Tüm alıştırma soruları

Soru 841Soru →

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

45 - \left[ 15 + \left\{ 24 \div \left( 3 + \frac{1}{2} \text{ of } \left( 16 - \overline{8 - 2} \right) \right) \right\} \right]

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The simplified numerical value of the given expression is 27.

Evaluating step-by-step using VBODMAS order of precedence (Vinculum -> Brackets -> Of -> Division -> Multiplication -> Addition -> Subtraction) yields the simplified result 27.

Adım Adım Çözüm

1

Evaluate the vinculum (bar) operation

\overline{8 - 2} = 6

According to VBODMAS, operations under a vinculum take precedence over standard bracket operations.

2

Simplify the innermost subtraction inside the round brackets

16 - 6 = 10

Subtracted the value obtained from the vinculum within the innermost parentheses.

3

Evaluate the 'of' operator

\frac{1}{2} \text{ of } 10 = 5

The 'of' operator represents multiplication with priority immediately following brackets.

4

Complete the addition in the round brackets

3 + 5 = 8

Evaluated the remaining addition within the round brackets.

5

Perform division inside the curly brackets

24 \div 8 = 3

Divided 24 by the simplified result of the round brackets.

6

Perform addition inside the square brackets

15 + 3 = 18

Added 15 to the result from the curly brackets.

7

Perform final subtraction

45 - 18 = 27

Subtracted the square bracket result from 45 to obtain the final simplified value.

Anahtar Kavram

Order of Operations (VBODMAS Rule)

Tahmini Süre:1m 30s

Soru 842Soru →

What is the unit digit of the expression $S = 2^{84} + 4^{97} \times 3^{54} - 9^{45}$ ?

3

8

2

1

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The unit digit of the expression is 3.

The unit digit of

2^{84}

is 6 (since 84 is a multiple of 4, corresponding to the 4th position in cyclicity). The product

4^{97} \times 3^{54}

has a unit digit of

(4 \times 9) \pmod{10} = 6

. The unit digit of

9^{45}

is 9. Substituting these into the original expression yields

6 + 6 - 9 = 3

.

Adım Adım Çözüm

1

Find the unit digit of

2^{84}

Unit digit is 6

The cyclicity of

2

is

4

(

2, 4, 8, 6

). Since

84 \div 4 = 21

with remainder

0

, we take the 4th term in the cyclicity pattern, which is

6

.

2

Find the unit digit of

4^{97}

and

3^{54}

Unit digit of

4^{97}

is 4; Unit digit of

3^{54}

is 9

The cyclicity of

4

is

2

(

4

for odd powers,

6

for even powers). Since

97

is odd,

4^{97}

ends in

4

. The cyclicity of

3

is

4

(

3, 9, 7, 1

). Since

54 = 4 \times 13 + 2

, remainder is

2

, giving

3^2 = 9

.

3

Find the unit digit of the product

4^{97} \times 3^{54}

Unit digit of product is 6

Multiplying the unit digits:

4 \times 9 = 36

, so the unit digit is

6

.

4

Find the unit digit of

9^{45}

Unit digit is 9

The cyclicity of

9

is

2

(

9

for odd powers,

1

for even powers). Since

45

is odd,

9^{45}

ends in

9

.

5

Combine the unit digits following the algebraic operations

Unit digit is 3

Evaluating

6 + 6 - 9 = 12 - 9 = 3

.

Anahtar Kavram

Unit Digit and Cyclicity of Exponential Numbers

Soru 843Soru →

If $x = \sqrt{11 + 4\sqrt{6}} - \sqrt{11 - 4\sqrt{6}}$ and $\left(\frac{1}{16}\right)^{x^2 - 15} = 64^{y + 1}$ , what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of

y

is

1

.

Simplifying

\sqrt{11 \pm 4\sqrt{6}}

as

\sqrt{8} \pm \sqrt{3}

yields

x = 2\sqrt{3}

and

x^2 = 12

. Substituting

x^2

gives

(1/16)^{-3} = 16^3 = 2^{12}

. Matching exponents with

64^{y+1} = 2^{6y+6}

yields

12 = 6y + 6

, giving

y = 1

.

Adım Adım Çözüm

1

Simplify the nested surds for

x

x = 2\sqrt{3}

Express

11 \pm 4\sqrt{6}

as

11 \pm 2\sqrt{24} = (\sqrt{8} \pm \sqrt{3})^2

, giving

\sqrt{11 + 4\sqrt{6}} - \sqrt{11 - 4\sqrt{6}} = (2\sqrt{2} + \sqrt{3}) - (2\sqrt{2} - \sqrt{3}) = 2\sqrt{3}

.

2

Evaluate

x^2

x^2 = 12

Squaring

x = 2\sqrt{3}

yields

x^2 = 4 \times 3 = 12

.

3

Substitute

x^2 = 12

into the exponent of the given equation

\left(\frac{1}{16}\right)^{-3} = 64^{y + 1}

The exponent

x^2 - 15

evaluates to

12 - 15 = -3

.

4

Convert both sides to a common prime base (base 2)

2^{12} = 2^{6y + 6}

\left(\frac{1}{16}\right)^{-3} = 16^3 = (2^4)^3 = 2^{12}

and

64^{y + 1} = (2^6)^{y + 1} = 2^{6y + 6}

.

