All practice questions

If $y = \cos(3x) + e^x$ , what is $\frac{dy}{dx}$ ?

-3\sin(3x) + e^x

3\sin(3x) + e^x

-\sin(3x) + e^x

\sin(3x) + e^x

Show answer & explanation

Answer: $-3\sin(3x) + e^x$

Answer

-3\sin(3x) + e^x

Differentiating

y = \cos(3x) + e^x

term-by-term yields

\frac{dy}{dx} = -3\sin(3x) + e^x

. By the chain rule,

\frac{d}{dx}\cos(3x) = -3\sin(3x)

, while the derivative of

e^x

is simply

e^x

.

Step-by-Step Solution

1

Differentiate the trigonometric term

\cos(3x)

using the chain rule

\frac{d}{dx}(\cos(3x)) = -\sin(3x) \cdot \frac{d}{dx}(3x) = -3\sin(3x)

The derivative of

\cos(u)

is

-\sin(u) \cdot u'

2

Differentiate the exponential term

e^x

\frac{d}{dx}(e^x) = e^x

The exponential function

e^x

is its own derivative

3

Combine the results using the sum rule

\frac{dy}{dx} = -3\sin(3x) + e^x

The derivative of a sum of functions is the sum of their individual derivatives

Key Concept

Differentiation of trigonometric and exponential functions using the chain rule

Question 7222Question →

What is the simplified form of the expression $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} - \sqrt{3}$ ?

2

2 + 2\sqrt{3}

\sqrt{3} - 1

2\sqrt{3}

Show answer & explanation

Answer: $2$

Answer

The simplified form of the expression is

2

.

Multiplying the numerator and denominator of

\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}

by the conjugate

(\sqrt{6} + \sqrt{2})

gives

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

. Subtracting

\sqrt{3}

yields

2

.

Step-by-Step Solution

1

Rationalize the denominator of

\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}

Multiply numerator and denominator by the conjugate

(\sqrt{6} + \sqrt{2})

to get

\frac{(\sqrt{6} + \sqrt{2})^2}{(\sqrt{6})^2 - (\sqrt{2})^2}

.

Rationalizing eliminates the surd from the denominator using the difference of two squares identity.

2

Expand the numerator and simplify the fraction

Numerator:

(\sqrt{6})^2 + 2\sqrt{12} + (\sqrt{2})^2 = 6 + 4\sqrt{3} + 2 = 8 + 4\sqrt{3}

. Denominator:

6 - 2 = 4

. Fraction simplifies to

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

.

Simplifying radical factors (

\sqrt{12} = 2\sqrt{3}

) allows division by the common denominator.

3

Subtract

\sqrt{3}

from the rationalized term

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

.

Subtracting like surd terms cancels out

\sqrt{3}

leaving the rational constant.

Key Concept

Rationalization of Binomial Denominators and Surd Simplification

Question 7223Question →

A circle has a circumference of $44\text{ cm}$ . What is the area of the circle? (Take $\pi = \frac{22}{7}$ )

154\text{ cm}^2

308\text{ cm}^2

616\text{ cm}^2

44\text{ cm}^2

Show answer & explanation

Answer: $154\text{ cm}^2$

Answer

The area of the circle is

154\text{ cm}^2

.

First, find the radius from the circumference:

C = 2\pi r \implies 44 = 2 \times \frac{22}{7} \times r \implies r = 7\text{ cm}

. Then compute the area:

A = \pi r^2 = \frac{22}{7} \times 7^2 = 154\text{ cm}^2

. Therefore, the value

154\text{ cm}^2

is correct.

Step-by-Step Solution

1

Calculate the radius of the circle using the circumference formula

r = 7\text{ cm}

The formula for circumference is

C = 2\pi r

. Substituting

C = 44

and

\pi = \frac{22}{7}

yields

44 = 2 \times \frac{22}{7} \times r

, which simplifies to

\frac{44}{7}r = 44

, so

r = 7\text{ cm}

.

2

Calculate the area using the area formula for a circle

Area =

154\text{ cm}^2

The area formula is

A = \pi r^2

. Substituting

r = 7

and

\pi = \frac{22}{7}

gives

A = \frac{22}{7} \times 7^2 = 22 \times 7 = 154\text{ cm}^2

.

Key Concept

Relationship between circumference and area of a circle

Question 7224Question →

If $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}$ is expressed in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

3

2

5

1

Show answer & explanation

Answer: 3

Answer

The value of

a + b

is 3.

Multiplying both numerator and denominator by the conjugate

(\sqrt{6} + \sqrt{2})

yields

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

. Comparing this to

a + b\sqrt{3}

gives

a = 2

and

b = 1

, so

a + b = 3

.

