All practice questions

Question 1Question →

In a sequence of arithmetic operations, a candidate calculates a value of $-82$ . If this integer is then divided by $9$ , determine the least non-negative remainder.

Show answer & explanation

Answer: 8

Answer

8

Applying the division algorithm,

-82

can be expressed in the form

dq + r

as

9 \times (-10) + 8

. Since the remainder

r

must be greater than or equal to

0

and strictly less than the divisor

9

, the least non-negative remainder is

8

.

Step-by-Step Solution

1

Identify the dividend and the divisor.

Dividend is

-82

, divisor is

9

.

To apply the formal mathematical definition of division with remainders.

2

Find the largest integer multiple of

9

that is less than or equal to

-82

.

The correct multiple is

-90

(since

-90 \leq -82

).

The remainder must be a strictly non-negative value, meaning the multiple we subtract from the dividend must not exceed the dividend.

3

Calculate the difference between the dividend and this multiple to find the remainder.

-82 - (-90) = 8

.

This difference represents the least non-negative remainder required by the problem.

Key Concept

Remainder Theorem and Negative Dividends

Question 2Question →

Determine the positive remainder when the value of $36^{43}$ is divided by $37$ .

Show answer & explanation

Answer: 36

Answer

36

By the properties of modular arithmetic,

36

is congruent to

-1

modulo

37

. When we raise both sides to the power of

43

, we obtain

36^{43} \equiv (-1)^{43} \pmod{37}

. Since an odd power of

-1

is still

-1

, the expression leaves a remainder of

-1

. To find the standard positive remainder, we simply add the divisor to this result:

-1 + 37 = 36

.

Step-by-Step Solution

1

Express the base as a negative remainder modulo the divisor.

When

36

is divided by

37

, the remainder can be expressed as

-1

because

36 = 37 \times 1 - 1

. Therefore, we can state that

36 \equiv -1 \pmod{37}

.

Using a negative remainder with an absolute value of

1

makes calculating large exponentiations trivial.

2

Apply the exponent to the negative remainder and convert back to a positive value.

Calculate

(-1)^{43}

. Since

43

is an odd number,

(-1)^{43} = -1

. To find the valid positive remainder, add the divisor:

-1 + 37 = 36

.

The rules of modular arithmetic allow us to raise the remainder to the given power. However, the final answer must always be a non-negative integer strictly less than the divisor.

Key Concept

Negative Remainders for Large Powers

Alternative Method

You can use Fermat's Little Theorem, which states

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

for any prime

p

(as long as

a

is not divisible by

p

). Here,

p = 37

, so

36^{36} \equiv 1 \pmod{37}

. We can split the expression:

36^{43} = 36^{36} \times 36^7 \equiv 1 \times 36^7 \pmod{37}

. Using the negative remainder trick for the remaining part gives

(-1)^7 = -1

. Adding

37

yields the positive remainder of

36

.

Estimated Time:30s

Question 3Question →

Find the next term in the following number sequence:

3, 5, 8, 13, 22, 39, 72, \dots

Show answer & explanation

Answer: 137

Answer

The next number in the given series is 137.

Computing second-level differences between terms reveals a doubling sequence (

1, 2, 4, 8, 16

). The next second-level difference is

32

. Adding

32

to the last first-level difference (

33

) yields

65

. Finally, adding

65

to the last term (

72

) gives the correct value of

137

.

Step-by-Step Solution

1

Compute the first-order differences between consecutive terms in the series.

The first-order differences are

2, 3, 5, 9, 17, 33

.

Analyzing differences helps determine whether the growth rate follows an arithmetic, geometric, or polynomial rule.

2

Compute the second-order differences of the first-order difference sequence.

The second-order differences are

1, 2, 4, 8, 16

.

Because the first-order differences do not show an obvious linear pattern, evaluating higher-order differences is necessary.

3

Identify the underlying rule governing the second-order differences.

The second-order sequence doubles with each step (

2^0, 2^1, 2^2, 2^3, 2^4, \dots

). The next second-order difference is

16 \times 2 = 32

.

Recognizing geometric doubling (

2^n

) allows for projecting the next step accurately.

4

Reconstruct the next first-order difference and calculate the required next sequence value.

Next first-order difference

= 33 + 32 = 65

. Next series term

= 72 + 65 = 137

.

