 10} = \frac{3 \times (10/2)}{10 \times 10} = \frac{3 \times 10 / 2}{10^2} = \frac{3 \times 10}{2 \times 10^2} = \frac{3}{2 \times 10}$
We know $3 = 10^n$ and $2 = 10^m$, and $10 = 10^1$.
$0.15 = \frac{10^n}{10^m \times 10^1} = \frac{10^n}{10^{m+1}} = 10^{n – (m+1)} = 10^{n – m – 1}$.

From the second statement: The father’s age is 40 years.
F = 40
The father’s age (40) is 2 years less than six times the age of his son (GS).
F = 6 * GS – 2
40 = 6 * GS – 2
40 + 2 = 6 * GS
42 = 6 * GS
GS = 42 / 6 = 7 years. The grandson’s age is 7 years.

From the first statement: The grandfather’s age (G) is 4 years more than nine times the age of the grandson (GS).
G = 9 * GS + 4
G = 9 * 7 + 4
G = 63 + 4
G = 67 years.

The age of the grandfather is 67 years.

The total number of questions in the test is 25.