Tüm alıştırma soruları

Soru 121Soru →

The table below presents the organic horticulture harvest (in metric tonnes) across four agricultural zones in a state for the years 2022 to 2025:

Agricultural Zone	2022	2023	2024	2025
Zone A	320	360	400	480
Zone B	450	500	550	620
Zone C	280	310	350	420
Zone D	550	630	700	780

Based on the table, what is the percentage share of Zone B in the total horticulture harvest across all four agricultural zones combined in the year 2024?

Cevabı ve açıklamayı göster

Cevap: 27.5

Cevap

The percentage share of Zone B in the total horticulture harvest across all four zones combined in 2024 is

27.5\%

.

The total harvest across all four zones in 2024 is

400 + 550 + 350 + 700 = 2000

metric tonnes. Zone B contributed

550

metric tonnes in 2024. Dividing

550

by

2000

and multiplying by

100

yields exactly

27.5\%

.

Adım Adım Çözüm

1

Calculate total harvest for all zones in 2024

Total harvest in 2024 =

400 + 550 + 350 + 700 = 2000

metric tonnes

To find the baseline total harvest across all zones for the target year.

2

Locate the specific harvest for Zone B in 2024

Zone B harvest in 2024 =

550

metric tonnes

To isolate the target numerator specified in the question.

3

Compute the percentage share

Percentage share =

\left( \frac{550}{2000} \right) \times 100 = 27.5\%

Dividing the specific zone harvest by the total harvest yields the fractional share, which is converted to percentage by multiplying by 100.

Anahtar Kavram

Tabular Percentage Share Calculation

Tahmini Süre:1m 30s

Soru 122Soru →

A positive integer $N$ is divisible by $10$ . The sum of all distinct prime factors of $N$ is $10$ , and $N$ has exactly $12$ positive factors. What is the smallest possible value of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The smallest possible value of

N

is

60

.

Because

N

is divisible by

10

, its prime factorization must contain

2

and

5

. The sum of all distinct prime factors is

10

, which requires

2 + 5 + p = 10

, giving

p = 3

. Hence,

N = 2^a \times 3^b \times 5^c

with

a, b, c \ge 1

. The number of factors is

(a+1)(b+1)(c+1) = 12

. The unique product representation of

12

using three integers

\ge 2

is

3 \times 2 \times 2

, meaning the exponents are one

2

and two

1

s. Minimizing

N

requires assigning the highest exponent

2

to the smallest prime

2

, giving

N = 2^2 \times 3^1 \times 5^1 = 60

.

Adım Adım Çözüm

1

Identify the distinct prime factors of

N

The distinct prime factors are

2, 3,

and

5

.

Divisibility by

10

implies

2

and

5

are prime factors of

N

. The sum of distinct prime factors is

10

, so the remaining prime factor is

10 - (2 + 5) = 3

.

2

Determine the exponents in the prime factorization of

N

The set of exponents

\{a, b, c\}

must be

\{2, 1, 1\}

.

The total number of factors is

(a+1)(b+1)(c+1) = 12

. Since

a, b, c \ge 1

, the only integer partition of

12

into three factors

\ge 2

is

3 \times 2 \times 2

.

3

Assign exponents to minimize the value of

N

N = 2^2 \times 3^1 \times 5^1 = 60

.

To minimize

N

, the largest exponent

2

must be assigned to the smallest prime base

2

.

Anahtar Kavram

Prime factorization, distinct prime factor properties, and total factor count formula

Soru 123Soru →

The table below presents the operational and processing metrics for four municipal water treatment plants in a coastal state for FY 2025–26:

Plant Location	Total Raw Water Processed (Million Liters)	Purified Water Output (%)	Operational Cost per Million Liters (₹ Thousand)
Northern Zone	450	80%	12
Southern Zone	600	85%	10
Eastern Zone	500	78%	15
Western Zone	750	92%	14

Based on the table above, calculate the average volume of purified water output (in Million Liters) produced per plant across the four zones.

Cevabı ve açıklamayı göster

Cevap: 487.5

Cevap

The average volume of purified water output per plant is

487.5

Million Liters.

The purified water volume outputs for Northern, Southern, Eastern, and Western zones are

360

,

510

,

390

, and

690

Million Liters respectively. Summing these values gives a total output of

1950

Million Liters. Dividing this total by the

4

plants gives an average purified water output of

487.5

Million Liters per plant.