5

Equate powers and solve for

y

y = 1

Since the bases are identical,

12 = 6y + 6 \implies 6y = 6 \implies y = 1

.

Anahtar Kavram

Nested surd evaluation via perfect square conversion combined with solving exponential equations using index rules.

Soru 844Soru →

In a state public health drive conducted across three administrative zones—Zone X, Zone Y, and Zone Z—a total of $12,000$ vaccines were administered. Zone X received $35\%$ of the total vaccines administered. In Zone Y, the total number of vaccines administered was $800$ more than that in Zone X, and the ratio of vaccines administered to children versus adults was $2 : 3$ . The remaining vaccines were administered in Zone Z, where $60\%$ of the recipients were adults and the rest were children. If $40\%$ of the recipients in Zone X were children, what is the total number of children who received vaccines across all three zones combined?

5,360

4,800

7,920

5,800

Cevabı ve açıklamayı göster

Cevap: $4,800$

Cevap

4,800

children received vaccines across all three zones combined.

The total vaccines administered across the three zones equal

12,000

. Zone X accounts for

35\%

(

4,200

vaccines), of which

40\%

are children (

1,680

). Zone Y receives

800

more than Zone X (

5,000

vaccines), with children making up

2/5

of this amount (

2,000

). Zone Z receives the remaining

2,800

vaccines, where

40\%

are children (

1,120

). Adding these together (

1,680 + 2,000 + 1,120

) yields

4,800

.

Adım Adım Çözüm

1

Calculate total vaccines and child recipients in Zone X

Total in Zone X =

0.35 \times 12,000 = 4,200

. Children in Zone X =

0.40 \times 4,200 = 1,680

.

Zone X receives

35\%

of total vaccines, and

40\%

of Zone X recipients are children.

2

Calculate total vaccines and child recipients in Zone Y

Total in Zone Y =

4,200 + 800 = 5,000

. Children in Zone Y =

\frac{2}{2 + 3} \times 5,000 = \frac{2}{5} \times 5,000 = 2,000

.

Zone Y receives

800

more vaccines than Zone X, and children make up

2

parts out of

5

total parts.

3

Calculate total vaccines and child recipients in Zone Z

Total in Zone Z =

12,000 - (4,200 + 5,000) = 2,800

. Children percentage in Zone Z =

100\% - 60\% = 40\%

. Children in Zone Z =

0.40 \times 2,800 = 1,120

.

Zone Z receives the remainder of total vaccines, and children constitute the remaining

40\%

of Zone Z recipients.

4

Sum child recipients across all three zones

Total children =

1,680 + 2,000 + 1,120 = 4,800

.

Adding the child counts from Zone X, Zone Y, and Zone Z yields the final combined total.

Anahtar Kavram

Multi-step quantitative extraction, ratio division, and percentage base calculation from paragraph-based caselets.

Tahmini Süre:2m 30s

Soru 845Soru →

If January 10, 1896, was a Friday, what day of the week was January 10, 1908?

Saturday

Thursday

Friday

Sunday

Cevabı ve açıklamayı göster

Cevap: Friday

Cevap

Friday

The correct day is Friday because evaluating the 12-year interval from January 10, 1896, to January 10, 1908, yields exactly 14 odd days. This breaks down into 12 baseline days (one for each year) plus 2 extra days for the leap years explicitly crossed (1896 and 1904). Importantly, the year 1900 is not a leap year, and the leap day of 1908 is not reached. Because 14 is perfectly divisible by 7 (remainder 0), the day of the week does not shift.

Adım Adım Çözüm

1

Calculate the total number of years between the two dates.

The total span from January 10, 1896, to January 10, 1908, is exactly 12 years.

Each standard year contributes 1 odd day (since 365 mod 7 = 1) to the day of the week progression.

2

Identify the number of leap years crossed within this specific time frame.

The leap days crossed belong to the years 1896 and 1904. Total = 2 additional leap days.

1896 is a leap year, and we cross its February 29th because we start on January 10th. The year 1900 is a century year not divisible by 400, so it is a standard year (no leap day). 1904 is a leap year. Although 1908 is a leap year, the period ends on January 10th, so its February 29th is not crossed.

3

Calculate the total number of odd days and determine the final day of the week.

12 standard odd days + 2 leap odd days = 14 odd days. 14 divided by 7 leaves a remainder of 0. Friday + 0 days = Friday.

Adding the accumulated odd days to the original day provides the target day of the week. A remainder of 0 indicates a complete cycle.

Anahtar Kavram

Century leap year rules and date boundary logic in calendars

Soru 846Soru →

In a state disaster relief operation, a total of $1,200$ food packets were distributed among three relief camps: Camp A, Camp B, and Camp C. Camp A received $400$ packets, while Camp B received $150$ more packets than Camp A. The remaining food packets were distributed to Camp C. How many food packets were distributed to Camp C?

Cevabı ve açıklamayı göster

Cevap: 250

Cevap

The total number of food packets distributed to Camp C is 250.

Camp A receives 400 packets. Camp B receives 150 more than Camp A, which equals 400 + 150 = 550 packets. Combined, Camp A and Camp B receive 400 + 550 = 950 packets. Subtracting 950 from the total allocation of 1,200 packets leaves 250 packets for Camp C.

Adım Adım Çözüm

1

Identify the number of packets given to Camp A and calculate the packets given to Camp B.

