All practice questions

Question 881Question →

What is the simplified numerical value of the following mathematical expression when evaluated strictly using the BODMAS rule?

50 - \left[ 8 + \left\{ 16 \div \left( 8 - \overline{7 - 3} \right) \right\} \right]

38

44

50

46

Show answer & explanation

Answer: 38

Answer

The simplified numerical value of the expression is 38.

Evaluating step-by-step according to VBODMAS order gives: first vinculum 7 - 3 = 4, then round brackets 8 - 4 = 4, followed by curly bracket division 16 ÷ 4 = 4, then square bracket addition 8 + 4 = 12, and finally 50 - 12 = 38. The value 38 correctly demonstrates strict adherence to order of operations.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\overline{7 - 3} = 4

According to VBODMAS rules, operations under a vinculum bar have the highest priority.

2

Evaluate the expression inside the round brackets

8 - 4 = 4

Substitute the vinculum result into the round bracket.

3

Perform division inside the curly brackets

16 \div 4 = 4

Curly brackets are evaluated next after simplifying the inner round brackets.

4

Perform addition inside the square brackets

8 + 4 = 12

Resolve the remaining operations inside the outermost square brackets.

5

Perform final subtraction outside the brackets

50 - 12 = 38

Complete the evaluation by subtracting the bracket value from 50.

Key Concept

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

Question 882Question →

A positive integer $N$ has a prime factorization of the form $p^a \cdot q^b$ , where $p$ and $q$ are distinct prime numbers, and $a, b \ge 1$ . If the total number of positive factors of $N^2$ is $45$ and the total number of positive factors of $N^3$ is $91$ , what is the total number of positive factors of $N$ ?

12

15

18

20

Show answer & explanation

Answer: $15$

Answer

The total number of positive factors of

N

is

15

.

For a composite integer

N = p^a q^b

, the number of positive factors is

(a+1)(b+1)

. Given that

N^2 = p^{2a} q^{2b}

has

45

positive factors, we have

(2a+1)(2b+1) = 45

. Factoring

45

into odd pairs yields two possibilities for

(a, b)

:

(1, 7)

or

(2, 4)

. Using the second condition that

N^3 = p^{3a} q^{3b}

has

91

factors, we check

(3a+1)(3b+1) = 91

. For

(1, 7)

,

(4)(22) = 88 \neq 91

. For

(2, 4)

,

(7)(13) = 91

, which is correct. Therefore,

a=2

and

b=4

, so the number of positive factors of

N

is

(2+1)(4+1) = 15

.

Step-by-Step Solution

1

Set up the factor count equation for

N^2

Since

N = p^a q^b

,

N^2 = p^{2a} q^{2b}

. The number of positive factors of

N^2

is

(2a+1)(2b+1) = 45

.

For a prime factorization

p^x q^y

, the total number of positive factors is

(x+1)(y+1)

.

2

Find possible integer solutions for

(a, b)

The odd factor pairs of

45

are

(3, 15)

and

(5, 9)

. This gives two possible sets of exponents: Case 1:

2a+1=3 \implies a=1

and

2b+1=15 \implies b=7

. Case 2:

2a+1=5 \implies a=2

and

2b+1=9 \implies b=4

.

Since

a, b \ge 1

, both factors

2a+1

and

2b+1

must be odd integers greater than

1

.

3

Evaluate candidate solutions using the second condition for

N^3

For

N^3 = p^{3a} q^{3b}

, the number of factors is

(3a+1)(3b+1) = 91

. Testing Case 1 (

a=1, b=7

):

(3(1)+1)(3(7)+1) = 4 \times 22 = 88 \neq 91

. Testing Case 2 (

a=2, b=4

):

(3(2)+1)(3(4)+1) = 7 \times 13 = 91

, which matches.

The true exponent values must simultaneously satisfy both given conditions.

4

Calculate the total number of positive factors of

N

With

a=2

and

b=4

,

N = p^2 q^4

. The total number of positive factors of

N

is

(2+1)(4+1) = 3 \times 5 = 15

.