Step-by-Step Solution

1

Rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator,

(\sqrt{6} + \sqrt{2})

.

\frac{(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})}

Multiplying by the conjugate eliminates radicals from the denominator using the difference of squares.

2

Expand the numerator

(\sqrt{6} + \sqrt{2})^2

using

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

.

(\sqrt{6})^2 + 2\sqrt{12} + (\sqrt{2})^2 = 6 + 2(2\sqrt{3}) + 2 = 8 + 4\sqrt{3}

Simplifying

\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

converts compound surds to standard form.

3

Expand the denominator using the difference of two squares

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

.

(\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4

Squaring each square root leaves rational integers in the denominator.

4

Divide the expanded numerator by the denominator to find

a

and

b

.

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

Dividing each term by 4 gives

a = 2

and

b = 1

.

5

Compute the sum

a + b

.

2 + 1 = 3

Adding the coefficients

a

and

b

yields the required value.

Key Concept

Rationalisation of Binomial Denominators

Question 7225Question →

Two regular polygons, $P_1$ and $P_2$ , have $n$ sides and $2n$ sides respectively. If the sum of one interior angle of $P_1$ and one exterior angle of $P_2$ is $150^\circ$ , what is the total number of diagonals of polygon $P_2$ ?

9

54

66

108

Show answer & explanation

Answer: 54

Answer

The total number of diagonals of polygon

P_2

is 54.

The interior angle of an

n

-sided polygon is

180^\circ - \frac{360^\circ}{n}

and the exterior angle of a

2n

-sided polygon is

\frac{180^\circ}{n}

. Adding these gives

180^\circ - \frac{180^\circ}{n} = 150^\circ

, which yields

n = 6

. Polygon

P_2

therefore has

12

sides, and the number of diagonals is

\frac{12 \times 9}{2} = 54

.

Step-by-Step Solution

1

Express the interior angle of

P_1

and the exterior angle of

P_2

in terms of

n

.

Interior angle of

P_1 = 180^\circ - \frac{360^\circ}{n}

; Exterior angle of

P_2 = \frac{360^\circ}{2n} = \frac{180^\circ}{n}

.

The interior angle of an

n

-sided regular polygon is

180^\circ - \frac{360^\circ}{n}

, and the exterior angle of a

2n

-sided regular polygon is

\frac{360^\circ}{2n}

.

2

Set up and solve the angle sum equation.

\left(180^\circ - \frac{360^\circ}{n}\right) + \frac{180^\circ}{n} = 150^\circ \implies 180^\circ - \frac{180^\circ}{n} = 150^\circ \implies \frac{180^\circ}{n} = 30^\circ \implies n = 6$.

The sum of the two angles is given as

150^\circ

.

3

Determine the number of sides of polygon

P_2

.

Polygon

P_2

has

2n = 2(6) = 12

sides.

Polygon

P_2

has

2n

sides.

4

Calculate the total number of diagonals for a polygon with 12 sides using

D = \frac{k(k-3)}{2}

.

D = \frac{12(12 - 3)}{2} = \frac{12 \times 9}{2} = 54.

The formula for the number of diagonals in a polygon with

k

sides is

\frac{k(k-3)}{2}

.

Key Concept

Interior and exterior angles of regular polygons and the polygon diagonal count formula

Question 7226Question →

What is the value of $x$ that satisfies the equation $16^{x - 1} \times 4^{x + 2} = 8^{x + 3}$ ?

3

\frac{1}{3}

\frac{8}{3}

\frac{3}{2}

Show answer & explanation

Answer: $3$

Answer

The value of

x

is

3

.

Converting all terms to base 2 gives

2^{4(x-1)} \times 2^{2(x+2)} = 2^{3(x+3)}

. Expanding the exponents gives

2^{4x-4} \times 2^{2x+4} = 2^{3x+9}

, which simplifies to

2^{6x} = 2^{3x+9}

. Equating exponents yields

6x = 3x + 9

, giving the solution

x = 3

.

Step-by-Step Solution

1

Express all terms with a common base of 2.

16 = 2^4

,

4 = 2^2

, and

8 = 2^3

, so the equation becomes

(2^4)^{x-1} \times (2^2)^{x+2} = (2^3)^{x+3}

.

Converting all terms to a common prime base allows application of index laws.

2

Apply power of a power law

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

.

2^{4x - 4} \times 2^{2x + 4} = 2^{3x + 9}

.

Multiply exponents when raising a power to another power.

3

Apply multiplication law

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

on the left side.

2^{(4x - 4) + (2x + 4)} = 2^{6x}

.

Add exponents when multiplying powers with the same base.

4

Equate exponents of equal bases.

6x = 3x + 9 \implies 3x = 9 \implies x = 3

.

If

a^m = a^n

for

a > 0

and

a \neq 1

, then

m = n

.