Adım Adım Çözüm

1

Calculate the purified water volume for each of the four treatment plants

Northern Zone =

360

Million Liters, Southern Zone =

510

Million Liters, Eastern Zone =

390

Million Liters, Western Zone =

690

Million Liters

Purified water output volume is calculated by multiplying Total Raw Water Processed by the Purified Water Output percentage.

2

Sum the purified water volumes across all four zones

Total purified output =

1950

Million Liters

The aggregate sum is required to compute the mean value across all plants.

3

Divide the total purified output by the number of plants (4)

Average purified output =

487.5

Million Liters

Dividing the total sum by the count of entities yields the arithmetic average.

Anahtar Kavram

Tabular Data Extraction and Weighted Average Calculation

Soru 124Soru →

If $\frac{3^{2m + 1} \cdot 9^{m - 2}}{27^{m - 1}} = 81$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

m

is 4.

By converting all terms in the equation to base 3 and applying index rules (

a^x \cdot a^y = a^{x+y}

and

\frac{a^x}{a^y} = a^{x-y}

), the expression simplifies directly to

3^m = 3^4

, leading to

m = 4

.

Adım Adım Çözüm

1

Express all terms with a common base of 3

9^{m-2} = (3^2)^{m-2} = 3^{2m-4}

and

27^{m-1} = (3^3)^{m-1} = 3^{3m-3}

Rewriting powers with a single prime base allows for simplification using the laws of indices.

2

Simplify the numerator using the product rule of exponents

3^{2m+1} \cdot 3^{2m-4} = 3^{(2m+1) + (2m-4)} = 3^{4m-3}

According to the product rule

a^x \cdot a^y = a^{x+y}

, exponents with the same base are added during multiplication.

3

Apply the quotient rule of exponents to simplify the left-hand side

\frac{3^{4m-3}}{3^{3m-3}} = 3^{(4m-3) - (3m-3)} = 3^m

According to the quotient rule

\frac{a^x}{a^y} = a^{x-y}

, exponents are subtracted during division.

4

Express the right-hand side in base 3 and solve for

m

3^m = 81 = 3^4 \implies m = 4

Since the bases on both sides of the equation are equal, the exponents must also be equal.

Anahtar Kavram

Exponential Equations and Laws of Indices

Soru 125Soru →

If $x = 7 - 4\sqrt{3}$ , what is the value of $\sqrt{x} + \frac{1}{\sqrt{x}}$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

\sqrt{x} + \frac{1}{\sqrt{x}}

is 4.

Expressing

7 - 4\sqrt{3}

as

(2 - \sqrt{3})^2

allows taking the square root to get

\sqrt{x} = 2 - \sqrt{3}

. Rationalizing its reciprocal gives

\frac{1}{\sqrt{x}} = 2 + \sqrt{3}

. Adding these two values cancels the irrational component

\sqrt{3}

, leaving

2 + 2 = 4

.

Adım Adım Çözüm

1

Simplify the nested surd

\sqrt{x} = \sqrt{7 - 4\sqrt{3}}

\sqrt{x} = 2 - \sqrt{3}

Rewrite

7 - 4\sqrt{3}

as

2^2 + (\sqrt{3})^2 - 2(2)(\sqrt{3}) = (2 - \sqrt{3})^2

and take the principal square root.

2

Calculate the reciprocal

\frac{1}{\sqrt{x}}

by rationalizing the denominator

\frac{1}{\sqrt{x}} = 2 + \sqrt{3}

Multiply numerator and denominator of

\frac{1}{2 - \sqrt{3}}

by its conjugate

(2 + \sqrt{3})

.

3

Add

\sqrt{x}

and

\frac{1}{\sqrt{x}}

4

Sum

(2 - \sqrt{3}) + (2 + \sqrt{3})

so that the radical terms cancel out.

Anahtar Kavram

Simplification of surds of the form

\sqrt{a \pm \sqrt{b}}

and rationalization using conjugates

Soru 126Soru →

In a state handloom and textile development initiative, data was collected from three handicraft clusters: Cluster A, Cluster B, and Cluster C. A total of $1,200$ weavers are registered across these three clusters. The number of weavers in Cluster C is $300$ , of which $40\%$ are female. The number of weavers in Cluster A is $25\%$ more than the number of weavers in Cluster B. In Cluster B, the ratio of male to female weavers is $3:2$ . If the total number of female weavers across all three clusters is $480$ , what is the total number of male weavers registered in Cluster A?

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

The total number of male weavers registered in Cluster A is 300.

By determining the total weavers in Cluster A (500) and Cluster B (400), and finding the female counts in Cluster C (120) and Cluster B (160), we deduce that Cluster A has 200 female weavers. Subtracting 200 from the 500 total weavers in Cluster A leaves 300 male weavers.

