All practice questions

Question 141Question →

In the annual budget estimates of a state government, the following budgetary figures are reported for a given financial year:
- Total Revenue Receipts: $₹ 2,50,000$ crore
- Non-debt Capital Receipts: $₹ 15,000$ crore
- Revenue Expenditure: $�� 3,10,000$ crore
- Capital Expenditure: $₹ 75,000$ crore
- Interest Payments on past public debt: $₹ 45,000$ crore

Based on the provided figures, what is the Primary Deficit of the state government for that financial year (in $₹$ crore)?

Show answer & explanation

Answer: 75000

Answer

The Primary Deficit of the government is ₹ 75,000 crore.

The Primary Deficit reflects the net borrowing required to meet current year expenses excluding committed interest costs. It is derived as Fiscal Deficit minus Interest Payments. First, Total Expenditure (

3,10,000 + 75,000 = 3,85,000

crore) minus Total Non-Debt Receipts (

2,50,000 + 15,000 = 2,65,000

crore) gives a Fiscal Deficit of

₹ 1,20,000

crore. Subtracting the interest payments of

₹ 45,000

crore yields a Primary Deficit of

₹ 75,000

crore.

Step-by-Step Solution

1

Calculate Total Expenditure

₹ 3,85,000 crore

Total Expenditure includes both current operational expenses (Revenue Expenditure) and creation of assets (Capital Expenditure).

2

Calculate Total Non-Debt Receipts

₹ 2,65,000 crore

Non-Debt Receipts consist of tax/non-tax revenues and capital receipts that do not create future repayment obligations (such as loan recoveries).

3

Calculate Fiscal Deficit

₹ 1,20,000 crore

Fiscal Deficit measures the total borrowing requirement of the government, obtained by subtracting total non-debt receipts from total expenditure.

4

Calculate Primary Deficit

₹ 75,000 crore

Primary Deficit indicates the current fiscal imbalances by excluding interest burdens accumulated from past debt liabilities.

Key Concept

Primary Deficit and Fiscal Deficit Metrics in Public Finance

Estimated Time:1m 30s

Question 142Question →

According to a state economic survey on income distribution, the top $10\%$ of the population holds $52\%$ of the total gross income, whereas the bottom $40\%$ of the population accounts for $13\%$ of the total gross income. Based on the standard Palma ratio metric for income inequality, what is the calculated Palma ratio for this state?

Show answer & explanation

Answer: 4

Answer

The calculated Palma ratio for the state is

4.0

.

The Palma ratio is defined as the ratio of the richest

10\%

of the population's share of gross national income divided by the poorest

40\%'s

share. Substituting the given values (

52\%

divided by

13\%

) gives a Palma ratio of

4.0

.

Step-by-Step Solution

1

Extract the relevant income shares from the economic survey data.

Income share of top

10\% = 52\%

, Income share of bottom

40\% = 13\%

.

The Palma ratio specifically compares the income share of the top decile to that of the bottom four deciles.

2

Calculate the ratio of the top decile share to the bottom four deciles share.

\text{Palma Ratio} = \frac{52}{13} = 4.0

Applying the formula: Palma Ratio = (Income share of top 10%) / (Income share of bottom 40%).

Key Concept

Palma Ratio as an Inequality Metric

Question 143Question →

Consider the following numerical series: $4, 7, 12, 19, 28, \dots$ . What is the next number in this series?

Show answer & explanation

Answer: 39

Answer

The next number in the series is

39

.

The difference between consecutive terms increases by 2 each time (

+3, +5, +7, +9

). The next difference to add is

+11

, yielding

28 + 11 = 39

.

Step-by-Step Solution

1

Calculate the difference between each pair of consecutive terms.

The differences are

3, 5, 7, 9

.

Identifying the first-order difference sequence reveals the pattern of progression.

2

Find the next value in the difference sequence.

The next odd number after

9

is

11

.

The pattern of differences consists of consecutive prime/odd increments (specifically, odd numbers

+2

each step).

3

Add the calculated difference to the final given term of the series.

28 + 11 = 39

Applying the pattern to find the subsequent term.

Key Concept

Number series completion using a sequence of increasing odd differences.

Question 144Question →

Consider the numerical sequence: $5, 11, 24, 51, 106, 217, \dots$ . What is the value of the next term in this sequence?

Show answer & explanation

Answer: 440

Answer

The next term in the sequence is 440.