Key Concept

Laws of Indices and Solving Exponential Equations

Question 7227Question →

If $132_k + 43_k = 230_k$ , where $k$ represents a positive integer base, what is the value of $k$ ?

4

5

6

7

Show answer & explanation

Answer: 5

Answer

The value of the base

k

is 5.

Expanding

132_k

,

43_k

, and

230_k

into base 10 yields

(k^2 + 3k + 2) + (4k + 3) = 2k^2 + 3k

. Simplifying this gives the quadratic equation

k^2 - 4k - 5 = 0

, which factors as

(k - 5)(k + 1) = 0

. Since a number base must be a positive integer greater than the maximum digit present (which is 4), the correct base is 5.

Step-by-Step Solution

1

Expand each term in the equation

132_k + 43_k = 230_k

into base 10 using place-value notation.

132_k = 1 \cdot k^2 + 3 \cdot k^1 + 2 \cdot k^0 = k^2 + 3k + 2

43_k = 4 \cdot k^1 + 3 \cdot k^0 = 4k + 3

230_k = 2 \cdot k^2 + 3 \cdot k^1 + 0 \cdot k^0 = 2k^2 + 3k

Converting all terms to base 10 allows standard algebraic operations to be performed.

2

Set up the algebraic equation in base 10 and combine like terms.

(k^2 + 3k + 2) + (4k + 3) = 2k^2 + 3k

k^2 + 7k + 5 = 2k^2 + 3k

Equate the sum of the expanded left-hand side terms to the expanded right-hand side.

3

Rearrange the terms to form a standard quadratic equation

ak^2 + bk + c = 0

.

2k^2 + 3k - (k^2 + 7k + 5) = 0

k^2 - 4k - 5 = 0

Subtract

k^2 + 7k + 5

from both sides to set the quadratic expression equal to zero.

4

Factor the quadratic equation to find the possible values of

k

.

(k - 5)(k + 1) = 0 \implies k = 5 \text{ or } k = -1

Solve for

k

using factorization.

5

Select the valid base based on the domain of number bases.

k = 5

(since a number base must be a positive integer greater than the largest digit 4 present in the expression).

Base numbers cannot be negative or smaller than any of their constituent digits.

Key Concept

Number Bases and Conversions

Estimated Time:1m 30s

Question 7228Question →

How many distinct five-digit even numbers greater than $50,000$ can be formed using the digits $2, 3, 5, 6, 7,$ and $8$ if no digit may be repeated in any number?

240

288

480

40

Show answer & explanation

Answer: 240

Answer

The total number of distinct five-digit even numbers greater than 50,000 is 240.

The correct total is 240. Because the first digit requirement (greater than 50,000 requires 5, 6, 7, or 8) and the last digit requirement (even requires 2, 6, or 8) share common elements (6 and 8), the problem must be split into two mutually exclusive cases: when the first digit is odd (2 choices for

d_1

, 3 for

d_5

, and

^4P_3 = 24

for the middle digits, giving

2 \times 3 \times 24 = 144

) and when the first digit is even (2 choices for

d_1

, 2 for

d_5

, and

^4P_3 = 24

for the middle digits, giving

2 \times 2 \times 24 = 96

). Adding these yields

144 + 96 = 240

.

Step-by-Step Solution

1

Identify constraints on the positions of a five-digit number

d_1 d_2 d_3 d_4 d_5

.

For the number to be greater than 50,000,

d_1 \in \{5, 6, 7, 8\}

. For the number to be even,

d_5 \in \{2, 6, 8\}

.

The first digit determines the magnitude and the last digit determines parity.

2

Analyze Case 1 where the first digit

d_1

is odd (

5

or

7

).

There are 2 choices for

d_1

. All 3 even digits remain available for

d_5

(3 choices). The remaining 3 middle positions (

d_2, d_3, d_4

) can be arranged from the 4 remaining unused digits in

^4P_3 = 24

ways. Total for Case 1 =

2 \times 3 \times 24 = 144

.

Choosing an odd first digit does not reduce the pool of available even digits for the units position.

3

Analyze Case 2 where the first digit

d_1

is even (

6

or

8

).

There are 2 choices for

d_1

. Since one even digit is used in

d_1

, 2 even digits remain for

d_5

(2 choices). The 3 middle positions are filled from the 4 remaining digits in

^4P_3 = 24

ways. Total for Case 2 =

2 \times 2 \times 24 = 96

.

Selecting an even digit for the ten-thousands place directly reduces the available choices for the units place.

4

Sum the possibilities from both mutually exclusive cases.

Total valid numbers =

144 + 96 = 240

.

Applying the addition principle of counting over disjoint cases.