Working backward through the difference layers yields the exact value of the next term.

Key Concept

Double-Difference Series with Geometric Progression

Estimated Time:2m 0s

Question 4Question →

Consider the following numerical series:

4, 11, 30, 85, 248, \dots

What is the numerical value of the next term in this series?

Show answer & explanation

Answer: 735

Answer

735

Each term is generated by multiplying the preceding term by 3 and subtracting consecutive odd numbers starting from 1 (

1, 3, 5, 7, 9

). Therefore, the term following 248 is computed as

(248 \times 3) - 9 = 735

.

Step-by-Step Solution

1

Examine the multiplier and subtractor progression between consecutive terms.

Multiplier is constantly 3, while subtractors follow the odd number sequence

1, 3, 5, 7, \dots

Establishing a standard recurrence rule

T_n = 3 \cdot T_{n-1} - (2n - 3)

for

n \ge 2

.

2

Calculate the sixth term using the derived rule.

T_6 = (248 \times 3) - 9 = 744 - 9 = 735

The fifth term is 248, and the fifth odd integer to subtract is 9.

Key Concept

Mixed operation numerical series involving constant multiplication and arithmetic progression of subtractors

Estimated Time:2m 0s

Question 5Question →

In the following number sequence, a specific mathematical pattern is followed:

7, 14, 25, 42, 67, ?

What is the value of the missing term represented by the question mark?

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Answer: 102

Answer

The missing term in the series is 102.

The correct answer is 102. The sequence is governed by a double difference pattern. The consecutive differences between terms are 7, 11, 17, and 25. The differences of these differences are 4, 6, and 8, which increase by 2 at each step. Thus, the next second difference is 10, the next term difference is 35, and the next term in the sequence is 67 + 35 = 102.

Step-by-Step Solution

1

Find the first-order differences between consecutive terms.

The differences are 7, 11, 17, and 25.

Analyzing first differences helps determine if the growth rate follows a recognizable secondary pattern.

2

Find the second-order differences.

The differences between the consecutive first differences are 4, 6, and 8.

Since the first differences are non-linear, second-order differences reveal the underlying constant acceleration (+2).

3

Project the next first-order difference using the second-order pattern.

The next second-order difference is 10 (since 8 + 2 = 10), making the next first-order difference 25 + 10 = 35.

Extrapolating the arithmetic pattern of the second differences yields the correct next increment.

4

Add the projected difference to the last term of the sequence.

67 + 35 = 102.

Applying the calculated increment to the last term completes the series.

Key Concept

Second-order arithmetic sequence (Double Difference Series)

Question 6Question →

A field researcher starts from a campsite and walks $10\text{ km}$ due North. She then turns right and walks $7\text{ km}$ due East. Next, she turns left and walks $6\text{ km}$ due North. Finally, she turns right and walks $5\text{ km}$ due East. What is the shortest straight-line distance (in km) from her final position to the campsite?

Show answer & explanation

Answer: 20

Answer

The shortest straight-line distance from the final position to the campsite is 20 km.

The researcher ends up

16\text{ km}

North and

12\text{ km}

East of her starting position. Using the right-triangle distance formula

\sqrt{\text{North}^2 + \text{East}^2}

, we get

\sqrt{16^2 + 12^2} = \sqrt{400} = 20\text{ km}

.

Step-by-Step Solution

1

Sum up the individual directional movements along the North-South axis.

Total North displacement =

10\text{ km} + 6\text{ km} = 16\text{ km North}

.

Both vertical moves are towards the North, so their distances add together directly.

2

Sum up the individual directional movements along the East-West axis.

Total East displacement =

7\text{ km} + 5\text{ km} = 12\text{ km East}

.

Both horizontal moves are towards the East, so their distances add together directly.

3

Calculate the hypotenuse using the Pythagorean theorem.

Shortest distance =

\sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20\text{ km}

.

The net North and East displacements form a right-angled triangle where the shortest distance is the hypotenuse.

Key Concept

Displacement Vector Summation and Pythagorean Distance Calculation

Estimated Time:50s

Question 7Question →

What is the unit digit of the expression $7^{43} + 9^{27}$ ?

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Answer: 2

Answer

The unit digit of the given expression

7^{43} + 9^{27}

is 2.