Camp A = 400 packets; Camp B = 400 + 150 = 550 packets.

The paragraph states Camp B received 150 more packets than Camp A.

2

Calculate the total number of packets distributed to Camp A and Camp B together.

Total for Camp A and Camp B = 400 + 550 = 950 packets.

Adding the individual allocations of Camp A and Camp B.

3

Subtract the combined allocation of Camp A and Camp B from the overall total packets.

Camp C = 1200 - 950 = 250 packets.

The remaining packets out of the total 1,200 were given to Camp C.

Anahtar Kavram

Data extraction and step-by-step arithmetic computation from narrative caselets

Soru 847Soru →

Two business partners, $A$ and $B$ , invested in a joint venture. What is the total profit earned by the business at the end of one year?

Statement I: Partner $A$ invested $\$6,000$ for the entire year, while Partner $B$ invested $\$9,000$ for $8$ months.
Statement II: Partner $A$ 's share of the annual profit is $\$2,400$ .

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Cevabı ve açıklamayı göster

Cevap: Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Cevap

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating Statement I alone gives the profit-sharing ratio between Partner A and Partner B as

(6000 \times 12) : (9000 \times 8) = 1 : 1

, which is insufficient by itself to find the monetary profit. Statement II alone gives Partner A's profit share as

\$2,400

, which is also insufficient without knowing the proportion. Combining both statements shows that both partners receive equal shares, so total profit is

\$2,400 \times 2 = \$4,800

. Therefore, both statements together are necessary and sufficient.

Adım Adım Çözüm

1

Evaluate Statement I alone

Profit sharing ratio of A to B =

(6000 \times 12) : (9000 \times 8) = 72,000 : 72,000 = 1 : 1

.

Profit is distributed in proportion to the product of capital invested and time period. Since no dollar amounts of profit are given, total profit cannot be determined from Statement I alone.

2

Evaluate Statement II alone

Partner A's profit share =

\$2,400

.

Without knowing Partner B's share or the ratio between their shares, the total profit cannot be calculated from Statement II alone.

3

Evaluate Statement I and Statement II together

Since ratio of shares is

1:1

and A's share is

\$2,400

, B's share is also

\$2,400

. Total profit =

\$2,400 + \$2,400 = \$4,800

.

Combining the ratio from Statement I and the monetary value from Statement II gives a unique answer to the question.

Anahtar Kavram

Partnership profit distribution ratio and data sufficiency evaluation

Soru 848Soru →

What is the simplified numerical value of the mathematical expression given below when evaluated strictly according to the VBODMAS rule?

15\% \text{ of } 200 - \left[ 36 \div \left\{ 18 - \left( 2 \times \overline{7 - 4} \right) \right\} \right] \times 3.5

19.5

94.5

14.25

27.5

Cevabı ve açıklamayı göster

Cevap: $19.5$

Cevap

The simplified numerical value of the expression is

19.5

.

Following the VBODMAS hierarchy, we first solve the bar term

\overline{7-4} = 3

, giving

(2 \times 3) = 6

inside the inner bracket. Next, the curly bracket simplifies to

18 - 6 = 12

. The square bracket simplifies to

36 \div 12 = 3

. Evaluating

15\% \text{ of } 200 = 30

and the product

3 \times 3.5 = 10.5

, we get

30 - 10.5 = 19.5

.

Adım Adım Çözüm

1

Evaluate the expression under the Vinculum (bar)

\overline{7 - 4} = 3

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

2

Evaluate the innermost round bracket

( )

2 \times 3 = 6

Resolve the multiplication inside the round brackets.

3

Evaluate the curly bracket

\{ \}

18 - 6 = 12

Perform subtraction inside the curly brackets.

4

Evaluate the square bracket

[ ]

36 \div 12 = 3

Perform division inside the square brackets.

5

Calculate the percentage portion

15\% \text{ of } 200 = \frac{15}{100} \times 200 = 30

'Of' represents multiplication in percentage evaluation.

6

Perform multiplication before final subtraction

3 \times 3.5 = 10.5

Multiplication takes precedence over final subtraction.

7

Perform final subtraction

30 - 10.5 = 19.5

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

Anahtar Kavram

Order of Operations (VBODMAS Rule)

Tahmini Süre:1m 30s

Soru 849Soru →

What is the unit digit of the numerical expression $M = 3^{105} \times 7^{82} + 8^{96}$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The unit digit of the given expression is 3.

The unit digit of an exponential term is determined by its base's cyclicity cycle and the exponent modulo 4. For

3^{105}

,

105 \pmod 4 = 1 \implies 3^1 = 3

. For

7^{82}

,

82 \pmod 4 = 2 \implies 7^2 = 49 \implies 9

. The product of these terms has a unit digit of

(3 \times 9) \pmod{10} = 7

. For

8^{96}

,

96 \pmod 4 = 0

, which indicates the 4th position in the cyclicity pattern of 8 (8, 4, 2, 6), yielding a unit digit of 6. Finally, adding the unit digits gives

(7 + 6) \pmod{10} = 3

.

Adım Adım Çözüm

1

Determine the cyclicity and remainder for

3^{105}

Unit digit is 3

The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent by 4 gives

105 \pmod 4 = 1

, corresponding to

3^1 = 3

.