Key Concept

Conditional permutations with non-disjoint position restrictions

Question 7229Question →

A piecewise function

f(x)

is defined by

f(x) = \begin{cases} \frac{2x^2 - 5x - 3}{x - 3}, & \text{for } x \neq 3 \ a^2 - 2, & \text{for } x = 3 \end{cases}

If

f(x)

is continuous at

x = 3

and

a > 0

, what is the numerical value of

a

?

Show answer & explanation

Answer: 3

Answer

The numerical value of a is 3.

For

f(x)

to be continuous at

x = 3

, the defined value

f(3) = a^2 - 2

must equal

\lim_{x \to 3} f(x)

. Factoring the numerator gives

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

for

x \neq 3

. Taking the limit as

x \to 3

yields

2(3) + 1 = 7

. Setting

a^2 - 2 = 7

leads to

a^2 = 9

, which gives

a = 3

under the constraint

a > 0

.

Step-by-Step Solution

1

Evaluate the limit of

f(x)

as

x \to 3

Factor the numerator

2x^2 - 5x - 3 = (2x + 1)(x - 3)

. For

x \neq 3

,

f(x) = 2x + 1

. Thus,

\lim_{x \to 3} f(x) = 2(3) + 1 = 7

.

Direct substitution gives an indeterminate form

\frac{0}{0}

, so canceling the common factor

(x - 3)

allows direct evaluation of the limit.

2

Apply the definition of continuity at

x = 3

f(3) = a^2 - 2 = 7

.

For a function to be continuous at a point

c

, the function value

f(c)

must equal the limit

\lim_{x \to c} f(x)

.

3

Solve for the parameter

a

a^2 = 9 \implies a = 3

(since

a > 0

).

Solving

a^2 = 9

gives solutions

3

and

-3

. The condition

a > 0

specifies the positive root.

Key Concept

Continuity of a Piecewise Function at a Point

Question 7230Question →

If $\log_x 64 - 2\log_4 x = 1$ , what is the sum of all possible real values of $x$ ?

\frac{33}{8}

\frac{31}{8}

4

-2

Show answer & explanation

Answer: $\frac{33}{8}$

Answer

The sum of all possible real values of

x

is

\frac{33}{8}

.

Using the change of base formula,

\log_x 64 = \frac{6}{\log_2 x}

and

2\log_4 x = \log_2 x

. Letting

u = \log_2 x

yields

\frac{6}{u} - u = 1

, which rearranges to

u^2 + u - 6 = 0

. Factoring gives

u = 2

or

u = -3

, yielding solutions

x = 2^2 = 4

and

x = 2^{-3} = \frac{1}{8}

. Summing these values gives

4 + \frac{1}{8} = \frac{33}{8}

.

Step-by-Step Solution

1

Apply the change of base formula to express all logarithmic terms in base 2.

\log_x 64 = \frac{\log_2 64}{\log_2 x} = \frac{6}{\log_2 x}

and

\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{\log_2 x}{2}

, so

2\log_4 x = \log_2 x

.

Converting all terms to a common base allows substitution into a simple algebraic equation.

2

Substitute

u = \log_2 x

into the original equation.

\frac{6}{u} - u = 1.

This simplifies the logarithmic equation into a single-variable algebraic form.

3

Clear the fraction by multiplying through by

u

and rearrange into standard quadratic form.

6 - u^2 = u \implies u^2 + u - 6 = 0.

Multiplying by

u

(where

u \neq 0

) forms a quadratic equation.

4

Factor and solve the quadratic equation for

u

.

(u + 3)(u - 2) = 0 \implies u = 2 \text{ or } u = -3.

Finding the roots of the quadratic equation provides the possible values for

\log_2 x

.

5

Solve for

x

using the exponential form

x = 2^u

.

For

u = 2

:

x = 2^2 = 4

. For

u = -3

:

x = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}

.

Both

x = 4

and

x = \frac{1}{8}

are positive real numbers not equal to 1, so both are valid logarithmic bases.

6

Calculate the sum of all valid solutions for

x

.

4 + \frac{1}{8} = \frac{32}{8} + \frac{1}{8} = \frac{33}{8}.

The question asks for the sum of all real values of

x

satisfying the equation.

Key Concept

Logarithms and Change of Base

Estimated Time:2m 0s

Question 7231Question →

The table below shows the frequency distribution of marks obtained by a group of students in a short test:

Mark ( $x$ )	1	2	3	4
Frequency ( $f$ )	2	3	3	2

What is the mean deviation of the distribution?

0.9

1.0

1.05

2.25

Show answer & explanation

Answer: $0.9$

Answer

The mean deviation of the distribution is

0.9

.