The unit digit of

7^{43}

is determined by dividing 43 by 4, giving a remainder of 3, so

7^3

ends in 3. The unit digit of

9^{27}

is determined by taking 9 to an odd power, which ends in 9. Adding

3 + 9 = 12

gives a final unit digit of 2.

Step-by-Step Solution

1

Determine the unit digit of

7^{43}

using cyclicity rules.

The unit digit of

7^{43}

is 3.

The base 7 follows a 4-step cyclicity pattern (7, 9, 3, 1). Dividing exponent 43 by 4 leaves remainder 3, giving

7^3 \pmod{10} = 3

.

2

Determine the unit digit of

9^{27}

using cyclicity rules.

The unit digit of

9^{27}

is 9.

The base 9 follows a 2-step cyclicity pattern (9 for odd powers, 1 for even powers). Since 27 is odd, the unit digit is 9.

3

Combine the resulting unit digits.

The unit digit of the sum is 2.

Summing the individual unit digits gives

3 + 9 = 12

, whose unit digit is 2.

Key Concept

Unit Digit and Cyclicity

Question 8Question →

A land surveyor begins at a base camp $O$ . She walks $12\text{ m}$ due East to reach point $A$ . From point $A$ , she turns $90^\circ$ clockwise and walks $16\text{ m}$ to point $B$ . Facing South at point $B$ , she turns $135^\circ$ anti-clockwise and walks $10\sqrt{2}\text{ m}$ to point $C$ . Finally, she turns due West and walks $14\text{ m}$ to stop at point $D$ . What is the shortest distance (in meters) between the base camp $O$ and point $D$ ?

Show answer & explanation

Answer: 10

Answer

The shortest distance between the base camp

O

and point

D

is

10\text{ meters}

.

Tracking relative coordinates gives point

A(12,0)

, point

B(12,-16)

, point

C(22,-6)

, and final point

D(8,-6)

. The straight-line distance from the origin

O(0,0)

to

D(8,-6)

is

\sqrt{8^2 + (-6)^2} = 10\text{ meters}

.

Step-by-Step Solution

1

Establish coordinate system

Base camp

O = (0, 0)

Assigning the origin to the start point allows vector tracking.

2

Calculate displacement to point A and B

Point

A = (12, 0)

, Point

B = (12, -16)

East increases x-coordinate by 12; South decreases y-coordinate by 16.

3

Decompose angular movement from B to C

Point

C = (22, -6)

Turning

135^\circ

anti-clockwise from South points to North-East.

\Delta x = 10\sqrt{2} \times \frac{1}{\sqrt{2}} = 10\text{ m}

East,

\Delta y = 10\sqrt{2} \times \frac{1}{\sqrt{2}} = 10\text{ m}

North.

4

Determine final coordinates at point D

Point

D = (8, -6)

Walking

14\text{ m}

West reduces the x-coordinate from 22 to 8.

5

Apply distance formula for straight-line displacement

Distance

OD = 10\text{ m}

OD = \sqrt{(8 - 0)^2 + (-6 - 0)^2} = \sqrt{64 + 36} = 10\text{ m}

.

Key Concept

Vector displacement decomposition and 2D Cartesian plane distance calculation

Question 9Question →

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

12.5\% \text{ of } 480 - \left[ 3 \frac{3}{4} \div \left\{ 1.25 + \frac{1}{2} \text{ of } \left( 3.6 - \overline{1.8 - 0.7} \right) \right\} \right] \times \frac{8}{15}

Show answer & explanation

Answer: 59.2

Answer

The evaluated numerical value of the expression is 59.2.

Following strict BODMAS hierarchy—evaluating the vinculum (1.1), parentheses (2.5), 'of' operation (1.25), curly brackets (2.5), square bracket division (1.5), multiplication (0.8), percentage term (60), and final subtraction—gives 60 - 0.8 = 59.2.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar.

1.8 - 0.7 = 1.1

The vinculum has the highest priority and acts as an implicit innermost bracket.

2

Evaluate the terms inside the round brackets (parentheses).

3.6 - 1.1 = 2.5

Round brackets must be resolved after the vinculum.

3

Evaluate the 'of' operation inside the curly brackets.

(1/2) * 2.5 = 1.25

'Of' precedes addition and multiplication in the standard BODMAS order.