Adım Adım Çözüm

1

Determine total weavers in Clusters A and B combined

900 weavers

Subtract Cluster C weavers (300) from the overall total of 1,200 weavers.

2

Calculate weavers in Cluster B and Cluster A

Cluster B = 400 weavers, Cluster A = 500 weavers

Set Cluster A = 1.25 × Cluster B. Then 2.25 × Cluster B = 900, yielding Cluster B = 400 and Cluster A = 500.

3

Determine female weavers in Cluster C and Cluster B

Cluster C females = 120, Cluster B females = 160

40% of 300 is 120 for Cluster C. Based on the 3:2 male to female ratio in Cluster B, females account for 2/5 of 400, which is 160.

4

Calculate female weavers in Cluster A

200 female weavers

Subtract known females in B (160) and C (120) from the total 480 female weavers: 480 - (120 + 160) = 200.

5

Calculate male weavers in Cluster A

300 male weavers

Subtract female weavers in Cluster A from total weavers in Cluster A: 500 - 200 = 300.

Anahtar Kavram

Data extraction, multi-step ratio calculation, and percentage breakdown from paragraph caselets.

Soru 127Soru →

If $2^{x+1} + 2^{x-1} = 320$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is 7.

Factoring out

2^{x-1}

from the expression gives

2^{x-1}(2^2 + 1) = 320

, which simplifies to

5 \cdot 2^{x-1} = 320

. Dividing by 5 yields

2^{x-1} = 64 = 2^6

. Equating exponents gives

x - 1 = 6

, so

x = 7

.

Adım Adım Çözüm

1

Rewrite terms using exponent rules to factor out the common power

2^{x-1}

2^{x-1}(2^2 + 1) = 320

Using the product rule

a^{m+n} = a^m \cdot a^n

, we have

2^{x+1} = 2^{x-1} \cdot 2^2

.

2

Simplify the numerical factor inside the parentheses

5 \cdot 2^{x-1} = 320

2^2 + 1 = 4 + 1 = 5

.

3

Isolate the exponential term by dividing by 5

2^{x-1} = 64

Dividing

320

by

5

gives

64

.

4

Write 64 as a power of 2 and solve for

x

2^{x-1} = 2^6 \implies x - 1 = 6 \implies x = 7

Equating exponents when bases are equal.

Anahtar Kavram

Solving Exponential Equations by Factoring Common Powers

Soru 128Soru →

What is the remainder when the sum

22^{2026} + 39^{2026} + 56^{2026} + 73^{2026} + 90^{2026}

is divided by

17

?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

By reducing all bases modulo 17, the sum becomes five identical terms of

5^{2026}

, which simplifies to

5^{2027}

. Using Fermat's Little Theorem (

5^{16} \equiv 1 \pmod{17}

), the exponent 2027 is reduced modulo 16 to 11. Finally, calculating

5^{11} \pmod{17}

yields

-40

, which corresponds to a positive remainder of 11.

Adım Adım Çözüm

1

Reduce each base in the expression modulo 17.

The expression simplifies to

5^{2026} + 5^{2026} + 5^{2026} + 5^{2026} + 5^{2026} \pmod{17}

.

Using the property

(A \pmod M)^k \pmod M = A^k \pmod M

, we find that 22, 39, 56, 73, and 90 all leave a remainder of 5 when divided by 17.

2

Combine the identical terms into a single expression.

The sum becomes

5 \times 5^{2026} = 5^{2027} \pmod{17}

.

Since there are exactly 5 identical terms, adding them together is equivalent to multiplying the term by 5, which conveniently increases the exponent by 1.

3

Apply Fermat's Little Theorem to reduce the large exponent.

Fermat's Little Theorem states

5^{16} \equiv 1 \pmod{17}

. Dividing the exponent 2027 by 16 yields

2027 = 16 \times 126 + 11

.

Because 17 is a prime number and does not divide 5, we can use

a^{p-1} \equiv 1 \pmod p

to eliminate full cycles of 16 in the exponent.

4

Simplify the remaining expression

5^{11} \pmod{17}

.

The expression reduces to

-6 \pmod{17}

.

We break down the power:

5^2 = 25 \equiv 8

,

5^4 \equiv 8^2 = 64 \equiv -4

, and

5^8 \equiv (-4)^2 = 16 \equiv -1

. Thus,

5^{11} = 5^8 \times 5^2 \times 5^1 \equiv (-1) \times 8 \times 5 = -40

. Since

-40 = -3 \times 17 + 11

, the remainder is 11 (or

-6

).