The sequence follows the recurrence pattern where each term is twice the previous term plus an increasing integer offset:

5 \times 2 + 1 = 11

,

11 \times 2 + 2 = 24

,

24 \times 2 + 3 = 51

,

51 \times 2 + 4 = 106

,

106 \times 2 + 5 = 217

. Following this pattern, the next term is

217 \times 2 + 6 = 440

.

Step-by-Step Solution

1

Examine the relationship between successive terms.

The differences between successive terms are

6, 13, 27, 55, 111

, and the second-level differences are

7, 14, 28, 56

(doubling each step). Alternatively, each term is multiplied by 2 and increased by an incrementing integer.

Analyzing both direct operations and higher-order differences establishes the recurrence rule

a_{n+1} = 2a_n + n

.

2

Determine the required operation for the next step in the sequence.

Multiply the 6th term (217) by 2 and add 6.

The additive component increases by 1 at each term (

+1, +2, +3, +4, +5 \rightarrow +6

).

3

Calculate the value of the next term.

217 \times 2 + 6 = 440

.

217 \times 2 = 434

, and adding 6 yields 440.

Key Concept

Recurrent series with multiplicative factor and sequential linear increment

Estimated Time:1m 30s

Question 145Question →

A surveyor mapping a nature reserve starts from a base station. He walks $14\text{ km}$ due East, turns $90^\circ$ to his right, and walks $20\text{ km}$ . Next, he turns $135^\circ$ to his left and walks $10\sqrt{2}\text{ km}$ . Finally, he turns $45^\circ$ to his left and walks $3\text{ km}$ due North to reach an observation point. What is the shortest straight-line distance (in km) between the base station and the observation point?

Show answer & explanation

Answer: 25

Answer

The shortest straight-line distance between the base station and the observation point is 25 km.

Resolving each displacement into Cartesian coordinates gives a net horizontal displacement of

\Delta x = 14 + 10 = 24\text{ km}

(East) and a net vertical displacement of

\Delta y = -20 + 10 + 3 = -7\text{ km}

(South). Applying the Pythagorean theorem yields

\sqrt{24^2 + (-7)^2} = \sqrt{625} = 25\text{ km}

.

Step-by-Step Solution

1

Establish a Cartesian coordinate system with the base station as the origin

(0,0)

.

Initial coordinates are

(0,0)

facing East.

Assigning coordinates simplifies multi-turn vector displacement tracking.

2

Calculate coordinates after moving

14\text{ km}

East, turning

90^\circ

right (facing South), and walking

20\text{ km}

.

Position is

(14, -20)

facing South.

Moving East adds

14

to the x-coordinate, and moving South subtracts

20

from the y-coordinate.

3

Resolve the

10\sqrt{2}\text{ km}

walk after a

135^\circ

left turn from South into components.

Facing North-East, displacement components are

+10\text{ km}

East and

+10\text{ km}

North, yielding coordinates

(24, -10)

.

Turning

135^\circ

left from South points directly North-East (

45^\circ

north of east). The components are

10\sqrt{2}\cos(45^\circ) = 10\text{ km}

East and

10\sqrt{2}\sin(45^\circ) = 10\text{ km}

North.

4

Update coordinates after turning

45^\circ

left from North-East (facing North) and walking

3\text{ km}

.

Final position is

(24, -10 + 3) = (24, -7)

.

A

45^\circ

left turn aligns the direction due North, adding

3\text{ km}

to the y-coordinate.

5

Calculate the straight-line displacement from

(0,0)

to

(24, -7)

using the distance formula.

Distance

d = \sqrt{24^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25\text{ km}

.

The shortest distance between origin and final point is given by

d = \sqrt{\Delta x^2 + \Delta y^2}

.

Key Concept

Direction and Distance Test involving multi-turn vector resolution, angular rotations, and Pythagorean straight-line distance calculation.

Question 146Question →

Consider the following numerical sequence:

3, 11, 38, 102, 227, \dots

What is the value of the next term in this sequence?

Show answer & explanation

Answer: 443

Answer

The next term in the sequence is 443.

Each term increases by the cube of consecutive natural numbers starting from 2: +2³, +3³, +4³, +5³, and +6³. Adding 6³ (which equals 216) to the preceding term 227 yields 443.

Step-by-Step Solution

1

Find the difference between consecutive terms in the given sequence.

The differences are 8, 27, 64, and 125.

Analyzing first-order differences reveals the rate of increase between terms.