Apply the total factor formula

(a+1)(b+1)

to

N

.

Key Concept

Calculating factor counts of a prime-factored number from equations involving factor counts of its powers

Estimated Time:2m 0s

Question 883Question →

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?

Show answer & explanation

Answer: 27.5

Answer

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\%

.

Step-by-Step Solution

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.

Key Concept

Tabular Percentage Share Calculation

Estimated Time:1m 30s

Question 884Question →

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 $N$ ?

Show answer & explanation

Answer: 60

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Question 885Question →

What is the simplified value of the following mathematical expression when evaluated strictly using the VBODMAS rule?

15 + 2.5 \times \left[ 16 - \left\{ 10 \div \left( 3.5 - \overline{2.1 - 0.6} \right) \right\} \right]

42.5

192.5

23.75

50

Show answer & explanation

Answer: 42.5

Answer

The simplified value of the expression is 42.5.

Evaluating strictly according to VBODMAS rules, we first simplify the expression under the vinculum bar (2.1 - 0.6 = 1.5), then the round brackets (3.5 - 1.5 = 2), followed by the curly brackets (10 ÷ 2 = 5), and then the square brackets (16 - 5 = 11). Finally, executing multiplication before addition yields 15 + (2.5 × 11) = 15 + 27.5 = 42.5.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{2.1 - 0.6} = 1.5

According to VBODMAS, the vinculum takes highest priority over standard brackets.

2

Evaluate the terms inside the innermost parentheses (round brackets)

3.5 - 1.5 = 2.0

Parentheses are evaluated next after resolving the vinculum.

3

Evaluate the expression inside the curly brackets

10 \div 2.0 = 5

Division within the curly brackets takes priority before moving outward.

4

Evaluate the expression inside the square brackets

16 - 5 = 11

Resolve the remaining operations within the square brackets.

5

Perform multiplication before addition

15 + (2.5 \times 11) = 15 + 27.5 = 42.5

Multiplication takes precedence over addition according to BODMAS rules.

Key Concept

Order of Operations (VBODMAS Rule)

Question 886Question →

The table below presents the operational metrics for four Inland Freight Terminals in a state for FY 2025–26:

Freight Terminal	Total Cargo Handled (in thousand metric tonnes)	Rail Transshipment Share (%)	Average Dwell Time per Container (days)	Revenue Generated per Tonne (₹)
North	450	40%	4.5	800
South	600	65%	3.0	750
East	350	50%	5.0	900
West	500	30%	6.0	850

What is the total revenue generated from cargo moved specifically via rail transshipment across all four freight terminals combined during FY 2025–26?

₹7.215 crore

₹72.15 crore

₹82.85 crore

₹71.69 crore

Show answer & explanation

Answer: ₹72.15 crore

Answer

₹72.15 crore

To find the total rail transshipment revenue, compute the rail-moved tonnage for each terminal: North (1.80 lakh tonnes), South (3.90 lakh tonnes), East (1.75 lakh tonnes), and West (1.50 lakh tonnes). Multiplying each by its respective revenue per tonne yields ₹14.40 crore, ₹29.25 crore, ₹15.75 crore, and ₹12.75 crore. Summing these four values gives exactly ₹72.15 crore.

Step-by-Step Solution

1

Convert cargo volumes from thousand metric tonnes to tonnes for each terminal

North: 4,50,000 tonnes; South: 6,00,000 tonnes; East: 3,50,000 tonnes; West: 5,00,000 tonnes.

The table header specifies cargo handled in 'thousand metric tonnes'.

2

Calculate the rail transshipment volume for each terminal

North: 4,50,000 × 40% = 1,80,000 tonnes
South: 6,00,000 × 65% = 3,90,000 tonnes
East: 3,50,000 × 50% = 1,75,000 tonnes
West: 5,00,000 × 30% = 1,50,000 tonnes

Rail transshipment share percentage is applied to total cargo volume of each terminal.