5

Convert any negative remainder to a valid positive remainder.

-6 + 17 = 11

. The final remainder is 11.

A standard remainder must be a positive integer strictly less than the divisor.

Anahtar Kavram

Fermat's Little Theorem and Modulo Arithmetic Reductions

Soru 129Soru →

What is the smallest positive integer that has exactly $21$ positive divisors?

Cevabı ve açıklamayı göster

Cevap: 576

Cevap

The smallest positive integer with exactly 21 positive divisors is 576.

The number 576 factors completely into

2^6 \times 3^2

. According to the divisor formula, we take the exponents, add 1 to each, and multiply them:

(6+1) \times (2+1) = 7 \times 3 = 21

. Because 2 is the smallest prime and is paired with the largest exponent, 576 is the absolute minimum positive integer that can produce exactly 21 divisors.

Adım Adım Çözüm

1

Use the divisor counting formula.

The number of divisors of

N = p^a \times q^b

is

(a+1)(b+1)

. We set this equal to

21

.

This establishes the relationship between prime exponents and the given divisor count.

2

Find the factor pairs of 21.

The ways to factor 21 are

21

(as

21 \times 1

) or

7 \times 3

.

This determines the possible combinations of

(a+1)

and

(b+1)

.

3

Test the first case (single prime factor).

a+1 = 21 \implies a = 20

. The smallest number is

2^{20} = 1,048,576

.

We must evaluate all cases to ensure we find the absolute minimum.

4

Test the second case (two prime factors).

(a+1)(b+1) = 7 \times 3 \implies a = 6, b = 2

. The number form is

p^6 \times q^2

.

This evaluates the only other valid prime exponent combination.

5

Minimize the two-prime case.

Assign

p=2

and

q=3

to get

2^6 \times 3^2 = 64 \times 9 = 576

.

Assigning the smallest available prime to the largest exponent minimizes the product.

6

Compare the results of both cases.

576 < 1,048,576

.

To conclusively identify the smallest positive integer.

Anahtar Kavram

Prime Factorization and the Divisor Counting Formula

Soru 130Soru →

A delivery cyclist starts from a warehouse and rides $12\text{ km}$ towards the South. He then turns left and rides $9\text{ km}$ . What is the shortest direct distance, in kilometers, between his current location and the warehouse?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The shortest direct distance between the current location and the warehouse is

15\text{ km}

.

The straight-line displacement forms the hypotenuse of a right triangle with perpendicular sides of length

12\text{ km}

and

9\text{ km}

. Applying the Pythagorean theorem yields

\sqrt{12^2 + 9^2} = \sqrt{225} = 15\text{ km}

.

Adım Adım Çözüm

1

Determine the cardinal directions of each movement segment

First displacement is

12\text{ km}

South. Turning left while facing South directs the cyclist towards East, so the second displacement is

9\text{ km}

East.

A 90-degree left turn from South points towards East.

2

Set up the right-angled displacement triangle

The two perpendicular legs of the right triangle are

12\text{ km}

(South) and

9\text{ km}

(East).

South and East form a perpendicular angle (

90^\circ

).

3

Calculate the straight-line distance using the Pythagorean theorem

Distance

= \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15\text{ km}

.

The direct distance corresponds to the hypotenuse:

c = \sqrt{a^2 + b^2}

.

Anahtar Kavram

Shortest direct distance calculation using the Pythagorean theorem

Tahmini Süre:45s

Soru 131Soru →

A field researcher starting at a base station walks $15\text{ km}$ towards the North. He then turns $135^\circ$ clockwise and walks $8\sqrt{2}\text{ km}$ . Next, he turns $90^\circ$ counter-clockwise and walks $7\sqrt{2}\text{ km}$ . From this location, he turns West and walks $3\text{ km}$ , and finally turns South and walks $9\text{ km}$ to reach an observation post. What is the shortest straight-line distance (in km) between the observation post and the base station?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The shortest straight-line distance between the observation post and the base station is

13\text{ km}

.

Resolving each directional segment into horizontal and vertical components gives a net final coordinate of

(12, 5)

relative to the starting origin

(0, 0)

. Applying the Pythagorean theorem

\sqrt{12^2 + 5^2} = \sqrt{169} = 13\text{ km}

yields the exact straight-line distance.

Adım Adım Çözüm

1

Represent the trajectory using 2D coordinate vectors starting from origin (0, 0).

Initial position = (0, 0). After moving 15 km North, position = (0, 15).