2

Identify the logical pattern governing the sequence of differences.

The differences correspond to consecutive perfect cubes: 2³, 3³, 4³, and 5³.

Recognizing that 8 = 2³, 27 = 3³, 64 = 4³, and 125 = 5³ establishes the cubic growth rule.

3

Calculate the next term by extending the identified pattern.

Next difference = 6³ = 216, resulting in next term = 227 + 216 = 443.

Following the consecutive cubic sequence, the fifth difference must be 6³.

Key Concept

Cube-Difference Series Progression

Question 147Question →

What is the simplified value of the following mathematical expression?

45 \div 5 \times 3 + [18 - \{14 - (8 - 3)\}]

Show answer & explanation

Answer: 36

Answer

36

Following the BODMAS rule step by step:
1. Innermost parenthesis:

8 - 3 = 5

.
2. Curly braces:

14 - 5 = 9

.
3. Square brackets:

18 - 9 = 9

.
4. Left-to-right division and multiplication:

45 \div 5 \times 3 = 9 \times 3 = 27

.
5. Addition:

27 + 9 = 36

.

Step-by-Step Solution

1

Evaluate the innermost round brackets

8 - 3 = 5

According to the BODMAS rule, operations inside the innermost brackets must be solved first.

2

Evaluate the expression inside the curly braces

14 - 5 = 9

Substitute the result from the round brackets into the curly braces.

3

Evaluate the expression inside the square brackets

18 - 9 = 9

Substitute the result from the curly braces into the square brackets.

4

Perform Division and Multiplication from left to right

45 / 5 * 3 = 9 * 3 = 27

Division and multiplication have equal precedence and must be evaluated strictly from left to right.

5

Add the result of the bracket terms to the result of the arithmetic terms

27 + 9 = 36

Final addition yields the simplified result.

Key Concept

BODMAS Rule (Brackets, Orders, Division and Multiplication left-to-right, Addition and Subtraction left-to-right)

Question 148Question →

The table below presents the freshwater fish production and pricing metrics across four regional inland lake zones managed by the State Fisheries Development Board for the year 2025:

Lake Zone	Total Catch (in Tonnes)	Export Grade Share (%)	Local Market Price (in ₹ per kg)	Export Market Price (in ₹ per kg)
Lake Alpha	400	30%	150	350
Lake Beta	500	40%	160	400
Lake Gamma	350	20%	140	300
Lake Delta	600	25%	180	420

Note: $1\text{ Tonne} = 1,000\text{ kg}$ and $1\text{ Lakh} = 100,000$ .

Based on the table, calculate the total revenue (in ₹ Lakhs) generated by Lake Beta from both local and export market sales combined.

Show answer & explanation

Answer: 1280

Answer

The total revenue generated by Lake Beta from local and export market sales combined is ₹1,280 Lakhs.

Lake Beta produces

500\text{ tonnes} = 500,000\text{ kg}

of fish.

40\%

of this (

200,000\text{ kg}

) is exported at ₹

400/\text{kg}

, yielding ₹

800\text{ Lakhs}

. The remaining

60\%

(

300,000\text{ kg}

) is sold locally at ₹

160/\text{kg}

, yielding ₹

480\text{ Lakhs}

. Combining both gives ₹

1,280\text{ Lakhs}

.

Step-by-Step Solution

1

Convert the total catch of Lake Beta to kilograms.

500\text{ tonnes} = 500,000\text{ kg}

.

Prices are given per kilogram, so total mass must be in kilograms.

2

Partition the total catch into export grade and local market quantities.

Export quantity =

40\% \times 500,000 = 200,000\text{ kg}

; Local quantity =

60\% \times 500,000 = 300,000\text{ kg}

.

Different market prices apply to export grade vs local grade catch.

3

Calculate monetary revenue for each grade and find their sum in ₹ Lakhs.

Export revenue =

200,000 \times 400 = ₹80,000,000 = ₹800\text{ Lakhs}

; Local revenue =

300,000 \times 160 = ₹48,000,000 = ₹480\text{ Lakhs}

; Total =

800 + 480 = ₹1,280\text{ Lakhs}

.

Summing both revenue streams gives the total earnings for Lake Beta.

Key Concept

Weighted revenue calculation from multi-column tabular data

Estimated Time:1m 30s

Question 149Question →

What is the unit digit of $7^{82}$ ?

Show answer & explanation

Answer: 9

Answer

The unit digit of

7^{82}

is 9.