2

Determine the cyclicity and remainder for

7^{82}

Unit digit is 9

The unit digits of powers of 7 repeat in a cycle of 4 (7, 9, 3, 1). Dividing the exponent by 4 gives

82 \pmod 4 = 2

, corresponding to

7^2 = 49

(unit digit 9).

3

Calculate the unit digit of the product

3^{105} \times 7^{82}

Unit digit is 7

Multiply the unit digits of the two factors:

(3 \times 9) = 27

, which has a unit digit of 7.

4

Determine the cyclicity and remainder for

8^{96}

Unit digit is 6

The unit digits of powers of 8 repeat in a cycle of 4 (8, 4, 2, 6). The exponent 96 is completely divisible by 4 (

96 \pmod 4 = 0

). When remainder is 0, we use the 4th power in the cycle (

8^4 = 4096

), which ends in 6.

5

Sum the unit digits of the two main terms

Unit digit is 3

Adding the unit digits gives

(7 + 6) = 13

, which has a unit digit of 3.

Anahtar Kavram

Determining the unit digit of composite exponential expressions using power cyclicity.

Soru 850Soru →

If $p = \sqrt{12 + \sqrt{12 + \sqrt{12 + \dots}}}$ and $q = \sqrt{12 - \sqrt{12 - \sqrt{12 - \dots}}}$ , such that $2^{p+q-z} = \frac{1}{64}$ , what is the value of $z$ ?

1

5

13

-1

Cevabı ve açıklamayı göster

Cevap: $13$

Cevap

The value of

z

is

13

.

Solving

p = \sqrt{12 + p}

yields the quadratic equation

p^2 - p - 12 = 0

, giving

p = 4

. Solving

q = \sqrt{12 - q}

yields

q^2 + q - 12 = 0

, giving

q = 3

. Adding

p

and

q

gives

p + q = 7

. Substituting this into

2^{p+q-z} = \frac{1}{64}

produces

2^{7-z} = 2^{-6}

. Equating the indices gives

7 - z = -6

, which solves to

z = 13

.

Adım Adım Çözüm

1

Evaluate the first infinite nested surd

p

p = 4

Let

p = \sqrt{12 + p}

. Squaring both sides yields

p^2 = 12 + p \implies p^2 - p - 12 = 0

. Factoring gives

(p-4)(p+3) = 0

. Since

p > 0

,

p = 4

.

2

Evaluate the second infinite nested surd

q

q = 3

Let

q = \sqrt{12 - q}

. Squaring both sides yields

q^2 = 12 - q \implies q^2 + q - 12 = 0

. Factoring gives

(q+4)(q-3) = 0

. Since

q > 0

,

q = 3

.

3

Substitute

p

and

q

into the exponent expression

p + q = 7

, giving

2^{7-z} = \frac{1}{64}

Summing

p

and

q

gives

4 + 3 = 7

.

4

Solve the exponential equation for

z

z = 13

Express

\frac{1}{64}

as a base

2

power:

\frac{1}{64} = 2^{-6}

. Equating exponents gives

7 - z = -6 \implies z = 7 + 6 = 13

.

Anahtar Kavram

Evaluation of infinite nested surds and solving exponential equations using laws of indices

Tahmini Süre:2m 0s

Soru 851Soru →

What is the simplified value of the expression $\left(\frac{64}{125}\right)^{-\frac{2}{3}}$ ?

\frac{25}{16}

\frac{16}{25}

-\frac{25}{16}

\frac{5}{4}

Cevabı ve açıklamayı göster

Cevap: $\frac{25}{16}$

Cevap

\frac{25}{16}

The expression

\left(\frac{64}{125}\right)^{-\frac{2}{3}}

is simplified by first converting the negative power to positive by inverting the fraction to

\left(\frac{125}{64}\right)^{\frac{2}{3}}

. Taking the cube root yields

\frac{5}{4}

, and squaring that result gives

\frac{25}{16}

.

Adım Adım Çözüm

1

Apply the negative exponent rule

a^{-n} = \frac{1}{a^n}

to invert the fraction

\left(\frac{64}{125}\right)^{-\frac{2}{3}} = \left(\frac{125}{64}\right)^{\frac{2}{3}}

A negative exponent indicates taking the reciprocal of the base.

2

Express the base numbers as perfect cubes

125 = 5^3

and

64 = 4^3

, so

\frac{125}{64} = \left(\frac{5}{4}\right)^3

Rewriting bases into prime factors with power multiples simplifies fractional exponents.

3

Apply the power of a power rule

(a^m)^n = a^{m \times n}

\left(\left(\frac{5}{4}\right)^3\right)^{\frac{2}{3}} = \left(\frac{5}{4}\right)^{3 \times \frac{2}{3}} = \left(\frac{5}{4}\right)^2

Multiplying the inner exponent

3

by the outer exponent

\frac{2}{3}

yields

2

.

4

Square the fraction

\left(\frac{5}{4}\right)^2 = \frac{25}{16}

Square both the numerator and the denominator to get the final numerical value.