North movement aligns with the positive y-axis.

2

Decompose the 135-degree clockwise turn vector (South-East displacement).

Displacement is (+8, -8), placing the researcher at (8, 7).

A 135-degree clockwise turn from North faces South-East (45 degrees below positive x-axis).

3

Decompose the 90-degree counter-clockwise turn vector (North-East displacement).

Displacement is (+7, +7), placing the researcher at (15, 14).

A 90-degree counter-clockwise turn from South-East faces North-East (45 degrees above positive x-axis).

4

Apply cardinal direction adjustments for West and South movements.

Moving 3 km West and 9 km South gives final coordinates (12, 5).

West subtracts from x-coordinate, and South subtracts from y-coordinate.

5

Calculate net displacement using Pythagorean theorem.

Distance = sqrt(12^2 + 5^2) = 13 km.

The direct line between (0, 0) and (12, 5) forms a right-angled triangle with sides 12 and 5.

Anahtar Kavram

Vector resolution of cardinal and angular displacements using Cartesian coordinates and the Pythagorean theorem

Tahmini Süre:3m 0s

Soru 132Soru →

A maintenance engineer at a solar power facility begins an inspection from the central monitoring unit facing East. He first walks $15\text{ m}$ straight ahead, then turns $90^\circ$ clockwise and walks $20\text{ m}$ . Next, he turns $135^\circ$ anti-clockwise and walks $10\sqrt{2}\text{ m}$ . He then turns $45^\circ$ clockwise and walks $5\text{ m}$ . Finally, he turns $90^\circ$ anti-clockwise and walks $10\text{ m}$ to reach his final inspection point. What is the shortest distance (in meters) between his final position and the central monitoring unit?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The shortest distance between the final position and the central monitoring unit is

30\text{ m}

.

By resolving each directional move into Cartesian coordinates

(x, y)

, the initial segment gives

(15, 0)

, the second segment gives

(15, -20)

, the third segment along North-East adds

(10, 10)

to reach

(25, -10)

, the fourth segment adds

(5, 0)

to reach

(30, -10)

, and the final segment adds

(0, 10)

to land exactly at

(30, 0)

. The straight-line distance from

(0,0)

to

(30,0)

is

30\text{ m}

.

Adım Adım Çözüm

1

Set up a 2D Cartesian coordinate system with the starting central monitoring unit at

(0,0)

facing East along the positive x-axis.

Initial position after

15\text{ m}

East is

(15, 0)

.

Tracking displacement using coordinates avoids directional confusion.

2

Determine facing direction and displacement after a

90^\circ

clockwise turn.

Facing direction becomes South. Position after moving

20\text{ m}

South is

(15, -20)

.

Turning

90^\circ

clockwise from East points directly South.

3

Calculate component displacements for a

135^\circ

anti-clockwise rotation from South.

Facing direction becomes North-East (

45^\circ

). Moving

10\sqrt{2}\text{ m}

adds

+10\text{ m}

to x and

+10\text{ m}

to y, yielding position

(25, -10)

.

Anti-clockwise rotation from South (

270^\circ

) by

135^\circ

leads to

45^\circ

(North-East), where

10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10\text{ m}

along each axis.

4

Apply the next movement of

5\text{ m}

after a

45^\circ

clockwise turn.

Facing direction turns from North-East back to East. Position becomes

(25 + 5, -10) = (30, -10)

.

Turning

45^\circ

clockwise from North-East realigns the path to due East.

5

Apply final turn of

90^\circ

anti-clockwise and movement of

10\text{ m}

.

Facing direction becomes North. Position becomes

(30, -10 + 10) = (30, 0)

.

Turning

90^\circ

anti-clockwise from East points North, adding

+10\text{ m}

to the y-coordinate.

6

Compute net displacement from starting point

(0,0)

to final point

(30,0)

.

Shortest distance =

\sqrt{(30-0)^2 + (0-0)^2} = 30\text{ m}

.

Applying the Euclidean distance formula gives the straight-line distance.

Anahtar Kavram

Multi-leg spatial vector addition using 2D Cartesian coordinates and trigonometric angular conversions.

Soru 133Soru →

In a region with a total population of $500,000$ , a socio-economic survey revealed that $75,000$ individuals live below the officially designated poverty line. What is the Head Count Ratio (HCR) of poverty in this region, expressed as a percentage?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The Head Count Ratio (HCR) of poverty in this region is 15%.