The unit digits of powers of 7 repeat in a pattern of 4 (7, 9, 3, 1). Dividing the exponent 82 by 4 yields a remainder of 2. The second number in the cyclic pattern is 9, making 9 the unit digit of

7^{82}

.

Step-by-Step Solution

1

Find the cyclicity of the base number 7

The unit digits follow a repeating 4-step sequence: 7, 9, 3, 1.

Powers of 7 cycle every 4 powers because

7^1=7

,

7^2=49

,

7^3=343

, and

7^4=2401

.

2

Divide the exponent by the cyclicity period

82 \div 4 = 20

with a remainder of 2.

The remainder determines the equivalent power position within the 4-step cycle.

3

Determine the unit digit from the remainder

The unit digit of

7^2

is 9.

A remainder of 2 corresponds to

7^2

, giving 9.

Key Concept

Unit Digit Cyclicity

Question 150Question →

Calculate the exact numerical value of the following mathematical expression by applying the standard order of operations (BODMAS):

36.4 - \left[ 12.8 + \left\{ 6.5 \times \left( 4.4 - \overline{2.6 + 0.8} \right) \right\} \div 1.3 \right]

Show answer & explanation

Answer: 18.6

Answer

The simplified value of the mathematical expression is 18.6.

Strict application of BODMAS requires simplifying grouping symbols from the innermost vinculum outward, followed by resolving division prior to addition within brackets, yielding an exact answer of 18.6.

Step-by-Step Solution

1

Evaluate the expression underneath the vinculum (bar line)

\overline{2.6 + 0.8} = 3.4

The vinculum functions as an innermost bracket with highest evaluation priority.

2

Evaluate the terms inside the round brackets

4.4 - 3.4 = 1.0

Process operations inside round brackets next.

3

Perform multiplication within the curly braces

6.5 \times 1.0 = 6.5

Complete operations within the curly braces.

4

Execute division inside the square brackets

6.5 \div 1.3 = 5.0

Division takes precedence over addition according to BODMAS rules.

5

Perform addition within the square brackets

12.8 + 5.0 = 17.8

Finish evaluating all operations contained within the square brackets.

6

Perform final subtraction from left to right

36.4 - 17.8 = 18.6

Complete the outermost subtraction to arrive at the final simplified value.

Key Concept

BODMAS Rule with Vinculum and Decimals

Question 151Question →

What is the simplified value of $\sqrt{7 + 4\sqrt{3}} - \sqrt{3}$ ?

Show answer & explanation

Answer: 2

Answer

The simplified value is 2.

Expressing

7 + 4\sqrt{3}

as

(2 + \sqrt{3})^2

allows the square root to simplify directly to

2 + \sqrt{3}

. Subtracting

\sqrt{3}

leaves the exact numerical answer 2.

Step-by-Step Solution

1

Rewrite the expression under the square root as a perfect square of a binomial.

7 + 4\sqrt{3} = 2^2 + (\sqrt{3})^2 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2

Using the identity

(a+b)^2 = a^2 + b^2 + 2ab

, setting

a = 2

and

b = \sqrt{3}

yields

a^2 + b^2 = 4 + 3 = 7

and

2ab = 4\sqrt{3}

.

2

Evaluate the square root of the perfect square.

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

The principal square root of a positive squared expression

\sqrt{x^2}

is

x

.

3

Perform the subtraction indicated in the stem.

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

The radical terms

\sqrt{3}

and

-\sqrt{3}

cancel out, leaving the integer 2.

Key Concept

Simplification of Nested Surds

Question 152Question →

Find the exact numerical value of the following mathematical expression evaluated strictly according to the VBODMAS rule:

75\% \text{ of } 160 - \left[ 3.5 \times 8 + \left\{ 48 \div \left( 14 - \overline{5 \times 2} \right) \right\} \right]

Show answer & explanation

Answer: 80

Answer

The simplified numerical value of the expression is

80

.

Evaluating the expression following strict VBODMAS hierarchy: bar expression

\overline{5 \times 2} = 10

, round brackets

14 - 10 = 4

, curly brackets

48 \div 4 = 12

, square brackets

3.5 \times 8 + 12 = 40

, and percentage 'of' term

75\% \text{ of } 160 = 120

. Finally,

120 - 40 = 80

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{5 \times 2} = 10

According to VBODMAS, operations grouped under a bar take highest priority.