The mean of the data is

\bar{x} = \frac{25}{10} = 2.5

. Calculating the sum of weighted absolute deviations yields

2(1.5) + 3(0.5) + 3(0.5) + 2(1.5) = 9.0

. Dividing by total frequency

10

gives the mean deviation as

0.9

.

Step-by-Step Solution

1

Calculate the mean (

\bar{x}

) of the frequency distribution.

\sum f = 2 + 3 + 3 + 2 = 10

,

\sum fx = (2 \times 1) + (3 \times 2) + (3 \times 3) + (2 \times 4) = 2 + 6 + 9 + 8 = 25

. Therefore,

\bar{x} = \frac{25}{10} = 2.5

.

The mean is required to determine the deviations of each score value.

2

Find the absolute deviation

|x - \bar{x}|

for each score value.

For

x = 1

:

|1 - 2.5| = 1.5

; for

x = 2

:

|2 - 2.5| = 0.5

; for

x = 3

:

|3 - 2.5| = 0.5

; for

x = 4

:

|4 - 2.5| = 1.5

.

Mean deviation measures average distance from the mean, ignoring signs.

3

Multiply each absolute deviation by its corresponding frequency and sum them.

\sum f|x - \bar{x}| = (2 \times 1.5) + (3 \times 0.5) + (3 \times 0.5) + (2 \times 1.5) = 3 + 1.5 + 1.5 + 3 = 9.0

.

Frequencies reflect how many times each deviation occurs in the dataset.

4

Compute the mean deviation by dividing the weighted sum by the total frequency

\sum f

.

\text{Mean Deviation} = \frac{\sum f|x - \bar{x}|}{\sum f} = \frac{9.0}{10} = 0.9.

The mean deviation is the average of these absolute deviations across all observations.

Key Concept

Mean Deviation for Discrete Frequency Distribution

Question 7232Question →

Let the universal set $\mathcal{U} = \{x \in \mathbb{Z}^+ : x \le 60\}$ . Subsets $A$ , $B$ , and $C$ of $\mathcal{U}$ are defined as:

$A = \{x \in \mathcal{U} : x \text{ is a multiple of } 3\}$
$B = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\}$
$C = \{x \in \mathcal{U} : x \text{ is a factor of } 60\}$

What is the cardinality of the set $(A \Delta B) \setminus C'$ , where $A \Delta B$ represents the symmetric difference of $A$ and $B$ , and $C'$ is the complement of $C$ in $\mathcal{U}$ ?

6

8

12

19

Show answer & explanation

Answer: 6

Answer

6

Using the set identity

X \setminus Y' = X \cap Y

, the expression

(A \Delta B) \setminus C'

simplifies directly to

(A \Delta B) \cap C

. The set

C

consists of all factors of 60:

\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}

. The symmetric difference

A \Delta B

consists of numbers that are divisible by 3 or 4, but not both (excluding multiples of 12). Checking the 12 factors of 60 against this rule, the numbers 3, 6, 15, and 30 are multiples of 3 but not 4, while 4 and 20 are multiples of 4 but not 3. The numbers 12 and 60 are multiples of both 3 and 4, so they are excluded. Thus, the resulting set is

\{3, 4, 6, 15, 20, 30\}

, which contains exactly 6 elements.

Step-by-Step Solution

1

Simplify the set expression using complement and relative difference identities.

(A \Delta B) \setminus C' = (A \Delta B) \cap (C')' = (A \Delta B) \cap C

By definition of set difference,

X \setminus Y = X \cap Y'

. Replacing

Y

with

C'

gives

X \cap (C')' = X \cap C

.

2

List all elements in set

C

(factors of 60).

C = \{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}

60 prime factorizes as

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

, giving

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

positive factors.

3

Recall the definition of symmetric difference

A \Delta B

.

A \Delta B = (A \cup B) \setminus (A \cap B)

An element belongs to

A \Delta B

if it is a multiple of 3 or a multiple of 4, but NOT a multiple of both 3 and 4 (i.e. not a multiple of 12).

4

Evaluate which elements of

C

belong to

A \Delta B

.

The elements in

C

that are multiples of 3 only (not 12) are

\{3, 6, 15, 30\}

. The elements in

C

that are multiples of 4 only (not 12) are

\{4, 20\}

. Elements

\{12, 60\}

are multiples of 12, so they are excluded.

Testing each factor of 60 against the criteria for symmetric difference.

5

Form the set

(A \Delta B) \cap C

and find its cardinality.

(A \Delta B) \cap C = \{3, 4, 6, 15, 20, 30\}

, so

n((A \Delta B) \cap C) = 6

.

Counting the elements in the intersection.