The Head Count Ratio (HCR) quantifies the proportion of a population that falls below the poverty line. By dividing the number of poor individuals (75,000) by the total population (500,000) and multiplying by 100, we get 15%.

Adım Adım Çözüm

1

Identify the given values from the survey data

Total population (

P

) = 500,000; Population below the poverty line (

N_p

) = 75,000

These inputs represent the total population sample and the sub-group living in absolute poverty.

2

Apply the Head Count Ratio formula for poverty measurement

HCR = \left(\frac{N_p}{P}\right) \times 100

The Head Count Ratio measures the proportion of the total population living below the poverty threshold.

3

Calculate the numerical percentage

HCR = \left(\frac{75,000}{500,000}\right) \times 100 = 15\%

Dividing 75,000 by 500,000 gives 0.15, which equals 15% when multiplied by 100.

Anahtar Kavram

Head Count Ratio (HCR) as a primary metric for measuring poverty incidence

Tahmini Süre:45s

Soru 134Soru →

During a financial year, a state government records a total Revenue Expenditure of ₹85,000 crore and total Revenue Receipts of ₹70,000 crore. If grants-in-aid released by the state for the creation of capital assets total ₹5,000 crore, what is the Effective Revenue Deficit of the state in ₹ crore?

Cevabı ve açıklamayı göster

Cevap: 10000

Cevap

The Effective Revenue Deficit of the state is ₹10,000 crore.

The Effective Revenue Deficit measures the difference between Revenue Deficit and grants-in-aid used for generating capital assets. First, find the Revenue Deficit (

85,000 - 70,000 = 15,000

crore). Subtracting the grants for capital assets (

15,000 - 5,000

) yields an Effective Revenue Deficit of ₹10,000 crore.

Adım Adım Çözüm

1

Calculate the standard Revenue Deficit.

Revenue Deficit = ₹15,000 crore.

Revenue Deficit represents the excess of total revenue expenditure over total revenue receipts:

85,000 - 70,000 = 15,000

.

2

Deduct the grants-in-aid earmarked for the creation of capital assets from the Revenue Deficit.

Effective Revenue Deficit = ₹10,000 crore.

Effective Revenue Deficit excludes grants given for capital asset formation as they lead to asset creation rather than pure consumption:

15,000 - 5,000 = 10,000

.

Anahtar Kavram

Effective Revenue Deficit

Soru 135Soru →

For a given financial year, the budgetary estimates of the Union Government of India are recorded as follows:
- Total Expenditure: $\text{₹}45,00,000$ crore
- Revenue Receipts: $\text{₹}22,50,000$ crore
- Non-Debt Capital Receipts (Recovery of Loans and Disinvestment): $\text{₹}75,000$ crore
- Interest Payments: $\text{₹}10,00,000$ crore

Based on these budgetary figures, what is the Gross Fiscal Deficit of the government for the financial year (in ₹ crore)?

Cevabı ve açıklamayı göster

Cevap: 2175000

Cevap

The Gross Fiscal Deficit of the government for the financial year is ₹21,75,000 crore.

Gross Fiscal Deficit measures the net borrowing requirement of the government. It is calculated as the difference between Total Expenditure and Total Non-Debt Receipts (Revenue Receipts + Non-Debt Capital Receipts). Substituting the given values: ₹45,00,000 crore - (₹22,50,000 crore + ₹75,000 crore) = ₹21,75,000 crore.

Adım Adım Çözüm

1

Calculate Total Non-Debt Receipts

₹23,25,000 crore

Non-debt receipts consist of Revenue Receipts plus Non-Debt Capital Receipts (such as recovery of loans and disinvestment proceeds).

2

Compute Gross Fiscal Deficit

₹21,75,000 crore

Gross Fiscal Deficit represents the total borrowing requirements of the government, defined as the excess of Total Expenditure over Total Non-Debt Receipts.

Anahtar Kavram

Gross Fiscal Deficit Calculation

Tahmini Süre:1m 30s

Soru 136Soru →

During a financial year, a State Government's budgetary accounts reveal the following figures:

- Total Revenue Receipts: ₹2,40,000 crore
- Total Revenue Expenditure: ₹2,85,000 crore
- Total Capital Expenditure: ₹65,000 crore
- Non-Debt Capital Receipts (Recovery of Loans and Disinvestment Proceeds): ₹25,000 crore
- Total Interest Payments on Past Debt: ₹32,000 crore

Based on the given budgetary data, what is the Primary Deficit of the State Government in ₹ crore?

Cevabı ve açıklamayı göster

Cevap: 53000

Cevap

The Primary Deficit of the State Government is ₹53,000 crore.