2

Evaluate inside the innermost round brackets

(14 - 10)

14 - 10 = 4

Perform subtraction inside the parentheses.

3

Evaluate inside the curly brackets

\{48 \div 4\}

48 \div 4 = 12

Divide the number outside the round bracket by the result of the round bracket.

4

Evaluate inside the square brackets

[3.5 \times 8 + 12]

28 + 12 = 40

Perform multiplication (

3.5 \times 8 = 28

) before adding

12

.

5

Calculate the percentage term

75\% \text{ of } 160

\frac{75}{100} \times 160 = 120

Evaluate the 'of' operation before final subtraction.

6

Perform final subtraction

120 - 40

80

Subtract the total result of the bracketed expression from the percentage term.

Key Concept

Order of Operations (VBODMAS Rule)

Question 153Question →

Evaluate the following numerical expression strictly according to the VBODMAS rule:

2.5 \times 12 - \left[ 8 + \left\{ 15 \div \left( 4.5 - \overline{2.1 + 0.9} \right) \right\} \right]

What is the final calculated value?

Show answer & explanation

Answer: 12

Answer

12

Evaluating the expression following strict VBODMAS priority yields: Vinculum (2.1 + 0.9 = 3) -> Round brackets (4.5 - 3 = 1.5) -> Curly brackets (15 / 1.5 = 10) -> Square brackets (8 + 10 = 18) -> Multiplication (2.5 * 12 = 30) -> Subtraction (30 - 18 = 12).

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

2.1 + 0.9 = 3

According to the VBODMAS rule, operations under a vinculum take highest priority.

2

Evaluate the innermost round brackets

4.5 - 3 = 1.5

Perform subtraction inside the round brackets after resolving the vinculum.

3

Perform division within the curly brackets

15 / 1.5 = 10

Division inside curly brackets must be calculated before simplifying the outer bracket.

4

Perform addition within the square brackets

8 + 10 = 18

Complete the evaluation of the square bracket section.

5

Perform multiplication outside the brackets

2.5 * 12 = 30

Multiplication takes precedence over final subtraction.

6

Perform final subtraction

30 - 18 = 12

Complete the expression by subtracting the bracket result from the product.

Key Concept

VBODMAS Rule (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Question 154Question →

Find the total number of positive factors of the integer $75$ .

Show answer & explanation

Answer: 6

Answer

The total number of positive factors of 75 is 6.

The prime factorization of 75 is

3^1 \times 5^2

. The number of positive factors is obtained by adding 1 to each exponent in the prime factorization and multiplying the results:

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

. The complete list of positive factors is 1, 3, 5, 15, 25, and 75.

Step-by-Step Solution

1

Express 75 in terms of its prime factors.

75 = 3^1 \times 5^2

Prime factorization breaks down the number into prime components essential for counting factors.

2

Apply the standard formula for total positive factors of a number

N = p^a \times q^b

.

Total factors

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

Each factor is formed by choosing a power of 3 (2 choices:

3^0, 3^1

) and a power of 5 (3 choices:

5^0, 5^1, 5^2

).

Key Concept

Total Number of Positive Factors from Prime Factorization

Question 155Question →

What is the total number of positive integer factors of $120$ ?

Show answer & explanation

Answer: 16

Answer

The total number of positive integer factors of

120

is

16

.

By prime factorizing

120

, we get

2^3 \times 3^1 \times 5^1

. Applying the factor count formula

(a+1)(b+1)(c+1)

, we obtain

(3+1)(1+1)(1+1) = 4 \times 2 \times 2 = 16

.

Step-by-Step Solution

1

Find the prime factorization of

120

120 = 2^3 \times 3^1 \times 5^1

To calculate the total number of factors systematically, express the number as a product of prime factors.

2

Apply the total factor count formula

Number of factors =

(3 + 1)(1 + 1)(1 + 1)

If a number

N = p^a \times q^b \times r^c

, the total number of positive factors is

(a+1)(b+1)(c+1)

.

3

Compute the product

4 \times 2 \times 2 = 16

Multiplying the incremented prime exponents gives the total count of factors.

Key Concept

Number of Positive Factors from Prime Factorization

Question 156Question →

If $\frac{2^{x+3} \cdot 4^{2x-1}}{8^{x+1}} = 64$ , what is the value of $(x + 2)^2$ ?

Show answer & explanation

Answer: 36

Answer

The value of

(x + 2)^2

is 36.