4

Complete the addition within the curly brackets.

1.25 + 1.25 = 2.5 = 5/2

Brackets must be fully simplified from inside out.

5

Perform division inside the square brackets.

(15/4) / (5/2) = (15/4) * (2/5) = 3/2 = 1.5

Square brackets are evaluated after inner curly brackets.

6

Multiply the square bracket result by 8/15.

1.5 * (8/15) = (3/2) * (8/15) = 4/5 = 0.8

Multiplication takes precedence over final subtraction.

7

Calculate the initial percentage 'of' term.

12.5% of 480 = 0.125 * 480 = 60

'Of' operation on percentage takes precedence before subtraction.

8

Subtract the bracketed product from the initial term.

60 - 0.8 = 59.2

Subtraction is performed last according to BODMAS.

Key Concept

BODMAS Order of Operations with Vinculum and Nested Brackets

Estimated Time:2m 0s

Question 10Question →

What is the unit digit of the expression $S = (1! + 2! + 3! + \dots + 99!) + 3^{99}$ ?

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Answer: 0

Answer

The unit digit of the given expression is 0.

The unit digit of a sum is determined by the sum of the unit digits of its individual terms. For the factorial sum

(1! + 2! + 3! + \dots + 99!)

, terms from

5!

onward end in 0 because

5! = 120

. Summing the first four terms yields

1 + 2 + 6 + 24 = 33

, contributing a unit digit of 3. For

3^{99}

, the cyclicity sequence of base 3 is 3, 9, 7, 1 (length 4). Dividing 99 by 4 leaves a remainder of 3, corresponding to

3^3 = 27

, which contributes a unit digit of 7. Summing the two unit digits yields

3 + 7 = 10

, giving a final unit digit of 0.

Step-by-Step Solution

1

Evaluate the unit digit contribution of the factorial terms

Unit digit of

(1! + 2! + 3! + \dots + 99!)

is 3

Since

n!

ends in 0 for every

n \ge 5

, only

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

affects the unit digit.

2

Find the unit digit of

3^{99}

using pattern cyclicity

Unit digit of

3^{99}

is 7

The cyclicity of base 3 is 4. Since

99 \pmod 4 = 3

, the unit digit corresponds to

3^3 = 27

.

3

Combine the unit digits of both parts

Unit digit of

S

is 0

Adding the unit digits gives

3 + 7 = 10

, making the final unit digit 0.

Key Concept

Unit digit evaluation using factorial terminal zeros and exponential cyclicity

Question 11Question →

Let $N$ be the smallest positive integer that leaves a remainder of $3$ when divided by $11$ , a remainder of $5$ when divided by $13$ , and is exactly divisible by $7$ . What is the remainder when $N^4 + N^2 + 1$ is divided by $17$ ?

Show answer & explanation

Answer: 3

Answer

The remainder when

N^4 + N^2 + 1

is divided by

17

is 3.

Using the Chinese Remainder Theorem, the smallest integer meeting all divisibility constraints is determined to be

N = 707

. Reducing

707

modulo

17

gives

10

. Substituting

10

into the expression

N^4 + N^2 + 1

modulo

17

yields

(10^4 + 10^2 + 1) \pmod{17} = (4 + 15 + 1) \pmod{17} = 20 \pmod{17} = 3

.

Step-by-Step Solution

1

Express the given conditions as a system of linear congruences

N \equiv 3 \pmod{11}

,

N \equiv 5 \pmod{13}

, and

N \equiv 0 \pmod{7}

.

Translates the problem statement into standard modular arithmetic notation.

2

Solve the simultaneous congruences for modulo 11 and modulo 13

Let

N = 11k + 3

. Substituting into

11k + 3 \equiv 5 \pmod{13}

gives

11k \equiv 2 \pmod{13} \implies -2k \equiv 2 \pmod{13} \implies k \equiv 12 \pmod{13}

. Thus,

N = 11(13m + 12) + 3 = 143m + 135

.

Combines two modular constraints into a single general expression using the Chinese Remainder Theorem principle.

3

Apply the divisibility condition for 7 to determine the smallest value of

N

143m + 135 \equiv 0 \pmod{7} \implies 3m + 2 \equiv 0 \pmod{7} \implies 3m \equiv 5 \pmod{7} \implies m \equiv 4 \pmod{7}

. For the smallest positive integer, set

m = 4

, so

N = 143(4) + 135 = 707

.