Primary Deficit is defined as Gross Fiscal Deficit minus Interest Payments. Gross Fiscal Deficit is calculated as Total Expenditure (Revenue Expenditure + Capital Expenditure) minus Total Non-Debt Receipts (Revenue Receipts + Non-Debt Capital Receipts). Substituting the given values: Total Expenditure = ₹2,85,000 crore + ₹65,000 crore = ₹3,50,000 crore; Total Non-Debt Receipts = ₹2,40,000 crore + ₹25,000 crore = ₹2,65,000 crore. Therefore, Gross Fiscal Deficit = ₹3,50,000 crore - ₹2,65,000 crore = ₹85,000 crore. Subtracting Interest Payments of ₹32,000 crore gives a Primary Deficit of ₹53,000 crore.

Adım Adım Çözüm

1

Calculate Total Budgetary Expenditure

₹3,50,000 crore

Total expenditure includes both operational revenue spending and long-term capital creation: Revenue Expenditure (₹2,85,000 crore) + Capital Expenditure (₹65,000 crore).

2

Calculate Total Non-Debt Receipts

₹2,65,000 crore

Non-debt receipts represent government income that does not create future repayment obligations: Revenue Receipts (₹2,40,000 crore) + Non-Debt Capital Receipts (₹25,000 crore).

3

Calculate Gross Fiscal Deficit

₹85,000 crore

Gross Fiscal Deficit measures total borrowing requirements: Total Expenditure (₹3,50,000 crore) - Total Non-Debt Receipts (₹2,65,000 crore).

4

Calculate Primary Deficit

₹53,000 crore

Primary Deficit isolates current financial year fiscal imbalance by subtracting past debt servicing burdens (Interest Payments of ₹32,000 crore) from the Gross Fiscal Deficit (₹85,000 crore).

Anahtar Kavram

Derivation of Fiscal Deficit and Primary Deficit from public budget aggregates

Soru 137Soru →

In a sample economy comprising $5$ individuals, the daily per capita consumption expenditures are recorded as $₹20$ , $₹30$ , $₹40$ , $₹50$ , and $₹110$ . The official poverty line for this economy is established at $₹50$ per day. What is the Poverty Gap Index ( $PGI$ ) of this economy, expressed as a percentage?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

The Poverty Gap Index of the economy is 24%

The Poverty Gap Index (

PGI

) is calculated as the average of normalized poverty gaps across the entire population:

PGI = \frac{1}{N} \sum_{i=1}^{N} \left(\frac{Z - y_i}{Z}\right)

for all

y_i < Z

. For this population (

N=5

,

Z=50

), the normalized gaps are

0.60

,

0.40

,

0.20

,

0

, and

0

. Taking the mean gives

1.20 / 5 = 0.24

, which equals

24\%

.

Adım Adım Çözüm

1

Identify poor individuals and determine consumption shortfalls relative to the poverty line

Three individuals have expenditures below

₹50

, with shortfalls of

₹30

,

₹20

, and

₹10

.

Poverty gap measures depth of poverty, considering only individuals below or at the poverty line (

y_i \le Z

).

2

Calculate the normalized poverty gap for each individual

Normalized gaps are

0.60

,

0.40

,

0.20

,

0.00

, and

0.00

.

Normalizing by the poverty line (

Z

) expresses individual shortfalls as proportions of

Z

.

3

Sum normalized gaps and average over total population N

PGI = \frac{1}{5} (0.60 + 0.40 + 0.20 + 0 + 0) = 0.24

or

24\%

.

The Poverty Gap Index formula is

PGI = \frac{1}{N} \sum_{i=1}^{N} \frac{\max(0, Z - y_i)}{Z}

.

Anahtar Kavram

Poverty Gap Index (PGI) calculation and intensity of poverty measurement

Soru 138Soru →

According to the budget estimates of a state government for the financial year 2025–26, total expenditure is estimated at ₹ $1,20,000$ crore, while total receipts excluding borrowings are estimated at ₹ $85,000$ crore. If the state's revenue receipts are ₹ $75,000$ crore, revenue expenditure is ₹ $95,000$ crore, and interest payments on previous debts account for ₹ $12,500$ crore, what is the Primary Deficit of the state for the financial year 2025–26 (in ₹ crore)?

Cevabı ve açıklamayı göster

Cevap: 22500

Cevap

The Primary Deficit of the state for the financial year 2025–26 is ₹22,500 crore.