Anahtar Kavram

Negative and Fractional Indices Rules

Soru 852Soru →

Under a state renewable energy initiative, a total capacity of $15,000\text{ kW}$ of solar panels was installed across three districts: District Alpha, District Beta, and District Gamma. District Alpha was allocated $40\%$ of the total capacity. The capacity installed in District Beta was $\frac{3}{4}$ of that in District Alpha, while the remaining capacity was installed in District Gamma. In District Alpha, $60\%$ of the installed capacity consists of rooftop solar panels, and the remainder consists of ground-mounted panels. In District Beta, the ratio of rooftop solar capacity to ground-mounted solar capacity is $2 : 3$ . In District Gamma, the ground-mounted solar capacity exceeds the rooftop solar capacity by $1,500\text{ kW}$ .

Based on the provided information, which of the following statements regarding the solar panel capacity distribution are correct?

Geçerli olan tümünü seçin

The total rooftop solar capacity installed across all three districts is

6,900\text{ kW}

.

The ground-mounted solar capacity in District Gamma is exactly twice its rooftop solar capacity.

District Beta accounts for

35\%

of the total solar panel capacity installed across all three districts.

The ratio of total ground-mounted solar capacity to total rooftop solar capacity across all three districts is

27 : 23

.

Cevabı ve açıklamayı göster

Cevap: The total rooftop solar capacity installed across all three districts is $6,900\text{ kW}$ .; The ground-mounted solar capacity in District Gamma is exactly twice its rooftop solar capacity.; The ratio of total ground-mounted solar capacity to total rooftop solar capacity across all three districts is $27 : 23$ .

Cevap

The statements confirming that total rooftop solar capacity is

6,900\text{ kW}

, ground-mounted capacity in District Gamma is twice its rooftop capacity, and the ratio of total ground-mounted to rooftop capacity is

27 : 23

are all correct.

Evaluating the data step by step shows that District Alpha has

3,600\text{ kW}

rooftop and

2,400\text{ kW}

ground-mounted capacity; District Beta has

1,800\text{ kW}

rooftop and

2,700\text{ kW}

ground-mounted capacity; District Gamma has

1,500\text{ kW}

rooftop and

3,000\text{ kW}

ground-mounted capacity. This confirms that total rooftop capacity is

6,900\text{ kW}

, District Gamma's ground-mounted capacity is twice its rooftop capacity (

3,000\text{ kW} = 2 \times 1,500\text{ kW}

), and the overall ratio of ground-mounted to rooftop capacity is

8,100 : 6,900 = 27 : 23

.

Adım Adım Çözüm

1

Calculate total capacity allocation for each district.

Alpha =

40\% \text{ of } 15,000 = 6,000\text{ kW}

. Beta =

\frac{3}{4} \times 6,000 = 4,500\text{ kW}

. Gamma =

15,000 - (6,000 + 4,500) = 4,500\text{ kW}

.

Determining individual district totals is necessary to break down capacity by panel type.

2

Determine rooftop and ground-mounted capacities for District Alpha and District Beta.

District Alpha: Rooftop =

60\% \text{ of } 6,000 = 3,600\text{ kW}

, Ground =

2,400\text{ kW}

. District Beta: Ratio

2:3 \implies

Rooftop =

\frac{2}{5} \times 4,500 = 1,800\text{ kW}

, Ground =

\frac{3}{5} \times 4,500 = 2,700\text{ kW}

.

Applying given percentages and ratios yields the exact breakdown for Alpha and Beta.

3

Determine rooftop and ground-mounted capacities for District Gamma.

Let Rooftop =

R

and Ground =

G

. We have

G + R = 4,500

and

G - R = 1,500

. Solving gives

2G = 6,000 \implies G = 3,000\text{ kW}

and

R = 1,500\text{ kW}

.

Setting up simultaneous equations resolves the unknown components for Gamma.

4

Verify overall totals and evaluate each statement.

Total Rooftop =

3,600 + 1,800 + 1,500 = 6,900\text{ kW}

. Total Ground =

2,400 + 2,700 + 3,000 = 8,100\text{ kW}

. Gamma Ground/Rooftop ratio =

3,000 / 1,500 = 2

. Beta percentage of total =

4,500 / 15,000 = 30\%

. Ground to Rooftop ratio =

8,100 / 6,900 = 27 / 23

.

Direct comparison evaluates which statements are factually supported by the paragraph data.

Anahtar Kavram

Data extraction and simultaneous calculation from unstructured paragraph-based quantitative scenarios.

Soru 853Soru →

A vessel contains a mixture of milk and water. What is the initial total volume of the mixture in the vessel?

Statement I: The initial ratio of milk to water in the vessel is $3:2$ . When $10\text{ liters}$ of water is added to the mixture, the ratio of milk to water becomes $1:1$ .
Statement II: If $15\text{ liters}$ of the original mixture is removed and replaced with $15\text{ liters}$ of pure water, the quantity of milk remaining in the vessel is $27\text{ liters}$ .

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient, but Statement II alone is not sufficient.

Statement II alone is sufficient, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient.

Neither Statement I nor Statement II is sufficient.

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement I alone is sufficient, but Statement II alone is not sufficient.

Cevap

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement I alone gives a complete quantitative relationship: with initial quantities

3x

and

2x

, adding

10\text{ liters}

of water leads to

3x = 2x + 10

, yielding

x = 10

and a unique total initial volume of

50\text{ liters}

. Statement II alone leaves two unknown variables (initial milk fraction and total volume), so it is not sufficient.

Adım Adım Çözüm

1

Evaluate Statement I alone.

Initial volume is calculated as

50\text{ liters}

.