3

Compute revenue from rail transshipment for each terminal

North: 1,80,000 × ₹800 = ₹14,40,00,000 (₹14.40 crore)
South: 3,90,000 × ₹750 = ₹29,25,00,000 (₹29.25 crore)
East: 1,75,000 × ₹900 = ₹15,75,00,000 (₹15.75 crore)
West: 1,50,000 × ₹850 = ₹12,75,00,000 (₹12.75 crore)

Multiply rail transshipment tonnes by revenue per tonne.

4

Sum the rail transshipment revenues across all terminals

₹14.40 crore + ₹29.25 crore + ₹15.75 crore + ₹12.75 crore = ₹72.15 crore (₹72,15,00,000).

Total combined rail revenue is the sum of terminal-level rail revenues.

Key Concept

Multi-step weighted calculation from data grids involving unit scale factors and percentages.

Question 887Question →

If $5^{x+1} + 5^{x-1} = 650$ , what is the value of $2^{2x - 1}$ ?

32

64

16

8

Show answer & explanation

Answer: $32$

Answer

32

Factoring out

5^{x-1}

yields

26 \cdot 5^{x-1} = 650

, which simplifies to

5^{x-1} = 25

, giving

x = 3

. Substituting

x = 3

into

2^{2x - 1}

gives

2^5 = 32

.

Step-by-Step Solution

1

Factor out common exponential terms from the given equation.

5^{x-1}(5^2 + 1) = 650 \implies 5^{x-1}(25 + 1) = 650 \implies 26 \cdot 5^{x-1} = 650

Use the product law of indices

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

to rewrite the expression.

2

Solve for the unknown variable

x

.

5^{x-1} = \frac{650}{26} = 25 = 5^2 \implies x - 1 = 2 \implies x = 3

Since the bases are identical, equate the exponents.

3

Substitute

x = 3

into the expression

2^{2x - 1}

.

2^{2(3) - 1} = 2^{6 - 1} = 2^5 = 32

Evaluate the power after computing the exponent value.

Key Concept

Solving Exponential Equations using Laws of Indices

Question 888Question →

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.

Show answer & explanation

Answer: 487.5

Answer

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.

Step-by-Step Solution

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.

Key Concept

Tabular Data Extraction and Weighted Average Calculation

Question 889Question →

If $x = \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}}$ and $y = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}}$ , what is the value of $x^2 + y^2 + xy$ ?

142

143

144

145

Show answer & explanation

Answer: $143$

Answer

The value of

x^2 + y^2 + xy

is

143

.

Rationalizing both surds gives

x = 6 + \sqrt{35}

and

y = 6 - \sqrt{35}

. The sum

x + y = 12

and product

xy = 1

. Substituting into the algebraic identity

x^2 + y^2 + xy = (x + y)^2 - xy

gives

12^2 - 1 = 143

.

Step-by-Step Solution

1

Rationalize the denominator for

x

and

y

x = \frac{(\sqrt{7} + \sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{7 + 5 + 2\sqrt{35}}{2} = 6 + \sqrt{35}

, and similarly

y = 6 - \sqrt{35}

.

Eliminating radicals from denominators simplifies calculations.

2

Compute the sum

(x + y)

and product

xy

x + y = (6 + \sqrt{35}) + (6 - \sqrt{35}) = 12

, and

xy = (6 + \sqrt{35})(6 - \sqrt{35}) = 36 - 35 = 1

.

Using symmetric expressions simplifies evaluating degree 2 polynomials.

3

Express

x^2 + y^2 + xy

in terms of

(x + y)

and

xy

x^2 + y^2 + xy = (x + y)^2 - xy = 12^2 - 1 = 144 - 1 = 143

.

Applying the identity

x^2 + y^2 = (x + y)^2 - 2xy

gives

x^2 + y^2 + xy = (x + y)^2 - xy

.

Key Concept

Rationalization of surds and application of algebraic identities

Question 890Question →

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

Show answer & explanation

Answer: 4

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Exponential Equations and Laws of Indices

Question 891Question →

If $\left(\sqrt{\frac{3}{5}}\right)^{x + 1} = \left(\frac{125}{27}\right)^{x - 1}$ , what is the value of $x$ ?