Finds the specific smallest integer

N

that satisfies all three divisor requirements.

4

Reduce

N

modulo 17

707 = 17 \times 41 + 10 \implies N \equiv 10 \pmod{17}

.

Simplifies the base before evaluating higher powers.

5

Calculate the polynomial expression

(N^4 + N^2 + 1) \pmod{17}

N^2 \equiv 10^2 = 100 \equiv 15 \equiv -2 \pmod{17}

, and

N^4 \equiv (-2)^2 = 4 \pmod{17}

. Therefore,

N^4 + N^2 + 1 \equiv 4 + 15 + 1 = 20 \equiv 3 \pmod{17}

.

Uses modular exponentiation properties to efficiently evaluate the final remainder.

Key Concept

Chinese Remainder Theorem and Modular Exponentiation

Question 12Question →

Determine the unit digit of the composite exponential expression $E = 17^{18^{19}} + 18^{19^{20}} - 19^{20^{21}}$ .

Show answer & explanation

Answer: 8

Answer

8

The unit digit of

17^{18^{19}}

is 1 because

18^{19}

is divisible by 4, which corresponds to the 4th power

7^4 \equiv 1 \pmod{10}

. The unit digit of

18^{19^{20}}

is 8 because

19^{20} \equiv 1 \pmod 4

, giving

8^1 = 8

. The unit digit of

19^{20^{21}}

is 1 because

20^{21}

is an even exponent, giving

9^{\text{even}} \equiv 1 \pmod{10}

. Combining these gives

1 + 8 - 1 = 8

.

Step-by-Step Solution

1

Find the unit digit of

17^{18^{19}}

The unit digit is 1

Base 7 has cyclicity 4. Exponent

18^{19} = 2^{19} \equiv 0 \pmod 4

. When the remainder modulo 4 is 0, we take the 4th power in the cycle:

7^4 \equiv 1 \pmod{10}

.

2

Find the unit digit of

18^{19^{20}}

The unit digit is 8

Base 8 has cyclicity 4. Exponent

19^{20} \equiv (-1)^{20} \equiv 1 \pmod 4

. The 1st power in the cycle gives

8^1 = 8

.

3

Find the unit digit of

19^{20^{21}}

The unit digit is 1

Base 9 has cyclicity 2. The exponent

20^{21}

is even, so

9^{\text{even}} \equiv 1 \pmod{10}

.

4

Evaluate the combined expression

The final unit digit is 8

Combine the unit digits of the individual terms:

1 + 8 - 1 = 8

.

Key Concept

Cyclicity of numbers and modular arithmetic for tower exponents

Question 13Question →

What is the unit digit of the expression $E = (842^{324} \times 617^{153}) + 358^{217}$ ?

Show answer & explanation

Answer: 0

Answer

The unit digit of the expression

E = (842^{324} \times 617^{153}) + 358^{217}

is 0.

For

842^{324}

, exponent

324 \pmod 4 = 0

corresponds to the 4th power position in cyclicity (

2^4 = 16 \rightarrow 6

). For

617^{153}

,

153 \pmod 4 = 1

corresponds to

7^1 = 7

. The product of these terms ends in

6 \times 7 = 42 \rightarrow 2

. For

358^{217}

,

217 \pmod 4 = 1

corresponds to

8^1 = 8

. Adding the product's unit digit (2) and

358^{217}

's unit digit (8) gives

2 + 8 = 10 \rightarrow 0

.

Step-by-Step Solution

1

Find the unit digit of

842^{324}

Unit digit is 6

The unit digit of base 842 is 2, which repeats unit digits in a cycle of 4:

(2, 4, 8, 6)

. Dividing exponent 324 by 4 yields remainder 0, indicating the 4th position in the cycle,

2^4 = 16 \rightarrow 6

.

2

Find the unit digit of

617^{153}

Unit digit is 7

The unit digit of base 617 is 7, which repeats in a cycle of 4:

(7, 9, 3, 1)

. Dividing exponent 153 by 4 yields remainder 1, indicating the 1st position in the cycle,

7^1 = 7

.