Primary Deficit indicates the net borrowing requirement of the government to fulfill current fiscal commitments, excluding past debt obligations. Gross Fiscal Deficit is calculated as Total Expenditure minus Non-debt Receipts: ₹1,20,000 crore - ₹85,000 crore = ₹35,000 crore. Subtracting the interest payments of ₹12,500 crore yields a Primary Deficit of ₹22,500 crore.

Adım Adım Çözüm

1

Calculate the Fiscal Deficit of the state.

Fiscal Deficit = ₹1,20,000 crore - ₹85,000 crore = ₹35,000 crore.

Fiscal deficit measures the total borrowing requirements of the government, defined as the excess of total expenditure over total non-debt receipts.

2

Subtract Interest Payments from the Fiscal Deficit to compute the Primary Deficit.

Primary Deficit = ₹35,000 crore - ₹12,500 crore = ₹22,500 crore.

Primary deficit reflects the government's borrowing requirement for current expenses exclusive of past interest obligations.

Anahtar Kavram

Primary Deficit and Fiscal Deficit Metrics

Soru 139Soru →

In a financial year, the Union Government's Gross Tax Revenue is ₹ $35,000$ billion. From this total, ₹ $4,500$ billion is derived from cesses and surcharges, ₹ $500$ billion represents the cost of tax collection, and ₹ $1,000$ billion constitutes taxes collected from Union Territories without legislatures. If the Finance Commission mandates a $41\%$ vertical devolution rate of the net divisible pool to the states, what is the total amount (in ₹ billion) to be distributed among the states?

Cevabı ve açıklamayı göster

Cevap: 11890

Cevap

The total amount to be distributed among the states from the net divisible pool is ₹ 11,890 billion.

Under Article 270 of the Indian Constitution, the net proceeds of central taxes available for sharing (the Divisible Pool) exclude cesses, surcharges, tax collection costs, and tax revenues from Union Territories without legislatures. Subtracting ₹ 4,500 billion, ₹ 500 billion, and ₹ 1,000 billion from Gross Tax Revenue (₹ 35,000 billion) leaves a net pool of ₹ 29,000 billion. Multiplying this net pool by the 41% Finance Commission vertical devolution rate yields ₹ 11,890 billion.

Adım Adım Çözüm

1

Calculate the Net Divisible Pool of Union Taxes under Article 270.

Net Divisible Pool = ₹ 29,000 billion

Article 270 mandates that proceeds from cesses, surcharges, costs of collection, and taxes raised in Union Territories without legislatures are excluded from the divisible pool shared with states.

2

Apply the vertical tax devolution percentage mandated by the Finance Commission.

States' Total Devolution = ₹ 11,890 billion

The Finance Commission specifies vertical devolution (41%) as a percentage of the Net Divisible Pool.

Anahtar Kavram

Constitutional provisions under Article 270 for the calculation of the Net Divisible Pool of taxes and vertical tax devolution under Finance Commission recommendations.

Soru 140Soru →

In a socio-economic survey of a region, the total national income distribution across ten equal population deciles (from lowest to highest income group) is recorded as follows:

- Deciles 1 to 4 (poorest 40%): $2\%$ , $3\%$ , $4\%$ , and $6\%$ respectively
- Deciles 5 to 9 (middle 50%): $7\%$ , $9\%$ , $11\%$ , $13\%$ , and $15\%$ respectively
- Decile 10 (richest 10%): $30\%$

Based on this distribution, what is the Palma Ratio for this region?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The Palma Ratio for the region is 2.0.

The Palma Ratio measures economic inequality by dividing the share of national income held by the top 10% of the population by the share of national income held by the bottom 40%. Here, the share held by the top 10% (Decile 10) is 30%, while the cumulative share held by the bottom 40% (Deciles 1–4) is 2% + 3% + 4% + 6% = 15%. Dividing 30% by 15% yields a Palma Ratio of 2.0.

Adım Adım Çözüm

1

Identify the income share of the top 10% income group (Decile 10).

The top 10% population holds

30\%

of the total income.

The Palma Ratio specifically compares the richest 10% to the poorest 40%.

2

Sum the income shares for the poorest 40% of the population (Deciles 1 through 4).

Sum =

2\% + 3\% + 4\% + 6\% = 15\%

.

The bottom 40% share forms the denominator in the Palma Ratio calculation.

3

Divide the top 10% share by the bottom 40% share.

Palma Ratio =

\frac{30\%}{15\%} = 2.0

.

Applying the mathematical definition of the Palma Ratio.

Anahtar Kavram

Palma Ratio as an Inequality Metric

Tahmini Süre:1m 30s

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