\frac{5}{7}

\frac{7}{5}

\frac{1}{2}

\frac{1}{3}

Show answer & explanation

Answer: $\frac{5}{7}$

Answer

\frac{5}{7}

By converting both sides of the equation to the common base

\frac{3}{5}

, the left side becomes

\left(\frac{3}{5}\right)^{\frac{x+1}{2}}

and the right side becomes

\left(\frac{3}{5}\right)^{-3(x-1)}

. Equating exponents gives

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

, which simplifies directly to

x = \frac{5}{7}

.

Step-by-Step Solution

1

Express the left side of the equation using fractional exponents

\left(\sqrt{\frac{3}{5}}\right)^{x+1} = \left(\left(\frac{3}{5}\right)^{\frac{1}{2}}\right)^{x+1} = \left(\frac{3}{5}\right)^{\frac{x+1}{2}}

The square root of a quantity corresponds to an exponent of

\frac{1}{2}

.

2

Express the right side with base

\frac{3}{5}

\left(\frac{125}{27}\right)^{x-1} = \left(\left(\frac{5}{3}\right)^3\right)^{x-1} = \left(\left(\frac{3}{5}\right)^{-3}\right)^{x-1} = \left(\frac{3}{5}\right)^{-3(x-1)}

Since

125 = 5^3

and

27 = 3^3

,

\frac{125}{27} = \left(\frac{5}{3}\right)^3 = \left(\frac{3}{5}\right)^{-3}

.

3

Equate the exponents since the bases are identical

\frac{x+1}{2} = -3(x-1) \implies \frac{x+1}{2} = 3 - 3x

If

a^m = a^n

for

a > 0

and

a \neq 1

, then

m = n

.

4

Solve the linear equation for

x

x + 1 = 2(3 - 3x) \implies x + 1 = 6 - 6x \implies 7x = 5 \implies x = \frac{5}{7}

Standard algebraic simplification to isolate

x

.

Key Concept

Laws of Indices and Rational Base Equivalence

Question 892Question →

In a municipal solid waste management drive across three processing centers—Plant North, Plant South, and Plant East—a total of $10,000\text{ tonnes}$ of waste was processed in a month.

- Plant North processed $30\%$ of the total waste. Of the waste processed at Plant North, $60\%$ was biodegradable and the rest was non-biodegradable.
- Plant South processed $1.5$ times the amount of non-biodegradable waste processed by Plant North. The ratio of biodegradable to non-biodegradable waste processed at Plant South was $4 : 3$ .
- Plant East processed the remaining waste, with a ratio of biodegradable to non-biodegradable waste of $5 : 2$ .

Based on the information provided above, which of the following statements are correct?

Select all that apply

The total quantity of biodegradable waste processed across all three plants combined is

6,200\text{ tonnes}

.

The ratio of non-biodegradable waste processed by Plant South to that processed by Plant East is

9 : 4

.

Plant East processed

30\%

of the total waste collected across all three plants.

Plant South processed more non-biodegradable waste than the biodegradable waste processed by Plant North.

Show answer & explanation

Answer: The total quantity of biodegradable waste processed across all three plants combined is $6,200\text{ tonnes}$ .; The ratio of non-biodegradable waste processed by Plant South to that processed by Plant East is $9 : 4$ .

Answer

The statements confirming that total biodegradable waste across all plants is 6,200 tonnes and that the ratio of non-biodegradable waste between Plant South and Plant East is 9 : 4 are correct.

The total biodegradable waste is indeed

1,800 + 2,400 + 2,000 = 6,200\text{ tonnes}

, making the first statement correct. Furthermore, comparing non-biodegradable waste of Plant South (

1,800\text{ tonnes}

) and Plant East (

800\text{ tonnes}

) gives the ratio

1,800 : 800 = 9 : 4

, making the second statement correct.