Key Concept

Symmetric Difference and Set Complement Identities

Estimated Time:2m 30s

Question 7233Question →

A composite plane figure is formed by joining a major sector of a circle of radius $14\text{ cm}$ (having a central angle of $270^\circ$ and center $O$ ) to a square $OACB$ . The vertices $A$ and $B$ of the square lie on the circle such that $OA$ and $OB$ serve as the bounding radii of the major sector. What is the total area of the combined figure in $\text{cm}^2$ ? (Take $\pi = \frac{22}{7}$ )

350\text{ cm}^2

504\text{ cm}^2

560\text{ cm}^2

658\text{ cm}^2

Show answer & explanation

Answer: $658\text{ cm}^2$

Answer

658\text{ cm}^2

The total area of the composite figure is the sum of the non-overlapping regions: the major sector of central angle

270^\circ

(

462\text{ cm}^2

) and the square of side length

14\text{ cm}

(

196\text{ cm}^2

), giving

462 + 196 = 658\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the area of the major sector of central angle

270^\circ

and radius

14\text{ cm}

.

\text{Area}_{\text{sector}} = \frac{270^\circ}{360^\circ} \times \pi r^2 = \frac{3}{4} \times \frac{22}{7} \times 14^2 = \frac{3}{4} \times 616 = 462\text{ cm}^2

The major sector covers

\frac{270}{360} = \frac{3}{4}

of the full circle.

2

Calculate the area of the square

OACB

with side length

s = OA = 14\text{ cm}

.

\text{Area}_{\text{square}} = s^2 = 14^2 = 196\text{ cm}^2

The radii

OA

and

OB

form two adjacent sides of the square

OACB

of length

14\text{ cm}

.

3

Sum the areas of the major sector and the square to find the total composite area.

\text{Total Area} = 462\text{ cm}^2 + 196\text{ cm}^2 = 658\text{ cm}^2

The major sector and square share boundaries

OA

and

OB

without interior overlap.

Key Concept

Perimeter and Area of Plane Shapes

Estimated Time:2m 0s

Question 7234Question →

The time, $t$ hours, required to complete a road maintenance project varies inversely as the number of workers, $w$ , assigned to the project. If 8 workers can finish the project in 15 hours, calculate the time, in hours, required for 12 workers to finish the same project.

Show answer & explanation

Answer: 10

Answer

10 hours

Because the time

t

varies inversely as the number of workers

w

, the total worker-hours required for the project is constant:

k = 8 \times 15 = 120

worker-hours. Dividing this total work by 12 workers gives

\frac{120}{12} = 10

hours.

Step-by-Step Solution

1

Set up the inverse variation equation

t = \frac{k}{w}

, where

k

is the constant of variation.

Inverse variation implies that as the number of workers increases, the time required decreases proportionally.

2

Determine the value of the constant of variation

k

k = t \times w = 15 \times 8 = 120

.

Substitute the known pair of values (

w = 8, t = 15

) into the equation.

3

Compute the new value of

t

for 12 workers

t = \frac{120}{12} = 10

hours.

Substitute

w = 12

and

k = 120

into

t = \frac{k}{w}

.

Key Concept

Inverse Variation

Question 7235Question →

In a science competition, $3$ distinct prizes (first, second, and third place) are to be awarded to $3$ different students chosen from a group of $5$ finalists. In how many different ways can these $3$ prizes be awarded?

Show answer & explanation

Answer: 60

Answer

60 ways

Because the prizes are distinct, the order in which the finalists are selected matters. Calculating the number of arrangements of

3

students from

5

candidates is given by

^{5}P_{3} = 5 \times 4 \times 3 = 60

.

Step-by-Step Solution

1

Determine if order matters

Since the prizes are distinct (1st, 2nd, and 3rd place), the order of assignment matters, making this a permutation problem.

Assigning distinct positions to individuals requires calculating permutations rather than combinations.

2

Apply the permutation formula

^{n}P_{r} = \frac{n!}{(n-r)!}

^{5}P_{3} = \frac{5!}{(5-3)!} = \frac{5!}{2!}

There are

5

total candidates (

n = 5

) and

3

positions to fill (

r = 3

).

3

Calculate the numeric value

^{5}P_{3} = 5 \times 4 \times 3 = 60

Canceling

2!

from the numerator and denominator leaves

5 \times 4 \times 3 = 60

.

Key Concept

Linear permutation of r items selected from n distinct items

Estimated Time:45s

Question 7236Question →

If $x$ and $y$ are real numbers greater than $1$ satisfying the system of equations $\log_x y + \log_y x = \frac{5}{2}$ and $xy = 64$ with $x > y$ , find the value of $x - y$ .

Show answer & explanation

Answer: 12

Answer

The value of

x - y

is 12.