Let the initial volume of milk be

3x

liters and water be

2x

liters, making the total initial volume

5x

liters. Adding

10\text{ liters}

of water gives the equation

\frac{3x}{2x + 10} = \frac{1}{1}

, which simplifies to

3x = 2x + 10 \implies x = 10

. Thus, total initial volume

5x = 50\text{ liters}

. Hence, Statement I alone is sufficient.

2

Evaluate Statement II alone.

Insufficient to find total initial volume.

Let the total initial volume be

V

liters and the fraction of milk in the initial mixture be

f

. Removing

15\text{ liters}

of mixture removes

15f

liters of milk. Replacing it with water adds no milk. The remaining milk equation is

f(V - 15) = 27

. Since both

f

and

V

are unknown,

V

cannot be uniquely determined. Hence, Statement II alone is not sufficient.

Anahtar Kavram

Data Sufficiency evaluation of mixture ratios and algebraic equations

Tahmini Süre:1m 30s

Soru 854Soru →

What is the unit digit of the numerical value of the expression $N = 528^{144} + 363^{89} \times 244^{65} - 179^{51}$ ?

9

4

7

1

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

Evaluating each term's unit digit yields 6 for the first term, 3 for the second term, 4 for the third term, and 9 for the fourth term. Applying BODMAS rules: multiplication of 3 and 4 gives a unit digit of 2. Then adding 6 gives 8. Subtracting 9 from 8 gives a unit digit of 9 (since 18 - 9 = 9). Thus, the correct unit digit is 9.

Adım Adım Çözüm

1

Find the unit digit of

528^{144}

The unit digit of the base is 8. The cyclicity of 8 is 4. Dividing the exponent 144 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power:

8^4 = 4096

, which has a unit digit of 6.

Exponents divisible by the cyclicity length correspond to the 4th power in the cycle.

2

Find the unit digit of

363^{89}

The unit digit of the base is 3. The cyclicity of 3 is 4. Dividing 89 by 4 gives a remainder of 1. Thus, the unit digit is

3^1 = 3

.

The unit digit pattern for powers of 3 repeats every 4 powers.

3

Find the unit digit of

244^{65}

The unit digit of the base is 4. The cyclicity of 4 is 2 (powers alternate:

4^1 = 4, 4^2 = 6

). Since the exponent 65 is odd, the unit digit is 4.

Powers of 4 with odd exponents always end in 4.

4

Find the unit digit of

179^{51}

The unit digit of the base is 9. The cyclicity of 9 is 2 (powers alternate:

9^1 = 9, 9^2 = 1

). Since 51 is odd, the unit digit is 9.

Powers of 9 with odd exponents always end in 9.

5

Combine the unit digits using BODMAS rules

Unit digit of expression =

[6 + (3 \times 4) - 9] \pmod{10} = (6 + 2 - 9) \pmod{10} = (8 - 9) \pmod{10} = 9

.

Multiplication is performed before addition/subtraction, and a negative intermediate unit digit is adjusted by adding 10.

Anahtar Kavram

Unit digit calculation using cyclicity of digits and order of operations

Tahmini Süre:1m 30s

Soru 855Soru →

Consider the positive integer $N = 60$ . Which of the following statements regarding the factors and prime factorization of $N$ is/are correct?

Geçerli olan tümünü seçin

The total number of positive factors of

60

is

12

.

The prime factorization of

60

is

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

.

The sum of all distinct prime factors of

60

is

11

.

The number

60

has exactly

6

odd positive factors.

Cevabı ve açıklamayı göster

Cevap: The total number of positive factors of $60$ is $12$ .; The prime factorization of $60$ is $2^2 \times 3^1 \times 5^1$ .

Cevap

The total number of positive factors of 60 is 12, and the prime factorization of 60 is

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

.

The statement declaring that the total number of positive factors is

12

is correct because

(2+1)(1+1)(1+1) = 12

. Furthermore, the statement giving prime factorization as

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

is correct as it accurately breaks down

60

into prime bases.

Adım Adım Çözüm

1

Find the prime factorization of

60

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

Decomposing

60

into its prime components provides the exponents needed for factor analysis.

2

Calculate the total number of positive factors

Total factors =

(2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12

Adding

1

to each prime exponent and multiplying gives the total count of positive factors.

3

Evaluate the distinct prime factors and their sum

Distinct prime factors are

2, 3, 5

, and their sum is

2 + 3 + 5 = 10

Verifying distinct prime sum shows that claiming a sum of

11

is incorrect.

4

Calculate the number of odd factors

Odd factors =

(1+1)(1+1) = 4

Odd factors depend only on odd prime bases (

3^1 \times 5^1

), yielding

4

odd factors (

1, 3, 5, 15

).

Anahtar Kavram

Prime Factorization and Factor Counting Formula

Soru 856Soru →

If $x = \sqrt{10 + 2\sqrt{21}} - \sqrt{10 - 2\sqrt{21}}$ and $9^{x^2 - k} = \left(\frac{1}{243}\right)^{k - 6}$ , find the value of $k$ .

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

is 2.