Step-by-Step Solution

1

Calculate waste quantities for Plant North.

Total =

3,000\text{ tonnes}

; Biodegradable =

1,800\text{ tonnes}

; Non-biodegradable =

1,200\text{ tonnes}

.

Plant North processed

30\%

of

10,000\text{ tonnes} = 3,000\text{ tonnes}

. Biodegradable is

60\%

of

3,000 = 1,800\text{ tonnes}

, leaving

1,200\text{ tonnes}

non-biodegradable.

2

Calculate waste quantities for Plant South.

Non-biodegradable =

1,800\text{ tonnes}

; Biodegradable =

2,400\text{ tonnes}

; Total =

4,200\text{ tonnes}

.

Non-biodegradable at South is

1.5 \times 1,200 = 1,800\text{ tonnes}

. Given Bio : Non-bio ratio of

4 : 3

,

3\text{ parts} = 1,800\text{ tonnes} \implies 1\text{ part} = 600\text{ tonnes}

. Thus, Biodegradable =

4 \times 600 = 2,400\text{ tonnes}

.

3

Calculate waste quantities for Plant East.

Total =

2,800\text{ tonnes}

; Biodegradable =

2,000\text{ tonnes}

; Non-biodegradable =

800\text{ tonnes}

.

Remaining total waste =

10,000 - (3,000 + 4,200) = 2,800\text{ tonnes}

. Given Bio : Non-bio ratio of

5 : 2

, total parts =

7

.

1\text{ part} = 400\text{ tonnes}

. Biodegradable =

5 \times 400 = 2,000\text{ tonnes}

and Non-biodegradable =

2 \times 400 = 800\text{ tonnes}

.

4

Verify each statement against calculated values.

Total Biodegradable =

1,800 + 2,400 + 2,000 = 6,200\text{ tonnes}

(True). South Non-bio : East Non-bio =

1,800 : 800 = 9 : 4

(True). Plant East share =

2,800 / 10,000 = 28\%

(False). South Non-bio (

1,800

) vs North Bio (

1,800

) are equal (False).

Comparing calculated numerical values directly evaluates statement validity.

Key Concept

Caselet Data Extraction and Multi-Step Proportionate Allocation

Question 893Question →

Evaluate the two events described below to identify their causal relationship.

Event I: The metropolitan transit authority recently deployed a massive fleet of fully air-conditioned, highly subsidized electric buses across all major urban routes.
Event II: Private ride-hailing applications operating within the metropolitan area have reported an unprecedented 30% drop in passenger bookings over the last few weeks.

Which of the following logically explains the relationship between Event I and Event II?

Both Event I and Event II are separate effects resulting from independent causes.

Both Event I and Event II are parallel effects stemming from a common underlying cause.

Event II serves as the cause, and Event I is the logical effect of it.

Event I serves as the cause, and Event II is the logical effect of it.

Both Event I and Event II are independent causes of unstated phenomena.

Show answer & explanation

Answer: Event I serves as the cause, and Event II is the logical effect of it.

Answer

The first event serves as the cause, and the second event is the logical effect.

The deployment of a cheap, comfortable public transit alternative directly causes a shift in consumer behavior, reducing the demand for more expensive private ride-hailing services. Therefore, the first event is the direct catalyst for the second.

Step-by-Step Solution

1

Analyze Event I to determine its nature and potential market impact.

Event I involves the introduction of a highly subsidized, comfortable public transportation option.

Understanding the characteristics of the new service helps predict how consumers will react to a cheaper travel alternative.

2

Analyze Event II to determine its nature and possible triggers.

Event II involves a sudden decrease in demand for private ride-hailing services.

Identifying that Event II represents a negative demand shift prompts a search for a substitute service that may have attracted these customers.

3

Evaluate the chronological and logical link between the two events.

The deployment of the subsidized buses acts as a highly attractive substitute, which logically leads consumers to abandon more expensive private taxis, resulting in the booking drop.

A direct cause-and-effect relationship is confirmed when one event logically and directly accounts for the occurrence of the other.