3

Calculate the unit digit of the product

(842^{324} \times 617^{153})

Unit digit is 2

Multiplying the unit digits of the two terms gives

6 \times 7 = 42

, which has a unit digit of 2.

4

Find the unit digit of

358^{217}

Unit digit is 8

The unit digit of base 358 is 8, which repeats in a cycle of 4:

(8, 4, 2, 6)

. Dividing exponent 217 by 4 yields remainder 1, indicating the 1st position in the cycle,

8^1 = 8

.

5

Add the unit digit of the product to the unit digit of

358^{217}

Final unit digit is 0

Adding the unit digit of the product (2) and the unit digit of the final term (8) gives

2 + 8 = 10

, which ends in 0.

Key Concept

Unit Digit Computation using Exponents and Cyclicity

Question 14Question →

The following table shows the number of employees across three departments in a company over three years, with some missing data indicated by dashes:

Department	2021	2022	2023	Total
HR	40	—	50	140
IT	80	90	—	270
Finance	—	60	70	180
Total	170	200	220	590

What is the number of employees in the HR department in the year 2022?

Show answer & explanation

Answer: 50

Answer

The number of employees in the HR department in 2022 is 50.

The row total for HR is 140. Subtracting the known figures for 2021 (40) and 2023 (50) from 140 yields 50 employees for 2022.

Step-by-Step Solution

1

Examine the row corresponding to the HR department.

Known HR employees in 2021 = 40, 2023 = 50, and Total HR employees = 140.

The row total equals the sum of values across all three years.

2

Calculate the missing 2022 HR value.

140 - (40 + 50) = 50 employees.

Subtracting the sum of 2021 and 2023 values from the total gives the missing value for 2022.

Key Concept

Deriving missing data elements using row sums

Estimated Time:45s

Question 15Question →

An autonomous security robot at an industrial facility begins its inspection patrol from central gate $O$ facing East. It moves $14\text{ m}$ due East to reach checkpoint $A$ . From checkpoint $A$ , it turns $135^\circ$ anti-clockwise and travels $10\sqrt{2}\text{ m}$ to checkpoint $B$ . At checkpoint $B$ , it turns $90^\circ$ clockwise and travels $5\sqrt{2}\text{ m}$ to checkpoint $C$ . Next, it turns $90^\circ$ anti-clockwise and travels $4\sqrt{2}\text{ m}$ to reach checkpoint $D$ . Finally, it turns to face due South and moves $7\text{ m}$ to reach its final parking dock $E$ . What is the shortest straight-line distance (in meters) between central gate $O$ and final parking dock $E$ ?

Show answer & explanation

Answer: 13

Answer

The shortest straight-line distance between central gate

O

and final parking dock

E

is

13\text{ m}

.

By resolving each directional move into its respective orthogonal

(x, y)

components, the cumulative net position of the robot from origin

O(0, 0)

is found to be

(5, 12)

. Applying the distance formula gives

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

.

Step-by-Step Solution

1

Establish Cartesian Coordinates

Central gate

O = (0, 0)

, with

+x

as East,

-x

as West,

+y

as North, and

-y

as South.

Cartesian representation allows precise vector addition for multi-directional paths.

2

Calculate displacement to checkpoint A

Displacement is

(14, 0)

, so

A = (14, 0)

.

The robot travels

14\text{ m}

directly East along the positive x-axis.

3

Calculate displacement to checkpoint B

Displacement vector is

(-10, 10)

, so

B = (4, 10)

.

Turning

135^\circ

anti-clockwise from East points North-West (

135^\circ

). Distance

10\sqrt{2}\text{ m}

gives

\Delta x = -10\sqrt{2}\cdot\frac{1}{\sqrt{2}} = -10

and

\Delta y = 10\sqrt{2}\cdot\frac{1}{\sqrt{2}} = 10

.

4

Calculate displacement to checkpoint C

Displacement vector is

(5, 5)

, so

C = (9, 15)

.

Turning

90^\circ

clockwise from North-West points North-East (

45^\circ

). Distance

5\sqrt{2}\text{ m}

gives

\Delta x = 5

and

\Delta y = 5

.

5

Calculate displacement to checkpoint D

Displacement vector is

(-4, 4)

, so

D = (5, 19)

.

Turning

90^\circ

anti-clockwise from North-East points North-West (

135^\circ

). Distance

4\sqrt{2}\text{ m}

gives

\Delta x = -4

and

\Delta y = 4

.