Simplifying

x

yields

x = (\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

, which gives

x^2 = 12

. Substituting

x^2 = 12

into the exponential equation gives

9^{12 - k} = (1/243)^{k - 6}

. Expressing both sides with prime base 3 gives

3^{2(12 - k)} = 3^{-5(k - 6)}

, which simplifies to

3^{24 - 2k} = 3^{30 - 5k}

. Equating the exponents gives

24 - 2k = 30 - 5k

, which solves to

k = 2

.

Adım Adım Çözüm

1

Simplify the nested surds for

x

\sqrt{10 + 2\sqrt{21}} = \sqrt{7} + \sqrt{3}

and

\sqrt{10 - 2\sqrt{21}} = \sqrt{7} - \sqrt{3}

Nested surds of the form

\sqrt{a \pm 2\sqrt{b}}

simplify to

\sqrt{m} \pm \sqrt{n}

where

m + n = a

and

m \cdot n = b

.

2

Evaluate

x

and calculate

x^2

x = 2\sqrt{3}

and

x^2 = 12

Subtracting the two surds gives

(\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

. Squaring

2\sqrt{3}

gives

(2)^2 \times 3 = 12

.

3

Express both sides of the exponential equation with base 3

3^{24 - 2k} = 3^{30 - 5k}

Since

9 = 3^2

and

243 = 3^5

,

9^{12 - k} = (3^2)^{12 - k} = 3^{24 - 2k}

and

(1/243)^{k - 6} = (3^{-5})^{k - 6} = 3^{30 - 5k}

.

4

Equate the powers and solve for

k

k = 2

Since bases are equal (

3 > 0, 3 \neq 1

), setting exponents equal gives

24 - 2k = 30 - 5k \implies 3k = 6 \implies k = 2

.

Anahtar Kavram

Simplification of nested square root surds

\sqrt{a \pm 2\sqrt{b}}

combined with equating exponents using prime base factorization.

Soru 857Soru →

What is the exact value of the two-digit positive integer $N$ ?

Statement (I): The sum of the tens digit and the units digit of $N$ is equal to $12$ .
Statement (II): Reversing the digits of $N$ yields a new two-digit integer that is $36$ greater than $N$ .

Which of the following statements correctly describes the sufficiency of the given data to answer the question?

Statement (I) alone is sufficient, but Statement (II) alone is not sufficient.

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Either Statement (I) alone or Statement (II) alone is sufficient.

Neither Statement (I) nor Statement (II) is sufficient.

Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Cevabı ve açıklamayı göster

Cevap: Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Cevap

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating each statement independently reveals that neither statement alone narrows

N

down to a single value. Combining both equations (

x + y = 12

and

y - x = 4

) yields a single unique pair

x = 4

and

y = 8

, giving

N = 48

. Therefore, both statements together are required and sufficient.

Adım Adım Çözüm

1

Represent the two-digit integer algebraically.

Let

N = 10x + y

, where

x

is the tens digit (

1 \leq x \leq 9

) and

y

is the units digit (

0 \leq y \leq 9

).

Setting up standard place-value variables allows formal analysis of both statements.

2

Evaluate Statement (I) alone.

From Statement (I),

x + y = 12

. Possible pairs

(x,y)

are

(3,9), (4,8), (5,7), (6,6), (7,5), (8,4), (9,3)

. Thus,

N

could be

39, 48, 57, 66, 75, 84,

or

93

.

Since multiple valid values of

N

exist, Statement (I) alone is NOT sufficient.

3

Evaluate Statement (II) alone.

The reversed number is

10y + x

. Given

(10y + x) - (10x + y) = 36 \implies 9(y - x) = 36 \implies y - x = 4

. Possible pairs

(x,y)

are

(1,5), (2,6), (3,7), (4,8), (5,9)

, giving

N = 15, 26, 37, 48,

or

59

.

Since multiple valid values of

N

exist, Statement (II) alone is NOT sufficient.

4

Evaluate Statement (I) and Statement (II) together.

System of linear equations:

x + y = 12

and

y - x = 4

. Adding the two equations gives

2y = 16 \implies y = 8

. Substituting

y = 8

gives

x = 4

. Thus,

N = 48

uniquely.

Both statements combined provide a unique value for

N

, making them together sufficient.

Anahtar Kavram

Data Sufficiency in Digit-Based Integer Problems

Tahmini Süre:1m 30s

Soru 858Soru →

In a state sericulture development drive, three agricultural districts—District R, District S, and District T—produced a cumulative total of $3,000\text{ kg}$ of raw silk, consisting exclusively of Mulberry silk and Eri silk. Across all three districts combined, Mulberry silk accounted for $60\%$ of the total raw silk production. In District R, total raw silk production was $1,000\text{ kg}$ , of which $65\%$ was Mulberry silk. District S produced a total of $1,200\text{ kg}$ of raw silk with a Mulberry silk to Eri silk production ratio of $7 : 5$ . The remaining raw silk was produced in District T. What is the absolute difference between the quantity of Mulberry silk and the quantity of Eri silk produced in District T?

100\text{ kg}

150\text{ kg}

200\text{ kg}

50\text{ kg}

Cevabı ve açıklamayı göster

Cevap: $100\text{ kg}$

Cevap

The absolute difference between Mulberry silk and Eri silk produced in District T is

100\text{ kg}

.