Key Concept

Evaluating causal links through economic substitution and chronological feasibility.

Estimated Time:1m 15s

Question 894Question →

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

Show answer & explanation

Answer: 4

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Simplification of surds of the form

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

and rationalization using conjugates

Question 895Question →

What is the remainder when the value of the expression $2^{40} - 5 \times 3$ is divided by $10$ ?

9

1

3

6

Show answer & explanation

Answer: 1

Answer

1

The remainder when dividing by 10 is determined by the unit digit of the expression. First, evaluate the multiplication:

5 \times 3 = 15

. Next, find the unit digit of

2^{40}

. The powers of 2 follow a cyclicity of 4 (2, 4, 8, 6). Since 40 is perfectly divisible by 4, the unit digit corresponds to the 4th value in the cycle, which is 6. Substituting these into the expression gives

6 - 15 = -9

. To find the valid positive remainder modulo 10, the divisor 10 must be added to the negative result:

-9 + 10 = 1

.

Step-by-Step Solution

1

Evaluate the multiplication part of the expression.

The term

5 \times 3

equals

15

. The expression simplifies to

2^{40} - 15

.

The order of operations (BODMAS) requires multiplication to be performed before subtraction.

2

Determine the unit digit (remainder modulo 10) of

2^{40}

by identifying the cyclicity of base 2.

The unit digits of powers of 2 repeat in a cycle of 4: 2, 4, 8, 6.

Since dividing by 10 is equivalent to finding the unit digit, we use power cyclicity to simplify large exponents.

3

Divide the exponent 40 by the cyclicity length 4.

40 divided by 4 leaves a remainder of 0. A remainder of 0 corresponds to the 4th position in the cycle.

The remainder determines the position in the cyclicity sequence. A remainder of 0 means the sequence has completed a full cycle.

4

Substitute the determined unit digit back into the simplified expression.

The 4th power in the cycle ends in 6. Substituting this into the expression yields

6 - 15 = -9

.

Replacing the large power with its unit digit equivalent allows us to calculate the raw remainder.

5

Convert the negative remainder into a valid positive remainder modulo 10.

-9 + 10 = 1

. The final remainder is 1.

Remainders must be non-negative integers smaller than the divisor. Adding the divisor to a negative remainder provides the correct positive equivalent.

Key Concept

Applying cyclicity rules, the remainder theorem, and the order of operations to evaluate complex numerical expressions.

Question 896Question →

A state public transport corporation operates a total of $500$ buses across three urban divisions: Division X, Division Y, and Division Z. Division X accounts for $40\%$ of the total fleet, Division Y accounts for $30\%$ , and Division Z operates the remaining buses. In Division X, $25\%$ of the buses are electric. In Division Y, the ratio of electric buses to non-electric buses is $2 : 3$ . In Division Z, the number of electric buses is equal to $50\%$ of the number of electric buses in Division X. What is the ratio of the total number of non-electric buses to the total number of electric buses across all three divisions combined?

73 : 27

27 : 73

29 : 21

13 : 7

Show answer & explanation

Answer: 73 : 27

Answer

The ratio of the total number of non-electric buses to the total number of electric buses across all three divisions combined is 73 : 27.

Division X has 200 buses with 50 electric buses. Division Y has 150 buses with 60 electric buses (using the 2:3 ratio). Division Z has 150 buses with 25 electric buses (50% of 50). Total electric buses equal 135, leaving 365 non-electric buses. The simplified ratio of non-electric to electric buses is 365 : 135 = 73 : 27.

Step-by-Step Solution

1

Calculate the total fleet size for each division

Division X =

40\%

of

500 = 200

buses; Division Y =

30\%

of

500 = 150

buses; Division Z =

500 - (200 + 150) = 150

buses.

Establishing the total fleet per division is required to extract category values.

2

Calculate electric buses in each division

Division X electric =

25\%

of

200 = 50

buses. Division Y electric =

\frac{2}{2+3} \times 150 = 60

buses. Division Z electric =

50\%

of

50 = 25

buses.