Using the change of base identity

\log_y x = \frac{1}{\log_x y}

, the equation

\log_x y + \log_y x = \frac{5}{2}

converts to

u + \frac{1}{u} = \frac{5}{2}

for

u = \log_x y

. Solving the quadratic equation

2u^2 - 5u + 2 = 0

yields

u = \frac{1}{2}

or

u = 2

. Because

x > y > 1

, we must have

\log_x y < 1

, selecting

u = \frac{1}{2} \implies x = y^2

. Substituting into

xy = 64

gives

y^3 = 64 \implies y = 4

and

x = 16

. Therefore,

x - y = 16 - 4 = 12

.

Step-by-Step Solution

1

Apply the reciprocal change of base identity

Rewrite

\log_y x

as

\frac{1}{\log_x y}

, yielding

\log_x y + \frac{1}{\log_x y} = \frac{5}{2}

.

According to the change of base formula,

\log_y x = \frac{\log_x x}{\log_x y} = \frac{1}{\log_x y}

.

2

Solve the quadratic equation in terms of

u = \log_x y

Substituting

u = \log_x y

gives

u + \frac{1}{u} = \frac{5}{2} \implies 2u^2 - 5u + 2 = 0

, which factors into

(2u - 1)(u - 2) = 0

, yielding

u = \frac{1}{2}

or

u = 2

.

Multiplying through by

2u

clears fractions and forms a standard quadratic equation.

3

Select the valid root using given inequality constraints

Since

x > y > 1

, taking the logarithm base

x

yields

\log_x x > \log_x y \implies 1 > \log_x y

. Thus

u = \frac{1}{2}

, which means

y = x^{1/2}

or

x = y^2

.

The condition

x > y

restricts the logarithm of

y

base

x

to be strictly less than

1

.

4

Substitute into the product equation to find

x

and

y

Substituting

x = y^2

into

xy = 64

gives

y^3 = 64 \implies y = 4

. Consequently,

x = 4^2 = 16

.

Combining the relation

x = y^2

with

xy = 64

enables single-variable cubic solution.

5

Calculate the required difference

x - y

16 - 4 = 12

.

Direct subtraction of the derived values

x = 16

and

y = 4

.

Key Concept

Logarithmic Change of Base Reciprocal Property

Question 7237Question →

Find the equation of the locus of a point $P(x, y)$ that moves such that its distance from the fixed point $(3, 0)$ is always equal to its perpendicular distance from the vertical line $x = -3$ .

Show answer & explanation

Answer: y^2 = 12x; y^2 - 12x = 0; y^2=12x; y^2 - 12x = 0

Answer

The equation of the locus is

y^2 = 12x

(or

y^2 - 12x = 0

).

Equating the distance from

P(x, y)

to

(3, 0)

, which is

\sqrt{(x - 3)^2 + y^2}

, and the distance from

P(x, y)

to

x = -3

, which is

|x + 3|

, squaring both sides gives

x^2 - 6x + 9 + y^2 = x^2 + 6x + 9

. Subtracting

x^2 + 9

from both sides yields

y^2 = 12x

.

Step-by-Step Solution

1

Express the distance from

P(x, y)

to the point

(3, 0)

using the distance formula.

d_1 = \sqrt{(x - 3)^2 + (y - 0)^2} = \sqrt{(x - 3)^2 + y^2}

The distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

.

2

Express the perpendicular distance from

P(x, y)

to the line

x = -3

.

d_2 = |x - (-3)| = |x + 3|

The perpendicular distance from a point

(x, y)

to a vertical line

x = k

is given by

|x - k|

.

3

Set the two distance expressions equal according to the locus condition.

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

The locus condition states that the distance to

(3, 0)

is equal to the distance to line

x = -3

.

4

Square both sides and simplify to obtain the Cartesian equation.

(x - 3)^2 + y^2 = (x + 3)^2 \implies x^2 - 6x + 9 + y^2 = x^2 + 6x + 9 \implies y^2 = 12x

Squaring eliminates the square root and absolute value signs, leading to the algebraic representation of the locus.

Key Concept

Definition and equation of a parabola as the locus of a point equidistant from a fixed point (focus) and a fixed line (directrix)

Alternative Method

Recognize that the definition of a parabola is the locus of points equidistant from a focus

(a, 0)

and a directrix

x = -a

. Here

a = 3

, so the standard equation

y^2 = 4ax

directly gives

y^2 = 4(3)x = 12x

.

Estimated Time:1m 30s

Question 7238Question →

A sector of a circle of radius $6\text{ cm}$ has a total perimeter whose numerical value is equal to the numerical value of its area. What is the area of the sector, in $\text{cm}^2$ ?

Show answer & explanation

Answer: 18

Answer

The area of the sector is

18\text{ cm}^2

.

For a sector of radius

r = 6\text{ cm}

, its total perimeter is

P = 2(6) + s = 12 + s

, where

s

is the arc length. Its area is

A = \frac{1}{2}(6)s = 3s

. Setting

P = A

gives

12 + s = 3s \implies 2s = 12 \implies s = 6\text{ cm}

. Substituting

s = 6

into the area expression gives

A = 3(6) = 18\text{ cm}^2

.

Step-by-Step Solution

1

Express the perimeter and area of the sector in terms of the arc length

s

P = 2r + s = 12 + s

and

A = \frac{1}{2}rs = 3s

The total perimeter of a sector includes two straight radii plus the arc length, while its area is given by

\frac{1}{2}rs

.

2

Equate the numerical values of perimeter and area

12 + s = 3s

The question specifies that the numerical value of the total perimeter equals the numerical value of its area.

3

Solve for the arc length

s

2s = 12 \implies s = 6\text{ cm}

Subtracting

s

from both sides yields

2s = 12

, so

s = 6\text{ cm}

.

4

Calculate the sector area

A = 3(6) = 18\text{ cm}^2

Substituting

s = 6

into

A = 3s

yields an area of

18\text{ cm}^2

.

Key Concept

Perimeter and Area of a Circular Sector

Question 7239Question →

If $y = x^3(2x - 1)^4$ , find the value of $\frac{dy}{dx}$ at $x = 1$ .

Show answer & explanation

Answer: 11

Answer

The numerical value of

\frac{dy}{dx}

at

x = 1

is

11

.

Applying the product rule

\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}

along with the chain rule for the expression

(2x - 1)^4

gives

\frac{dy}{dx} = 8x^3(2x - 1)^3 + 3x^2(2x - 1)^4

. Evaluating this at

x = 1

gives

8(1)^3(1)^3 + 3(1)^2(1)^4 = 8 + 3 = 11

.

Step-by-Step Solution

1

Set up the product rule for

y = u \cdot v

, where

u = x^3

and

v = (2x - 1)^4

.

u = x^3

and

v = (2x - 1)^4

The function is expressed as the product of two algebraic terms.

2

Find the derivative of each function component.

\frac{du}{dx} = 3x^2

and

\frac{dv}{dx} = 8(2x - 1)^3

The power rule gives

\frac{du}{dx} = 3x^2

, and applying the chain rule to

(2x - 1)^4

yields

4(2x - 1)^3 \cdot 2 = 8(2x - 1)^3

.

3

Substitute components into the product rule formula

\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}

.

\frac{dy}{dx} = 8x^3(2x - 1)^3 + 3x^2(2x - 1)^4

Combining

u\frac{dv}{dx}

and

v\frac{du}{dx}

provides the full expression for the derivative.

4

Evaluate the derivative at

x = 1

.

\frac{dy}{dx}\Big|_{x=1} = 8(1)^3(2(1) - 1)^3 + 3(1)^2(2(1) - 1)^4 = 8 + 3 = 11

Substituting

x = 1

simplifies the terms to

8(1) + 3(1) = 11

.

Key Concept

Rules of Differentiation (Product and Chain Rules)

Question 7240Question →

The total energy loss $E$ in Joules per minute in a magnetic core circuit is partly constant and partly varies directly as the square of the frequency $f$ in Hz of the alternating current. Given that $E = 120\text{ J}$ when $f = 10\text{ Hz}$ , and $E = 360\text{ J}$ when $f = 20\text{ Hz}$ , what is the value of $E$ in Joules when $f = 15\text{ Hz}$ ?

Show answer & explanation

Answer: 220

Answer

220

The relation describes a partial variation model

E = c + k f^2

. Substituting the two given conditions (

f=10, E=120

and

f=20, E=360

) yields the system of equations

c + 100k = 120

and

c + 400k = 360

. Solving this system gives

k = 0.8

and

c = 40

. Evaluating

E = 40 + 0.8(15)^2

results in

220\text{ J}

.

Step-by-Step Solution

1

Write the general equation for partial variation involving a constant term and a term proportional to

f^2

E = c + k f^2

Partial variation consists of a sum of a constant component and a variable component.

2

Substitute the known conditions into the variation equation to create a system of linear equations

Equation 1:

c + 100k = 120

; Equation 2:

c + 400k = 360

Two pairs of values are provided to determine the two unknown constants

c

and

k

.

3

Solve the system of simultaneous equations for

k

and

c

k = 0.8

and

c = 40

Subtracting Equation 1 from Equation 2 eliminates

c

, allowing direct calculation of

k

, after which

c

is found by substitution.

4

Calculate the required value of

E

when

f = 15

E = 40 + 0.8(15^2) = 40 + 0.8(225) = 220

Applying the discovered constants to the target frequency yields the final energy loss value.

Key Concept

Partial Variation and Simultaneous Equations

Estimated Time:2m 0s

Answer

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