Converting all terms to base 2 simplifies the equation to

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

. Equating the powers gives

2x - 2 = 6

, so

x = 4

. Substituting

x = 4

into

(x + 2)^2

yields

(4 + 2)^2 = 36

.

Step-by-Step Solution

1

Convert all exponential terms to base 2

Numerator term

4^{2x-1} = 2^{4x-2}

, denominator term

8^{x+1} = 2^{3x+3}

, and right-hand side

64 = 2^6

.

To combine powers using index laws, all expressions must share a common base.

2

Simplify the left-hand side expression using exponent laws

\frac{2^{x+3} \cdot 2^{4x-2}}{2^{3x+3}} = \frac{2^{5x+1}}{2^{3x+3}} = 2^{(5x+1)-(3x+3)} = 2^{2x-2}

.

Apply product law

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

and quotient law

\frac{a^m}{a^n} = a^{m-n}

.

3

Solve for the variable x

2^{2x-2} = 2^6 \implies 2x - 2 = 6 \implies x = 4

.

When bases are equal, exponents must be equal.

4

Evaluate the requested target expression

(4 + 2)^2 = 6^2 = 36

.

Substitute the calculated value of

x = 4

into

(x + 2)^2

.

Key Concept

Laws of Indices and Exponential Equations

Estimated Time:1m 30s

Question 157Question →

Consider the number obtained by multiplying $48$ and $53$ . If this product is divided by $50$ , what is the resulting positive remainder?

Show answer & explanation

Answer: 44

Answer

44

Applying the properties of modular arithmetic,

48

is congruent to

-2 \pmod{50}

and

53

is congruent to

3 \pmod{50}

. Multiplying these gives a remainder of

-6 \pmod{50}

. To find the standard positive remainder, we add the divisor

50

to

-6

, resulting in

44

. Alternatively, direct calculation gives

48 \times 53 = 2544

, and

2544

divided by

50

yields a quotient of

50

with a remainder of

44

.

Step-by-Step Solution

1

Find the remainder of 48 divided by 50.

-2

Using a negative remainder simplifies the multiplication step.

2

Find the remainder of 53 divided by 50.

3

Individual remainders are needed to apply the product rule of modular arithmetic.

3

Multiply the individual remainders.

-2 \times 3 = -6

The remainder of a product is congruent to the product of the individual remainders.

4

Convert the negative remainder to a positive remainder.

50 - 6 = 44

A standard remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Remainder Theorem and Negative Modulus

Question 158Question →

A cryptography algorithm generates a continuous stream of security tokens based on a polynomial progression. The first five tokens generated by the system are $5$ , $12$ , $31$ , $68$ , and $129$ .

What is the numerical value of the sixth token generated by this algorithm?

Show answer & explanation

Answer: 220

Answer

220

The sequence follows the mathematical rule where the

n

-th term is equal to

n^3 + 4

. The first term is

1^3 + 4 = 5

, the second is

2^3 + 4 = 12

, up to the fifth which is

5^3 + 4 = 129

. Applying this rule to the sixth position yields

6^3 + 4 = 216 + 4 = 220

. Alternatively, solving via successive differences confirms that the constant third difference is 6, which accurately points to 220.

Step-by-Step Solution

1

Calculate the first-level differences between the given consecutive tokens.

The sequence of differences is 7, 19, 37, and 61.

Establishing the initial rate of change helps identify if a linear or higher-order polynomial pattern exists.

2

Calculate the second-level differences from the results of Step 1.

The differences between the differences are 12, 18, and 24.

Finding the differences of the differences reveals simpler underlying arithmetic progressions.

3

Identify the pattern in the second-level differences and extrapolate the next value.

The values (12, 18, 24) increase by exactly 6 each time. The next second-level difference is 24 + 6 = 30.

Extending this constant third-level difference (+6) is required to build the sequence forward.

4

Calculate the next first-level difference and the final sequence term.

Next first-level difference = 61 + 30 = 91. Next sequence term = 129 + 91 = 220.

Applying the extrapolated values back up the chain yields the target term in the main sequence.

Key Concept

Number series completion using the method of successive differences or cube offsets.

Estimated Time:1m 15s

Question 159Question →

If the equation $\sqrt{x} + \sqrt{x - 9} = 9$ holds true, what is the exact value of $x$ ?

Show answer & explanation

Answer: 25

Answer

25

The exact value of

x

is 25. By moving one radical to the right side and squaring both sides, we eliminate one square root. Simplifying and isolating the remaining square root allows us to square both sides a second time, revealing the final value. Alternatively, using the conjugate property of surds: multiplying both sides of the identity

(\sqrt{x})^2 - (\sqrt{x-9})^2 = 9

by their difference gives

(\sqrt{x} - \sqrt{x-9})(\sqrt{x} + \sqrt{x-9}) = 9

. Since the sum is 9, the difference must be 1 (i.e.,

\sqrt{x} - \sqrt{x-9} = 1

). Adding this back to the original equation yields

2\sqrt{x} = 10

, so

\sqrt{x} = 5

and

x = 25

.

Step-by-Step Solution

1

Isolate one of the square root terms on one side of the equation.

\sqrt{x} = 9 - \sqrt{x - 9}

Isolating a radical makes it easier to eliminate it by squaring both sides.

2

Square both sides of the equation and expand the right side.

x = 81 - 18\sqrt{x - 9} + (x - 9)

Squaring eliminates the isolated radical. The right side is expanded using the algebraic identity

(a - b)^2 = a^2 - 2ab + b^2

.

3

Simplify the equation by canceling

x

from both sides and combining constant terms.

x = x + 72 - 18\sqrt{x - 9} \Rightarrow 18\sqrt{x - 9} = 72

Combining like terms simplifies the equation, leaving only a single radical expression.

4

Divide by 18 and square both sides one final time to solve for

x

.

\sqrt{x - 9} = 4 \Rightarrow x - 9 = 16 \Rightarrow x = 25

Isolating the final radical and squaring removes the remaining root, yielding a simple linear equation for

x

.

Key Concept

Solving radical equations and applying algebraic identities with surds.

Question 160Question →

A language institute conducted a survey among $400$ civil service aspirants to assess their proficiency in three foreign languages: French, German, and Spanish. The survey revealed the following data:
- $180$ aspirants are proficient in French.
- $150$ aspirants are proficient in German.
- $160$ aspirants are proficient in Spanish.
- $40$ aspirants are proficient in exactly French and German, but not Spanish.
- $30$ aspirants are proficient in exactly German and Spanish, but not French.
- $50$ aspirants are proficient in exactly French and Spanish, but not German.
- $70$ aspirants are not proficient in any of these three languages.

What is the number of aspirants who are proficient in all three languages?

Show answer & explanation

Answer: 20

Answer

20

By determining the union of the three sets (

400 - 70 = 330

) and applying the inclusion-exclusion principle while correctly distinguishing between 'exactly two' and the full intersection of two sets, we find that

20

aspirants are proficient in all three languages.

Step-by-Step Solution

1

Determine the number of aspirants proficient in at least one of the three languages.

n(F \cup G \cup S) = 400 - 70 = 330

The total population consists of those who speak at least one language and those who speak none.

2

Set up an equation using the Principle of Inclusion-Exclusion for three sets. Let

x

be the number of aspirants proficient in all three languages.

n(F \cap G) = 40 + x

,

n(G \cap S) = 30 + x

, and

n(F \cap S) = 50 + x

The total intersection of any two sets includes those in exactly those two sets plus those in all three sets.

3

Substitute all values into the union formula.

330 = 180 + 150 + 160 - (40 + x) - (30 + x) - (50 + x) + x

The formula

n(F \cup G \cup S) = n(F) + n(G) + n(S) - n(F \cap G) - n(G \cap S) - n(F \cap S) + n(F \cap G \cap S)

accounts for all overlapping regions.

4

Simplify the equation and solve for

x

.

330 = 490 - 120 - 2x \implies 330 = 370 - 2x \implies 2x = 40 \implies x = 20

Basic algebraic simplification yields the final value for the intersection of all three sets.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Alternative Method

Instead of using the union formula, use a region-based approach in a Venn diagram. Let the central 'all three' region be

x

. Calculate the 'only one' regions in terms of

x

: Only French =

180 - (40 + 50 + x) = 90 - x

. Only German =

150 - (40 + 30 + x) = 80 - x

. Only Spanish =

160 - (50 + 30 + x) = 80 - x

. The sum of all disjoint regions inside the union is

(90 - x) + (80 - x) + (80 - x) + 40 + 30 + 50 + x = 370 - 2x

. Since the union is

400 - 70 = 330

, we have

370 - 2x = 330

, which gives

x = 20

.

Estimated Time:2m 0s

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