Extracting the electric bus count per division using the specified percentages and ratios.

3

Calculate total electric and non-electric buses combined

Total Electric =

50 + 60 + 25 = 135

buses. Total Non-Electric =

500 - 135 = 365

buses.

Summing values across all divisions to find overall totals.

4

Compute the required ratio of non-electric to electric buses

Ratio =

365 : 135 = 73 : 27

.

Simplifying the ratio by dividing both terms by their greatest common divisor,

5

.

Key Concept

Caselet Data Extraction and Multi-step Ratio & Percentage Calculations

Estimated Time:1m 30s

Question 897Question →

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?

Show answer & explanation

Answer: 300

Answer

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.

Step-by-Step Solution

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.

Key Concept

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

Question 898Question →

The table below presents the operational and storage metrics for the State Warehousing Corporation across four district hubs for FY 2024–25:

District Hub	Total Storage Capacity ('000 MT)	Actual Grain Stored ('000 MT)	Spoilage Rate (%)	Milling Processing Fee (₹/MT)
Hub Alpha	$250$	$200$	$4\%$	$150$
Hub Beta	$400$	$350$	$2\%$	$120$
Hub Gamma	$300$	$240$	$5\%$	$200$
Hub Delta	$500$	$450$	$3\%$	$100$

Note: $1\text{ MT} = 1\text{ Metric Tonne}$ ; figures in capacity and stored columns are in thousands of MT.

If the government procurement cost of grain is $₹20,000$ per Metric Tonne, what is the total financial loss (in $₹\text{ Lakhs}$ ) incurred due to grain spoilage across Hub Alpha and Hub Gamma combined?

₹4,000\text{ Lakhs}

₹5,000\text{ Lakhs}

₹1,600\text{ Lakhs}

₹4\text{ Lakhs}

Show answer & explanation

Answer: $₹4,000\text{ Lakhs}$

Answer

The total financial loss incurred due to grain spoilage across Hub Alpha and Hub Gamma combined is

₹4,000\text{ Lakhs}

.

To find the total financial loss due to spoilage across Hub Alpha and Hub Gamma, we calculate the spoiled quantity for each hub based on the actual grain stored (in MT):
- Hub Alpha:

200,000\text{ MT} \times 4\% = 8,000\text{ MT}

- Hub Gamma:

240,000\text{ MT} \times 5\% = 12,000\text{ MT}

Total spoiled quantity

= 8,000 + 12,000 = 20,000\text{ MT}

.
At

₹20,000

per MT, total loss

= 20,000 \times 20,000 = ₹400,000,000

. Expressed in Lakhs (

1\text{ Lakh} = 100,000

), this equals

₹4,000\text{ Lakhs}

.

Step-by-Step Solution

1

Calculate the actual quantity of spoiled grain in Hub Alpha

Spoiled grain in Alpha =

4\% \times 200,000\text{ MT} = 8,000\text{ MT}

Actual grain stored in Alpha is

200 \times 1,000 = 200,000\text{ MT}

and spoilage rate is

4\%

.

2

Calculate the actual quantity of spoiled grain in Hub Gamma

Spoiled grain in Gamma =

5\% \times 240,000\text{ MT} = 12,000\text{ MT}

Actual grain stored in Gamma is

240 \times 1,000 = 240,000\text{ MT}

and spoilage rate is

5\%

.

3

Find the combined total quantity of spoiled grain

Total spoiled grain =

8,000\text{ MT} + 12,000\text{ MT} = 20,000\text{ MT}

Combine the spoiled quantities from both hubs.

4

Compute the total financial loss in Rupees and convert to Lakhs

Total Loss =

20,000\text{ MT} \times ₹20,000/\text{MT} = ₹400,000,000 = ₹4,000\text{ Lakhs}

Multiply total spoiled MT by the cost per MT (

₹20,000

) and divide by

100,000

to express in Lakhs.