Total raw silk produced is

3,000\text{ kg}

, making total Mulberry silk

1,800\text{ kg}

(

60\%

) and total Eri silk

1,200\text{ kg}

. District R produces

650\text{ kg}

Mulberry and

350\text{ kg}

Eri silk. District S produces

700\text{ kg}

Mulberry and

500\text{ kg}

Eri silk. Consequently, District T produces

450\text{ kg}

Mulberry silk (

1,800 - 1,350

) and

350\text{ kg}

Eri silk (

1,200 - 850

). The difference between Mulberry and Eri silk in District T is

450 - 350 = 100\text{ kg}

.

Adım Adım Çözüm

1

Calculate overall production totals for Mulberry and Eri silk

Total Mulberry silk =

60\% \text{ of } 3,000\text{ kg} = 1,800\text{ kg}

. Total Eri silk =

3,000\text{ kg} - 1,800\text{ kg} = 1,200\text{ kg}

.

Establishing aggregate category totals provides the basis to determine missing District T values.

2

Determine breakdown for District R and District S

District R: Total =

1,000\text{ kg}

, Mulberry =

65\% \times 1,000 = 650\text{ kg}

, Eri =

350\text{ kg}

. District S: Total =

1,200\text{ kg}

, Mulberry =

\frac{7}{12} \times 1,200 = 700\text{ kg}

, Eri =

\frac{5}{12} \times 1,200 = 500\text{ kg}

.

Extract specific entity values from given percentage and ratio conditions.

3

Derive Mulberry and Eri silk quantities for District T

District T Mulberry =

1,800 - (650 + 700) = 450\text{ kg}

. District T Eri =

1,200 - (350 + 500) = 350\text{ kg}

.

Subtracting known district totals from overall state totals yields District T quantities.

4

Calculate the absolute difference for District T

Difference =

450\text{ kg} - 350\text{ kg} = 100\text{ kg}

.

Directly answers the final prompt query.

Anahtar Kavram

Data extraction, multi-entity category balance, and multi-step quantitative calculation from unstructured paragraph data.

Soru 859Soru →

What is the numerical value of the mathematical expression given below when evaluated strictly according to the VBODMAS rule?

18\% \text{ of } 250 + \left[ 34 - \left\{ 12 \times 2.5 - \left( 14 \div 2 + \overline{8 - 3} \right) \right\} \right]

61

51

66

29

Cevabı ve açıklamayı göster

Cevap: 61

Cevap

61

Following the VBODMAS priority rule, we first clear the vinculum

\overline{8-3}=5

. Next, within the round brackets, division precedes addition:

14 \div 2 + 5 = 7 + 5 = 12

. Inside the curly brackets, multiplication precedes subtraction:

12 \times 2.5 - 12 = 30 - 12 = 18

. The square bracket reduces to

34 - 18 = 16

. Evaluating

18\%

of

250

gives

45

. Adding

45 + 16

yields the correct final result of 61.

Adım Adım Çözüm

1

Evaluate the vinculum bar term

\overline{8 - 3} = 5

According to VBODMAS, operations under a vinculum (bar) must be simplified first.

2

Simplify the innermost round brackets

(14 \div 2 + 5) = 7 + 5 = 12

Division takes precedence over addition inside the parentheses.

3

Simplify the curly brackets

\{ 12 \times 2.5 - 12 \} = 30 - 12 = 18

Multiplication is performed before subtraction inside the brace group.

4

Simplify the square brackets

[ 34 - 18 ] = 16

Complete the bracket hierarchy by evaluating the outer square brackets.

5

Evaluate the 'of' percentage term

18\% \text{ of } 250 = \frac{18}{100} \times 250 = 45

The percentage 'of' operator is calculated as multiplication.

6

Perform the final addition

45 + 16 = 61

Combine the percentage term with the simplified bracket result.

Anahtar Kavram

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

Soru 860Soru →

Find the unit digit of the numerical expression $N = (1! + 2! + 3! + \dots + 50!)^{25} + 8^{88}$ .

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The unit digit of the given expression is 9.

All factorials from

5!

to

50!

end in 0, so the sum

(1! + 2! + 3! + \dots + 50!)

ends in

1 + 2 + 6 + 24 = 33

, which has a unit digit of 3. Raising 3 to the power 25 yields a unit digit of 3 because

25 \pmod 4 = 1

. For

8^{88}

, since 88 is divisible by 4 (

88 \pmod 4 = 0

), we take the 4th term of the cyclicity of 8, which is 6. Summing the unit digits yields

3 + 6 = 9

.

Adım Adım Çözüm

1

Determine the unit digit of the base sum

(1! + 2! + 3! + \dots + 50!)

.

Unit digit of base is 3.

Factorials

5!

and above all end in 0 (

5! = 120

), so only

1! + 2! + 3! + 4! = 33

contributes to the unit digit.

2

Calculate the unit digit of

3^{25}

.

Unit digit is 3.

The cyclicity pattern of 3 has period 4 (3, 9, 7, 1). Exponent

25 \pmod 4 = 1

, giving

3^1 = 3

.

3

Calculate the unit digit of

8^{88}

.

Unit digit is 6.

The cyclicity pattern of 8 has period 4 (8, 4, 2, 6). Exponent

88 \pmod 4 = 0

, so we take the 4th position in the cycle, which is 6.

4

Add the unit digits of both terms.

Final unit digit is 9.

3 + 6 = 9

, which gives a unit digit of 9.

Anahtar Kavram

Cyclicity of numbers and factorial unit digit properties

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