6

Calculate displacement to final dock E

Displacement vector is

(0, -7)

, so

E = (5, 12)

.

Moving

7\text{ m}

due South reduces the y-coordinate by

7

.

7

Compute shortest straight-line distance OE

Distance

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

.

Apply the Pythagorean theorem to net displacement components

x = 5

and

y = 12

.

Key Concept

Vector Addition of Angular Displacements and Pythagorean Theorem

Estimated Time:2m 0s

Question 16Question →

What is the final value of the expression $36 \div 4 \times 3 + (15 - 7) - 10$ ?

Show answer & explanation

Answer: 25

Answer

The correct answer is 25.

Applying the BODMAS order of operations strictly: first simplify the bracket

(15 - 7) = 8

. Next, evaluate multiplication and division from left to right:

36 \div 4 = 9

and

9 \times 3 = 27

. Finally, evaluate addition and subtraction left to right:

27 + 8 - 10 = 25

.

Step-by-Step Solution

1

Evaluate bracket expression

15 - 7 = 8

Brackets have the highest precedence in the BODMAS rule.

2

Perform division and multiplication from left to right

36 ÷ 4 = 9, then 9 × 3 = 27

Division and multiplication carry equal priority and are evaluated in order from left to right.

3

Perform addition and subtraction from left to right

27 + 8 - 10 = 25

Addition and subtraction are performed sequentially from left to right.

Key Concept

BODMAS Order of Operations

Estimated Time:45s

Question 17Question →

In basic arithmetic, the unit digit of a power of an integer repeats in a periodic pattern known as cyclicity. What is the unit digit of $3^{45}$ ?

Show answer & explanation

Answer: 3

Answer

The unit digit of

3^{45}

is 3.

The base number 3 has a cyclicity pattern of length 4: (3, 9, 7, 1). When the exponent 45 is divided by 4, the quotient is 11 and the remainder is 1. A remainder of 1 indicates the unit digit is equal to the first element of the cyclic pattern, which is 3.

Step-by-Step Solution

1

Identify the unit digit cyclicity pattern for base 3.

The unit digits repeat every 4 powers in the pattern (3, 9, 7, 1).

Calculating sequential powers of 3 yields last digits of 3, 9, 7, 1, 3, 9, 7, 1, establishing a cycle length of 4.

2

Divide the exponent by the cycle period of 4.

45 \div 4 = 11

with a remainder of

1

.

The remainder indicates the exact position within the repeating cycle.

3

Find the unit digit using the remainder.

Since the remainder is 1, the unit digit is

3^1 = 3

.

The first number in the cyclicity sequence (3, 9, 7, 1) is 3.

Key Concept

Unit Digit Cyclicity

Estimated Time:45s

Question 18Question →

Evaluate the following mathematical expression using the standard BODMAS rule:

40\% \text{ of } 150 + \left[ 72 \div \left\{ 18 - \left( 2\frac{1}{2} \times 4 + \overline{11 - 5} \right) \right\} \right]

What is the final simplified numerical value?

Show answer & explanation

Answer: 96

Answer

The simplified numerical value of the given expression is 96.

Applying the standard BODMAS rule: first evaluate the vinculum

\overline{11 - 5} = 6

. Next, evaluate inside the round brackets:

2.5 \times 4 + 6 = 16

. Then simplify inside the curly braces:

18 - 16 = 2

. Following that, perform division inside the square brackets:

72 \div 2 = 36

. Separately, evaluate the percentage term:

40\% \text{ of } 150 = 60

. Summing these two terms gives

60 + 36 = 96

.

Step-by-Step Solution

1

Evaluate the bar (vinculum) expression.

\overline{11 - 5} = 6

The vinculum (bar) has highest priority within bracket operations.

2

Simplify the terms within the round parentheses.

2\frac{1}{2} \times 4 + 6 = 10 + 6 = 16

Perform multiplication before addition inside parentheses.

3

Evaluate the expression inside the curly braces.

18 - 16 = 2

Subtract the result of the round brackets from 18.

4

Perform the division inside the square brackets.

72 \div 2 = 36

Divide 72 by the value obtained from curly braces.

5

Calculate the percentage component of the expression.

40\% \text{ of } 150 = \frac{40}{100} \times 150 = 60

The word 'of' indicates multiplication applied to the percentage value.

6

Perform the final addition.

60 + 36 = 96

Add the result of the percentage calculation to the result of the bracket evaluation.

Key Concept

Order of Operations (BODMAS Rule) with Vinculum, Nested Brackets, and Percentages

Estimated Time:1m 30s

Question 19Question →

During an urban mobility audit conducted among $400$ daily commuters in a metropolitan city, data was recorded regarding their regular use of three transit modes: Metro Rail ( $M$ ), Electric Bus ( $E$ ), and Shared Bicycle ( $B$ ). The survey revealed that $190$ commuters use Metro Rail, $160$ use Electric Bus, and $120$ use Shared Bicycle. Furthermore, $60$ commuters use both Metro Rail and Electric Bus, $40$ use both Electric Bus and Shared Bicycle, and $50$ use both Metro Rail and Shared Bicycle. If $20$ commuters utilize all three modes of transport, how many commuters do not use any of these three transit modes?

Show answer & explanation

Answer: 60

Answer

The number of commuters who do not use any of the three transit modes is 60.

Using the 3-set inclusion-exclusion principle, the total number of commuters using at least one mode is given by

|M \cup E \cup B| = (190 + 160 + 120) - (60 + 40 + 50) + 20 = 340

. Subtracting this from the total sample of 400 yields

400 - 340 = 60

commuters who use none of the three modes.

Step-by-Step Solution

1

Identify individual set cardinalities and intersections from the problem statement.

Total universe

N = 400

,

|M| = 190

,

|E| = 160

,

|B| = 120

,

|M \cap E| = 60

,

|E \cap B| = 40

,

|M \cap B| = 50

, and

|M \cap E \cap B| = 20

.

Establishing accurate set values is required before applying set formulas.

2

Apply the Principle of Inclusion-Exclusion formula for three overlapping sets to find the union

|M \cup E \cup B|

.

|M \cup E \cup B| = 190 + 160 + 120 - 60 - 40 - 50 + 20 = 340

.

Pairwise intersections are double-counted when summing individual sets and must be subtracted, while the triple intersection is subtracted thrice and must be added back.

3

Compute the complement of the union to determine commuters using none of the modes.

Neither mode

= N - |M \cup E \cup B| = 400 - 340 = 60

.

The complement set represents all members of the universe outside the three-set union.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Question 20Question →

Evaluate the following mathematical expression by strictly applying the BODMAS rule:

150\% \text{ of } 48 - \left[ 14.5 - \left\{ 4 \frac{1}{2} \times \left( 12.8 \div \overline{2.5 + 1.5} - 1 \frac{3}{5} \right) + 2.8 \right\} \right] \div 0.15

What is the simplified numerical value?

Show answer & explanation

Answer: 42

Answer

The simplified numerical value of the given mathematical expression is 42.

Following the strict BODMAS order of operations: Vinculum -> Round brackets -> Curly brackets -> Square brackets -> Division -> 'Of' -> Subtraction gives the exact result 42.

Step-by-Step Solution

1

Evaluate the term under the vinculum (bar)

2.5 + 1.5 = 4

The vinculum has highest priority among grouping symbols.

2

Evaluate operations inside the round brackets (parentheses)

12.8 / 4 - 1.6 = 3.2 - 1.6 = 1.6

Division inside parentheses precedes subtraction, and 1 3/5 converts to decimal 1.6.

3

Evaluate operations inside the curly brackets

4.5 * 1.6 + 2.8 = 7.2 + 2.8 = 10

Multiplication precedes addition inside curly brackets, with 4 1/2 converting to 4.5.

4

Evaluate operations inside the square brackets

14.5 - 10 = 4.5

Subtract the evaluated inner expression from 14.5.

5

Perform the division following the bracket result

4.5 / 0.15 = 30

Division operation takes precedence over final subtraction.

6

Calculate the percentage 'of' expression

150% of 48 = (150 / 100) * 48 = 72

'Of' operation is evaluated before basic addition and subtraction.

7

Compute the final subtraction

72 - 30 = 42

Subtract the result of the bracketed division term from the percentage evaluation.

Key Concept

Order of Operations (BODMAS / PEMDAS) with Vinculum and Nested Brackets

Answer

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