Key Concept

Data Interpretation from Tables: Multi-step calculation involving unit scale conversion, percentage bases, and aggregation

Question 899Question →

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

Show answer & explanation

Answer: 7

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Solving Exponential Equations by Factoring Common Powers

Question 900Question →

Five professional athletes (M, N, O, P, and Q) are assigned to five adjacent lockers, numbered 1 to 5 from left to right. Each athlete plays a different sport (Boxing, Golf, Rugby, Soccer, or Tennis) and is sponsored by a different brand (Alpha, Beta, Gamma, Delta, or Epsilon). The following conditions apply to their locker assignments:

1. The Rugby player's locker is immediately to the left of the Golf player's locker.
2. M's locker is immediately to the right of the locker belonging to the athlete sponsored by Gamma.
3. N plays Tennis and is assigned Locker 3.
4. The athlete sponsored by Delta is assigned a locker at one of the extreme ends.
5. Q's locker is positioned somewhere to the left of O's locker.
6. The Soccer player is sponsored by Beta, and this locker is immediately to the right of N's locker.
7. P plays Boxing.
8. N is not sponsored by Alpha.

Based on the rules provided, which of the following correctly identifies the athlete assigned to Locker 2, along with their sport and sponsor?

M, Golf, Alpha

O, Soccer, Beta

M, Golf, Epsilon

Q, Rugby, Gamma

Show answer & explanation

Answer: M, Golf, Alpha

Answer

The athlete assigned to Locker 2 is M, who plays Golf and is sponsored by Alpha.

By mapping all rules to a 5-position grid, Locker 1 holds Q/Rugby/Gamma, Locker 2 holds M/Golf/Alpha, Locker 3 holds N/Tennis/Epsilon, Locker 4 holds O/Soccer/Beta, and Locker 5 holds P/Boxing/Delta. The required combination for Locker 2 is therefore M, Golf, and Alpha.

Step-by-Step Solution

1

Place known exact positions.

Locker 3 is N (Tennis).

Directly stated in Rule 3.

2

Use relative positioning clues anchored to knowns.

Locker 4 is the Soccer player sponsored by Beta.

Rule 6 states the Soccer player (Beta) is immediately to the right of N (Locker 3).

3

Deduce remaining sports based on adjacent pair constraints.

Locker 1 is Rugby, Locker 2 is Golf, and Locker 5 is Boxing.

Rule 1 requires Rugby and Golf to be adjacent in that order. The only adjacent empty sport slots are 1 and 2. Therefore, Locker 1 = Rugby, Locker 2 = Golf. By elimination, Locker 5 must be the final sport, Boxing.

4

Assign remaining athletes based on sports and relative clues.

Locker 5 is P, Locker 1 is Q, Locker 4 is O, and Locker 2 is M.

Rule 7 states P plays Boxing, so P is Locker 5. Rule 5 states Q is left of O. The empty athlete slots are 1 and 4, so Q is Locker 1 and O is Locker 4. This leaves Locker 2 for M.

5

Assign sponsors based on positional constraints.

Locker 1 is Gamma, Locker 5 is Delta.

Rule 2 states M (Locker 2) is immediately right of Gamma. Thus, Gamma is Locker 1. Rule 4 places Delta at an extreme end. Since Locker 1 is Gamma, Delta must be Locker 5.

6

Finalize remaining sponsors using negations.

Locker 3 is Epsilon, Locker 2 is Alpha.

The remaining sponsors are Alpha and Epsilon for N (Locker 3) and M (Locker 2). Rule 8 states N is not Alpha, so N is assigned Epsilon. M is assigned Alpha.

Key Concept

Logical Deduction with Multiple Constraints and Spatial Ordering

Alternative Method

You can solve this efficiently by first ignoring the athletes and sponsors and purely solving the sports timeline: N is Tennis at 3; Soccer is right of N at 4; Rugby is left of Golf which must fit into 1 and 2; leaving Boxing for 5. Once the sports are locked in, tying the athletes and sponsors to the sports is straightforward.

Estimated Time:2m 0s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept